Mastering quantum light pulses

with nonlinear waveguide interactions

Kontrolle über Quantenlichtpulse

durch nichtlineare Interaktion in Wellenleitern

Der Naturwissenschaftlichen Fakultätder Friedrich-Alexander-Universität Erlangen-Nürnberg

zurErlangung des Doktorgrades Dr. rer. nat.

vorgelegt vonAndreas Eckstein

aus Altdorf b. Nürnberg

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultätder Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 1.3.2012Vorsitzender der Promotionskommission: Prof. Dr. Rainer FinkErstberichterstatterin: Prof. Dr. Christine SilberhornZweitberichterstatter: Prof. Dr. Uwe Morgner

Contents

1 Introduction 11.1 The EPR paradox and entangled quantum states . . . . . . . . . . . . . . . . . . . 21.2 Nonlinear medium polarization and three-wave mixing . . . . . . . . . . . . . . . 3

1.2.1 Sum- and difference frequency generation . . . . . . . . . . . . . . . . . . 41.2.2 Spontaneous parametric downconversion . . . . . . . . . . . . . . . . . . 4

1.3 Quantum light pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 A quantum pulse source and a quantum pulse gate . . . . . . . . . . . . . . . . . 7

2 Basic concepts 92.1 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Electromagnetic field quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Field quadratures and squeezed light . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Important classes of light states and their properties . . . . . . . . . . . . . . . . 12

2.4.1 Coherent states of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.2 Single mode squeezed vacuum states . . . . . . . . . . . . . . . . . . . . . 132.4.3 Two-mode squeezed vacuum states . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Ultrafast pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.1 Broadband mode operators . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.2 Functional orthogonality interval . . . . . . . . . . . . . . . . . . . . . . 162.5.3 Broadband modes in the temporal domain . . . . . . . . . . . . . . . . . 172.5.4 Pulse propagation and quantum mechanical phase . . . . . . . . . . . . . 17

2.6 Nonlinear optical interactions and three-wave-mixing . . . . . . . . . . . . . . . . 182.6.1 Emergence of frequency- and phase-matching conditions . . . . . . . . . 192.6.2 SPDC in a channel waveguide with discrete spatial mode spectrum . . . . 202.6.3 Time evolution of the SPDC output state . . . . . . . . . . . . . . . . . . . 222.6.4 Quasi-Phasematching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6.5 Classical undepleted SPDC pump . . . . . . . . . . . . . . . . . . . . . . . 232.6.6 Broadband mode structure and Schmidt decomposition . . . . . . . . . . 242.6.7 Effective mode number and spectral entanglement of a photon pair . . . . 252.6.8 Multiple squeezer excitation . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Modeling photon detection with binary detectors . . . . . . . . . . . . . . . . . . 272.7.1 Measurement operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7.2 Measuring the joint spectrum of a photon pair . . . . . . . . . . . . . . . 29

3 Spectral engineering 313.1 Pure heralded single photons and the two-mode squeezer . . . . . . . . . . . . . 323.2 The phasematching distribution Φ and group velocity matching . . . . . . . . . . 333.3 Critical phasematching through backward-wave SPDC . . . . . . . . . . . . . . . 36

i

3.4 Type I SPDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 Type II SPDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Survey of nonlinear waveguide materials for group velocity matching . . . . . . . 38

3.6.1 Lithium niobate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6.2 Lithium tantalate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.6.3 Potassium niobate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6.4 Potassium titanyl phosphate . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 A PP-KTP waveguide as parametric downconversion source 454.1 Single photon detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 The parametric downconversion source . . . . . . . . . . . . . . . . . . . . . . . 484.3 Phasematching contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Fiber spectrometer 555.1 Functional principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Experimental setup for photon pair spectrum measurement . . . . . . . . . . . . 575.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Spectral resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.5 The joint spectral intensity of photon pairs from the KTP source . . . . . . . . . . 625.6 Measurements beyond the perturbative limit 〈n〉 � 1 . . . . . . . . . . . . . . . . 635.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 Two-mode squeezed vacuum source 696.1 Mode-number and photon statistics of broadband squeezed vacuum states . . . . 696.2 The second order correlation function g(2) . . . . . . . . . . . . . . . . . . . . . 716.3 g(2) for broadband input states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.4 g(2) for the ultrafast multimode squeezer . . . . . . . . . . . . . . . . . . . . . . . 736.5 g(2) measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.6 Background event suppression and correction . . . . . . . . . . . . . . . . . . . . 756.7 Mean photon number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.8 Photon collection efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Quantum pulse manipulation 837.1 Beam-splitters, spectral filters and broadband mode selective filters . . . . . . . . 857.2 Broadband mode SFG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.3 Spectral engineering and the Quantum Pulse Gate . . . . . . . . . . . . . . . . . . 887.4 Critical group velocity matching and QPG mode-switching . . . . . . . . . . . . . 897.5 Experimental feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.6 The Quantum Pulse Shaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.7 Time ordering and strongly coupled three-wave-mixing . . . . . . . . . . . . . . . 947.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8 Conclusion and outlook 99

ii

Summary

Ultrafast quantum light pulses with durations of 1 ps and below show great promise as informationcarriers in quantum communication and computation. In future they may also be used to probephysical processes at ultrashort timescales with a resolution beyond the limits of Heisenberguncertainty. This thesis focuses on the creation and manipulation of quantum light pulses withsecond order nonlinear optical processes in optical waveguides.

In chapter “1. Introduction”, we lead the reader towards this work’s topic by giving a briefqualitative overview over the EPR paradox, quantum entanglement, three-wave-mixing andquantum light pulses.

Chapter “2. Basic concepts” familiarizes the reader with the physical concepts and mathematicaltools underlying this thesis.

In chapter “3. Spectral engineering”, we discuss the requirements to produce separable photonpair states with group velocity matching and examine several nonlinear materials widely usedfor optical waveguide inscription for their suitability to group velocity matching in the telecomwavelength regime.

In chapter “4. A PP-KTP waveguide as parametric downconversion source”, we give the basicspontaneous parametric downconversion source setup, as well as some initial measurements tocharacterize the single photon detectors, to demonstrate the production of correlated photons,and to determine the phasematching properties of the PP-KTP source.

In the following chapter, “5. Fiber spectrometer”, we present the single-photon fiber spectrometer[8].We discuss the experimental setup and calibration of the device, and measure the joint spectrumof photon pairs from our PP-KTP source. Finally, we investigate the behavior of a joint spectrummeasurement of a high mean photon number source with binary detectors.

In “6. Two-mode squeezed vacuum source”, we characterize spectral correlations of an ultrafastSPDC source with the second order correlation function g(2) . We present the g(2) measurementresults[42] and background substraction technique[43] and show that we can control the spectralcorrelations to produce a two-mode squeezed vacuum state of light. We then determine the meanphoton number and gain of the source.

“7. Quantum pulse gate”: While the previous chapters focus on the creation and characterizationof ultrafast quantum pulses of light, we now propose a way to manipulate the mode structureof a given quantum light state. We discuss the concept of an active optical filter sensitive tospectral/temporal pulse shape: The quantum pulse gate[41], and its reverse process, the quantumpulse shaper[21].

In the final chapter “8. Conclusion and outlook”, we recapitulate the main results of this thesisand provide a few pointers towards possibilities for future research building on it.

v

Zusammenfassung

Ultrakurze Quantenlichtpulse mit einer Pulsdauer von 1 ps und darunter sind vielversprechendeKandidaten als Informationsträger in der Quantenkommunikation und im Quantencomputing. InZukunft könnten sie auch dazu genutzt werden, physikalische Prozesse auf ultrakurzen Zeitskalenjenseits der Grenzen der Heisenberg-Unschärfe zu untersuchen. Diese Dissertation konzentriertsich auf die Erzeugung und Manipulation von Quantenlichtpulsen durch nichtlineare optischeProzesse zweiter Ordnung in optischen Wellenleitern.

In Kaptitel “1. Introduction” führen wir den Leser an das Thema der Arbeit heran, indem wireinen kurzen, qualitativen Überblick über das EPR-Paradoxon, Quantenverschränkung, Dreiwel-lenmischung und Quantenlichtpulse geben.

Kapitel “2. Basic concepts” macht den Leser mit den physikalischen Konzepten und mathemati-schen Werkzeugen vertraut, die dieser Arbeit zugrunde liegen.

In Kaptitel “3. Spectral engineering” diskutieren wir die Voraussetzungen, um separable Pho-tonenpaarzustände durch Gruppengeschwindigkeitsanpassung zu produzieren und wir untersu-chen einige nichtlineare Materialien, die häufig zur Produktion optischer Wellenleiter verwendetwerden, auf ihre Eignung für die Gruppengeschwindigkeitsanpassung im Bereich der Telekommu-nikationswellenlängen.

In Kaptitel “4. A PP-KTP waveguide as parametric downconversion source” zeigen wir dengrundlegenden Aufbau unserer parametrischen Fluoreszenz-Quelle im PP-KTP Wellenleiter auf,sowie einige vorbereitende Messungen, die die Einzelphoton-Detektorn charakterisieren, dieErzeugung korrelierter Photonen demonstrieren und die Phasenanpassungs-Eigenschaften derPP-KTP-Quelle bestimmen.

Im anschließenden Kapitel “5. Fiber spectrometer” präsentieren wir das Einzelphotonen-Faserspektrometer[8]. Wir erörtern den experimentellen Aufbau und die Kalibration des Gerätsund messen das Koinzidenz-Spektrum von Photonenpaaren aus unserer PP-KTP Quelle. Zuletztuntersuchen wir das Verhalten einer Messsung eines Koinzidenz-Spektrums einer Photonenpaar-Quelle mit hoher mittlerer Photonenzahl mit binären Detektoren.

In “6. Two-mode squeezed vacuum source” charakterisieren wir die spektralen Korrelationeneiner ultraschnell gepumpten SPDC-Quelle durch die Korrelationsfunktion zweiter Ordnung g(2) .Wir präsentieren die Ergebnisse der g(2) -Messung[42] und der Untergrund-Subtraktion[43] undzeigen, dass wir durch Kontrolle der spektralen Korrelationen einen zweimodigen gequetschtenVakuumszustand erzeugen können. Dann bestimmen wir die mittlere Photonenzahl und dieKonversionseffizienz unserer Quelle.

“7. Quantum pulse gate”: Während sich die vorangehenden Kapitel mit der Erzeugung undCharakterisierung ultraschneller Quantenlichtpulse mit einer auseinandersetzen, schlagen wirhier eine Methode vor, um die modale Struktur eines gegebenen Licht-Quantenzustands zumanipulieren. Wir erörtern das Konzept eines aktiven optischen Filters, der auf spektrale oderzeitliche Pulsformen sensitiv ist: Das Quantum Pulse Gate[41], und dessen Umkehrprozess, derQuantum Pulse Shaper[21].

vii

Im letzten Kapitel “8. Conclusion and outlook” rekapitulieren wir die Hauptergebnisse dieserDissertation und zeigen einige mögliche zukünftige Forschungsrichtungen auf.

1Introduction

The notion of a corpuscular nature of light was first put forward by Isaac Newton, who conjecturedthat different colors were the result of different sized light particles[94]. While this was a statementthat – with the benefit of hindsight – bears a remarkable similarity to modern optical concepts likephotons as fundamental excitations of the electromagnetic field and their optical wavelength, hiscontemporary scientific peers favored a wave-like description of light, as supported by diffractionexperiments. The idea was not unearthed again until 1900 when Max Planck succeeded to explainthe radiation emission spectrum of a black body by postulating light emission in discrete quanta.Together with Einstein’s discovery of the photoelectric effect it triggered the development of thetheory of quantum mechanics, unifying the wave and the particle aspects of light.

Although quantum mechanics has been around since the 1920s through the works of Dirac,Schrödinger, Born, Heisenberg, and others, and has been an extraordinarily successful theory,actual, direct technical applications of quantum effects were slower to emerge. One of the earliest(and most visible) of those comes from the field of optics: Lasers, developed in the late 1950s arean ubiquitous source of coherent radiation today, and are used in such diverse fields as opticalcommunication networks, medical applications and entertainment devices. But also today’s micro-electronics and computer technology would be unthinkable but for the understanding of electricconduction in solids afforded by quantum mechanics, which led to semiconductor diodes andtransistors, the basic building blocks of every microchip. And as a rather infamous exampleof quantum mechanical applications there is nuclear technology, both in military and civilianapplications.

Arguably, during the last sixty years the world has been profoundly shaped by quantum me-chanics, mostly by way of the nuclear arms race and information technology; but on the horizonthere are already new, potentially disruptive quantum technologies: Quantum communication,which holds the promise of unconditionally secure secret information exchange and authentica-tion between remote parties without complete physical control of the communication channels.Quantum computation, which can solve certain hard mathematical problems faster than any givenclassical algorithm, an example being the factorization of an arbitrary integer number, useful forbreaking classical asymmetric encryption schemes like the widely-used RSA protocol. And finallyakin to computation is quantum simulation, the accurate physical simulation of large quantum

2 1 Introduction

mechanical systems such as unordered solids or macro-molecules, which is sure to trigger greatadvances in material sciences, biophysics, and medicine.

The main reason those new applications have not (quite) materialized yet is decoherence: Ingeneral, after an interaction, two interacting quantum systems will share mutual information,called quantum entanglement. If an observer is able to monitor only one of the systems, heperceives the resulting loss of information as decoherence, as the gradual loss of all genuinelyquantum mechanical properties of the system. This problem exists in any experiment:

To observe quantum effects, we have to isolate our observed system, the experiment, from theenvironment, be it by setting it up in a vacuum, by working at cryogenic temperatures, or byobserving at time scales short enough for interactions with the environment to be negligible.

This aspect of system isolation is why quantum optics, the study of the properties and inter-actions of quantum states of light, is at the forefront of the implementation of new quantumtechnologies: A quantum light state interacts weakly with transparent materials and exists at timescales that make thermal interaction with the environment much less of a problem than for e. g.an electronic state in a solid, preserving its quantum nature.

1.1 The EPR paradox and entangled quantum states

It is a well known principle of quantum mechanics that the values of any two non-commutingobservables X and Y cannot be measured for a system at a given time with arbitrary precision.The product of their statistical variances will always have a lower boundary. For two conjugate

observables, i. e. observables with[X, Y

]= 1, the lower boundary is a state-independent

constant:

〈∆X2〉 〈∆Y2〉 ≥ 1

2(1.1)

One such pair of conjugate observables are the position operator x and the linear momentumoperator p of a massive particle.

In 1934, Einstein, Podolsky and Rosen (EPR) published a gedanken-experiment[46] to highlighta fundamental flaw in the then-young field of quantum mechanics. Let us consider the decay of aresting particle into two moving daughter particles 1 and 2. The direction in which the daughterparticles will move after the decay event is completely undetermined until measurement, butmomentum conservation causes the daughter particles to assume quantum mechanical statessuch that the result of a measurement of the linear momentum vectors gives results with oppositesigns such that 〈p1〉 + 〈p2〉 = 0: The measurement values, or more precisely their fluctuationsaround the mean value, are always exactly anti-correlated. Both particles are said to be entangled.Measuring the momentum observable p1 thus gives the exact value of the measurement of p2. It ison the other hand possible to measure the position observable x2 to arbitrary precision when wedon’t actually measure p2, but rather infer it from the momentum-correlated daughter particle.

So by measuring the observables p1 and x2, we gain knowledge of the values 〈p2〉 and 〈x2〉,and their variances are not subject to the uncertainty relation. From this surprising, seeminglyparadoxical result EPR argued that the description of reality according to quantum mechanicswas incomplete, since the momentum values were undetermined until measured, yet strictlyanti-correlated between both particles irregardless of distance or temporal order of measurement.This would imply what Einstein famously called a “spooky action at a distance” in order to ensurethe correlated measurement outcomes. However, the terminus entanglement was chosen overEinstein’s for this phenomenon.

1.2 Nonlinear medium polarization and three-wave mixing 3

Despite the initial skepticism, notably by EPR themselves as well as proponents of alternativehidden variable theories[15], it is well-accepted in the physics community today that entanglementexists and is a fundamental property of quantum mechanics, rather than a theoretical quirkexplained by hidden variables, a larger deterministic theory of which quantum mechanics is but astatistical approximation. This has to be attributed to John Bell[13] as well as Clauser et al.[36],who formulated quantitative boundaries for the validity of certain hidden variables theories, whichwere first shown to be violated in the pioneering experiments of Freedman et al.[51] and Aspect etal.[5, 6, 4].

1.2 Nonlinear medium polarization and three-wave mixing

Typically, optically transparent media are electric insulators, so electrons inside dielectric medium,e. g. glass, are not freely mobile as they are in metals. Yet applying a weak electric field ~E candisplace the electrons from their rest positions, giving the medium an overall polarization intothe opposite direction in response: ~P = ε0χ~E. The material-dependent polarizability constantχ reflects a linear dependency between electric field and charge carrier displacement. However,the electrons’ potential inside the medium can be considered harmonic for small displacementsonly. For higher field intensities, non-harmonic terms in the electrons’ potentials start to makethemselves felt, and we have to generalize the polarization response to

~P = ε0χ(1) ~E + ε0χ

(2) ~E ~E + ... (1.2)

The third rank tensor χ(2) describes the second order nonlinear polarization response ~P (2) =ε0χ

(2) ~E ~E which is quadratic in the electric field. Through polarization of the medium’s chargecarriers, an opposed electrical field is built up, and there is a back-action on the original electricfield. The associated potential energy V ∝ ~P (2) ◦ ~E is already cubic in ~E, and thus an opticalmedium’s second order nonlinearity mediates a third order self-interaction of an electric field.

This self-interaction is widely exploited for the frequency conversion of electromagneticwaves. In the simplest case, the incident field is just a harmonically oscillating wave ~E(~x, t) =~E0cos

(~k~x− ωt

). Now, the second order polarization term is also time-dependent

~P (2)(~x, t) ∝ χ(2) ~E(~x, t) ~E(~x, t) ∝ cos(~k~x− ωt

)2=

1

2cos(

2~k~x− 2ωt)

+1

2. (1.3)

and the quadratic dependency on the electric field gives rise to oscillations at the doubled fre-quency, the second harmonic of the incident wave. The process is consequently named secondharmonic generation (SHG). The constant term corresponds to an additional static polarizationterm. The oscillating polarization of the medium constitutes a continuous distribution of radiationemitting dipoles at frequency 2ω, but they emit at different phases, varying with position ~r propor-tional to the doubled wave vector 2~k. In general, this will cancel out any effective radiation outputof the second harmonic by destructive interference, unless the phase variation of the polarizationvector ~P is exactly the same as the in-medium phase variation of the second harmonic frequency,so that constructive interference allows the coherent build-up of a wave. This condition is knownas phase-matching, and can be expressed in terms of the associated wave vectors: ~k(2ω) = 2~k(ω).

The monochromatic field self-interacting through second order nonlinearity is just a specialcase though. Let the incident electric field ~E be composed of two oscillating fields ~E1 and ~E2 atdifferent frequencies:

~E(~x, t) = ~E1 (~x, t) + ~E2 (~x, t) = ~E1cos(~k1~x− ω1t

)+ ~E2cos

(~k2~x− ω2t

)(1.4)

4 1 Introduction

Besides the SHG waves, the quadratic dependency on the electric field gives rise to oscillations atadditional frequencies

~P (2)(t) ∝ 2 + cos(

2~k1~x− 2ω1t)

+ cos(

2~k2~x− 2ω2t)

+ cos(~k+~x− ω+t

)+ cos

(~k−~x− ω−t

)(1.5)

with ω± = ω1 ± ω2 and ~k± = ~k1 ± ~k2. The process generating ω+ is called sum frequencygeneration (SFG) and the corresponding process generating ω− is called difference frequencygeneration (DFG). Both are in principle always present as soon as two light beams at differentfrequencies overlap inside a χ(2) -nonlinear medium. But again, the coherent build-up of anoutput wave occurs only if the phase-matching condition ~kω± = ~k1 ± ~k2 is fulfilled. SHG is aspecial case of SFG with ω2 = ω1. The constant polarization contribution is the remnant of theDFG process with ω− → 0.

1.2.1 Sum- and difference frequency generation

Through canonical field quantization, we can express classical waves as superpositions of elemen-tary excitations of the electromagnetic field. Each light mode is modeled as a quantum mechanicalharmonic oscillator, and the modal excitations are photons. Fig. 1.1 is a representation of SFG andDFG on the single photon level, reminiscent of a Feynman diagram. In the SFG case depicted inFig. 1.1(a), two photons “fuse” to form a third photon at the sum frequency. Its frequency matchingcondition of SFG can, after multiplication with Planck’s constant, be read as energy conservationbetween two input photons and one output photon:

~ω1 + ~ω2 = ~ω+ (1.6)

Similarly, phase-matching can be re-interpreted as conservation of the photons’ crystal pseudo-momentum ~~k:

~~k1 + ~~k2 = ~~k+ (1.7)

For DFG, depicted in Fig. 1.1, the situation is more complicated: A photon with frequency ω1

is stimulated by the presence of light at a lower frequency ω2 to distribute its energy betweenanother photon of the same frequency ω2 and a third photon at the difference frequency ω− =ω1 − ω2. DFG not only generates the difference frequency output wave, but also amplifies thelower frequency input wave.

1.2.2 Spontaneous parametric downconversion

The second order polarization term that gives rise to DFG scales like the product of the inputwaves P (2) ∝ E1E2. Classically this means that when reducing the amplitude E2 to zero, no DFGwill take place. But the quantum vacuum still has non-zero vacuum field fluctuations, even if themean value of the electrical field is zero. If the DFG phasematching conditions ~k1 − ~k2 = ~k−

are fulfilled, those vacuum fluctuations are sufficient to excite the spontaneous decay of some ofthe high energy input photons into photons at ω2 and ω−. This purely quantum effect is calledspontaneous parametric downconversion (SPDC or PDC). One “pump” photon decays into one“signal” and one “idler” photon (Fig. 1.2). Since vacuum fluctuations for any frequency ω2 exist,PDC will in principle occur for all energy conserving combinations of ω2 and ω− at once. Butonly phase-matched processes will build up an output wave, all others will be suppressed bydestructive interference. But the creation of one photon pair is not the only possibility; rather,

1.3 Quantum light pulses 5

Figure 1.1: Three-wave-mixing on the single photon level. (a) SFG: Two photons “fuse” to a newphoton at the sum frequency. (b) DFG: One photon is stimulated by a lower energyphoton to distribute its energy between the stimulating photon mode and a differencemode.

as Fig. 1.3 illustrates, it must be considered to be a coherent superposition of the case of nointeraction, of one-pair-creation, two-pair-creation, and so on. This superposition is called asqueezed vacuum state. If signal and idler photons carry different polarizations, it implementsan EPR state in continuous variables. Only for small pair creation probabilities p � 1 we canconsider the probability to produce a pair of pairs p2 � p negligible in comparison and speak of aSPDC process as a probabilistic photon pair source.

The output field amplitude of SFG and DFG scales with the product of both input field ampli-tudes, but SPDC has only one input. Thus the output field amplitudes are a linear function of theinput field, making it a parametric process. While SFG and DFG efficiencies can be high enoughto deplete the energy of one of its input beams, for SPDC values of the order ηPDC ≈ 10−10 aretypical.

1.3 Quantum light pulses

The problem of localizing photons or, more generally speaking, localizing quantum states of light,has a long history[95] and stems from to the fact that standard field quantization as introduced byDirac in 1927[39] is based on monochromatic electromagnetic fields. This intrinsically impliescompletely de-localized plane wave solutions for the quantum fields. In 1966, Titulaer and Glauberintroduced broadband modes, polychromatic wave packets as single mode states by definingcontinuous superpositions of monochromatic modes[130].

Since time and frequency domain amplitudes of a wave are connected by the Fourier transfor-mation, a monochromatic state of light wave has a constant probability amplitude in the timedomain (c. f. Fig. 1.4a) and is therefore completely de-localized. A wave packet’s amplitude has afinite time duration (Fig. 1.4c) and thus forms a localized pulse. The pulse shape – the envelopefunction of a wave package – can be decomposed into broadband modes, a complete basis set oforthogonal weighting functions of the monochromatic modes. In this work, we are going to beconcerned with pulses with a duration of the order of 1 ps, right on the edge of what is generally

6 1 Introduction

Figure 1.2: Spontaneous parametric downconversion as a special case of difference frequencyconversion: The “stimulating” lower frequency field is the quantum vacuum.

Figure 1.3: SPDC process as a superposition of the creation of zero, one, two, three, ... photonpairs.

considered the ultrafast regime of femtosecond pulses.Generally we differentiate between classical and non-classical light states. Ultrafast pulsed lasers

can be under ideal conditions be considered sources of classical wave packets of light[56, 130].Experimentally, those were made available through the first mode-locked lasers in the 1960s[78],but only the relatively recent arrival of ultrafast self-mode-locked lasers[119, 103] in 1991 provideda reliable, high quality source of highly coherent femtosecond laser pulses. In particular thetitanium sapphire (Ti:Sa) solid state laser has become the “workhorse” laser source for ultrafastoptics experiments. In parallel, the first pulsed squeezed vacuum states, which are a class ofnon-classical two-partite light states, were generated with SPDC[114, 9]. Both developments,self-mode-locked lasers and pulsed wave mixing, paved the way for the first source of ultrafastsqueezed vacuum states[3] in 1995.

An aspect which sets apart ultrafast classical from the non-classical squeezed vacuum lightstates is their broadband mode structure. For the classical case, all sets of basis functions areequivalent. In contrast, squeezed vacuum states, or indeed all two-partite quantum states, carryin general an internal, well-defined, discrete bi-partite mode spectrum[49]. Even in the spectraldegree of freedom, which one would naturally consider to be continuous, this mode structure canbe found[67, 82], and it is determined by the spectral intra-correlations of the bi-partite squeezedvacuum. And since we match the intrinsic timescales of SPDC with ultrafast pulses, the effects ofthis internal structure will become much more distinct.

Those spectral correlations, and at the same time the intrinsic mode structure, can be manipu-

1.4 A quantum pulse source and a quantum pulse gate 7

Figure 1.4: Real spectral amplitudes (gray) and envelope functions (blue) of a monochromaticwave in (a) and (b) and of a wave packet superposition in (c) and (d), in time domainand frequency domain respectively.

lated by spectral engineering, that is by manipulation of the dispersion properties of the nonlineargeneration process as well as the spectral or spatial properties of its pump beam[60, 70, 28]. Itis even possible to eliminate the correlations altogether[59, 28]. In this special case only one ofthe bi-partite modes will be well defined, as photon pairs will be emitted into this mode only.All other modes are unpopulated and therefore degenerate. This way, the source emits quantumlight pulses, localized, single-mode ultrafast non-classical states of light. In general though, SPDCsource produces a whole ensemble of pairwise correlated quantum pulses[20]

In the single pair approximation for low pump pulse energies the quantum pulse source has theadditional advantage to produce two photons in completely uncorrelated, separable states, so thatby loss of one of the photons of the two-partite system its partner does not suffer from decoherence.By heralding one photon event in the signal arm we can generate pure single photons.

1.4 A quantum pulse source and a quantum pulse gate

In the course of this thesis we develop the means to efficiently generate EPR-entangled quantumlight pulses and manipulate their spectral mode structure. We implement a pulsed waveguideSPDC source in the telecom wavelength regime and demonstrate its high photon pair output, itsefficiency, and our control over its spectral entanglement that allows us to produce both spectrallysingle mode and multi-mode light pulses. The source advances its predecessor experimentpresented by Mosley et al. in 2008[93] considerably. By choosing a waveguide architecture overbulk crystal, we ensure emission into not only into one spectral, but also into one spatial signal andidler mode, implementing a source of genuine quantum pulses. Wave confinement of the pumpbeam over the waveguide length leads to a far greater interaction length so that both mean photonnumber and modal brightness of the output light at modest pump pulse energies far exceed priorexperiments.

Our source produces high quality continuous variable EPR states as well as pure heralded singlephotons at telecom wavelengths with multiple applications in discrete[75, 74] (i. e. single photon)and continuous variable quantum computing[84, 19] schemes. It is a particularly viable source ofresource states for continuous variable entanglement distillation[98, 37, 96], a protocol that can be

8 1 Introduction

used to reverse the deleterious effect of decoherence on the security of long distance quantumcommunication with a quantum repeater[25].

To manipulate the intrinsic spectral mode structure of a quantum state of light, we applythe same spectral engineering techniques we used for designing the SPDC waveguide source toSFG[105] and propose the quantum pulse gate (QPG). It is implemented by a mode-selective SFGprocess, i. e. a SFG that converts one well-defined broadband mode and transmits all orthogonalmodes. The mode to select is determined by the pump pulse form. This novel approach toquantum pulse manipulation allows to pick from an arbitrary SPDC squeezed vacuum state one ofits intrinsic quantum pulses and convert it to another wavelength, where it can be easily separatedfrom the rest with standard optical components. It can be used to de-multiplex an optical signalconsisting of several independent quantum pulses, thereby boosting the information channelcapacity for ultrafast, broadband pulses.

These tools, a source for creating genuine quantum pulses, and a device for manipulatingthem, will no doubt prove valuable to the emerging field of ultrafast quantum communication andcomputation.

2Basic concepts

In this chapter, we will briefly introduce the concepts necessary to understand the rationale andthe body of the theoretical and experimental work presented in this thesis.

2.1 Electromagnetic waves

In classical electrodynamics, we can describe monochromatic, linearly polarized electromagneticradiation idealized as a plane wave

~E(~x, t) = ~E0 cos(~k~x− ωt+ ϕ

)(2.1)

where ~k is the wave vector, ω = c∣∣∣~k∣∣∣ is the associated angular frequency, and ϕ is an arbitrary

phase. Throughout this work we will use angular frequency ω rather than plain frequency f = ω2π .

The electrical field amplitude vector ~E0 = ~eσE0 consists of the polarization direction unit vector~eσ and the field amplitude modulusE0. It is a solution to the homogeneous Maxwell equations thatgovern the dynamics of electric and magnetic fields in vacuum, and is completely characterizedby the unit vector of the polarization direction e, optical phase ϕ and wave vector ~k. ~k determines

its propagation direction k =~kk and frequency ω = ck. Any solution for a certain set of boundary

conditions is a light mode with respect to them, and the plane waves are monochromatic modes inunbounded vacuum conditions.

We can write for the (non-vectorial) electrical field strength

E(~x, t) = E0 cos(~k~x− ωt+ ϕ

)∝ aeı(~k~x−ωt) + a∗ e−ı(

~k~x−ωt) (2.2)

with the complex, dimensionless amplitude a ∝ E0eıϕ absorbing the phase ϕ. We now define the

amplitude quadrature X = Re[a] and the phase quadrature Y = Im[a] as a real representation ofthe field:

E(~x, t) ∝ Xcos(~k~x− ωt

)+ Y sin

(~k~x− ωt

)(2.3)

10 2 Basic concepts

Plane waves do not and cannot have an energy content assigned to them, since they expand overthe whole of space-time, and are as such unphysical. Nevertheless, superpositions of plane wavescan be used to describe finite, physical wave packages with a well defined energy. We can howeverdefine an energy density E for a monochromatic plane wave that is constant over space and time

E ∝ E20 +B2

0 ∝ X2 + Y 2 (2.4)

where E0 is the electrical and B0 the magnetic field strength of the wave. X and Y are theaforementioned field quadratures.

2.2 Electromagnetic field quantization

In the canonical quantization of the electromagnetic field, each plane wave light mode~k is modeledas a quantum mechanical harmonic oscillator. As such, its energy eigenstates will always containan integer number of field excitation quanta, or photons. The eigenstates span the Hilbert spaceof all possible states in this particular mode. The field quadratures X and Y take the roles ofdisplacement x and momentum p respectively from a mechanical oscillator. We then promotec-numbers to Hilbert space operators, and in the special case of real-valued physical quantities,to observables. The complex amplitude a and its complex conjugate a∗ are substituted with thenon-commuting photon annihilation and creation operators:

a→ a (2.5)

a∗ → a† (2.6)[a, a†

]= 1 (2.7)

For the field quadratures, we find

X = Re[a] =1

2(a∗ + a)→ X =

1

2

(a† + a

)(2.8)

Y = Im[a] =1

2ı (a∗ − a)→ Y =

ı

2

(a† − a

)(2.9)[

X, Y]

=ı

2(2.10)

Since both X and Y are Hermitian, they are observables, quantum mechanically measurable

properties such that X = 〈X〉 = Tr[ρX]

with ρ a general quantum state. However, since they do

not commute, they cannot both be measured to arbitrary precision at the same time. Finally, wecan express the electrical field observable in terms of the creation/annihilation operators

E(~x, t) ∝ a†e−ı(~k~x−ωt) + a eı(

~k~x−ωt) (2.11)

or alternatively, in terms of the quadrature operators

E(~x, t) ∝ Xcos(~k~x− ωt

)+ Ysin

(~k~x− ωt

). (2.12)

As their names suggest, the creation and annihilation operators create and destroy quanta ofthe electromagnetic field

a |n〉 =√n |n− 1〉 (2.13)

2.3 Field quadratures and squeezed light 11

a† |n〉 =√n+ 1 |n+ 1〉 (2.14)

where |n〉 is a pure state containing n quanta in mode ~k, and is called photon number state orFock state. Their normal-ordered product forms the Hermitian photon number operator for themode ~k

n = a†a |n〉 = n |n〉 (2.15)

with the eigenstates |n〉. They are identical to the energy eigenstates, and the Hamiltonian, i. e.the energy operator for a field mode describing free propagation is closely related to the photonnumber operator:

H =1

2~ω(

X2 + Y2)

=1

2~ω(

a†a + aa†)

= ~ω(

n +1

2

)(2.16)

It may be surprising at first glance that we can define a Hamiltonian operator instead of a Hamilto-nian density, when we could only give an energy density for a plane wave in classical electrody-namics. The reason for this is that in the classical case, we start from constant field amplitudes E0

and try to sum up their energy content over an infinite space, while the canonical quantizationimplicitly starts with the notion of finite energy quanta in one mode expanding over an arbitrarilylarge volume. For the quantized plane waves this results in infinitesimally small field amplitudes,and again, only wavepacket superpositions of plane waves describe physical light states with finiteamplitudes. These conflicting notions are reconciled by introducing a quantization volume towhich the radiation modes are confined, so that their energy is finite. A more modern take onfield quantization by Blow et al. [14] avoids this difficulty by using wavepackets from the outset.

2.3 Field quadratures and squeezed light

We have already defined the observables X and Y as a representation of an electromagnetic mode.In terms of the harmonic oscillator, they can be understood as analogues to displacement andmomentum of a harmonic pendulum in classical mechanics.

X =1

2

(a† + a

)(2.17)

Y =1

2ı(

a† − a)

(2.18)

They are the field quadrature operators[85, 12, 2] of a radiation mode and they are a pair ofcanonically conjugate observables, so their commutator is a non-zero constant[

X, Y]

=ı

2. (2.19)

As a consequence, it is not possible to measure both observables to arbitrary accuracy; therefore,their statistical variances 〈∆X2〉 , 〈∆Y2〉must obey a Heisenberg uncertainty relation

〈∆X2〉 〈∆Y2〉 ≥ 1

16. (2.20)

An arbitrary single-mode state of light ρ is fully described by its quadrature pseudo-probabilitydistribution in optical phase space, the Wigner function[109] W (X,Y ). This distribution allowsfor an intuitive illustration of certain properties of a quantum light state. To illustrate a few

12 2 Basic concepts

light states, we will make use of phase space diagrams which are a contour plot of the Wignerfunction W (X,Y ) = const. In classical electrodynamics, a monochromatic light field has awell defined field strength, and consequently sharp quadrature mean values 〈X〉 , 〈Y〉 with nostatistical uncertainty, a point in phase space. But owing to the Heisenberg relation 2.20, this isnot allowed in a quantum mechanical treatment: To fulfill it, both statistical variances 〈∆X2〉 and〈∆Y2〉must be non-zero, resulting in a distribution that is spread over phase space instead. Wecan understand this by looking at the so-called coherent light states.

2.4 Important classes of light states and their properties

2.4.1 Coherent states of light

Coherent states were first put forward by Erwin Schrödinger in 1928[111, 120] as a quantummechanical analog of the excitation states of a classical harmonic oscillator according to thecorrespondence principle. The original terminus for this class of state in Schrödinger’s work is“Wellengruppe” (wave group), which was meant to hint at the superposition of the energy Eigen-states that constitute each state rather than a multi-chromatic wave-packet in the spectroscopicsense.

The notion of coherent states of light was introduced by Roy Glauber starting in 1963[56], whenhe applied Schrödinger’s concept to the experimental findings of Hanbury Brown and Twiss [26, 27]who had shown that light from certain stellar sources exhibited different coherence properties thanfrom then-current (i. e. pre-laser) light sources on earth. Also in 1963, Sudarshan[123] showed thatany quantum state of light can be expressed as a superposition of coherent light states. Aroundthe same time, beginning with MASERs[57] in 1954 and later LASERs[88] in 1960, for the first timethere were practical sources of coherent light available.

Figure 2.1: Phase space representations of a coherent state (left) and the vacuum state (right)

A coherent state can be expressed as a coherent superposition of all Fock states |n〉

|α〉 = D(α) |0〉 = e−ı(αa†+α∗a) |0〉 = e−|α|2

2

∞∑n=0

αn√n!|n〉 . (2.21)

Its mean photon number 〈n〉 = |α|2 therefore exhibits a statistical spread, and so do its fieldstrength and quadrature values. The displacement operator D(α) generates a coherent state with〈X〉 = Re[α] and 〈Y〉 = Im[α] from a vacuum state and thus is a translation in phase space.

2.4 Important classes of light states and their properties 13

Coherent states are Gaussian minimum uncertainty states: Their Wigner function is a two-dimensional Gaussian distribution and they exactly fulfill 〈∆X2〉 〈∆Y2〉 = 1

16 , respectively. Also,their quadrature variances are equal 〈∆X2〉 = 〈∆Y2〉 = 1

4 , so that their phase space diagramsare always circular (Fig. 2.1 left). Under time evolution, they rotate counter-clockwise around thephase-space origin. The quantum vacuum |0〉 (Fig. 2.1 right) can be considered a special case ofthe coherent state with α = 0, located at the origin.

2.4.2 Single mode squeezed vacuum states

D. Stoler pointed out in 1970[121, 122] that coherent states are not the only class of Gaussianminimum uncertainty states. There are also the squeezed coherent states[29, 140], for whichthe equality of quadrature variances does not hold: 〈∆X2〉 6= 〈∆Y2〉. They derive their namefrom the elliptic shape of their phase space diagrams (Fig. 2.2 left), which can be imagined to besqueezed circular distributions of ordinary coherent states. For an extensive introduction, see e. g.[86]. In 1985, they were experimentally observed for the first time by Slusher et al.[115].

Figure 2.2: Left: Phase space diagram of a squeezed vacuum (solid) and the vacuum state (dotted).Right: Two-mode squeezed vacuum at t = 0 (dotted) and t > 0 (solid).

Formally, a squeezed coherent state is created by applying the squeezing operator Sa(ζ) to acoherent state α:

|α; ζ〉 = Sa(ζ) |α〉 = e−ı(ζ2(a†)

2+ ζ∗

2a2)|α〉 (2.22)

Remembering that the vacuum state is a special case of coherent states, we apply Sa(ζ) to vacuumto generate the single mode squeezed vacuum (SMSA) state

|ζa〉 = Sa(ζ) |0〉 =

√1− tanh(r)2

∞∑n=0

√(2n)!

n!

(− ı

2eıϕtanh(r)

)n|2n〉 (2.23)

with the squeezing parameter r = |ζ| and an optical phase ϕ = arg(ζ) that determines theorientation of the squeezing in phase space (Fig. 2.2 right). Surprisingly, the mean photon numberof the squeezed vacuum is not zero but rather 〈n〉 = sinh(r)2. The quadrature variances for ϕ = 0are 〈∆X2〉 = 1

4e±r and 〈∆Y2〉 = 1

4e∓r.

14 2 Basic concepts

Figure 2.3: Phase space diagram of both modes a and b of the two-mode squeezed vacuum state(dashed black shapes). Shot-for-shot comparison between quadrature measurementsfor both modes reveals correlated fluctuations (dotted and solid red shapes)

2.4.3 Two-mode squeezed vacuum states

Closely related are two-mode squeezed vacuum (TMSV) states

|ζa,b〉 = Sa,b(ζ) |0〉 = e−ı(ζa†b†+ζ∗ab) |0〉 =

√1− tanh(r)2

∞∑n=0

(−ıeıϕtanh(r))n |n〉 ⊗ |n〉 .

(2.24)A two-mode squeezed vacuum can be generated by mixing two identical SMSV states on a balancedbeamsplitter (represented by the unitary operator UBS):

Sc(ζ) Sd(ζ)→ UBSSc(ζ) Sd(ζ) U†BS = Sa,b(ζ) . (2.25)

This is reversible: Mixing both modes of a TMSV state on the same beamsplitter will result in twocompletely separable SMSV states. The TMSV state in contrast is entangled in photon number.Fig. 2.3 illustrates the effect of this entanglement on phase space measurements. For separatelymeasuring the phase space distribution of each mode, the result is a circular phasor around theorigin (black, dashed), with a larger radius than the vacuum state, i. e. it is not a minimumuncertainty state any more. It represents a thermal state, the partial trace over one mode of the

input state: ρa = Trb[|ζa,b〉〈ζa,b|

]∝∑

n tanh(r)2n |n〉〈n|. If each measurement from each mode

is compared to the corresponding measurement from its partner mode, a correlation between themeasurement results 〈Xa〉 and 〈Xb〉 as well as 〈Ya〉 and 〈Yb〉 becomes visible: Fluctuations fromthe mean values are correlated ( solid red shapes and dashed red shapes in 2.3, respectively). Thisnon-classical correlation between the quadratures of both modes of the two-mode squeezed statecan be exploited to implement the EPR experiment and explains why in the context of continuousvariable quantum optics the state |ζa,b〉 is also referred to as the EPR state[46, 9, 99].

2.5 Ultrafast pulses

A truly monochromatic light wave with one sharp frequency value, according to the Fourierrelationship to its temporal amplitude, must have an infinite duration in time. It is therefore, justas a plane wave, a theoretical construct and can be only achieved approximately in experiment.Finite light waves therefore exhibit a spectrum of frequency components, and their spectrumgrows broader as the pulses grow shorter. A first quantum optical description of broadband modeswas given by Titulaer and Glauber in [130], and [116] gives a good overview over the theoreticalchallenges of introducing broadband creation and destruction operators.

2.5 Ultrafast pulses 15

2.5.1 Broadband mode operators

Nevertheless, many quantum optical problems are treated in terms of monochromatic photonsa†(ω) |0〉 for simplicity’s sake. In ultrafast optics, with pulse lengths of 1 ps and below, thissimplification stops being viable. To account for a coherent continuum of frequencies, oneintroduces a spectral function ξ0(ω) to integrate over a monochromatic single photon statea†(ω) |0〉 (c. f. Fig. 2.4 left): ∫

dω ξ0(ω)[a†(ω) |0〉

]. (2.26)

The expression can also be read as a broadband operator A0 acting on the vacuum state[∫dω ξ0(ω) a†(ω)

]|0〉 = A†0 |0〉 . (2.27)

For the broadband operators to create physical states, we need to impose smoothness and square-

-1

-0.5

0

0.5

1

-2 0 2

u0(x)u1(x)u2(x)u3(x)

Figure 2.4: Left: An arbitrary square-integrable function ξ0(ω) as weighting function for thebroadband mode creation operator A†0. Right: The first four of the orthonormalHermite function basis {ui} with σ = 1.

integrability on the complex-valued spectral functions ξ0, otherwise we cannot guarantee thatthe created photons are of finite duration or carry a finite amount of energy. We further requirenormalized spectral functions for our convenience:∫

dω |ξ0(ω)|2 = 1 (2.28)

In this work, unless otherwise stated, all spectral functions are understood as normalized. Fora complete set of orthonormal functions {ξj} that forms a basis of the space of smooth, square-integrable functions, broadband operators obey the standard commutator relations for creationand annihilation operators of the harmonic oscillator[

Aj , A†k

]= δj,k. (2.29)

In general, the broadband mode operators A†j are related to the monochromatic frequency modesa†(ω) simply by a basis transformation between the discrete and the continuous basis:

A†j =

∫dω ξj(ω) a†(ω)

a†(ω) =∑j

ξ∗j (ω) A†j(2.30)

16 2 Basic concepts

We see that creation operators for different, orthogonal broadband modes commute, and conse-quently photons created by those orthogonal operators live in completely different Hilbert spacesand do not interact at all:

A†1A†2 |0〉 = |1〉1 ⊗ |1〉2 ∈ H1 ⊗H2 (2.31)

Any commutator[Af , A

†g

]between two non-orthogonal spectral modes f and g can be easily

calculated by using the completeness of the functional basis {ξj} and writing f(ω) =∑

j fjξj(ω)

and consequently Af =∑

j f∗j Aj (likewise for g) such that:[

Af , A†g

]=∑j

f∗j gj (2.32)

A broadband operator applied to the vacuum A†f |0〉 creates a single photon wavepacket[130],localized in time and space, and hence in frequency and momentum as well. Analogous to themonochromatic case, where photons interfere only if they are of the same frequency, broadbandphotons fully interfere only if they share the same spectral function. Any difference in spectrumwill result in diminished visibility of interference, to the point where photons with orthogonalspectral functions will not interfere at all. This applies for both classical and quantum- or Hong-Ou-Mandel-interference[66] between non-classical states.

2.5.2 Functional orthogonality interval

The orthogonality of two spectral functions depends on the interval [ω1, ω2] over which the overlapintegral is evaluated: ∫ ω2

ω1

dω f∗(ω) g(ω) = 0 (2.33)

We usually assume the full range of real numbers R = ]−∞,∞[. But for frequencies ω, andlikewise photon energies E = ~ω, the correct choice is obviously R+

0 = [0,∞[, since negativevalues for both are ill-defined and unphysical. Nevertheless we will continue to use the “incorrect”approach throughout this work as an approximation:∫ ∞

0dω f∗ g =

∫ ∞−∞

dω f∗ g −∫ 0

−∞dω f∗ g ≈

∫ ∞−∞

dω f∗ g (2.34)

The approximation works at high frequencies ω0 and relatively low spectral widths σ and becauseof the square-integrability requirement for our spectra. The former means σ � ω0, while the

latter means that every realistic spectral function must vanish at least as fast as(ω−ω0σ

)ne−

(ω−ω0)2

2σ2

(with n an arbitrary non-negative integer) far from the central frequency ω0. Since the polynomialterm grows much slower than the exponential term decreases, an overlap integral over the negativeinterval R−0 = ]−∞, 0] will be negligibly small.

Having established the spectral overlap integration interval as R, we now can use the Hermitefunctions as a popular choice of spectral basis orthogonal on this interval. The Hermite functionsuω0,σ,j(ω) are the weighted, normalized Hermite polynomials Hn around the central frequencyω0 with the spectral width σ [82, 143]:

uσ,j(ω − ω0) =1√√π2jj!σ

e−(ω−ω0)2

2σ2 Hj

(ω − ω0

σ

)(2.35)

2.5 Ultrafast pulses 17

The first four Hermite functions are plotted in Fig. 2.4(right). When the spectral width is apparentfrom the context, or assumed to be constant and its actual value of no consequence, we abbreviatethe Hermite functions as uj . They are orthonormal on R∫ ∞

−∞dω u∗j (ω)uk(ω) = δj,k (2.36)

and u0 is the Gaussian distribution.

2.5.3 Broadband modes in the temporal domain

It has already been stated that a broadband single photon state |1〉 = A†i |0〉 can be consideredwavepacket that is localized in both time and space. When we want to consider the time domainrepresentation of an arbitrary set of mode operators {A†i} defined as

A†i =

∫dω ξi(ω) a†(ω) (2.37)

with orthonormal spectral functions {ξi}, we substitute the monochromatic photon creationoperator a†(ω) with its Fourier transform 1√

2π

∫dτ e−ıωτ a†(τ) and find

A†i =

∫dω ξi(ω)

1√2π

∫dτe−ıωτ a†(τ)

=

∫dτ

[1√2π

∫dω e−ıωτξi(ω)

]a†(τ)

=

∫dτ ξi(τ) a†(τ) .

(2.38)

After switching order of the integrals in Eq. 2.38, we have applied the Fourier transform to thespectral amplitude ξi(ω) and obtained the temporal amplitude function ξj(τ). Like its frequencycounterpart, it is square-integrable and must therefore vanish sufficiently fast for large or smalltime values τ , i. e. it is localized around τ = 0 within a temporal standard deviation στ . The resultis the time domain representation of the broadband mode operator A†j , which is form invariantunder Fourier transform with respect to the frequency domain representation. The orthogonalityof the function set {ξi} carries over to the Fourier transformed set {ξi}, which then constitutes acomplete basis of temporal amplitude functions. We can consider the operators {A†i} not onlya set of spectral mode operators, but at the same time as operators that create one photon in atextitpulse mode with a localized temporal distribution.

2.5.4 Pulse propagation and quantum mechanical phase

For a full description of an optical network of interacting quantum light pulses, one has to takeinto account quantum mechanical phase, yet the creation operator for an ultrafast pulse A†i doesnot feature an explicit time dependence. One could argue that any phase dependence is alreadyimplicitly contained in its complex spectral amplitude function ξi(ω). For considering one modeor one beam path this is sufficient, but for a dynamical treatment concerning multiple lightmodes/paths, for instance to describe an interferometer, we need an explicit expression. Startingfrom an arbitrarily determined time t0 = 0, we define a broadband single photon state

A†i =

∫dωξi(ω) a†(ω) (2.39)

18 2 Basic concepts

-1

-0.5

0

0.5

1

ω0

Re[ξ

0(ω

)e-i ω

τ ]

Frequency ω [THz]

t=0.0pst=0.5pst=4.0ps

0

0.5

1

-8 -6 -4 -2 0 2 4 6 8

ξ 0(τ

-t)

Temporal walkoff τ [ps]

t=0.0pst=0.5pst=4.0ps

Figure 2.5: Left: Spectral function ξ0(ω) e−iωt (real part) for a propagating pulse of 1 ps dura-tion, corresponding to 3 nm spectral FWHM at 1550 nm central wavelength. Right:Corresponding temporal amplitude function ξ0(τ − t)

In the Heisenberg picture, a monochromatic photon creation operator transforms under freepropagation in vacuum like a†(ω) → e−ıωta†(ω). Substituting this into the above frequencydomain definition of a broadband operator, we obtain

A†i →∫

dωe−ıωtξi(ω) a†(ω) (2.40)

In the time domain, the translation is straightforward:

A†i →∫

dτ ξi(τ) a†(τ + t) ≡∫

dτ ξi(τ − t) a†(τ) (2.41)

So the effect of propagation on an optical broadband pulse can be thought of as a transformation ofthe mode itself; a phase term is multiplied with the initial spectral amplitude: ξi(ω)→ e−ıωtξi(ω).Fig. 2.5 exemplifies this with spectra for a picosecond pulse for relative temporal walk-offs of0 ps, 0.5 ps and 4 ps. On the left, the real part of the frequency spectral amplitude is plotted,on the right hand its Fourier transform, the temporal amplitude. The overlap between spectrais Fourier-invariant, it is the same in both domains. With a 0.5 ps relative temporal walk-offis still considerable, with a walk-off of 4.0 ps it is negligible. In time domain, the reason fordecreasing overlap is the relative translation between two peaks, in frequency domain it is therelative oscillation.

We will as a rule resort to the implicit time dependency of the optical pulses, unless statedotherwise.

2.6 Nonlinear optical interactions and three-wave-mixing

Since photons do not carry a charge, electromagnetic waves do not interact in vacuum (at lowenergies), they merely can interfere with each other. In transparent media, an indirect interactionis introduced via the electric susceptibility χ(1), which reflects the mobility of charge carriersinside the medium and their response to an incident field, as we have already explained in section

1.2. In the simplest case this response, the dielectric polarization vector ~P, is linear in the electricfield

~P = ε0χ

(1)~E. (2.42)

2.6 Nonlinear optical interactions and three-wave-mixing 19

However, many dielectric media exhibit higher order susceptibility terms

~P = ε0χ

(1)~E + ε0χ(2)~E

~E + ε0χ

(3)~E~E~E + ... (2.43)

where the according interaction Hamiltonian is given by the interaction between the polarizationand the field itself over an interaction volume V :

HI(t) =

∫V

d~r~P(~r, t) ◦ ~E(~r, t) (2.44)

The χ(n) are the n-th rank susceptibility tensors of a medium and they give rise to n+ 1-wave-mixing. Which of their elements are non-zero is determined by the medium’s microscopic sym-metries. For instance the three-wave-mixing tensor χ(2) implies a non-isotropic dipole structurewith a preferred direction, so for an isotropic medium like glass the elements of χ(2) will vanish.

2.6.1 Emergence of frequency- and phase-matching conditions

Crystals with broken rotational symmetries however may exhibit non-zero χ(2) elements and thuscan support three-wave-mixing. To illustrate how this interaction comes about, we assume the

field operator ~E consisting of a superposition of three plane wave modes in arbitrary directions ~kiat arbitrary frequencies ωi with arbitrary linear polarization directions σi

~E(~r, t) =

3∑i=1

E~ki,σi(~r, t) ∝3∑i=1

√ωi

(a~ki,σie

ı(~ki◦~r−ωit) + a†~ki,σie−ı(

~ki◦~r−ωit)). (2.45)

Considering the interaction Hamiltonian of a medium that features only second order susceptibilityχ(2)

H(t) ∝ χ(2)E(~r, t) E(~r, t) E(~r, t) (2.46)

we will, after expanding the triple product of the field operator in terms of mode operators, find inthe following form

H(t) ∝∑

i,j,k∈{1,2,3}

(χ

(2)ijk a†~ki,σi

a†~kj ,σja†~kk,σk

e−ı(Σ~k◦~r−Σωt) + h. c.

)

+∑

i,j,k∈{1,2,3}

(χ

(2)ijk a~ki,σi a

†~kj ,σj

a†~kk,σke−ı(∆~k◦~r−∆ωt) + h. c.

) (2.47)

with Σ~k = ~ki + ~kj + ~kk, Σω = ωi + ωj + ωk and ∆~k = ~ki − ~kj − ~kk, ∆ω = ωi − ωj − ωk. Thefirst sum contains terms that create or destroy three photons in the modes 1, 2, 3, the terms ofthe second sum destroy one photon and create two photons and vice versa. The time evolution

operator of a Hamiltonian process is defined as U(t1, t0) = e1ı~∫ t1t0

dtH(t), so we have to considerthe Hamiltonian integrated over the time interval t0, t1. Since Σω is always positive, all terms

proportional to e±ı(Σ~k◦~r−Σωt) are rapidly oscillating, leading to destructive interference under

the integration. In contrast, terms proportional to e±ı(∆~k◦~r−∆ωt) will interfere constructively, ifand only if the phasematching equation

∆~k = 0 (2.48)

20 2 Basic concepts

and the frequency matching equation∆ω = 0 (2.49)

hold. Then, the rotating wave approximation is applicable, and the rapidly oscillating terms of theHamiltonian can be neglected in favor of the slowly varying terms. Equations 2.49 and 2.48 can berewritten to explicitly show energy and linear momentum conservation between destroyed andcreated photons by multiplication with the Planck constant ~. While it is certainly not surprisingthat energy and momentum are preserved in three-wave-mixing – after all, Hamiltonian mechanicsis based on assuming overall energy conservation – it is instructive to see the conditions for bothfrequency- and phase-matching emerge. We note that while energy- and momentum conservationcan in principle be violated with the creation or destruction of two or even three non-matchedphotons, this is possible only on very short time scales of the order of 1

∆ω or on very small lengthscales of the order of 1

|∆~k| respectively, in keeping with the time-energy and position-momentum

uncertainty relations.Equation 2.47 does contain all possible combinations of polarization for pump, signal and idler

wave. For brevity’s sake we will now commit to one configuration with polarization modes a, band c and drop the sums over the polarization indices σi. After discarding all rapidly oscillatingterms of the Hamiltonian, we find

H(t) ∝ a†~kab†~kb

c~kc e−ı(∆~k◦~r−∆ωt) + h. c. (2.50)

It is however a simplification to assume the k-vectors as given, so we have to integrate over allpossible vectors and write more generally

H(t) =

∫∫∫d3~ka d3~kb d3~kc ζ

(~ka,~kb,~kc

)e−ı∆ω t+ı∆

~k ~x a†~kab†~kb

c~kc + h. c. (2.51)

where ζ is the nonlinear coupling strength of the three wave mixing. In the following subsectionswe will concentrate on spontaneous parametric downconversion (SPDC) as a special case of threewave mixing, but all arguments can be made similarly for sum frequency generation (SFG) anddifference frequency generation (DFG).

2.6.2 SPDC in a channel waveguide with discrete spatial mode spectrum

A waveguide is a guiding structure restricting the electromagnetical wave propagation to one (chan-nel waveguide) or two dimensions (slab waveguide) by introducing a reflecting boundary[108].This can be a literal channel with a reflective coating, or a zone of higher refractive index inside ablock of a transparent dielectric medium. Through total internal reflection, the traveling wavesare confined to the waveguide volume. If a waveguide is small enough in diameter, it supportsa discrete set of spatial modes whose form depends on the transversal boundary conditions forthe electrical field. For a rectangular cross-section area the modes resemble two-dimensionalHermite-Gaussian modes, for waveguides with radial symmetry, like most optical fibers have, onecan observe Laguerre-Gaussian modes instead. These modes become more easily observable fortightly confined beams in small waveguides, since it is much more probable to excite a highly puremode when coupling into the waveguide.

Fig. 2.6 shows a schematic drawing of a waveguide chip of the kind used in this thesis. Multipledielectric waveguides are inscribed on the surface of the nonlinear material. The waveguidevolume features a refractive index n2 greater than that of the surrounding material with n1 and of

2.6 Nonlinear optical interactions and three-wave-mixing 21

Figure 2.6: Nonlinear optical chip with multiple dielectric surface waveguides. The left blow-upshows the input facet area, and the right one illustrates wave guiding through totalinternal reflection.

the air boundary with n0, so that total internal reflection can take place. The effective k-vector orpropagation constant β = ~k ◦ ~ex of a ~k-mode is the projection of ~k on the propagation direction.~k-modes are guided if their transversal momentum vector ~k⊥ = ~k − β~ex fulfills the waveguide’sboundary conditions, causing a discretization of the bound transversal mode spectrum. If a mode’sangle of incidence on the boundary is greater than the critical angle of total internal reflection

θcritical = arcsin(n1n2

), then only part of the mode is reflected. Such leaky modes are subject to

exponential decay during propagation through the waveguide. If the critical angle allows exactlyone mode to be guided with low losses, the waveguide is single-mode, otherwise it is multi-mode.

According to geometrical optics, a waveguide with a quadratic cross-section and perfectlyreflecting boundaries of width d enforces as boundary conditions for the transverse k-vector to bepart of the waveguide’s square reciprocal lattice with lattice constant πd , that is

~k⊥ =mπ

d~ey +

nπ

d~ez (2.52)

with m and n positive integers. We can now write for the propagation constant

k2x,m,n = β2

m,n = ~k ◦ ~k −π2(m2 + n2

)d2

. (2.53)

For high mode numbers such that β2m,n < 0, this implies a sudden mode cut-off due to losses,

apart from the gradual cut-off caused by the break-down of total internal reflection: An imaginarypropagation constant in a propagation term eıβx causes exponential damping. This very simplisticmodel does not take into account evanescent waves or wavelength-dependent effects, but it issufficient to demonstrate the impact of waveguide mode confinement on wave dispersion. Thecanonical approach to dermine the guide’s modal dispersion properties is to solve the Helmholtzequation for the waveguide’s refraction index profile, a partial differential equation. Its solutionsdetermine the modes’ electrical field distribution.

Due to the connection between waveguide modes and wave dispersion, in a SPDC processwe have to allow for different waveguide modes as an additional degree of freedom for all threephotons. Each mode features its own dispersion relation and this influences any phasematchedprocess it is involved in. The SPDC phase-mismatch must be indexed accordingly to account forany triple of modes:

∆k(mp,np,ms,ns,mi,ni) = β(mp,np)p − β(ms,ns)

s − β(mi,ni)i (2.54)

22 2 Basic concepts

The corrections to the phase-matching term eı∆kz for each mode triple cause a shift in the spectraof the output photons. In a multi-mode waveguide, there can be thus several spectrally distinct,concurrent SPDC processes due to waveguide modes[11, 44, 34]. The spectral shift between modetriples can be used to isolate the processes and directly observe the signal and idler modes[92].

We will assume that the waveguide source used in the thesis is perfectly single-mode forpump, signal and idler. Our experimental results, in particular in section 6.5, will vindicate thisassumption. We now simplify the model of our three-wave-mixing processes for the single-modewaveguide[61, 132]. We apply the plane wave approximation, i. e. we assume that the sole effectof the waveguide on the transversal field distribution is a confinement to its cross-section, andthat the wave fronts of guided light are perfectly plane. Since the waveguide enforces collinearpropagation directions for all three fields, we replace the wave vector with a scalar and simplifythe Hamiltonian to

H(t) ∝ a†b†c eı∆ωt∫ L

0dz e−ı∆kz + h. c. (2.55)

where the waveguide is traversed in z-direction and L is the waveguide length. This effectivereduction to a one-dimensional problem does not take into account spatial mode discretizationand wavefront distortion at all, but will serve for now to study basic features of waveguided SPDC.Evaluating the integral along the length of the waveguide, we obtain∫ L

0dz eı∆kz = L sinc

(∆kL

2

)eı

∆kL2 . (2.56)

2.6.3 Time evolution of the SPDC output state

In quantum mechanics, non-dissipative temporal evolution – of a quantum state ρ in the Schrödingerpicture, or of an operator in the Heisenberg picture – is described by application of the unitarytime evolution operator

U(t1, t0) = T exp

[1

ı~

∫ t1

t0

dt H(t)

](2.57)

with T the time ordering operator. Time ordering can be neglected for small interactions thatcan be treated perturbatively. In section 7.7 we consider the case of a strongly coupled three-wavemixing process and the effects of time ordering. The evolution of a quantum state ρ from time t0to time t1 is described by the similarity transformation

ρ(t1) = U(t1, t0) ρ(t0) U†(t1, t0) . (2.58)

Since the time evolution operator U(t1, t0) contains the time integral of the Hamiltonian operator,we will from now on use the effective Hamiltonian

H =1

~

∫ t1

t0

dt H(t) (2.59)

such that U(t1, t0) = e−ıH. Time integration over e−ı∆ω t – representing the interaction timeinside the waveguide – will give a result similar to the length integration, but since we are workingin the ultrafast regime where all involved light pulses are much shorter than the integrationtime, we assume an infinite integration interval and find the result converging towards a deltadistribution

∫∞−∞dt eı∆ωt = δ(∆ω), enforcing an exact frequency matching between signal, idler

and pump photon: ωp = ωs + ωi.

2.6 Nonlinear optical interactions and three-wave-mixing 23

2.6.4 Quasi-Phasematching

The occurrence of three-wave mixing between three plane waves depends on frequency- andphase-matching, putting severe constraints on the experimentally viable combinations of inputund output frequencies and polarizations. While there is basically no way around frequency-matching, since it is equivalent to energy conservation, the phase-matching conditions can berelaxed up to a point, by introducing a periodic inversion into the crystal domain structure. Thistechnique called quasi-phase-matching[108] causes the crystal medium itself to absorb part ofthe pump photon momentum, or conversely to contribute to the overall momentum of the outputphoton pair. Physically this can be permanently achieved for some materials like lithium niobate(LiNbO3) by applying a strong electrical, spatially modulated field. The domain structure ofother materials, such as potassium titanyl phosphate (KTiOPO4 ) can be manipulated by an ionexchange process. We model this by modulating the second order nonlinear tensor element with aperiodic sign flip function g(z) = sign

(sin(

2πzΛ

)). The effect of the sign flip function can be most

readily understood by expressing it by its Fourier series

g(z) =∑±m∈odd

1

meımkΛz (2.60)

with kΛ = 2πΛ the quasi-phase-matching vector and m the phase-matching order. When the

substitution χ(2) → g(z)χ(2) is introduced into the Hamiltonian, the phase-mismatch ∆k gainsan additional term kΛ representing the momentum exchange with the periodically poled crystalstructure

∆km = kp − ks − ki −mkΛ (2.61)

and phase-matching ∆k = 0 can be achieved for each order m ∈ {1,−1, 3,−3, 5,−5, ...}separately. Therefore, the effective Hamiltonian is a superposition of all possible orders m:

H ∝∫

dt a†b†c eı∆ωt∑m

1

m

∫ L

0dz e−ı∆kmz + h. c.. (2.62)

It must also be observed that each phase-matching order m contributes a factor 1m to the effective

coupling constant ζm = ζm , so that in the weak coupling regime anmth order process will generate

a 1m2 fraction of the photon pairs that a first order process (m = ±1) emits; thus the conversion

efficiency for higher order phase-matching decreases like 1m2 .

2.6.5 Classical undepleted SPDC pump

As a three-wave-mixing process, SPDC is a coupling of a populated light mode c to two vacuummodes a and b. The initial quantum state then reads

|ψ0〉 = |0〉a ⊗ |0〉b ⊗ |Ψ〉c . (2.63)

When we assume |Ψ〉c to be a coherent state |α〉 with a spectral amplitude α(ω) – justly so if wepump the process with a classical laser – so we can apply the left-hand eigenvalue equation ofthe mode’s annihilation operator c(ω) |α〉 = α(ω) |α〉. For a bright classical state with a photonnumber 〈n〉 = |α|2 � 1, we can use for the creation operator the following approximation

c†(ω) |α〉 = c†(ω) D(α) |0〉 = D(α)(

c†(ω) + α∗)|0〉 = D(α) (|1〉+ α∗ |0〉) ≈ α∗ |α〉 . (2.64)

24 2 Basic concepts

Applying the Hamiltonian to the SPDC initial state then gives

H |ψ0〉 ∝[α(ω1 + ω2) Φ(ω1, ω2) a†(ω1) b†(ω2) + h. c.

]|ψ0〉 (2.65)

allowing us to drop all quantum mechanical operators acting on the pump state, and to treat it likea classical, static object. This is justified in situations where the pump is undepleted, i. e. wherethe actual photon down-conversion rate is low, and only a minuscle part of the pump beam isconverted. Accordingly, the now static c-mode is usually dropped from all equations, and theinitial SPDC state is written |ψ0〉 = |0〉a ⊗ |0〉b. Also by now, the two output modes are correlatedonly in frequency. This correlation is described by the joint spectral amplitude (JSA)

f(ω1, ω2) =1

Nα(ω1 + ω2) Φ(ω1, ω2) , (2.66)

which we will assume to be normalized:∫∫

dω1 dω2 |f(ω1, ω2)|2 = 1. The normalization constantN is absorbed by the Hamiltonian’s coupling constant ζ .

2.6.6 Broadband mode structure and Schmidt decomposition

If we want to express our classically pumped three-wave-mixing process in broadband modes,we have to decompose the joint amplitude f(ω1, ω2) into a superposition of separable pairs ofpulse forms. We choose two arbitrary spectral basis sets { ξi(ω) } and { ψj(ω) } and calculate theoverlap between all mode pairs and the amplitude function:

cij =

∫∫dω1 dω2 ξ

∗i (ω1) ψ∗j (ω2) f(ω1, ω2) (2.67)

Conversely, we can now re-construct the amplitude function from the spectral mode functionsand the overlap constants cij :

f(ω1, ω2) =∞∑i=0

∞∑j=0

cij ξi(ω1) ψj(ω2) (2.68)

The complex matrix c with an infinite number of rows and columns is a representation of thefunction f(ω1, ω2) with respect to the chosen basis sets. Among those there is exactly one specialcase that allows for a simpler representation. We can arrive at this special case by applying asingular value decomposition to c: Any matrix with a non-degenerate system of Eigen-vectors canbe expressed as

c = UcV† (2.69)

where U and V are unitary matrices and c is a diagonal matrix. Thus, by defining two new basissets { ξi } and {ψj } with

ξj(ω) =∑k

U∗kj ξk(ω)

ψj(ω) =∑k

Vjkψk(ω)(2.70)

we gain the diagonal representation of the joint spectrum:

f(ω1, ω2) =∞∑j=0

cj ξj(ω1)ψj(ω2) . (2.71)

2.6 Nonlinear optical interactions and three-wave-mixing 25

where cj ≡ cjj . This method allows to express any bivariate, smooth, square integrable functionf(ω1, ω2) ∈ L2 in a diagonal representation in terms of two complete orthonormal function sets{ ξj } and {ψj }. The choice of basis functions is unambiguous up to a complex phase. The matrixelements cjj are chosen to be real positive values, as any phase can be moved to the basis functions.For a normalized function f , they also obey the normalization condition

∑j |cj |

2 = 1.In terms of quantum mechanics this means that any physical, pure bi-partite system that

is correlated in one degree of freedom can be decomposed[67, 48, 82] into a superposition ofmutually orthogonal, separable bi-partite states:

|ΨAB〉 =∑j

cj |ξAj 〉 ⊗ |ψBj 〉 . (2.72)

This is usually referred to as the Schmidt decomposition[110] with the Schmidt modes { ξj } and{ψj } and the Schmidt coefficients cj .

2.6.7 Effective mode number and spectral entanglement of a photon pair

The number of separable superpositions necessary to express the entangled bi-partite state |ΨAB〉is at most countable infinite. The amount of entanglement between both partite systems A and Bcan be characterized by the number of superimposed states. For an infinite number of differentlyweighted superimposed modes, the effective mode number or cooperativity parameter or Schmidtnumber K is used:

K−1 =

∫dω1

∫dω2

∫dω3

∫dω4 f

∗(ω1, ω2) f∗(ω3, ω4) f(ω1, ω4) f(ω3, ω2)

=∑j

|cj |4(2.73)

For a separable state |ΨABsep 〉 = |ΨA

0 〉⊗|ΨB0 〉 and f(ω1, ω2) = f1(ω1) f2(ω2) it assumes its minimal

value K = 1. Bell states for instance have K = 2, and for a (unphysical) state in the limit whereall cj are equal and infinitesimally small, it diverges: K →∞. Bell states are often described asmaximally entangled states, but this is true only if there are two dimensions available, as is e. g.the case for polarization entangled photon pairs. A Schmidt decomposition of the spectral degreeof freedom yields in general an infinite number of modes, so two-partite systems can containmore entanglement, i. e. values K > 2 are allowed.

Here the bi-partite system is a two photon state produced – in first-order approximation –by SPDC in a waveguide with weak coupling |ζ| � 1. Both output photons are assumed to beentangled in the frequency degree of freedom only. The joint spectral amplitude is decomposedinto two complete sets of basis functions, and the output photon pair is written as a superpositionof broadband mode pairs

|Ψ〉 =e−ıH |0〉

≈(1− ıH

)|0〉

= |0〉 − ıζ∫∫

dω1 dω2 f(ω1, ω2) a†(ω1) b†(ω2) |0〉

= |0〉 − ıζ∑j

cj |ξj〉 ⊗ |ψj〉

(2.74)

26 2 Basic concepts

where |ξj〉 =∫

dω ξj(ω) a†(ω) |0〉 and |ψj〉 =∫

dω ψj(ω) b†(ω) |0〉. Alternatively, we can usethe broadband mode operators A†j =

∫dω ξj(ω) a†(ω) and B†j =

∫dω ψj(ω) b†(ω) to write the

photon pair state as|ψAB〉 = ζ

∑j

cjA†jB†j |0〉 . (2.75)

In Fig. 2.7, we show a graphical representation of such a two-partite state with Hermite modesas Schmidt modes. Product-state pairs of corresponding modes are superimposed to form thespectrally correlated photon pair state.

Figure 2.7: A superposition of Schmidt pairs forming a spectrally correlated two-photon state

2.6.8 Multiple squeezer excitation

One can write the effective Hamiltonian of the SPDC process as the sum over all Schmidt modepairs, and likewise the unitary time evolution operator of the waveguided SPDC can be expressed

as a product of broadband two-mode squeezing operators Sj(ζ) = e−ı(ζcjA

†jB†j+ζ

∗c∗j AjBj

):

UPDC = e−ıH = e−ı∑j

(ζcjA

†jB†j+ζ

∗c∗j AjBj

)=∏j

Sj(ζcj) . (2.76)

The additive Schmidt decomposition at single photon level corresponds to a decomposition intoa tensor product of the modes in the full description of the phenomenon, and accordingly, wecan separate the overall Hilbert space of output states into a tensor product of subspaces of thesqueezers Sj(ζ)

H = H0 ⊗H1 ⊗ ... (2.77)

so that we can view the according two-mode squeezed states as physically independent from eachother[20]. This decomposition of the Hilbert space for a multi-mode squeezer is also known asthe Bloch-Messiah reduction of the process.

2.7 Modeling photon detection with binary detectors 27

The resulting multi-mode squeezed vacuum state is then a product of orthogonal two-modesqueezed states

|Ψ〉 =∞∏j=0

Sj |0〉 =∞⊗j=0

√1− |λj |2

∞∑nj=0

(λjA

†jB†j

)nnj !

|0〉 (2.78)

with λj = tanh(|ζcj |) eıarg(ζcj). Interestingly, other than the approximated photon pair state in Eq.2.74, this full multi-mode squeezed vacuum contains no spectral entanglement, since the sub-statesof the individual Schmidt modes are in a product state, not in a superposition. Only through thenon-Gaussian process of photon detection, and thus removal of the vacuum contribution, we canobserve entanglement. Thus, when one speaks of the spectral entanglement of the two-modesqueezed state, one usually means that of the photon pair state.

Each squeezing operator

Sj = e−ıHj = e−ıζjA†jB†j+h. c. (2.79)

is parametrized by a coupling constant ζj = cjζ that determines the mean photon number nj ofthe generated light:

nj = 〈nj〉 = sinh(|ζj |)2 (2.80)

The squared modulus of the parameter λj is connected to the mean photon number of mode jthrough

|λj |2 = tanh(|ζj |)2 =sinh(|ζj |)2

cosh(|ζj |)2 =sinh(|ζj |)2

sinh(|ζj |)2 + 1=

njnj + 1

(2.81)

[85].The coupling of each single squeezing operator scales with pump beam power like ζj ∝

√Pp.

From their super-linear scaling of the mean photon number nj it follows that for high pump powervalues, the stronger squeezers begin to “out-shine” the weaker ones, because through strongerself-stimulation of the downconversion process their photon production relatively increases.

2.7 Modeling photon detection with binary detectors

We now consider the probability for a photon detection event with binary detectors, such asavalanche photo detectors or photo multiplier tubes. A binary detection event indicates the arrivalof one or more photons, but it is impossible to determine how many photons were present withoutan ensemble measurement.

2.7.1 Measurement operator

For an arbitrary single mode state of light ρ, the detection probability is

p = Tr[ρµ] (2.82)

where µ is the measurement operator of a binary detector, or in general a convex combination ofprojectors onto the basis subspaces of the Hilbert space of the problem. As input states we willconsider only pure states ρ = |Ψ〉〈Ψ|, since in this work we assume that we produce pure statesthat degenerate into mixed states through optical losses on the path from the light source to the

28 2 Basic concepts

ideal detector. Any such loss can be modeled by introducing a virtual beam-splitter, which in turncan be concatenated with the ideal detector into a lossy detector.

If only one single photon at one time is assumed to arrive at the detector, the correspondingmeasurement µ|1〉,ω operator is simply a projector on a single photon:

µ|1〉,ω = |1〉 〈1| = a†(ω) |0〉〈0| a(ω) . (2.83)

However this definition is specific to a single frequency ω. In order to measure photons of anyfrequency or frequency distribution, we have to integrate over all frequencies. Since in practicedetectors do not function with unit probability, we add as a weighting function its quantumefficiency ηQE(ω) which assumes real values between 0 and 1. If the spectral width of the lightstates in question are well within the quantum efficiency profile of the SPD, sometimes the quantumefficiency is assumed to be constant, and for ideal detectors one sets ηQE = 1. The measurementoperator for a lossy detector in the presence of at most one photon thus reads

µ|1〉 =

∫dω ηQE(ω) a†(ω) |0〉〈0| a(ω) . (2.84)

In a matrix representation, this operator has only diagonal elements, all coherences – all off-diagonal elements – have been washed out. This is not generally the case, but we here can write themeasurement operator like this because we implicitly assume a measurement time much longerthan the duration of our photon states[89]. For very short measurement times, the operator wouldrather have a form like

∫∫dω dω′ F (ω, ω′) a†(ω) |0〉〈0| a(ω′) where off-diagonal elements ω 6= ω′

can be different from zero.Sometimes it is more convenient, when dealing with ultrafast pulses, to express the measurement

operator in a broadband mode basis {ξj(ω)}. We make use of the basis transform in Eq. 2.30 thatallows us to rewrite the measurement operator for an ideal SPD as

µ|1〉 =∑j

A†j |0〉〈0| Aj . (2.85)

This simple single-photon approach however falls short when there may be more than one photonat a time impinging on the detector. Surely if there are two photons |2〉〈2|, the probability to detecteither or both of them must be higher than for detecting one single photon. Instead, since Fockstates are mutually orthogonal, we find p = Tr

[|2〉〈2| µ|1〉

]= 0. So we have to add projectors for

every Fock state except the vacuum state |0〉〈0| to catch any number of photons possible that cantrigger an detection event. The identity operator in photon number basis is just the sum of allFock state projectors 1 =

∑∞n=0 |n〉〈n|, and so we can write

µ =

∞∑n=1

|n〉〈n| = 1− |0〉〈0| (2.86)

This term is easily generalized for non-ideal SPDs in the multi-photon case by multiplying eachterm of the sum with the probability of a detection event. If n photons arrive at the detector, theprobability of a detection event and the probability of detecting no photon at all (1− ηQE)n mustadd up to unity, hence[113]

µ =

∞∑n=1

(1− (1− ηQE)n) |n〉〈n| = 1−∞∑n=0

(1− ηQE)n |n〉〈n| (2.87)

2.7 Modeling photon detection with binary detectors 29

If one wants to measure the detector click probability in the broadband regime where the quantumefficiencies ηQE(ω) for different photons vary significantly, one would have to allow for distinctfrequency variables per photon, so that the probability to detect no photon at all was rather∏ni=1 (1− ηQE(ωi)). However for sufficiently narrow-band light pulses we can consider ηQE(ω) a

slowly varying function around their central frequency and approximate it as a constant.

2.7.2 Measuring the joint spectrum of a photon pair

The measurement operator µc for coincidence events between two detectors is simply the productof two single detector operators in mode 1 and 2:

µc = µ1 ⊗ µ2 (2.88)

In order to spectrally resolve a photon pair created in a weakly pumped, ultrafast type II SPDCprocess we assume frequency dependent, lossy detectors, and introduce very narrow frequencyfilters with δ-peak transmissions at ω1 into the signal arm in front of the detector, and at ω2 intothe idler arm. For the detector efficiencies, we can therefore write

ηi(ω) = ηiδ(ωi − ω) (2.89)

with i ∈ {1, 2}. For low coupling strength ζ � 1, we neglect multiple pair creation and thereforeconveniently use as measurement operators µ|1〉,ωi and finds

µc =η1η2 a†(ω1) |0a〉〈0a| a(ω1) b†(ω2) |0b〉〈0b| b(ω2)

=η1η2 a†(ω1) b†(ω2) |0〉〈0| a(ω1) b(ω2)(2.90)

The infinitesimal probability to detect a coincidence event at frequencies ω1 and ω2 is thus

p(ω1, ω2) =Tr[µc |Ψ〉〈Ψ|]

=η1η2

∣∣∣∣〈0| a(ω1) b(ω2) ζ

∫∫dω′1 dω′2 f

(ω′1, ω

′2

)a†(ω′1)

b†(ω′2)|0〉∣∣∣∣2 (2.91)

By substituting the standard commutation relation[a(ω) , a†(ω′)

]= δ(ω − ω′), we further sim-

plify top(ω1, ω2) = η1η2 |ζ|2 |f(ω1, ω2)|2 (2.92)

For low squeezing ζ or low mean photon number, the infinitesimal probability p(ω1, ω2) to detecta coincidence event with signal frequency ω1 and idler frequency ω2 is proportional to the jointspectral intensity value |f(ω1, ω2)|2. Conversely, this means that by measuring spectrally resolvedcoincidence clicks, we can reconstruct the JSI of a PDC process. For higher photon numbers, therelationship between detection probability and joint spectrum is less straight-forward. We willinvestigate this matter in section 5.6.

To calculate an actual detection probability P , we have to define a frequency range for bothsignal and idler photon and integrate over them:

P =

∫ ω′s

ωs

dω1

∫ ω′i

ωi

dω2 p(ω1, ω2) (2.93)

In an actual measurement of the joint spectrum, one obtains the probability P for a range of“pixels” with size [ωs, ω

′s] × [ωi, ω

′i], and for a sufficiently small pixel size one can approximate

P[ωs,ω′s]×[ωi,ω′i]∝ |f(ωs, ωi)|2.

3Spectral engineering

The purpose of spectral engineering of a three-wave mixing process is to shape the spectralcorrelations between the participating light beams. In the SPDC case, these determine the jointspectral amplitude of the output photon pair. For frequency-degenerate SPDC sources, usingultrafast pump beams with a broadband spectrum can increase indistinguishably between signaland idler photon[60, 70], making for higher visibility in signal-idler quantum interference[38, 58,22].

In contrast, the idler photons from two identical but separate photon pair sources are necessarilycompletely identical in their spectral and temporal properties. Yet in general, they do not interferewith full visibility; quantum interference between two photons depends not only on spectral andtemporal overlap, but also on the coherence of the single photons. Since the mutual informationbetween entangled bi-photons introduces decoherence in the reduced systems of both singlephotons, we find optimal interference visibility between photons from two sources if those emituncorrelated, separable pairs.

To generate spectrally separable photon pairs from a ultrafast pumped SPDC source, onechooses a nonlinear optical material where the pump photon’s group velocity value is betweensignal and idler group velocity. For a pump photon with the right spectral – and thus temporal –variance, the output photon pair’s temporal – and thus spectral – inter-correlation is washed out.Since typically the group velocity is a monotonous function of wavelength in the visible regimeand beyond, we need a birefringent crystal and a wave mixing process involving both polarizationsto fulfill this condition called group velocity matching[59] (GVM).

It is well suited for separable photon pair generation in nonlinear waveguides[133], as it doesnot utilize the spatial distribution of the pump beam to shape the output spectrum[28, 136]. Inthe near infrared range, the nonlinear crystal KH2PO4 (potassium dihydrogen phosphate, KDP)has been proposed[134] as a candidate for a group velocity matched photon pair source, andalso has been implemented[93] in bulk crystal. For separable telecom wavelength photon pairs,KTiOPO4 (potassium titanyl phosphate, KTP) has been proposed[134]. GVM is also applicable tofour-wave-mixing in χ(3)-nonlinear media such as photonic crystal fibers[63, 117] or even standardoptical fibers[118].

32 3 Spectral engineering

3.1 Pure heralded single photons and the two-mode squeezer

One goal of this thesis was the realization of an ultrafast SPDC source of pure heralded singlephotons. Heralding single photons is achieved by first employing a weakly coupled PDC process(|ζ| � 1) to probabilistically produce photon pairs

||Ψ〉〉 = e−ıH ≈ |0〉 − ıζ |Ψs,i〉 (3.1)

where |Ψs,i〉 is photon pair which is created with a probability of approximately |ζ|2. Pairs ofphoton pairs are produced with a probability |ζ|4 � |ζ|2, and can be neglected for small |ζ|, socan even higher order contributions to |Ψ〉.

For a general collinear SPDC process, the photon pairs |Ψs,i〉 are frequency-entangled, andtherefore may be written in terms of broadband mode operators according to their Schmidtdecomposition:

|Ψs,i〉 =

∫∫dω1 dω2 f(ω1, ω2) a†(ω1) b†(ω2) |0〉 =

∑j

cjA†jB†j |0〉 (3.2)

In order to herald a single idler photon, we detect its corresponding signal photon with an idealsingle photon detector (SPD) module. The corresponding measurement operator µs is simply aprojector on a single signal photon:

µs = |1〉 〈1| =∫

dω a†(ω) |0〉〈0| a(ω) =∑j

A†j |0〉〈0| Aj (3.3)

To calculate the resulting reduced quantum state, we take the normalized, partial trace of the inputstate:

ρi =Trs[µs |Ψ〉〈Ψ|]Tr[µs |Ψ〉〈Ψ|]

=∑j

|cj |2 B†j |0〉〈0| Bj . (3.4)

The signal photon state ρi is a statistical mixture of the spectrally orthogonal pure single photonstates B†j |0〉〈0| Bj . This is problematic if the heralded single photon is to be used for any kind ofquantum optical experiment involving Hong-Ou-Mandel (HOM) interference[66, 85]. For HOMinterference, two single photons are overlapped at a balanced beam splitter. If their wave functionsoverlap perfectly, both will leave the beamsplitter through the same output port, an effect knownas bunching. For their wave functions to overlap, all their physical properties such as frequencydistribution, arrival time and transverse spatial distribution must be the same, such that, if onlyone photon travels through the beam splitter, it would be impossible for an observer to decidefrom which input port it had entered. The core of the problem here is that a state that can bedescribed by one wave function is necessarily a pure state, while mixed states have to be describedby an incoherent sum of pure states.

As an example, we consider a beam splitter where each input port is supplied with an identicalphoton ρs. As has been already stated, the broadband mode components |cj |2 B†j |0〉〈0| Bj aremutually orthogonal, and thus do not overlap at all. The probability for two photons to overlapand bunch is then simply the sum probability of two equal pure state photons coinciding, whichis at the same time the visibility V of the quantum interference and the inverse of the initialphoton pairs’ Schmidt number K (Eq. 2.73): pHOM = V =

∑j |cj |

4 = 1K . The cooperativity

parameter is a measure for the spectral entanglement in the initial photon pair |Ψs,i〉. A high

3.2 The phasematching distribution Φ and group velocity matching 33

amount of entanglement leads to a high value for K, a highly mixed heralded single photonstate ρi and a low HOM interference probability because of poor mode overlap. On the otherhand, the absence of spectral entanglement means that the Schmidt decomposition of the photonpair state yields c0 = 1 and for j > 0 : cj = 0, meaning that the photon pair is separable|Ψs,i〉 = |Ψs〉 ⊗ |Ψi〉, and the resulting heralded single photon is in a pure state ρi = B†0 |0〉〈0| B0.The cooperativity parameter assumes its minimum value at K = 1, and photon bunching occurswith unit probability pHOM = 1 due to perfect mode overlap.

3.2 The phasematching distribution Φ and group velocity matching

In the previous chapter we examined the connection between bi-photon spectral entanglement andheralded single photon purity. This entanglement is the consequence of the spectral correlationsin the underlying SPDC effective Hamiltonian’s joint spectral amplitude f(ω1, ω2), and photonpair separability translates directly into the separability of the amplitude function: f(ω1, ω2) =ψ0(ω1)ϕ0(ω2). It is also, up to normalization, the product of the pump spectral amplitude andthe phasematching function f(ω1, ω2) = α(ω1 + ω2) Φ(ω1, ω2). When we make a Gaussian

approximation for the phasematching Φ(ω1, ω2) ∝ sinc(

∆kL2

)≈ e−γ(

∆kL2 )

2

, the numericalconstant γ = 0.193 adapts the width of the Gaussian curve to that of the central peak of the sincfunction. We then do a Taylor expansion of the phase-mismatch ∆k(ω1, ω2) = kp(ω1 + ω2) −ks(ω1)− ki(ω2)− kΛ around the central frequencies for signal and idler beams, and find that the0th order vanishes, since we assume phase-matching. The first order is affine linear in signal andidler frequency

∆k(ω1, ω2) ≈ k′p(ω1 + ω2) (ω1 − ω1 + ω2 − ω2)− k′s(ω1) (ω1 − ω1)− k′i(ω2) (ω2 − ω2) (3.5)

with k′µ(ω) = ∂∂ωkµ(ω). Now the approximated Φ(ω1, ω2) is a two-dimensional Gaussian function,

just as the pump spectrum in terms of signal and idler frequency α(ω1 + ω2) for an ultrafastpulsed laser, and one can find a graphic understanding of bi-photon entanglement and separabilityby plotting the contours of both functions and of their product in the (ω1, ω2)-plane.

Figure 3.1: Frequency- and phase-matching plots for a SPDC process with negative phasematchingslope.

In Fig. 3.1, the multiplication of a negatively correlated pump function α and an also negativelycorrelated phasematching function Φ results in an elliptic shape with a negative correlation. Thisis very typical for SPDC bi-photon amplitude distributions, since α is always negatively correlated

34 3 Spectral engineering

with an angle of−45◦ due to frequency matching or energy conservation ωp = ω1 + ω2, and thephasematching curve normally has a negative slope as well: The phasematching angle Θpm of the

phasematching function Φ(ωo, ωi) ≈ e−γ(∆kL

2 )2

can be easily calculated: Φ is maximal on theline ∆k = 0, and the slope of this line at central frequencies ωs, ωi is according to Eq. 3.5

tan(Θpm

)= −

k′p(ωs + ωi)− k′i(ωi)k′p(ωs + ωi)− k′s(ωs)

. (3.6)

The group velocity v of a wave is related to its wave vector through v = ∂ω∂k ≈

1k′ , an approximation

that is exact for a wave in a bulk medium and still reasonable for a narrow waveguide. Withoutdiscussing its accuracy here, it can provide us with a better qualitative understanding of thephasematching slope’s physical significance. We can now express it alternatively as

tan(Θpm

)= −

v−1p − v−1

i

v−1p − v−1

s. (3.7)

meaning that we get a positive slope only if the pump group velocity has a value between signaland idler group velocities. We can reduce the spectral correlation by choosing a SPDC process

Figure 3.2: SPDC process with positive phasematching slope.

Figure 3.3: SPDC process with positive phasematching slope and matched pump width.

with a positively sloped phasematching curve, as depicted in Fig. 3.2. By additionally adjustingthe spectral width of the pump beam in Fig. 3.3, we can find an uncorrelated spectrum. Itis important to notice that this is only possible for a non-negative phasematching slope, or

3.2 The phasematching distribution Φ and group velocity matching 35

0◦ ≤ Θpm ≤ 90◦. With a negative slope, there is always residual correlation. In the corner cases ofa horizontal or vertical slope, pump width has to go to infinity to create a completely separabletwo-photon spectrum. Only for a positive slope, it is possible to find a finite pump width forspectral separability. In terms of frequency detunings νi = ωµ − ωµ we can find a formula todetermine if this is possible:

f(ν1, ν2) = α(ωo − ωi) Φ(ωo, ωi) ≈ exp

[−(ν1 + ν2)2

2σ2− γ∆k(ν1, ν2)2 L2

4

](3.8)

In order for f to be separable, any term of the exponential expression proportional to the productν1ν2, and therefore frequency correlated, has to vanish:

− ν1ν2

σ2− γL

2

4

(k′p − k′s

) (k′p − k′i

)ν1ν2 = 0 (3.9)

⇒σ =1

L

[−γ

2

(k′p − k′s

) (k′p − k′i

)]− 12

(3.10)

Since γ is a positive constant, the expression Eq. 3.10 has a real solution only if the outcomes of(k′p − k′s

)and

(k′p − k′i

)have different signs. This condition is equivalent to a positive phase-

matching slope, so that a spectral pump width for separability can only be found if the pumpgroup velocity is between signal and idler group velocity

vs ≤ vp ≤ vi (3.11)

and is therefore known as group velocity matching (GVM), even if exact matching is not necessary.For a given configuration, the product of crystal length and separability width is constant: If aspectral width σ for two-photon separability can be found for a crystal length L, then a crystalwith length 2L has a separability width σ

2 .We distinguish a special case[77] of solution to Eq. 3.9, where

vs = vp < vi. (3.12)

This condition we will refer to as critical group velocity matching. Its phasematching angle Θpm iseither zero or 90◦, depending on the interchangeable signal/idler labeling of the output beams.For a pump width σ significantly greater than the phasematching width σpm, the resulting jointamplitude shows only minimal correlations, as demonstrated in Fig. 3.4. The KDP bulk crystalsource implemented by Mosley et al.[93] utilized critical GVM.

It has to be remembered that we applied two approximations here, namely linearizing thephase-mismatch ∆k around the SPDC central frequencies and approximating the “sinc”-profile ofthe phasematching function with a Gaussian curve. For the exact case, the phasematching curve∆k(ωs, ωi) = 0 is curved instead of strictly linear, and the sinc profile causes additional maxima,so called side lobes, in addition to the main maximum of the bi-photon spectrum, potentiallyintroducing additional frequency correlation. Fig. 3.5 (left) illustrates the approximation of thesinc function by a Gaussian, and its impact on the two-photon spectral intensity (middle andleft). However, if one is using QPM to achieve phase-matching, it is possible to introduce aGaussian modulation to the periodic poling in order to suppress the sinc beatings and physicallyapproximate the phasematching function to a Gaussian distribution[17].

The effects of the curved phasematching contour can be largely avoided by choosing longercrystals, so that the spectral width of signal and idler photons, which scales with the inverse

36 3 Spectral engineering

Figure 3.4: SPDC process with zero phasematching slope, narrow phasematching width and aspectrally broad pump.

-0.5

0

0.5

1

-30 -20 -10 0 10 20 30Signal frequency

Idle

r fr

equency

Signal frequency

Idle

r fr

equency

Figure 3.5: Left: The function sinc(x) (blue) and the Gaussian e−γx2

(violet) to approximate itscentral peak. Middle: A joint spectral intensity function with the Gaussian approxi-mation for the phasematching Φ. Right: An exact joint spectral intensity function withthe first pair of “sinc” side lobes visible.

interaction length 1L , is small when compared to the radius of curvature. The sinc side lobes are

approximately perpendicular to the phasematching curve’s orientation, so for critical GVM thefrequency correlation they cause is mitigated for a spectrally broad pump.

3.3 Critical phasematching through backward-wave SPDC

We specifically chose as a source of squeezed light a nonlinear waveguide to enforce exactly onepropagation direction of the output states. There is however one more possible direction: Inwhat is sometimes called backward-wave SPDC[64], either one or both of the daughter photonstravel into the opposite propagation direction of the pump photon. The energy conservationωp − ωs − ωi = 0 still holds, but the phasematching equation – and every equation depending onthe photon pseudo momenta – has to be generalized from the collinear case to

kp ∓ ks ∓ ki − kΛ = 0 (3.13)

where we introduced a sign-flip for ks or ki, depending on whether either photon or both travelbackwards. Considering a degenerate type I SPDC process where the signal beam is generated as a

3.4 Type I SPDC 37

backward wave, we write down the adapted expression for the phasematching slope

tan(Θpm

)= −

k′p(ωs + ωi)− k′i(ωi)k′p(ωs + ωi) + k′s(ωs)

. (3.14)

where we substituted −k′s(ωs) → +k′s(ωs). The phasematching slope for collinear SPDC isexactly −1, since for type I we have signal and idler in the same polarization mode and thusk(ω) = ks(ω) = ki(ω). For the backward signal beam case, the slope is the quotient of thedifference and the sum of two numbers that are of the same order of magnitude:

tan(Θpm

)= −

k′p(ωs + ωi)− k′(ωi)k′p(ωs + ωi) + k′(ωs)

. (3.15)

Consequently, the smaller the difference between k′p(ωs + ωi) and k′(ωi) is, the smaller thephasematching slope tan

(Θpm

)becomes. For small angles, we approach critical GVM, so that

with a spectrally wide pump, we expect the generation of near-separable photon pairs frombackward-wave SPDC, in a basically material-independent way[102, 32]. The drawback in thisscheme is that the phase-mismatch kp + ks − ki = kΛ = 2πm

Λ is significantly higher than in thecollinear case. To compensate for it, one either has to produce very small QPM poling periods Λ,or use a high QPM order m. The former is just beyond the reach of today’s technology, while thelatter diminishes the produced photon pair flux by 1

m2 .

3.4 Type I SPDC

We have seen that in order to generate separable photon pairs from SPDC, we need to achieveGVM: A configuration where the pump beam’s group velocity is between signal and idler groupvelocity. For the materials considered here, the group velocity typically is a monotonous functionof wavelength or frequency in the visible to telecom regime. They exhibit an extremal groupvelocity at the first “zero dispersion wavelength” (i. e. group velocity dispersion and the related k′′

vanish). Since signal and idler wavelengths are always greater than the pump wavelength, GVMis simply not possible at wavelengths lower than the first zero dispersion wavelength, if all threebeams follow the same chromatic dispersion function k(ω). In Fig. 3.6, this situation is illustratedfor lithium niobate and a degenerate type Ia downconversion process: Pump, signal and idler arelinearly polarized along the “fast” axis, and their group velocities are plotted against wavelengthfor signal/ idler and against double wavelength for the pump beam. Their intersection marks adegenerate type Ia SPDC process for which exact group velocity matching vp = vs = vi is possible:

1300 nm→ 2600 nm + 2600 nm (3.16)

But if evaluated naively, the expression for the phasematching slope from Eq. 3.7 yieldstanϕexact

pm = 00 and in order to calculate a well-defined value, we have to apply the theorem

of l’Hospital and once more derive both numerator and denominator with respect to frequency:

tan(

Θexactpm

)= −

k′′p(ωs + ωi)− k′′i (ωi)

k′′p(ωs + ωi)− k′′s (ωs)(3.17)

Assuming neither beam is at a zero dispersion point where k′′{p,s,i}(ω{p,s,i}

)= 0, we then have

thanks to ks = ki a slope of tan(

Θexactpm

)= −1. This is the case for every wavelength-degenerate

38 3 Spectral engineering

0.3

0.4

0.5

500 1500 2500 3500 4500

Gro

up v

elo

city

[c]

Wavelength [nm]

signal/idler (fast)pump (fast, x-axis x2)

Figure 3.6: Group velocity matching in lithium niobate for type-I SPDC

type I SPDC process, independently of phasematching. We can understand this from simplesymmetry considerations: The phasematching curve ∆k(ωs, ωi) = 0 for a type I process must besymmetric in its arguments ωs, ωi; after all, “signal” and “idler” are just labels for two photonswith the same polarization, and thus the same material dispersion. Also, the curve must becontinuously differentiable. For both conditions to hold around the point ∆k(ω, ω) = 0, it mustintersect with the symmetry axis ωs = ωi at a right angle, and thus it must have slope−1 in itsvicinity.

As a physical consequence, neither degenerate type Ia nor type Ib SPDC processes, even if groupvelocity matched, can generate spectrally separable photon pairs, as the intersection of pump andphasematching distribution will always result in a negatively correlated joint spectral distribution.Separability is however in principle achievable for frequency non-degenerate type I processes, butsince the goal of this work was a degenerate source for the telecom regime, this direction is notfurther pursued here.

3.5 Type II SPDC

Birefringent materials do not only exhibit polarization-dependent refractive indices, but in turnalso polarization dependent group velocities. The existence of one “fast” and one “slow” polar-ization axis (see Fig. 3.7) gives rise to new possibilities for GVM: In Fig. 3.8, we see that thegroup velocity curve for the fast pump beam (blue) – again plotted against double wavelength forconvenience – intersects with the slow output beam’s curve (green) at lower wavelengths than withthe fast output beam (red). Both figures plot the group velocity of a fast- and a slow-polarized lightwave in lithium niobate. Between the intersecting points, the GVM condition Eq. 3.11 is fulfilled,the phasematching slope is positive, and we can find a finite spectral width for the pump beam tocreate spectrally separable SPDC photon pairs, where one “fast” photon is converted to a “fast”and a “slow” one.

3.6 Survey of nonlinear waveguide materials for group velocity match-ing

Choosing one nonlinear crystal of a length L, a polarization type, and an operating temperature,it is now possible to plot the separability width for a SPDC process in dependence of signal and

3.6 Survey of nonlinear waveguide materials for group velocity matching 39

0.3

0.35

0.4

0.45

0.5

500 1000 1500 2000 2500 3000 3500 4000G

roup v

elo

city

[c]

Wavelength [nm]

fast axisslow axis

Figure 3.7: Group velocities of the fast (red) and slow (green) polarization in lithium niobate

0.3

0.4

0.5

500 1500 2500 3500 4500

Gro

up

velo

city

[c]

Wavelength [nm]

signal/idler (fast)signal/idler (slow)

pump (fast, x-axis x2)pump (slow, x-axis x2)

Figure 3.8: Group velocity matching in lithium niobate for type-II SPDC

idler central wavelengths. Since it is our goal to implement a waveguide SPDC source for separablephoton pairs, we can restrict ourselves to collinear processes. Also, among the nonlinear crystalswidely used for frequency conversion, not all can be used to produce wave-guiding structures.The most common material for nonlinear waveguide devices by far is lithium niobate (LiNbO3),but also the structurally similar lithium tantalate (LiTaO3) is used. In recent years, potassiumtitanyl phosphate (KTiOPO4 or KTP for short) has seen increased adoption, not the least due tofavorable group velocity matching properties for infrared applications.

In the following sections, we will provide an overview of the possibility of group velocitymatched, frequency degenerate and collinear type II processes by solving equation 3.10 for differentcrystals with varying working parameters. The Sellmeier equations to determine crystal dispersionare the standard versions for bulk crystal at room temperature. As interaction length we assumedL = 10 mm throughout. If a real-valued solution σ exists, a PDC process pumped with a Gaussianspectrum with variance σ or with FWHM 2.35σ will, in the low power limit, produce photon pairswith an approximately separable joint spectral amplitude f(ωs, ωi). Each process is characterizedby the nonlinear coefficient dxy, since besides giving the coupling strength, it unambiguouslyidentifies the polarizations of pump, signal and idler. Owing to the crystal properties of eachmaterial, we can discard some processes with dxy = 0 right away.

Our objective is to identify a suitable low-loss nonlinear crystal to produce waveguides forspectrally separable photon pairs with feasible working parameters, such as pump spectral width,crystal temperature and QPM poling period. The degenerate signal/idler wavelength should be

40 3 Spectral engineering

in the telecom regime, specifically at 1550 nm. Operating at this wavelength allows using andintegrating with existing technologies: low-loss optical standard fibers are commercially available,as well as efficient single photon detectors and photo diodes, and a large array of optical devices.

3.6.1 Lithium niobate

0

20

40

60

80

100

1500 2500 3500 4500

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

d24d34

Figure 3.9: Left: GVM in lithium niobate (LiNbO3). Right: Comparison between d24 and d34process.

One of the most common versatile materials in modern optics is the uni-axial crystal lithiumniobate. With its strong χ(2) nonlinearity and its electro-optical properties it is used in manythree-wave-mixing applications as well as for devices like electro-optical modulators (EOMs).Due to its ferroelectric properties, its domain structure can be manipulated by applying strongelectrical fields, which is exploited to create periodic poling structures for quasi-phasematchinginside bulk crystal samples.

In Fig. 3.9, we calculate the pump FWHM necessary to create separable, frequency degeneratephoton pairs at signal/idler wavelength λ for the PDC processes corresponding to the nonlinearcoefficient d24 (red) and d34 (green). Both have cross-polarized signal- and idler-photons, butthe former’s pump is y-polarized, while the latter’s is z-polarized. For d34, separability becomespossible for a signal/idler wavelength of 1615 nm, which is close enough to the telecom wavelenghtsat 1550 nm to be of interest. However, the tensor element d34 is zero due to the crystal symmetryof lithium niobate. For d24 the separability range begins around 2700 nm, which is too far into theinfrared for any kind of single photon detector to see. Yet we now investigate wether manipulatingexperimental parameters can improve the situation.

The Sellmeier equations of lithium niobate[45] show a large temperature dependence. In Fig.3.10 (left), the GVM spectral width for d24 is plotted for temperatures between 193 K and 493 K,or between −80 ◦C and 220 ◦C. The lowest GVM wavelength rises with the temperature. Evenwith severe cooling, GVM starts only at 2690 nm, assuming that the Sellmeier equations are stillvalid in this temperature regime. Often lithium niobate is heated to temperatures over 393 ◦C tolower the light absorption and emission from color centers in the crystal structure.

Inside a waveguide, the field confinement changes wave dispersion and group velocity. Thus,the lower GVM threshold is also sensitive on waveguide size. In the simplified waveguide modelbased on geometric optics (c. f. section 2.6.2), we define an effective dispersion relation keff(ω) =

β =√k(ω)2 − 2π

2

b2, where k is the bulk crystal dispersion and b is the width and height of a

rectangular, perfectly internally reflecting waveguide. Substituting keff into equation 3.10 has a

3.6 Survey of nonlinear waveguide materials for group velocity matching 41

0

20

40

60

80

100

2680 2690 2700 2710 2720

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

T=193KT=293KT=393KT=493K

0

20

40

60

80

100

2000 2200 2400 2600 2800

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

bulk crystal6 µm5 µm4 µm

3 µm

Figure 3.10: Left: d24 process GVM in bulk PPLN for different temperatures. Right: d24 process at200K for different waveguide sizes.

noticeable effect in GVM: In Fig. 3.10 (left), the bulk crystal situation is compared to waveguidesizes down to 3µm, and the lower GVM threshold drops to 2050 nm for the smallest waveguidesize. One should keep in mind that this result though that this result is based on geometric optics,so for small structure in the order of the target wavelength they will quantitatively differ fromthe results of a rigorous solution of the Maxwell equations to determine waveguide dispersion.The qualitative trend however is clear enough: Small diameter PPLN waveguides can produceseparable two photon pairs at significantly lower wavelengths than bulk PPLN, yet the wavelengthsare still too high to be of practical use to us.

3.6.2 Lithium tantalate

0

50

100

150

200

250

300

350

400

2400 2500 2600 2700

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

d24

Figure 3.11: GVM in lithium tantalate (LiTaO3).

Lithium tantalate (LiTaO3) chemically differs from LiNbO3 by only one atom from the samegroup of elements in the negative ion, and shares the same crystal symmetry group (trigonal 3m),so their mechanical, chemical, and electrical properties are similar. However there is a markeddifference in optical dispersion, which impacts on the GVM situation. In Fig. 3.11, we see thatGVM is possible only in a small small spectral range around 2500 nm.

Fig. 3.12 shows that while cooling and heating the crystal does little to improve the situation,waveguide size has a large impact on GVM and shifts the threshold down to 1900 nm for 3µmside length. This still falls short of the goal of telecom wavelengths, and along with the enormous

42 3 Spectral engineering

0

50

100

150

200

250

300

350

400

2450 2500 2550 2600

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

T=193KT=293KT=393KT=493K

0

50

100

150

200

250

300

350

400

1800 2000 2200 2400 2600

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

bulk crystal6 µm5 µm4 µm

3 µm

Figure 3.12: Temperature dependence (left) and waveguide size dependence (right) of GVM inlithium tantalate (LiTaO3).

spectral pump width of around 50 nm to 100 nm needed for photon pair separability, it makesLiTaO3 unsuitable for our purposes.

3.6.3 Potassium niobate

0

5

10

15

20

1500 2000 2500 3000

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

d24

Figure 3.13: GVM in bulk potassium niobate (KNbO3).

Potassium niobate (KNbO3) is a bi-axial nonlinear crystal that has not been in wide use inintegrated quantum optics, although the bulk crystal is used as a PDC source[100] and waveguidesfor SHG have been produced[50]. Care must be taken when handling this material, as mechanicalstress or even vibration can lead to the formation of unwanted crystal domains. With Fig. 3.13, weidentified one type-II processes of interest: d24 with propagation direction x, with GVM startingto be possible at roughly 1500 nm.

We investigate the temperature and waveguide size dependence of the dispersion in Fig. 3.14. Asbefore, the GVM threshold rises with both temperature and waveguide size, so that we can expecta waveguide at room temperature to emit spectrally separable photon pairs with a pump width of2 nm.

3.7 Conclusion 43

0

5

10

15

20

1300 1500 1700 1900 2100 2300

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

T=193KT=293KT=393KT=493K

0

5

10

15

20

1400 1450 1500 1550 1600

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

bulk crystal6 µm5 µm4 µm

3 µm

Figure 3.14: Temperature dependence (left) and waveguide size dependence (right) of GVM inpotassium niobate (KNbO3).

0

5

10

15

20

1250 1550 1850 2150 2450

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

d24

0

5

10

15

20

1150 1350 1550

Pum

p s

pect

rum

FW

HM

[nm

]

Signal/Idler wavelength [nm]

bulk crystal6 µm5 µm4 µm

3 µm

Figure 3.15: Left: GVM in bulk potassium titanyl phosphate (KTiOPO4). Right: GVM withwaveguide dispersion

3.6.4 Potassium titanyl phosphate

Finally we come to potassium titanyl phosphate (KTiOPO4, or KTP), a mechanically robust, bi-axial nonlinear crystal, and a popular choice for nonlinear and quantum optics applications, oftenas a source for SPDC photon pairs (see e. g. [3, 135, 79, 44, 17]). As we see in Fig. 3.15 for the d24

process in bulk crystal, GVM occurs at room temperature from 1200 nm up to almost 2500 nm,and has indeed been shown for photon pairs at 1580 nm[79]. From Fig. 3.15 (right), we ascertainthat waveguide dispersion does not introduce drastic changes for the telecom wavelengths around1550 nm. The necessary pump width for separability ranges, depending on waveguide size, from2 nm to 2.5 nm at 775 nm central wavelength, which is equivalent to pulse lengths of about 1 ps.

3.7 Conclusion

We have discussed spectral engineering by group velocity matching in collinear geometries andgeneralized it to anti-collinear geometries, and have investigated the suitability of several popularnonlinear materials for the implementation of a source of frequency-uncorrelated photon pairs attelecom wavelengths. To generate spectrally separable photon pairs in the telecom wavelengthregime, both KNbO3 and KTP work as bulk crystal and waveguide sources at room temperature.

44 3 Spectral engineering

LiNbO3 shows promise for small waveguide sizes, if such waveguides can be produced and are stillable to guide light modes at 1550 nm without high losses, while LiTaO3 is completely unsuitablesince GVM is possible only around signal and idler wavelengths of 2500 nm. We decided tobuild a KTP waveguide source based on the d24 process mainly due the wide spectral range inwhich separable photon pair production is possible, but also due to the commercial availability ofwaveguide chips, and prior experience with the material[44].

4A PP-KTP waveguide as parametric

downconversion source

For the generation of photon pairs, PDC sources are an established standard. Recent works haveshown that source engineering[59, 134, 79] is capable of producing separable two-photon states|ψ〉 = |ψs〉 ⊗ |ψi〉 from PDC [93]. This allows the preparation of pure heralded single photonsneeded in single photon linear optical quantum computing schemes[74].

Going beyond the single photon pair approximation, we find that in general PDC can be under-stood as a source of squeezed states of light[144, 141]. First observed in a four-wave-mixing processby Slusher et al.[115], squeezed states originally garnered interest due to the noise reduction in oneof their quadrature observables X, Y below the classical shot noise level, and found application inquantum-enhanced interferometry[29]. The availability of mode locked laser systems allowed thegeneration of pulsed squeezed states[114], albeit multimode ones[97], and a strong trend towardsminiaturization and integration of PDC squeezing sources can be seen[3, 69, 104].

In more recent developments, the non-classical character of squeezed states has been harnessedas the basic resource in continuous variable (CV) quantum information processing protocolssuch as CV teleportation[52, 16], CV entanglement swapping[125] as well as advanced metrologyapplications[131]. A long standing goal of CV quantum information processing was entanglementdistillation[98, 37, 96] which can overcome transmission losses in wide area quantum commu-nication networks, as it is an essential building block of quantum repeaters[25], and has beenrecently demonstrated in experiment[124]. It has been shown that distillation needs non-Gaussianoperations[47, 54, 30], most commonly implemented by photon counting with avalanche photodiodes (APDs). Their inability to discriminate between “neighboring” spectral modes introducesmixedness and masks the quantum characteristics of a multimode squeezed state, which can neverbe completely compensated by narrow spectral filtering, as it cannot restore phase coherencebut always introduces additional losses[106]. The straight-forward solution is to use a two-modesqueezed state in the first place.

The outlined applications for a two-mode squeezer call for a compact, robust, and very brightsource, as the mean photon number generated is related to the amount of usable squeezing.We find these favorable properties in nonlinear waveguide sources[3, 128], which are superior

46 4 A PP-KTP waveguide as parametric downconversion source

to bulk crystal sources in several ways: All downconverted light is emitted in tightly definedwaveguide modes, mode confinement allows the whole waveguide length to contribute to photonpair generation (pump is “always in focus”), and gain induced diffraction[72] is suppressed[3].

In this chapter we present an ultrafast waveguided type II PDC-based two-mode squeezer sourcein the telecom wavelength regime. We perform a g(2) correlation function measurement[129, 7]to characterize the phase coherence of our source’s output beams, demonstrating near thermalphoton statistics and thus full two-photon wave function separability. We show that our sourceemits two single mode light pulses with an extraordinarily high mean photon number andmoreover features the gain signature of a true two-mode squeezer. It generates separable two-photon states in the low pump power regime or generally two mode squeezed vacuum states bycombining the high nonlinearity of χ(2) processes with the enormous gains possible in nonlinearwaveguides and the modal control made possible by spectral engineering. It advances modalsource brightness over several orders of magnitude with respect to prior experiments[93, 79].

In general, we can describe a type II PDC process as a multimode squeezer in terms of broadbandfrequency modes with its interaction Hamiltonian

HPDC =∑j

Hj = ζ∑j

cj

(A†jB

†j + AjBj

)(4.1)

Aj and Bj are two sets of orthogonal broadband modes, and the Hj describe a set of non-interacting two-mode squeezers with coupling strength ζcj . For the special case where all coeffi-cients cj but c0 vanish. We now produce exactly one mode in each output beam, and thereforethe effective mode number of the source is minimal with K = 1, so we have a perfect broadbandtwo-mode squeezer:

H0 = ζc0

(A†0B†0 + A0B0

)(4.2)

4.1 Single photon detectors

As single photon detectors we use the idQuantique id201 InGaAs avalanche photo diodes, andsince they are crucial components for all of our experiments, their properties and idiosyncrasiesbear closer inspection[33].

Avalanche photo diodes can be thought of as the solid state advancement of classic photo-multiplier tubes; instead of a vacuum tube, a semiconductor diode is the underlying device.Depending on the target operation wavelength, different diode materials are used, from siliconfor visible light and germanium at 1064 nm to indium gallim arsenide (InGaAs) at telecomwavelengths. To this diode a high reverse bias voltage (smaller than its breakthrough voltage), isapplied. If an electrons is freed within the diode’s depletion layer by an impinging photon via thephoto-electric effect, the bias voltage accelerates it and causes it to release further electrons viaimpact ionization. Those electrons are also accelerated and free even more electrons; an electronavalanche is excited. The resulting current flowing in the direction of the bias voltage indicates animpinging photon. There is always the possibility of a thermally excited election giving rise to anavalanche, causing a “dark count” event, therefore most APDs are actively cooled. Also after anavalanche event, the APD’s detection current must be quenched, that is the diode’s depletion layermust be emptied of free charge carriers. Imperfect quenching leads to an increased probabilityfor dark counts due to remaining electrons from the previous avalanche, an effect known asafter-pulsing. While a higher bias voltage promises higher quantum efficiency of the APD, it will

4.1 Single photon detectors 47

also increase both thermal dark counts and after-pulsing, so that one has to find a balance betweendetector efficiency and detector noise for each experiment. To reduce noise from dark counts,one can apply gating to the APD, that is to apply a bias voltage only when an incoming photon isexpected. To obviate after-pulsing, the time between gating events must be big enough for anyfree charge carriers to disperse or re-combine. Alternatively, a “dead time” can be introduced, atime interval in which the detector is shut down after each avalanche event.

Figure 4.1: id201 APD quantum efficiency.

The id201 by idQuantique is an actively gated APD, and is specified for gating signal frequenciesof up to 4 MHz. Gate widths vary from 2.5 ns to 100 ns. Although the bias voltage can be freelychosen, as a rule it is configured with four pre-calibrated settings for the quantum efficiency of10%, 15%, 20%and 25%, to be understood for the telecom wavelength of 1550 nm. Fig. 4.1 showsthe full quantum efficiency curve as specified by the maker. The id201 electrical inputs and outputscan be given and are available respectively both as TTL and NIM pulses, and an incoming gatingpulse can be electronically delayed for up to 25 ns.

First we tested the dark count level depending on the trigger rate for the two lowest gate widthsof 2.5 ns and 5 ns and different dead times, as depicted in Fig. 4.2. The APD’s light input isblocked, and an external trigger signal from a pulse generator supplies trigger pulses with avariable frequency, and the gain is set to the 25%level. On the left hand, with a 2.5 ns gate, wecan observe an approximately linear rise in dark counts, which for gating frequencies lower than1.2 MHz is dead time independent. Above 1.2 MHz, the low dead time measurement curves risefaster, suggesting an onset of after-pulsing. On the right hand, the gating width is 5 ns. Here wesee for gating frequencies over 1 MHz a dramatic increase of detection events when a low deadtime is used. Since we use in our PDC experiments a mode-locked pump laser with virtually notiming jitter as a source, and the down-converted light pulse lengths are in the order of magnitudeof 1 ps, we can easily use the lowest APD gate widths and no dead time, as long as we stay with lowenough laser pulse rates. We therefore employ a pulse picker after our laser source to emit pumppulses at a 1 MHz frequency, and can expect an APD dark count rate of about 50 Hz.

To examine the APDs’ after-pulsing characteristics, we coupled a heavily attenuated CW beamfrom a 1550 nm laser diode into the APD, and monitored the detection event rate in dependence

48 4 A PP-KTP waveguide as parametric downconversion source

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2kcounts / second

trigger frequency [MHz]

Gating width: 2.5 ns Detector efficiency 25%

Deadtime: 0 µs Deadtime: 1 µs Deadtime: 2 µs Deadtime: 5 µs Deadtime: 10 µs

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

kcounts / second

trigger frequency [MHz]

Gating width: 5 ns Detector efficiency 25%

Deadtime: 0 µs Deadtime: 1 µs Deadtime: 2 µs Deadtime: 5 µs Deadtime: 10 µs

Figure 4.2: Dark counts of id201 APDs with no incident light.

of the gating frequency. Since photons are impinging on the detector at every time, the count rateshould in first order approximation be proportional to the overall open gate time in absence ofafter-pulsing. Fig. 4.3 confirms this for a 0 ns dead time and a trigger frequency lower than 1 MHz.Above this level, a slight super-linear growth can be observed which we attribute to after-pulsing.For higher dead times, we can observe a drop in detection rate as soon as the time between triggerpulses becomes smaller than the configured dead time value, as the gating pulse after a successfuldetection is lost. Again, this measurement shows that the APDs’ can be operated at 25%efficiencywith 0 ns dead time, if we restrict our experiments to gating frequencies of 1 MHz or lower.

4.2 The parametric downconversion source

The core of all experiments presented here is the waveguide chip employed as the PDC source. Itis a 10 mm long KTiOPO4 chip with channel waveguides manufactured by AdvR Inc. In Fig. 4.4,the input facet is shown in 50x magnification. Each waveguide is roughly 4µm× 6µm in crosssection. Since the waveguides inscribed into the material by a diffusive proton exchange process,the waveguides seem less clearly defined further from the surface, as an influx of particles fromthe surface decays exponentially with penetration depth. To the left and right, and also at the topdue to the material-air-boundary, they are sharply defined. The KTP chip is made from a z-cutwafer (z direction is depth) and contains seven groups of seven waveguides in x direction. Theyexhibit a periodic domain poling of Λ = 104µm, designed to support the type-II downconversionprocess with nonlinear coefficient d24 from a P-polarized pump at 775 nm to P-polarized signal

4.2 The parametric downconversion source 49

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

kcounts / second

trigger frequency [MHz]

Gating width: 2.5 ns Detector efficiency 25%

Deadtime: 0 µs Deadtime: 1 µs Deadtime: 2 µs Deadtime: 5 µs Deadtime: 10 µs

Figure 4.3: After-pulsing of id201 APDs with weak coherent input light at 1550 nm.

Figure 4.4: The waveguide chip’s input facet under 50x magnification with a light microscope.Three waveguides with a 25µm spacing are clearly visible.

and S-polarized idler beams at 1550 nm. Production imperfections cause a shift from these idealtelecom wavelength values, but this does not affect our experiments negatively in any way. Twoneighboring waveguides of the same group are set 25µm apart. Between groups the distance is50µm. The waveguides show stark differences in quality, dispersion and spatial mode properties;we determined during experimentation – and will discuss later – that for our purposes waveguide#6 gives the best results. Unless stated otherwise, we used this waveguide.

Figure 4.5: An ultrafast pumped KTiOPO4 waveguide chip as PDC source

The basic setup of our PDC experiments is simple: A beam from the pump laser is coupled intothe waveguide, where the downconversion takes place, and the three outgoing beams are separated.Signal and idler beam are fed separately into single photon detectors to test if the desired processis actually occurring.

In Fig. 4.5, we discuss this in more detail: The pump laser is a Coherent MIRA titanium sapphiremode locked system. In its default configuration, it produces femtosecond laser pulses between710 nm to 980 nm with an autocorrelation length as short as 200 fs. It features an alternate

50 4 A PP-KTP waveguide as parametric downconversion source

cavity configuration that allows for picosecond pulses. The average output power is specifiedas 650 mW, and the repetition rate is frep = 76 MHz. In order to not over-saturate detectorsand data acquisition logic, we use an APE PulseSwitch accusto-optical pulse picker (AOM) toscale down the pulse frequency to 1 MHz. After passing through a power control consisting ofhalf-wave plate HWP1 and the polarizing beam-splitter PBS1, the pump beam’s polarization can becontrolled by HWP2. It is then coupled into one of the chip’s waveguides using 10x microscopeobjective MO, where the P-polarized component of each pump pulse excites a P-polarized signalpulse at 1544 nm and a S-polarized idler pulse at 1528 nm in a quasi-phasematched type-II PDCprocess. It is mounted on a Elliot Gold Series Professional Workstation which allows besidesadjusting waveguide height and z-rotation also a translation in y-direction of 25 mm for easywaveguide selection. The outgoing beams are collimated with lens L1, an NIR-antireflection-coatedQ940 lens sold by Thorlabs. The pump beam is split off by a Semrock RazorEdge LP02-808RSdichroic mirror DM and directed to power-meter PM to measure the pump power available insidethe waveguide. Signal and idler beam are finally split by PBS2 into separate paths, collimatedwith NIR-coated lenses L2 and L3 (Thorlabs C430) and coupled into SMF28 single-mode fibersconnected to avalanche photo diode (APD) single photon detectors. The detectors are activelygated id201 InGaAs APDs by idQuantique and have been described in the previous section. Theirgating signal is supplied by the monitor signal of the AOM that is triggered every time it couplesout a laser pulse. To prevent signal degradation and reflexion through a passive T-junction elementin the electrical line, it is then multiplexed and by the Stanford Research DG645 pulse generator.The same device is used to delay it in order to synchronize the detectors’ gate time with incominglight pulses from the PDC source.

To determine if photon pair generation is actually occurring, the APD detection signals have tobe checked for a correlation of detection events. In particular, the level of coincident events hasto exceed the expectation value of accidentally occurring coincidences. Important experimentalfigures in this context are the Klyshko efficiencies[73] that determine the overall fraction of thePDC produced photons being detected by the single photon detectors. Let Rp be the rate of pairsproduced in the waveguide chip, and η1, η2 the absolute single photon detection efficiencies ofeach arm of the setup, including losses due to maladjustment of the beam-path, transmissionlosses of optical elements and detector quantum efficiency. If we assume only single photonpairs are produced and R1 and R2 is the rate of detection events of APD1 and APD2 in Fig. 4.5respectively, andRc is the rate of coincidence detection events between APD1 and APD2, then theyare connected to Rp by:

R1 = η1Rp (4.3)

R2 = η2Rp (4.4)

To the rate of coincidences Rc, losses from both setup arms have to be applied:

Rc = η1η2Rp (4.5)

Solving for the efficiencies yields

η1 =RcR2

(4.6)

η2 =RcR1

(4.7)

We now allow for the production of multiple photon pairs in the PDC and assume as its outputa two-mode squeezed vacuum state |Ψ(r)〉. For detectors far from saturation, the rate Ri is

4.2 The parametric downconversion source 51

connected to the detection probability pi simply through the pump laser repetition rate frep viaRi = freppi. The Klyshko efficiencies, in dependence of the squeezing operator r, now read

η1(r) =pc(r)

p2(r)

η2(r) =pc(r)

p1(r).

(4.8)

The single click probabilities p1 and p2 can be calculated with the measurement operator µifrom Eq. 2.87 :

pi = 〈Ψ| µi |Ψ〉 =ηitanh2r

1− (1− ηi) tanh2r(4.9)

The measurement operator µc for the coincidence click probability pc is simply the product µ1⊗µ2,and the measurement evaluates to

pc = 〈Ψ| µc |Ψ〉 = 1− 1

1 + η1sinh2r− 1

1 + η2sinh2r+

1

1 + (η1 + η2 − η1η2) sinh2r(4.10)

For low squeezing values r → 0, we use the small angle approximation sinh(r) ≈ tanh(r) ≈ r.Then, we apply the l’Hospital theorem twice to find

η1(0) ≈[∂2rpc(r)

∂2rp2(r)

]r=0

= η1

η2(0) ≈[∂2rpc(r)

∂2rp1(r)

]r=0

= η2.

(4.11)

For small squeezing values r, the Klyshko efficiencies coincide with the absolute setup efficienciesη1, η2. For moderate squeezing with r < 1, they scale approximately linearly, and this detectionefficiency gain can be simply ascribed to the fact that more photon pairs at once are generatedand single photon detectors will detect multiple photons with higher probability than a singlephoton. When we compare this to the Klyshko efficiencies that arise from any two uncorrelatedphoton sources with single event detection probabilities q1, q2 and coincidence event probabilityqc = q1q2, we see the crucial difference illustrated in Fig. 4.6: For low pump powers P , the Klyshkoefficiencies η1 = qc

q1= q2 for the uncorrelated sources go towards zero for small squeezing (violet

curve), while the efficiencies for the PDC source approach finite positive values η1, η2 (red andblue fit curves). Thus we can easily verify the existence of correlated photon events.

Also in Fig. 4.6, we see experimentally gained Klyshko efficiencies from a PDC process in oursetup according to Fig. 4.5. The laser is set to a 768 nm pump wavelength, and the wave plateHWP1 inside the power control block is rotated to tune pump power. For each pump power P ,Klyshko efficiencies for signal and idler beam are recorded with a 15 s measurement time. Abovea pump power of 2µW the efficiencies show a linear growth as expected for a correlated photonpair source. From axis-intercept of the trend lines, we can estimate the absolute photon collectionefficiencies for this measurement: η1 = 0.068 for the signal beam and η2 = 0.035 for idler. Belowthis power threshold, η1 and η2 drastically change their slope and quickly fall to zero. We canexplain this by the presence of a constant background of dark count events in the APDs. As soonas their number is comparable the number of genuine single photon detection events, they willmake their presence felt on the Klyshko efficiencies. If both the dark count probability pdark and

52 4 A PP-KTP waveguide as parametric downconversion source

the background-free APD click probabilities are small, we can neglect all quadratic terms andwrite approximately:

η′1 =

pc(r)

p2(r) + pdark(4.12)

η′2 =

pc(r)

p1(r) + pdark(4.13)

These expressions go towards zero for small pump powers and small squeezing parameters r andotherwise show the expected linear behavior, in accordance with the measured efficiencies in Fig.4.6.

0

0.05

0.1

0.15

0 5 10 15 20

Kly

shko

effi

ciency

Pump power P [µW]

η1 exp. dataη2 exp. data

uncorrelated sources

Figure 4.6: Experimental Klyshko efficiencies from our PDC source (red and blue) and theoreticalKlyshko efficiency for two uncorrelated sources (violet)

In conclusion this measurement proves not only that correlated photon pairs are produced inour setup, but also gives an estimate of its overall photon collection efficiency.

4.3 Phasematching contour

Figure 4.7: Experimental setup for measuring PDC signal and idler photon marginal spectra withthe MicroHR monochromator.

As has been discussed in section 2.6.1, the PDC phasematching function Φ(ωo, ωi) describesthe restrictions by momentum conservation on the distribution of pump photon energy betweensignal and idler photons. We have shown that it must exhibit a non-negative slope to allow for

4.4 Conclusion 53

group velocity matching (GVM), and thus for the heralding of pure single photons. We will nowtrace the phasematching contour ∆k = 0 of the phasematching function Φ(ωo, ωi) by measuringmarginal spectra of signal and idler photons at different pump wavelengths and plotting theirmaxima.

0

100

200

300

400

1520 1560 1600

APD

count

rate

[H

z]

Signal wavelength [nm]

100

200

300

400

500

600

1480 1520 1560 1600

APD

count

rate

[H

z]

Idler wavelength [nm]

Figure 4.8: Marginal spectra of signal (left) and idler (right) photons. Equal colors label correspond-ing signal and idler peaks. The left-most peaks correspond to a pump wavelength λpof 750 nm.

The setup in Fig. 4.7 is used to measure marginal spectra of signal and idler. Pump pulses atcentral wavelength λp are coupled into waveguide #6 , and the output light then travels throughsome polarization optics before being coupled into a single mode fiber. The half-wave plate HWPis rotated to select one of the type-II PDC output polarizations to be transmitted through the PBS,and then is spectrally filtered by the fiber-coupled Horiba MicroHR. The MicroHR is a motorizedmonochromator, and features the “510 16 X36NJ” line grating for the spectral range from 1000 nmto 2500 nm with 600 lines

mm and a 1500 nm blazing. The filtered output is detected by an id201 APD.For λp between 750 nm to 790 nm, we measured in steps of 5 nm the marginal spectra as

presented in Fig. 4.8. Each peak in the signal and idler plots respectively, corresponds to onepump wavelength, increasing from left to right. Plotting the peaks’ central wavelengths against thepump central wavelength λp (Fig. 4.9 left), we are able to determine the waveguide’s degeneracypoint: Signal and idler wavelengths are equal for λp = 786 nm. The shape of Φ(ωo, ωi) is obtainedby plotting signal against idler central frequencies in Fig. 4.9 (right). We note the positive slope ofthe measured phasematching contour that in good approximation can be fitted against a linearfunction with a positive slope, corresponding to a phasematching angle of Θpm = 59.2◦ andconclude that successful GVM is indeed possible.

4.4 Conclusion

We have implemented a type-II SPDC source in a KTP waveguide, and proven the production ofcorrelated, cross-polarized photon pairs by measuring non-zero Klyshko efficiencies for smallpump powers. We then have mapped the phasematching contour ∆k = 0 from the centralwavelengths of the marginal spectra for signal and idler measured with a monochromator and

54 4 A PP-KTP waveguide as parametric downconversion source

1480

1500

1520

1540

1560

1580

750 760 770 780 790

Sig

nal/Id

ler

wavele

ng

th [

nm

]

Pump wavelength [nm]

1.18

1.2

1.22

1.24

1.26

1.28

1.18 1.2 1.22 1.24 1.26

Idle

r fr

eq

uency

[PH

z]

Signal frequency [PHz]

Figure 4.9: Left: Central wavelengths for signal (red) and idler (blue) beams against pump centralwavelength on the x-axis. Right: Phasematching contour from plotting correspondingsignal and idler central frequencies as data points.

single photon detectors. As expected, both photons are in the telecom wavelength regime for apump beam at 775 nm. For 786 nm, the photons are wavelength degenerate at 1572 nm.

The presented method of obtaining the marginal spectra of PDC output photons does havemajor drawbacks though: Due to very high losses within the MicroHR monochromator, the photonfluxes observed are quite small, making measurement either slow or inaccurate. This problem willbe greatly exacerbated when we try to adapt it to measure the joint spectrum of signal and idler:The setup’s transmissivity, already much smaller than 1, will be squared when counting coincidentphotons filtered by a monochromator in each PDC output beam. We therefore we developed themore efficient, cheaper alternative free of moving parts alternative: The fiber spectrometer.

5Fiber spectrometer

The great majority of spectrometers follows the same basic principle in combining a dispersiveoptical element with an ideally wavelength-agnostic optical detector[108, 89]. The dispersiveelement typically introduces a wavelength-spatial mode correlation and uses a simple movableslit to select one spatial mode to be measured. This is usually implemented with prisms orgratings. Instead of a slit and one detector, one can use a detector array, and reach higher detectionefficiencies by avoiding the loss of all photons filtered out by the slit. Grating spectrometers withCCD arrays as detectors make use of this method and provide a cheap, sensitive device withoutmoving parts.

When spectrally characterizing a single photon source, detection efficiency is of paramountimportance; due to typically low photon fluxes, measurements take a relatively long time, andlosses would add to that, so an array of detectors would be preferable to a movable-slit-and-detector setup. But single-photon sensitive detectors, such as avalanche photo diodes (APDs)or recent approaches like cryogenic detectors such as nanowire single photon detectors[62], aremuch more expensive than standard photo diodes or CCD sensors, and an array of them mightnot be affordable to the physicist on a budget.

In order to solve this dilemma, we introduce the fiber spectrometer[8]: We transfer the spectro-meter principle from the spatial to the time domain: With the help of chromatic dispersion wecreate a spectral-temporal correlation in our photons[137, 24, 10], and by measuring their arrivaltime at the detector, we are able to reconstruct their wavelength.

5.1 Functional principle

The fiber spectrometer setup is straightforward: A beam of single photons travels through along stretch of dispersive optical fiber, picks up a wavelength-dependent group delay through thefiber’s chromatic dispersion and its arrival time at a single photon detector is measured. A graphdepicting the wavelength-time correlation of such a chirped photon – as might be measured witha SPIDER apparatus[68] – can be seen in Fig. 5.1. On the left, a Fourier-limited pulse exhibits notime-wavelength correlation, while on the right, the pulse after has been transmitted through the

56 5 Fiber spectrometer

fiber and clearly shows arrival time depending on wavelength. A calibration of the spectrometergives the exact magnitude of this dependence λ(τ),

but in general will need to be modeled with a higher order polynomial, if a larger wavelengthrange is considered. Any intrinsic temporal bandwidth of the single photons is assumed to bemuch smaller than the effects of the fiber dispersion, and therefore will be neglected.

Figure 5.1: Time-Wavelength correlation graphs of a broadband photon signal before (left) andafter (right) fiber transmission

We aim to measure the spectra of photons in the telecom wavelength regime. As dispersivemedium we use dispersion compensating fiber (DCF) modules originally designed to counterchromatic dispersion of standard telecommunication fibers. Their dispersion is ten times thenegative value of SMF28 fiber at 1550 nm, so that for 10 km of this type, 1 km of the DCF is neededto counter the accumulated chirp of the telecom signals. Our detectors are again id201 InGaAsAPDs by idQuantique.

We have made an implicit assumptions about our single photon source here that is necessary forthis kind of spectrometer to work, namely the existence of a clock signal to measure the photonarrival time against. A PDC source pumped by light pulses from a mode-locked laser providessuch a time reference naturally, but also a CW-pumped PDC source is capable of this: By heraldingthe photon whose spectrum is to be measured with its partner that does not travel through adispersive medium, we also obtain a clock.

Figure 5.2: Fiber spectrometer working principle: The incoming singe photon pulse is elongatedinside the DCF fiber coil, and its arrival time against a clock reference signal ismeasured.

Fig. 5.2 illustrates the basic working principle of the fiber spectrometer. From the measureddetection events we obtain a single photon arrival time statistic, that with the help of a calibrationcurve directly translates into a single photon spectral intensity. Timing errors inherent in the setupinherent are restricting the spectrometer’s resolution: Jitter of the clock and the detection signal,as well as the temporal uncertainty of the initial single photon pulses themselves. With initial

5.2 Experimental setup for photon pair spectrum measurement 57

pulse lengths in the order of 1 ps, several nanometers of spectral width, and a fiber dispersionof the DCF coils used of −862 ps

nm at 1550 nm, we can neglect this last source of errors for allapplications within this work.

Great care has to be taken to keep constant (or at the very least to account for changes of) theoptical and electrical runtime of all signals involved, as they have a direct impact on the crucialtiming difference between clock and photon arrival. While it is relatively trivial to keep all opticalelements of the setup fixed to avoid longer or shorter photon paths, there are more subtle ways tointroduce timing errors: We found that changing the wavelength of a mode locked laser sourceoffset its output pulses with respect to its electrical trigger signals used as clock. Also, any kindof equipment to delay or multiplex electrical signals is prone to react with altered intrinsic timedelays to changes of their configuration, often in unexpected and undocumented ways.

5.2 Experimental setup for photon pair spectrum measurement

Nonlinear optical processes such as PDC can generate photon pairs entangled in, depending onperspective, time, energy or frequency. This frequency entanglement is visible as a correlationin the photon pair’s joint spectral intensity (JSI), but will not feature in either single photonspectrum of the pair’s constituent photons. To correctly measure the JSI, one has to use a dedicatedspectrometers for each photon, and additionally post-select only coincident detection eventsbetween both spectrometers. Such a setup combines the PDC source from 4.5 with two fiberspectrometers and is shown in Fig. 5.3: A type II PDC KTP waveguide is pumped by a Ti:Saph laserat 768 nm to generate a photon pair in the telecom regime. The signal and idler photons are splitup by polarization and fed into individual DCF coils, and their arrival times detected separatelyby the time-to-digital (TDC) module TDC-GPX by acam. But only if two photons arrive in thesame clock cycle, the event is considered in the joint arrival time statistic. An early version of thisexperiment used one DCF coil and separated photons after that. For this to work, the polarizationrotation by the DCF has to be compensated before a PDC can be employed to split up signal andidler. In general though, they have different wavelengths, and thus their polarization rotation isdifferent and cannot be undone by the same QWP-HWP combination in all cases, only for specialsignal/idler wavelength combinations. It would be possible to separate by wavelength insteadof polarization if sufficiently different signal and idler wavelengths with an appropriate dichroicmirror. Here, we sidestep this problem by simply using one DCF coil per output arm.

Figure 5.3: Fiber spectrometer for measuring the joint spectral intensity of PDC photon pairs

A measured temporal distribution of photon pair detection events from the waveguide sourcecan be seen in Fig. 5.4. Each pixel represents a measurement interval of 30 s during whichcoincidence events from the APD pair are counted. The APD detection windows of 2.5 ns lengthare delayed by the x- or the y-value of the pixel respectively. Pump wavelength is 768 nm and

58 5 Fiber spectrometer

579 581 583 585

Signal arm delay τs [ns]

719

721

723

725

Idle

r arm

dela

y τ

i [ns]

Figure 5.4: Statistical distribution of coincident signal and idler photon arrival times

pump FWHM is 0.7 nm. Signal and idler central wavelengths are as of yet undetermined, and wecannot extract any information from the central arrival times 581.5 ns and 722 ns without makingassumptions about signal and idler beam paths.

5.3 Calibration

Calibration of the two-photon fiber spectrometer was done against the commercial Horiba Mi-croHR motorized grating monochromator with the 510 16 X36NJ etched grating for near infrared.A broadband pulse from a OPA system is filtered down to the spectral resolution of the monochro-mator and then sent through our DCF coils. Against the OPA trigger signal as clock, an arrivaltime of the pulses at the APD detectors for a range of wavelengths is recorded.

Figure 5.5: Fiber spectrometer calibration setup against MicroHR grating spectrometer

The experimental setup is depicted in Fig. 5.5. The OPA system generates broadband lightpulses in the telecom wavelength regime with a repetition rate of 120 kHz. This is filtered to aspectral width of 0.6 nm around wavelength λ by the MicroHR monochromator, coupled into twosubsequent Lucent WBDK-25 DCF coils with a specified dispersion of−431 ps

nm each. The fiberis directly connected to a idQuantique id201 APD. For each OPA light pulse, a trigger signal is

5.3 Calibration 59

sent to the Stanford Research DG645 pulse generator, where it is delayed for a time τ , and thenforwarded as a gating signal to the APD. If the electrical signal delay τ and the optical delay inthe fiber for the pulse at wavelength λ match, the detector registers photon events, otherwise onlydark counts. By scanning through the delay time τ , we measure a peak in detection count rate thatis displaced when we tune the wavelength λ also. In Fig. 5.6 we show the results of this calibrationmeasurement for λ = 1400 nm to λ = 1600 nm in 5 nm steps. Each peak corresponds to onewavelength, with wavelengths increasing from right to left.

0

50

100

150

3500 3550 3600 3650

APD

count

rate

[kH

z]

Time delay for APD trigger [ns]

Figure 5.6: APD response against OPA trigger delay for Horiba MicroHR monochromator settingsfrom 1400 nm (leftmost peak) to 1600 nm (rightmost peak) in 5 nm steps

1350

1400

1450

1500

1550

1600

1650

3500 3550 3600 3650

Wavele

ngth

[nm

]

Photon detection time [ns]

Group velocity dispersion

Figure 5.7: Each data point (red triangle) corresponds to one measurement from Fig. 5.6; the fitpolynomial (blue) serves as calibration curve of the spectrometer

In Fig. 5.7, we map the center of each peak to its associated wavelength. With a least-squaremethod we fit a third-order polynomial to our data. The resulting calibration curve is

λ(τ)

1 nm= −5.3056× 10−7τ3

1 ns3+ 3.2986× 10−3τ2

1 ns2− 4.5908× τ

1 ns+ 0.1732 (5.1)

60 5 Fiber spectrometer

It is important to keep in mind though that this curve is only applicable if the relative time delaybetween optical pulse and electrical gating signal in the setup is not changed. Such a temporaloffset can be caused by changes to the optical path, by electrical cables of different length, by adifferent laser configuration or different device settings of the DG645 and the APD. In that case, wehave to adjust the calibration curve accordingly by introducing a temporal offset. For our photonpair source we need two time delays τµ with µ ∈ {s, i} for signal and idler photon detection arm,respectively. The calibration function now reads:

λµ(τ) := λ(τ + τµ) (5.2)

In our further experimental program, we use the same waveguide #6 on our KTP chip at theconstant pump wavelength λp = 768 nm at room temperature. We use the phasematching contourobtained independently from spectral measurements with a standard grating spectrometer in Fig.4.9 (left). The signal and idler temporal maxima τs,max and τi,max of any temporal distribution fromthe same pump wavelength measured with the fiber spectrometer is identified with λs = 1544 nmfor signal and to λi = 1528 nm for idler, according to the phasematching contour. The calibrationtime delays for each arm are τs = λ−1(1545 nm) − τs,max and τs = λ−1(1528 nm) − τs,max.Applying this calibration to the measurement results pictured in Fig. 5.4, we obtain the bi-photon

(a) Pump FWHM 0.70nm

1541 1544 1547Signal wavelength [nm]

15

26

15

29

15

32

Idle

r w

avele

ngth

[nm

]

Figure 5.8: Joint spectral intensity of photon pairs generated by a pulsed pump at 768 nm

spectrum shown in Fig. 5.8. Each data point for arrival times (τs, τi) has been converted to a datapoint at (λs(τs) , λi(τi)).

5.4 Spectral resolution

Arguably the most important figure of merit for any spectrometer is its spectral resolution. Thefiber spectrometer’s resolution is determined by the steepness of its calibration curve and thetiming error of the photon detection events:

∆λ =∂λ(τ)

∂τ

∣∣∣∣τ0

∆τ (5.3)

5.4 Spectral resolution 61

Source Symbol Value DescriptionPulseSwitch ∆τPulseSwitch 50 ps Jitter of the pump pulse pickerDG645 ∆τDG645 80 ps Jitter of the delay generatorid201 ∆τid201 500 ps Jitter of the APD detection signalid201 ∆τgate 2.5 ns APD gate width

Figure 5.9: Fiber spectrometer timing error sources

All active elements of the setup can be sources of time jitter, but most prominently the electricaldevices are prone to this. The sources of electrical signal jitter in the fiber spectrometer setup withthe vendor specified values are compiled in Fig. 5.9.

For our MIRA laser system itself we neglected any timing error, because the timing jitter of amode-locked femto-second laser cavity must be much smaller than the pulse length, or otherwisethe laser would fall out of lock. Each of the electrical components in the setup contributes withtheir vendor-specified signal jitter. The time correlation between a photon detection event andthe id201 APD’s electrical output pulse is not documented by the vendor. So in the worst case, wecannot time-resolve photon arrivals within the id201 APD gate width, and have to consider it atiming error as well. Assuming Gaussian error propagation, the squared timing error of a photondetection event is the squared sum of all single timing errors. Using the vendor specified timejitter values and the APD gate width, we make a conservative estimation of the overall time error∆τ .

∆τ2 = ∆τ2DG645 + ∆τ2

id201 + ∆τ2gate + ∆τ2

pulse picker = (2.55 ns)2 (5.4)

As we can see, the id201 gate width ∆τgate is the dominant term here. From the fiber calibra-tion curve λ(τ) we can now estimate the wavelength error and thus the resolution of the fiberspectrometer at 1550 nm⇒ ∆τ = 2.55 ns⇒ ∆λ = 3.0 nm.

0

100

200

300

400

500

600

41 41.5 42 42.5

Sin

gle

event

count

rate

[kH

z]

Arrival time [ns]

SignalIdler

Signal half-maximumIdler half-maximum

Figure 5.10: “Marginal spectra” of the two-photon state measured without DCF coils to determinethe temporal jitter of the experimental setup.

This spectral resolution width seems rather high, but then again we used a worst-case as-sumption for the APD gate width. To get a more realistic idea about the timing uncertainty of

62 5 Fiber spectrometer

our setup, we conduct a measurement of the “marginal distributions” of a photon pair with thesetup depicted in Fig. 5.3, but with the DCF coils replaced with 1 m long standard telecom fiberpatch cords. This way, the optical pulse length of the detected photons is not elongated throughchromatic dispersion, but rather of the order of 1 ps. Since all other elements of the setup are inplace, the temporal spread of the measured distributions is only due to the overall timing jitter∆τ . The resulting curves for signal and idler arm are plotted in Fig. 5.10. The idler distribution iswider than the signal distribution, reflecting differences in the electrical and timing properties ofthe id201 APDs. As an experimental measure of the timing jitter of the fiber spectrometer setupwe take the FWHM ∆τ = 1.56 ns of the idler distribution. The resulting spectral resolution at1550 nm is the significantly lower ∆λ = 1.84 nm. The reason for this discrepancy must comefrom the highest contribution to the jitter,Dtgate. As the compounded jitter ∆τ is even lower than the specified id201 gate width of∆τgate = 2.5 ns we used, either the effective gate width is shorter than 1.56 ns, or there is somecorrelation between the exact time of a photon detection event and the electrical detection signalfrom the APD.

5.5 The joint spectral intensity of photon pairs from the KTP source

764 766 768 770 772

Pum

p inte

nsi

ty [

a. u.]

Pump wavelength [nm]

Figure 5.11: Pump spectra with a FWHM of 0.70 nm (red), 1.95 nm (green) and 4.0 nm (blue),respectively

Having developed and tested the fiber spectrometer, we now use it to demonstrate control overthe spectral correlations over photon pairs generated by our source. As experimental setup wecontinue to use the setup depicted in Fig. 5.3. With the 4f grating setup, we can manipulate thewidth of the ultrafast pump pulses, and thus the correlation of the photon pair joint spectra[59, 134].We verified this by measuring the joint spectral intensity of generated photon pairs at differentspectral pump widths, as depicted in Fig.5.11. APD efficiency was set to 25%, the AOM releasedpump pulses at 1 MHz. Fig. 5.12 shows the results for spectral pump FWHM of 0.70 nm and 26µW,1.95 nm and 81 muW and lastly 4.0 nm and 51µW: We observe negative spectral correlation,an uncorrelated spectrum, and a positive spectral correlation between signal and idler photons,respectively. This shows that we can control spectral correlations in our source by spectrallyfiltering the pump beam, and that we can expect minimal spectral correlation of photon pairsaround 1.95 nm pump FWHM.

5.6 Measurements beyond the perturbative limit 〈n〉 � 1 63

(a) Pump FWHM 0.70nm

1541 1544 1547Signal wavelength [nm]

15

25

15

28

15

31

Idle

r w

avele

ng

th [

nm

]

(b) Pump FWHM 1.95nm

1541 1544 1547Signal wavelength [nm]

15

25

15

28

15

31

(c) Pump FWHM 4.00nm

1541 1544 1547Signal wavelength [nm]

15

25

15

28

15

31

Figure 5.12: Joint spectra from setup 6.2(b) with pump width above (c), equal to (b) and below(a) separability width at 1.95 nm FWHM.

5.6 Measurements beyond the perturbative limit 〈n〉 � 1

If with a non-negligible probability more than one photon pair is produced, then we cannotassume with certainty any more that the coincident detection events are caused by photons incorresponding Schmidt modes. Consequently the spectral correlation apparent in the JSI at lowmean photon numbers is weakened, and we arrive at a “rounder”, less correlated spectral shape. Itcannot be viewed as a true two-photon spectrum any more, as multiple pairs are involved. Thiseffect will inevitably appear at higher mean photon numbers, and is independent of detectorsaturation. If the multi-pair state suffers optical losses, there is again no guarantee that thesurviving photons are in corresponding Schmidt modes, so attenuation of the SPDC output beamsin front of the detectors cannot prevent it.

To illustrate the effect, we pump waveguide #6 on our KTP chip with pulses at 768 nm with0.35 nm spectral width. We use the fiber spectrometer to measure joint spectra for a pump powerof 10µW (Fig. 5.13 left) and for a significantly higher power of 160µW. The result is a matrix ofcoincidence event counts, and its indices refer to discrete signal and idler photon arrival timesat the detectors. With a calibration function C(t), we can translate these arrival time values tophoton wavelengths or frequencies. We will however refrain from doing so for now, since thestructure of the Schmidt decomposition itself does not change under this kind of transformation,as it is applied to signal and idler separately.

We now analyze the effect quantitatively by measuring photon coincidence detection events witha pair of lossy binary detectors with quantum efficiencies η1 and η2. The use of the monochromaticmode operators a(ω1) and b(ω2) models sharp spectral filtering at the frequencies ω1, ω2, as ifwe introduced spectral filters with a δ-function shaped transmission spectrum in front of both

64 5 Fiber spectrometer

325 328 331 334

Signal arm delay τs [ns]

440

443

446

449Id

ler

arm

dela

y τ

i [ns]

325 328 331 334

Signal arm delay τs [ns]

440

443

446

449

Idle

r arm

dela

y τ

i [ns]

Figure 5.13: Joint spectra with 10µW pump power (left) and 160µW pump power (right)

detectors. We apply the measurement operators (c. f. section 2.7)

µ1 =

∞∑n=0

1

n!(1− (1− η1)n) a†(ω1)n |0〉〈0| a(ω1)n

µ2 =∞∑n=0

1

n!(1− (1− η2)n) b†(ω2)n |0〉〈0| b(ω2)n

(5.5)

to signal and idler arm of a general multi-mode squeezed vacuum state from type-II SPDC (cf.section 2.6.8)

|Ψ〉 =

∞∏j=0

Sj |0〉

=∞⊗j=0

√1− |λj |2

∞∑nj=0

(λjA

†jB†j

)njnj !

|0〉

(5.6)

with λj = tanh(|ζcj |) eıarg(ζcj), the coupling constant ζ and the Schmidt coefficients cj . Werewrite the state |Ψ〉 as a sum over all integer non-negative series {nj}. Each series correspondsto one possible mode occupation configuration.

|Ψ〉 =∑{nj}

∞∏j=0

√1− |λj |2

(λjA

†jB†j

)njnj !

|0〉 . (5.7)

The probability differential to detect a coincidence detection event at the sharp frequencies (ω1, ω2)is

p(ω1, ω2) = Tr[µ1 ⊗ µ2 |Ψ〉〈Ψ|] (5.8)

and the probability to detect a coincidence event in a finite frequency area is the integral of thedifferential probability over this area.

5.6 Measurements beyond the perturbative limit 〈n〉 � 1 65

The squeezed state is expressed in broadband modes Aj and {Bj}, while the measurementoperators are built from monochromatic mode operators a(ω1) and b(ω2). We can express thelatter ones in terms of broadband modes using the completeness theorem for orthogonal functionsand write

a(ω) =∑j

ξj(ω) Aj

b(ω) =∑j

ψj(ω) Bj

(5.9)

where ξj and ψj are the spectral mode functions corresponding to Aj and Bj , respectively. Wethen re-write the powers of the monochromatic mode operators as

a(ω)n =

∑j

ξj(ω) Aj

n

=∑

∑kj=n

∞∏j=0

(ξj(ω) Aj

)kj(5.10)

where∑∑

kj=nis the sum over all integer, non-negative series {kj} that total to n.

The form of both state vector and measurement operators was specifically chosen such that theproblem of normal-ordering would not arise in further calculations. Substituting the broadbanddefinitions into Eq. 5.8, we find

p(ω1, ω2) = 〈Ψ| µ1 ⊗ µ2 |Ψ〉

=

∞∑m=0

∞∑n=0

1

m!n!(1− (1− η1)m) (1− (1− η2)n)

∞∏j=0

(1− |λj |2

)×

∣∣∣∣∣∣∑{mj}

∑∑kj=m

∑∑lj=m

∞∏j=0

λ∗jmjξ∗(ω1)kj ψ∗(ω2)lj

mj !〈0|(

AjBj

)mj (A†j

)kj (B†j

)lj|0〉

∣∣∣∣∣∣2

(5.11)

which can be simplified considerably.

The vacuum expectation value 〈0|(

AjBj

)mj (A†j

)kj (B†j

)lj|0〉 evaluates almost trivially to∏∞

j=0 (mj !)2 δmj ,kjδmj ,lj so that the series {mj}, {kj} and {lj} have to be identical. Since we

required∑

j kj = m and∑

j lj = n, this infers m = n and we can write for the product of all

vacuum expectation values δm,n∏∞j=0 (mj !)

2 δmj ,kjδmj ,lj . Three of the five sums in Eq. 5.11 thuscollapse and we are left with the rather more compact expression

p(ω1, ω2) =∞∑n=0

1

(n!)2 (1− (1− η1)n) (1− (1− η2)n)

×

∣∣∣∣∣∣∑

∑nj=n

∞∏j=0

nj ! (λjξj(ω1)ψj(ω2))nj

∣∣∣∣∣∣2 (5.12)

For low coupling constants ζ � 1, we have λj = tanh(|ζcj |) eıarg(ζcj) ≈ ζcj . Also, we canapproximate λj � λjλk ≈ 0 and neglect all terms with photon number n > 1. The infinitesimal

66 5 Fiber spectrometer

coincidence event detection probability thus simplifies to the expression for a single photon pairwe already encountered in Eq. 2.92

p(ω1, ω2) =1∑

n=0

1

(n!)2 (1− (1− η1)n) (1− (1− η2)n)

∣∣∣∣∣∣ζ∞∑j=0

cjξj(ω1)ψj(ω2)

∣∣∣∣∣∣2

=η1η2 |ζ|2 |f(ω1, ω2)|2(5.13)

where f(ω1, ω2) is the normalized joint spectral amplitude function of the SPDC process. Here,the coincidence click probability is proportional to the photon pair joint spectral amplitude.

But in the general case it is not so easy any more to find a physical property of the squeezingsource that corresponds with p(ω1, ω2) from Eq. 5.12, apart from the obvious one of the coinci-dence click probability. The spectral correlations of a high photon number multi-mode SPDC stateare still governed by the amplitude function f(ω1, ω2), but it does not directly map to the jointspectral measurement any more. This is in stark contrast to the situation for classical, coherentpulse spectra. Here, any measured spectrum for any mean photon number is proportional to themodulus squared of the amplitude function |fclassical(ω)|2.

r=0.1, η=1.0 r=0.6, η=1.0 r=1.6, η=1.0

r=0.1, η=0.01 r=0.6, η=0.01 r=1.6, η=0.01

Figure 5.14: Calculated coincidence click spectra from Eq. 5.12, with a mode cutoff after j = 9mode and a multi photon pair cutoff after n = 12.

Fig. 5.14 qualitatively illustrates the impact on the joint spectral measurement of a multi-modeSPDC source states with squeezing values r = 0.1, r = 0.6 and r = 1.6, corresponding to meanphoton numbers 〈n〉 of 0.1, 0.4 and 5.6 respectively (roughly, since we have a multi-mode state).The case r = 0.4 is comparable to the measurement for 10µW in Fig. 5.13 (left), r = 1.6 to thehigher powered measurement with 1.6µW Fig. 5.13 (right). The joint spectra are plotted forperfect detectors in the first row, and for detectors with a quantum efficiency η = 0.01 in the

5.7 Conclusion 67

second row. In the first column, there is virtually no difference between the plots. Since thereis in the vast majority of cases no more than one photon pair present, the coincidence eventsare triggered by photons in corresponding Schmidt modes. The second column for r = 0.6 isalready affected by multi-pair contributions: The central peak for low detection efficiencies isslightly lower and rounder. The trend continues in the third column for r = 1.6. While theupper spectral distribution is slightly narrower than the other plots in the first row, the spectralcorrelation lossy-detector distribution is noticeably weaker than both in the upper plot and theplots for smaller squeezing.

This simulation of the fiber spectrometer coincidence measurement, together with the estimatedspectrometer resolution of 1.86 nm at 1550 nm, qualitatively explains the measurement resultsfrom Fig. 5.13. It also shows that spectrally resolved coincidence measurements with binarydetectors give accurate measurements of the SPDC joint spectral intensity |f(ωs, ωi)|2 only ifthe production rate of photon pairs is low. The presence of multiple photon pairs in the samemeasurement cycle will distort the resulting spectrum noticeably, even if optical losses preventdetector saturation. SPDC sources with an arbitrarily high photon pair output do not and cannothave a well defined two photon spectrum, because of the ambiguity that multiple pairs create.The joint spectral amplitude function f(ωs, ωi) still governs the spectral structure of the two-mode squeezed vacuum, but loses its immediate physical interpretation as an actual, measurablebi-photon spectrum.

5.7 Conclusion

We have described the setup and calibration of an optical fiber based spectrometer to measurethe spectrum of single photon states. The spectral spread of a measured photon is translated intoa temporal spread by a long dispersive fiber. By measuring the arrival time of the photon at theend of the fiber with a time-resolving binary detector, we gather temporal statistics from whichwe gain the spectral distribution of the photon. The fiber spectrometer’s spectral resolution is1.84 nm and is mainly caused by the detection gate length of our binary detectors that limits timeresolution. To measure the joint spectrum of a photon pair, we use a fiber spectrometer for eachphoton and take statistics only from coincident measurement events. Finally we show that SPDCstates from the same source but with a higher mean photon number lead to results with seeminglyless spectral correlation, and argue that the results, while still indicating spectral correlations inthe photon pairs, do not faithfully give the joint spectral intensity |f(ωs, ωi)| any more.

6Two-mode squeezed vacuum source

For a genuine two-mode squeezing source, exhibiting an uncorrelated joint spectral intensity forphoton pairs generated at low pump power is necessary but not sufficient. Since the joint intensityis proportional to the modulus square of the complex joint amplitude |f(ω1, ω2)|2 of the photonpair, all phase information is lost in an intensity measurement, so that any phase entanglementbetween signal and idler is undetected. In order to demonstrate full photon pair separabilityand the resulting two-mode character of our source, we need to measure an additional quantitysensitive to its mode number. In this chapter, we use the second order correlation function g(2) todiscriminate between beams with thermal (g(2) = 2) and Poissonian photon statistics (g(2) = 1)from a two-mode squeezing source[129] and ensure that the the power-dependency of the source’sphoton pair output coincides with that of a two-mode squeezer.

6.1 Mode-number and photon statistics of broadband squeezed vac-uum states

It has been shown early on in the experimental exploration of squeezing that PDC producessqueezed states of light[144]. In photon number representation, a two-mode squeezed vacuumstate has the form

|Ψ〉 = Sa,b |0〉 = eıHa,b |0〉 =

√1− |λ|2

∑n

λn |n, n〉 (6.1)

where a and b are two orthogonal modes, Sa,b is the two-mode squeezing operator, and Ha,b isits effective Hamiltonian. It is a coherent superposition of strictly photon number correlated Fockstate pairs, and exhibits thermal photon statistics in both modes a and b, i. e. the probability togenerate n photons in each mode scales like (const)n:

pn =(

1− |λ|2)|λ|2n (6.2)

The photon number correlation between both modes allows for the heralding of pure Fock statesusing a photon number resolving detector, in the simplest case for pure heralded single photons

70 6 Two-mode squeezed vacuum source

with binary detectors. However, the underlying bilinear effective Hamiltonian Ha,b = ζa†b†+ h. c.describes only a special case of PDC. In general though, the effective PDC Hamiltonian has aricher spatio-spectral structure; additional to the broadband spectrum there can exist also acontiuous spectrum of ~k-modes, or a discrete spectrum of waveguide modes for pump, signaland idler respectively. The effective Hamiltonian must reflect this by summation over all possiblecombinations of mode triples, potentially resulting in hyper-entangled states[34]. This mostgeneral case cannot be decomposed into a set of independent two-mode squeezers any more, andalso goes against our goal to engineer a two-mode squeezer, or separable photon pairs at lowpowers. Assuming a single mode waveguide, the effective Hamiltonian reads

HPDC = ζ

∫∫dω1 dω2 f(ω1, ω2) a†(ω1)b†(ω2) + h. c.. (6.3)

It generates a generalized version of the two-mode squeezed vacuum in Eq. 6.1; its output beamsare spectrally correlated. The coupling constant ζ determines the strength of this interaction,while spectral correlations between photons of the pairs produced are governed by the normalizedjoint spectral amplitude

f(ω1, ω2) ∝ α(ω1 + ω2) Φ(ω1, ω2) (6.4)

where α(ω) is the spectral amplitude of the pump beam and Φ(ω1, ω2) is the phasematchingfunction that depends on the nonlinear medium’s dispersion properties.

For low pump powers, PDC is in good approximation a probabilistic source of photon pairs. Byapplying a Schmidt decomposition to the pairs’ joint amplitude[82] f(ω1, ω2) =

∑j cjξj(ω1)ψj(ω2),

we obtain two orthonormal basis sets ξj(ω1) and ψj(ω2) and a set of weighting coefficients cjwith

∑j |cj |

2 = 1. Now the PDC Hamiltonian can be expressed in terms of broadband modes

HPDC =∑j

Hj = ζ∑j

cj

(A†jB

†j + AjBj

). (6.5)

Each broadband mode operator Aj , Bj is defined as superposition of monochromatic annihilationoperators a (ω) , b (ω) weighted with a function from the Schmidt basis: A†j :=

∫dωξj(ω) a† (ω)

and B†j :=∫

dωψj(ω) b† (ω). Since the effective Hamiltonians Hj do not interact with each

other (i. e.[Hj , Hl

]= 0), we see that the PDC time evolution operator is in fact an ensemble of

independent two-mode squeezing operators S = UPDC = eıHPDC = SA0,B0 ⊗ SA1,B1 ⊗ ... wherethe coefficients cj determine the relative strength of all squeezers as well as spectral correlationbetween signal and idler beams[82]. This only an approximation for a small coupling constant|ζ| � 1 and low probability of multiple photon pair creation, since it implicitly assumes thatmultiple photon pairs per pump pulse are created independently. This is not true, since the bosoniccharacter of photons leads to higher photon pair creation probabilities into already populatedmodes. This interaction leads to a deformation of the broadband mode system, the multimodesqueezer system however can still be decomposed into orthogonal two-mode squeezers[20, 141, 87],but with increasing coupling strength their spectral modes will increasingly deviate from theSchmidt modes of the joint spectral amplitude f(ω1, ω2). We will ignore this effect for now, andreserve a deeper investigation for section 7.7. The strength of correlation between the outputbeams of such a source is characterized by its effective mode number K = 1∑

j |cj |4 . For c0 = 1

6.2 The second order correlation function g(2) 71

and all other cj = 0, K assumes its minimum value of 1, and the PDC process can be described asa two-mode squeezer according to Eq. 6.1.

A multimode SPDC source will not exhibit thermal photon statistics, although each of itsconstituent two-mode squeezers – each one corresponding to one Schmidt mode pair – will. Theoverall statistics is a convolution of their thermal statistics and for a high effective mode numberK it converges towards Poissonian photon statistics, where the n-photon probability scales like(const)n

n! .

6.2 The second order correlation function g(2)

The second order or Glauber correlation function is defined as the normal-ordered correlation ofthe field intensity (proportional to a†a ) with itself at a different time:

g(2)(t1, t2) =〈a†(t1) a†(t2) a(t1) a(t2)〉〈a†(t1) a(t1)〉 〈a†(t2) a(t2)〉

(6.6)

The value of g(2)(0, 0) is characteristic for states of light with certain well-defined photon statistics.A coherent state with Poissonian statistics exhibits g(2)(0, 0) = 1, a state with thermal statisticswill result in g(2)(0, 0) = 2. A Fock state |n〉 has g(2)(0, 0) = 1 − 1

n . The difference in secondorder correlation function values for different photon statistics is maximal for t1 = t2, so we willconsider only g(2)(0, 0) here.

For light states with low photon number, i. e. 〈n〉 � 1, g(2)(0, 0) is readily measured with aHanbury-Brown-Twiss type experiment[26], with the setup depicted in Fig. 6.1: The input lightstate ρ comprises of an arbitrary state ρa in mode a and a vacuum state |0〉〈0| in mode b that aremixed on a balanced beam-splitter. The output modes c and d impinge on a pair of single photondetectors, and single and coincident detection events are recorded. In the low photon number limit,the mean photon number goes linear with the detector click probability: pi ≈ ηi 〈n〉, where ηi isthe detector’s quantum efficiency. Accordingly, we can approximate the detectors’ measurementoperators µ1 and µ2 with the photon number operators in both beam-splitter output modes, c†cand d†d, respectively. The coincidence measurement operator – again, for a low photon numberapproximation – is their product µc = µ1 ⊗ µ2 = η1η2c†cd†d. In each case, the probability pi isgiven by 〈µi〉 = Tr[µiρ] with i ∈ {1, 2, c}.

Figure 6.1: Hanbury-Brown-Twiss interferometer to measure g(2)

To understand how this is a measurement of the second order correlation function g(2)(0, 0) ,we consider the quotient pc

p1p2of coincidences over single click events, and apply the beam-splitter

72 6 Two-mode squeezed vacuum source

transformations c, d→ 1√2

(a± ıb

)to express the measurement operators in terms of the input

modes:

pcp1p2

=η1η2 〈µc〉

η1 〈µ1〉 η2 〈µ2〉=〈c†d†cd〉〈c†c〉 〈d†d〉

=〈(

a† − ıb†)(

a† + ıb†)(

a + ıb)(

a− ıb)〉

〈(

a† − ıb†)(

a + ıb)〉 〈(

a† + ıb†)(

a− ıb)〉

(6.7)

Since the statistical average 〈.〉 is a linear operation, both numerator and denominator can beexpanded into averages of normal ordered products of the mode operators a and b. Mode b is inthe vacuum state, and thus all expressions containing b or b† vanish, and we are left with

pcp1p2

=Tr[a†a†aa ρ

]Tr[a†a ρ]

2 ≡ g(2)(0, 0) . (6.8)

6.3 g(2) for broadband input states

The derivation of g(2)(0, 0) in the last section implicitly assumes single photon detectors withperfect time resolution. This may be a fair assumption when working with very fast detectors andlight states with a long duration, such as a CW laser beam. The situation for our ultrafast SPDCsource is very much different however[129]: Our id201 APD modules are periodically triggeredwith a 2.5 ns gate width, and consequently their time resolution is also of that order of magnitude.We are pumping with ultrafast laser pulses with pulse lengths of about 1 ps, and the SPDC theduration of the output pulses is of the same order. We need to modify the mathematical model ofthe g(2) measurement procedure to accommodate this experimental situation[35].

The probabilities p1, p2 are in fact time-averaged mean values of the time resolved expressionsover a detection window with length 2T

pi =1

2T

∫ T

−Tdt 〈µi(t)〉 (6.9)

for i ∈ {1, 2}. The coincidence event probability depends on two events and hence needs to betime-averaged twice.

pc =1

4T 2

∫ T

−Tdt1

∫ T

−Tdt2 〈µc(t1, t2)〉 (6.10)

In terms of the ultrafast light pulse we set out to measure, the detection window has a durationof the order of 103 temporal variances. As the pulse amplitude decays like a Gaussian function,we can approximate it to be zero at the edges of the detection window. Therefore, it is a goodapproximation to assume an infinite detection window. For the ultrafast second order correlationfunction which we will simply refer to g(2) from now on, this leads to

g(2) = limT→∞

pcp1, p2

= limT→∞

∫ T−Tdt1

∫ T−Tdt2 〈µc(t1, t2)〉∫ T

−Tdt1 〈µ1(t1)〉∫ T−Tdt2 〈µ2(t2)〉

(6.11)

From this point we can apply the same set of transformations that led from Eq. 6.7 to Eq. 6.8 andfind

g(2) =

∫dt1∫

dt2 〈a†(t1) a†(t2) a(t1) a(t2)〉(∫dt 〈a†(t) a(t)〉

)2 (6.12)

6.4 g(2) for the ultrafast multimode squeezer 73

Invoking the Fourier transformation a(t) =∫

dω a(ω) eıωt allows us to switch to a frequency-based view of the problem.

g(2) =

∫dω1

∫dω2 〈a†(ω1) a†(ω2) a(ω1) a(ω2)〉(∫

dω 〈a†(ω) a(ω)〉)2 (6.13)

The subsequent basis transform a(ω) =∑

j ξj(ω) Aj gives us the measurement in the arbitrarilychosen spectral basis {ξj(ω)}.

g(2) =

∑j,k 〈A

†jA†kAjAk〉(∑

j 〈A†jAj〉

)2 (6.14)

In this form, the g(2) function reveals an alternate view on its physical meaning. Rather thanintensity correlations between different times, Eq. 6.14 suggests the measurement of intensitycorrelations between different broadband modes.

6.4 g(2) for the ultrafast multimode squeezer

Now we can calculate g(2) for a pure multimode vacuum squeezed state |Ψ〉 = S |0〉 = e−ıHPDC |0〉with relative ease. The mean photon number of an individual squeezer mode j for a smallsqueezing parameter ζ is

nj = 〈Ψ| A†jAj |Ψ〉 = sinh(|ζcj |)2 ≈ |ζcj |2 . (6.15)

The correlation term is most easily calculated in the Heisenberg picture. The Bogoliubov transfor-mation associated with the squeezer with Hamiltonian HPDC is

S†Aj S = S†Aj ,Bj Aj SAj ,Bj = cosh(|ζcj |) Aj − sinh(|ζcj |) B†j . (6.16)

Applying this to the correlation term gives

〈Ψ| A†jA†kAjAk |Ψ〉 = 〈0| S†A†j SS†A†kSS†Aj SS†AkS |0〉

= 〈0|(

cosh(|ζcj |) A†j − sinh(|ζcj |) Bj

)(cosh(|ζcj |) A†k − sinh(|ζcj |) Bk

)×(

cosh(|ζcj |) Aj − sinh(|ζcj |) B†j

)(cosh(|ζcj |) Ak − sinh(|ζcj |) B†k

)|0〉

(6.17)

which is a vacuum expectation value of normal ordered mode operators Aj and anti-normalordered mode operators Bj . Upon expansion, all terms containing the normal-ordered Aj vanish,leaving only

〈Ψ| A†jA†kAjAk |Ψ〉 = sinh(|ζcj |)2 sinh(|ζck|)2 〈0| BjBkB

†jB†k |0〉

= sinh(|ζcj |)2 sinh(|ζck|) (1 + δj,k).(6.18)

Substituting equations 6.18 and 6.15 into 6.14 finally results in

g(2) =

∑j,k sinh(|ζcj |)2 sinh(|ζck|) (1 + δj,k)(∑

j sinh(|ζcj |)2)2 (6.19)

74 6 Two-mode squeezed vacuum source

For a small squeezing parameter ζ , we can use the small angle approximation sinh(|ζcj |) ≈ |ζcj |and the normalization condition of the Schmidt coefficients

∑j |cj |

2 = 1 to simplify Eq. 6.19 to

g(2) = 1 +∑j

|cj |4 = 1 +1

K(6.20)

where K is the effective mode number, or cooperativity parameter of the two mode squeezer. Soapparently for two-mode squeezed vacuum states, the time-averaged g(2) correlation function forlarge detection times is connected in this very elegant manner to its mode number.

We try to understand this mathematical connection in physical terms: As has been noted above,type II PDC can in general be seen as an ensemble of broadband two-mode squeezers, each ofthem emitting a two mode squeezed vacuum state with thermal photon statistics. All broadbandmodes Aj or Bj of this decomposition share one polarization mode, a and b respectively. Astandard single photon detector cannot resolve them. It “sees” a convolution of the thermalphoton statistics of all broadband modes, and in the limit of a large number of modes, this is aPoissonian distribution[7]. If, on the other hand, there is only one mode per polarization to beginwith (which is only true for a two-mode squeezer), the detector receives a thermal distributionof photon numbers. Therefore, with the assumption that PDC emits a pure state, we can inferfrom g(2) = 2 measured in either output beam a two-mode squeezer source. In the presence ofadditional background events, we measure again a convolution of different photon statistics. Thiswill always reduce the experimentally obtained value of g(2) further towards 1.

6.5 g(2) measurement

In Fig. 6.2 (c) we illustrate the g(2) measurement: The KTP waveguide is pumped with ultrafastlaser pulses at 768 nm and a duration of the order of 1 ps. After the SPDC source, the idler beamis discarded, and the signal beam split by a 50/50 beamsplitter. The output modes are fed intoid201 APDs with a 2.5 ns gate width. Single (p1, p2) and coincidence (pc) click probabilities fordifferent spectral pump widths are recorded. In the previous sections we have shown that the timeaveraged second order correlation function g(2) can be reconstructed from these event probabilitymeasurements:

g(2) ≈ pcp1p2

. (6.21)

Since frequency correlations between signal and idler beam and thus squeezer mode numbercan be controlled by manipulation of the spectral width of the PDC pump beam, we see in Fig.6.4 (left) measurement results that show a maximum g(2) value at 1 .95 nm pump FWHM, inaccordance with the uncorrelated joint spectrum in Fig. 6.3 (left). When departing from theoptimum pump width, we see g(2) drop towards 1 as expected. Due to residual backgroundevents from waveguide material fluorescence and detector dark counts, we find a maximum ofg(2) = 1.80, and g(2) = 1.95 after background correction, corresponding to a cooperativityparameter of K = 1.05. This result demonstrates the next-to-perfect two-mode character of ourPDC squeezing source, and the degree of control we exact over the mode number and photonstatistics of the system.

6.6 Background event suppression and correction 75

Figure 6.2: Experimental setup: (a) Squeezed light source: A PP-KTP waveguide, spatially singlemode at 1550 nm, is pumped with a mode locked Ti:Sa laser emitting ultrafast pulseswith 8 nm FWHM. An accusto-optic modulator (AOM) reduces full repetition rate of76 MHz to 1 MHz, a HWP+PBS combination controls pump beam power. We adjustpump spectral width with a 4f spectral filter setup (SF-4f) and monitor it with a gratingspectrometer (GS). Pump light coupled through the waveguide is then separated fromthe generated signal and idler beams with a dichroic mirror (DM) and its powermeasured (PM). (b) g(2) measurement: Background light is removed from the signalbeam with a 12 nm FWHM spectral filter (SF12), then split at a 50/50 BS and eachoutput arm fed into APDs. Single, coincidence and trigger event rates are recorded.

6.6 Background event suppression and correction

In the JSI measurements, apart from the main SPDC peak, we detect a low intensity backgroundflux in a wide spectral range. In Fig. 6.4 (right), we plot the marginal spectrum of the signal beamof our source with a pump wavelength of 768 nm with setup 6.2. The blue spectral function wasmeasured with, the red one without a spectral band-pass filter specified at 1550 nm with 12 nmwidth (SF12) in front of the APD in-coupling. Both spectral measurements ran back-to-back, withidentical experimental parameters apart from the filter. The filter transmission at the centralwavelength of the peak (which is at 1544 nm rather than 1550 nm) is almost perfect at 98%. Thebackground we see here can originate from several sources. Our waveguide source supports inpropagation direction along x more PDC processes than the phase-matched type II process weare utilizing, namely the type I processes d32 and d33. Being not phase-matched in the telecomwavelength range, they produce a non-resonant, spectrally flat background in our detectionwindow. The most significant source of background is PDC coupling to radiation modes[65]: Notall photons propagate in waveguide modes, sometimes either signal or idler are created in anunguided radiation mode. As there are no boundary conditions for those, they form a continuumrather than a discrete spectrum, so that phasematching for such a guided-unguided pair is mucheaser to obtain. The radiation modes then allow for a wide range of energy distribution in the

76 6 Two-mode squeezed vacuum source

Pump FWHM 1.95nm

1541 1544 1547

Signal wavelength [nm]

15

25

15

28

15

31

0 5

00

0 1

00

00

1541 1544 1547Inte

nsi

ty [

counts

/s]

Signal wavelength [nm]

exptheo.

0 5

00

0 1

00

00

1525 1528 1531Inte

nsi

ty [

counts

/s]

Idler wavelength [nm]

exptheo.

Figure 6.3: Left: Joint spectrum from setup 6.2(b) with pump width at separability width 1.95 nmFWHM. Right: Marginal spectra.

photon pair, such that the guided half of these pairs show up as uncorrelated, spectrally broadbandbackground such as we observe. Other fluorescence processes besides PDC, caused by color centersor faults in the nonlinear crystal, are an unlikely source, since one would expect distinctly differentarrival time for their photons, due to long decay times compared to the time frames of our ultrafastPDC process. We were unable, with the equipment at our disposal allowing for timing resolutionof ca. 500 ps, to observe any significant arrival time difference between signal beam photons andbackground photons. Whatever the photon statistics of these processes are, in combination withthe thermal statistics of either beam of a perfect two-mode squeezed vacuum state they result in a“less thermal” distribution, i. e. g(2) will drop below a value of 2 for a two-mode squeezer sourcewith background noise.

Detector dark counts are most easily dealt with: We apply the least possible bias voltage to ourAPDs and use the smallest possible gate width of 2.5 ns to arrive at a dark count probability of5× 10−5 per measurement cycle.

Background photons degrade our experimental results most noticeably. However, most ofthem can be removed from the signal beam by using a suitable bandpass filter. After introducingspectral filter SF12 in setup 6.2(b), our detection event count dropped by a factor of five and wewere able to see g(2) values significantly greater than unity with an event probability of 2× 10−2.

This leaves the background photons that are transmitted by the bandpass filter SF12. Assumingidentically efficient single photon detectors (p = p1 = p2), we add the background photons as astatistically independent source of detector events with probability q at each detector. Coincidencedetection events from these uncorrelated background photons happen with probability q2. Wesubstitute p→ p+ q − pq and pc → pc + q2 − pcq2 and find

g(2) ≈ pc + q2 − pcq2

(p+ q − pq)2 (6.22)

In order to estimate q, we compare the unfiltered spectral distribution of the signal beam to thesame distribution with and without a spectral filter SF12 applied (Fig. 6.4 right). We see that thefiltered spectrum quickly falls to the level of detector dark counts outside the main peak, while the

6.7 Mean photon number 77

1.6

1.7

1.8

1.9

2

1 2 3 4

g(2

)

Pump spectrum FWHM [nm]

(a) (b) (c)

0

5

10

15

20

1540 1544 1548

Dete

ctio

n e

vent

rate

[kH

z]

Signal wavelength [nm]

FilterNo filter

Figure 6.4: Left: g(2) values from setup 6.2(c) for a variable pump FWHM. (red) experimentalvalues. (blue) Theory curve according to Eq. 6.22 with R = 1

20 . (violet) Backgroundcorrected theory curve according to Eq. 6.21. Right: Signal beam spectrum withbackground events in unfiltered spectrum (blue) and dark count events only inbackground-filtered spectrum (red).

unfiltered beam maintains a background event level 450 counts over the dark count level of 50counts over a wide range. We can expect those to be present in the main peak in both instances,and the signal-to-noise ratio R = q

p of detection events to background events equals the ratio ofthe areas of the background level “under the main peak” and the main peak itself. From the graphin Fig. 6.4 (right), we estimate R = 1

20 and thus find with Eq. 6.22 the theory curve in Fig. 6.4(left) in excellent agreement with our experimental data. We use the background corrected curveto predict g(2) = 1.95 in the absence of background at an optimal pump FWHM of 1.95 nm. Fromthe relationship between photon statistics and effective mode number in Eq. 6.20 we determinethe mode numberK = 1

g(2)−1= 1.05 of our source. In a pure single photon heralding experiment

we can therefore expect a photon purity of P = 1K = 0.95 (c. f. section 3.1).

6.7 Mean photon number

We now investigate the power-dependent photon pair flux of the two-mode squeezer source.We still pump the SPDC process with the optimal pump width of 1.95 nm at 768 nm centralwavelength and 1 MHz repetition rate and use the experimental setup from Fig. 6.2 (a). Thecorresponding separable joint spectrum is pictured in Fig. 6.3 (b). We feed the two-mode squeezedbeams into the setup according to Fig. 6.5, where signal or idler are separated with a PBS. Eitherbeam is cleaned of most of its background events with a spectral filter and coupled into the id201APDs. Single event probabilities ps and pi and the coincidence click probability pc are recorded.For signal we use again the filter SF12. The idler arm is at 1528 nm, and therefore we use as filterSF12b a 12 nm wide band-pass filter at 1530 nm. Its transmission of the idler beam is lower thanthe SF12’s transmission of the signal beam, which causes a significantly lower count rate. Fig.

78 6 Two-mode squeezed vacuum source

Figure 6.5: Setup for measuring single and coincidence photon detection events.

6.6 (left) holds the measurement results. Pump pulse energies are tuned with the half-wave-plate(HWP) in the SPDC source (c. f. Fig. 6.2a) up to 75 pJ.

A source of two-mode squeezed vacuum states with squeezing parameter r has in each arm amean photon number of

〈n〉 = sinh2r. (6.23)

The squeezing parameter r grows linear with pump field amplitude. Therefore it is proportionalto the square root of the pump intensity, and also of the pump pulse energy Ep. With a fittinglydefined proportionality constant B such that r = B

√Ep, we write

〈n〉 = sinh2B√Ep. (6.24)

For small pump energies there is a regime of linear growth of the photon number: 〈n〉 ≈ B2Ep.The mean photon number of a multi-mode SPDC source is simply the sum of the mean photonnumbers of all its modes. For a multi-mode state with Schmidt coefficients cj , this means

〈nMM〉 =∑j

sinh2cjB√Ep. (6.25)

A highly multi-mode state with K →∞ has infinitesimally small Schmidt coefficients, so that wecan apply the small angle approximation sinh(x) ≈ x:

〈nK→∞〉 =∑j

(cjB

√Ep

)2=∑j

c2jB

2Ep = B2Ep. (6.26)

In the limit of a very large effective mode number K, the mean photon number is expectedto increase linearly with pump pulse energy, since none of its squeezer modes will be pumpedstrongly enough to leave the linear regime. Already in Fig. 6.6 (left), we can see a departurein linear growth for both signal and idler, as can be expected for a nearly two-mode squeezedvacuum source. However, since we are using APDs, i. e. binary detectors, rather than intensitydetectors, we will see the profile of the two-mode squeezer mean photon number only for smallsqueezing values. For higher squeezing, and higher click probabilities, detector saturation willcause deviations, so that we cannot directly fit sinh(r)2 against the signal and idler measurements,and instead an expression for the APD click probability is needed.

6.8 Photon collection efficiency 79

0

0.05

0.1

0.15

0.2

0 25 50 75

Dete

ctio

n e

vent

pro

babili

ty [

1]

Pump pulse energy Ep [pJ]

SignalIdler

Coincidences

0

5

10

15

0 5 10 15 20 25

Kly

shko

effi

ciency

[%

]

Pump pulse energy Ep [pJ]

SignalIdler

Figure 6.6: Left: Click event rate of the KTP source pumped with pump spectrum FWHM 1.95 nmfor signal (red) and idler (blue) arm and coincidence events (violet). Right: Signal andidler Klyshko efficiencies with a linear fit each in the linear regime.

Since we have approximately single-mode signal and idler beams, we can use the result from Eq.4.9 and write for the single click probabilities

px(Ep) =ηxtanh

(B√Ep)2

1− (1− ηx) tanh(B√Ep)2 (6.27)

where ηx is the absolute detector efficiency and x ∈ {s, i}. Because of r = B√Ep, we have two

unknown parameters, ηx and the constant B, which must be the same for signal and idler. First,we separately determine the efficiencies from the Klyshko efficiencies plotted in Fig. 6.6 (right). Aswe have explained in section 4.2, ηs is the Ep = 0 intercept of a fit of the linear region of the signalarm click probability curve, and likewise for idler. In our case, this is the region from 5 pJ pumpenergy to 25 pJ. For smaller energies, detector dark counts suppress the Klyshko efficienciestowards 0 and for higher energies, the super-linear growth in mean photon number starts to benoticeable. From the linear fit lines in Fig. 6.6 (right), we read ηs = 7.0% and ηi = 3.5% andnow can fit the parameter B. The fit curves in Fig. 6.6 (left) follow Eq. 6.27 with the measuredefficiencies ηs, ηi for signal and idler respectively, and B = 0.149 1√

pJfor both arms. The fact that

both fits return the same value for B further confirms the fit results.

In Fig. 6.7 (left), we finally can plot the mean number of photon pairs 〈n〉 = sinh(

0.149 1√pJ

√Ep

)2

generated in the KTP source. For the maximal pump pulse energy of Ep = 75 pJ, we find〈n〉 = 2.8. This corresponds to a two-mode squeezing parameter r = 1.29 or a logarithmictwo-mode squeezing value of 20r

ln(10)dB = 11.2 dB. Since one pump pulse contains an average

of pJ 752π~

768 nm

= 75 pJ1.62×10−6 pJ

= 4.63 × 107 photons, the mean photon number 2.8 corresponds to a

conversion efficiency from pump photon to signal and idler photon pair of 6.0× 10−8.

80 6 Two-mode squeezed vacuum source

0

0.5

1

1.5

2

2.5

3

0 25 50 75

Mean p

hoto

n n

um

ber

<n>

Pump pulse energy Ep [pJ]

Signal, idlerLinear

0

4

8

12

0 25 50 75

Two-m

od

e s

queezi

ng [

dB

]

Pump pulse energy Ep [pJ]

Figure 6.7: Left: Mean photon number for signal and idler beams (red) of the KTP source pumpedwith pump spectrum FWHM 1.95 nm and a linear curve (violet) for comparison. Right:Corresponding two-mode squeezing values.

6.8 Photon collection efficiency

To determine the photon collection efficiency of our experiment, the fraction of photons that arenot lost due to optical losses between source and detector, we need to measure the overall quantumefficiency of the setup and correct it against detector losses. After very carefully adjusting oursetup (c. f. Fig. 6.5) for optimal coupling of the idler beam to its fiber-coupled id201 APD, we repeatthe power-dependent photon counting measurement from the previous section. The pump centralwavelength is 768 nm and its spectral FWHM is 1.95 nm. The APD efficiencies are set to 20%. Theresults can be seen in Fig. 6.8: The intercepts for the signal and idler Klyskho efficiency curves areηs = 1.3% and ηi = 13.7%, respectively. The signal efficiency obviously suffered in comparisonto idler, since we optimized the out-coupling lens after the waveguide source for maximal couplinginto the idler arm detector. This hints at small differences in numerical aperture at the waveguideout-coupling facet between the p-polarized signal and the s-polarized idler mode, as duringsimultaneous signal and idler in-coupling optimization we always had to compromise between ηsand ηi by maximizing coincidence counts. Here, we opted for one-sided optimization of the idlerarm instead to get a measure of the mode quality of the waveguide with respect to coupling intostandard telecom single-mode fiber. With a low power Klyshko efficiency ηi = 13.7% and 20%detector efficiency we estimate a photon collection efficiency of 13.7%

20% = 68.5%.

6.9 Conclusion

In this chapter, we discuss the photon statistics of a twin-beam source, and their connection withthe second order correlation function g(2) and effective mode number K. We then present ameasured g(2) value of 1.80, or 1.95 after background correction, demonstrating the near-perfecttwo-mode character of our waveguide source. A mean photon number per pump pulse of up to2.8 at 75 pJ pulse power is measured, corresponding to 11.2 dB two mode squeezing. The photoncollection efficiency has been measured with 68%, indicating a good modal overlap between the

6.9 Conclusion 81

0

0.05

0.1

0.15

0.2

0 25

Dete

ctio

n e

vent

pro

babili

ty [

1]

Pump pulse energy Ep [pJ]

SignalIdler

Coincidences

0

5

10

15

20

25

30

0 5 10 15 20 25

Kly

shko

effi

ciency

[%

]

Pump pulse energy Ep [pJ]

SignalIdler

Figure 6.8: Left: Click event rate of the KTP source pumped with pump spectrum FWHM 1.95 nmfor signal (red) and idler (blue) arm and coincidence events (violet). Right: Signal andidler Klyshko efficiencies with a linear fit each in the linear regime.

waveguide output modes and the spatial mode of the SMF28 standard telecom fibers guiding lightto the detectors. This is to our knowledge the first waveguide implementation of a two-modesqueezing source[42], and it out-performs its predecessor earlier bulk-crystal based two-modesqueezing sources[136, 93, 79] in terms of modal brightness, that is mean photon number permode, by several orders of magnitude. In terms of single-modedness, it surpasses a contemporaryfiber two-mode squeezing experiments[117, 118].

7Quantum pulse manipulation

With applications in secure quantum key distribution[40] and high precision positioning andclock synchronization protocols[55], ultrashort pulses of light play an ever-increasing role inmodern quantum information and communications. In recent years there has been increasedinterest in a finer control over the rich temporal and spectral structure of quantum light pulses,for applications such as ultrafast probing of the temporal wave function of photon pairs[101], orefficiently coupling single photons to trapped atoms. Given direct access, this structure could alsobe utilized to encode more information into or extract more quantum information from one pulseof light.

As we have discussed in section 2.6.6, it is possible to decompose any pulse amplitude into anycomplete set of orthogonal basis functions, or broadband modes[130]. Thus it can be consideredto be made up of an infinite number of temporally overlapping but independent pulses. While forclassical light all basis sets are equivalent, for quantum light there may be one special, intrinsicbasis choice[90]. For photon pair states this choice determined by a Schmidt decompositionof their bi-photon spectral amplitude into two correlated basis sets of broadband pulse forms,the Schmidt modes[82]. Heralding one of those photons by detecting the other with a singlephoton detector (SPD), this correlation results in the preparation of a photon in a mixed state ofall Schmidt modes present[59]. But with a SPD sensitive to a certain Schmidt mode, it opens upthe possibility to prepare pure single photons in the correlated Schmidt mode. Typically though,SPDs and optical detectors in general exhibit very broad spectral response and are not able todiscern between different pulse forms.

To compensate for the detectors’ shortcomings, one needs to include a filter operation sensitiveto broadband modes. It has already been shown that ordinary spectral filters cannot fulfill thisrole[106, 18]: They always transmit part of all impinging broadband modes at once, and thuscannot be matched to a single broadband mode. A sufficiently narrow spectral filter can be usedto select a monochromatic mode, however this way, high purity heralded quantum states areimpossible[80, 18]. Also, most of the original beam’s brightness as well as its pulse characteristicsare lost.

The idea of using broadband modes as quantum information carriers is compelling because oftheir natural occurrence in ultrafast pulses, and their stability in transmission: Centered around

84 7 Quantum pulse manipulation

Figure 7.1: Quantum Pulse Gate schema: Gating with a pulse in spectral broadband mode ujconverts only the corresponding mode from the input pulse to a Gaussian wave packetat sum frequency.

one frequency within a relatively small bandwidth typically, they allow for optical componentsthat are highly optimized for a small spectral range. Since all broadband modes experience thesame chromatic dispersion in optical media, they exhibit the same phase modulation and thusstay exactly orthogonal to each other. So a light pulse’s broadband mode structure is resilientto the effects of chromatic dispersion, making a multi-channel protocol based on them ideal foroptical fiber transmission. Additionally they allow for high transmission rates, as they inheritthe ultrashort properties of their ’carrier’ pulse, when compared to the ’long’ pulses used forclassical, narrow-band frequency multiplexing techniques. However, it is extremely challengingto actually access them in a controlled manner: Ordinary spectral filters and standard opticaldetectors destroy the mode structure of a beam. A homodyne detector with an ultrafast pulsedlocal oscillator beam is able to select a single broadband mode by spectral overlap, but only at thecost of consuming the whole input beam.[146, 112]

For discrete spatial modes, complete control of a beam’s multi-mode structure can be accom-plished with linear optics, as combining them to synthesize multi-mode beams and separatingconstituents without losses is possible[83, 131, 81, 145]. In order to exploit the pulse form degree offreedom, we must be able to exact similar control over broadband modes.

An important step towards this goal is to selectively target a single broadband mode for in-terconversion into a more accessible channel, for instance to shift it to another frequency withSFG. On the single photon level, in the SFG process two single photons “fuse” into one photon attheir sum frequency inside a χ(2)-nonlinear material. Well known in classical nonlinear optics,in recent years it has seen increasing adoption in quantum optics for efficient NIR single photondetection[107, 1, 138, 127], all-optical fast switching[139], super high resolution time tomographyof quantum pulses[79], quantum information erasure[126], and for translating non-classical statesof light to different frequencies[91]. Moreover, combined with spectral engineering[59, 134, 93], itenables a new type of quantum interference between photons of different color[105].

Here, we introduce the Quantum Pulse Gate (QPG)[41]: A device based on spectrally engineeredSFG to extract photons in a well-defined broadband mode from a light beam. We overlap anincoming weak, multi-mode input pulse with a bright, classical gating pulse inside a nonlinearoptical material (Fig. 7.1). Spectral engineering ensures that only the fraction of the input pulse

7.1 Beam-splitters, spectral filters and broadband mode selective filters 85

which follows the gating pulse form is converted. The residual pulse, orthogonal to the gating pulse,is ignored. An input quantum light pulse’s quantum properties can be preserved in conversion bymode-matching the gating pulse to its intrinsic mode structure.

SFG conversion efficiency can be tuned with gating pulse power, and unit efficiency is inprinciple reachable. Thus the QPG is able to unconditionally filter broadband modes from arbitraryinput states, and to convert them into a well-defined Gaussian wave packet at the sum frequency.By pulse-shaping the gating pulse we are able to switch between different target broadband modesduring the experiment. By superimposing gating pulses for two different broadband modes, wecreate interference between those previously orthogonal pulses. In combination with a standardsingle photon detector we are able to herald pulsed, pure, single-mode single photons from amulti-mode photon pair source.

7.1 Beam-splitters, spectral filters and broadband mode selective fil-ters

Our goal here is to develop the notion of a broadband mode selective filter, a filter that ideallytransforms one broadband mode from a beam and lets all others pass through unchanged. Thesimplest and without a doubt most frequently used mode transformer is the standard beam-splitter,operating on spatial rather than spectral modes. Assuming a ideal wavelength-independent beam-splitter, the effective Hamiltonian reads

HBS =

∫dω θa(ω) c†(ω) + θ∗a†(ω) c(ω) (7.1)

where θ is the beam-splitter angle that governs its transmittivity T = cos2θ and reflectivityR = sin2θ. To express the Hamiltonian in an arbitrary broadband mode basis {ψj(ω)} such that

Aj =

∫dω ξ∗j (ω) a(ω) (7.2)

Cj =

∫dω ψ∗j (ω) c(ω) (7.3)

we once again use the completeness relation∑

j ψ∗j (ω)ψj(ω

′) = δ(ω − ω′) and derive the reversetransformation

a(ω) =∑j

ξj(ω) Aj (7.4)

c(ω) =∑j

ψj(ω) Cj . (7.5)

Substituting this into the beam-splitter Hamiltonian HBS we calculate

HBS = θ∑j

AjC†j + A†jCj (7.6)

and see that the beam-splitter transforms each input broadband mode into a mode with thesame spectral function ψj(ω) but in a different spatial mode c. It acts on all broadband modesindiscriminately, however. As we required it to act wavelength-independent from the outset, this

86 7 Quantum pulse manipulation

result is not uprising. Yet we want to achieve a single-mode transformation with a Hamiltonianthat must have the form

Hsinglemode = θ(

A0C†0 + A†0C0

)(7.7)

but the simple model of the ideal beam-splitter does not provide us with any parameters we cantune to engineer this outcome.

If we now replace the constant θ in the ideal beam-splitter Hamiltonian in Eq. 7.1 with thefrequency dependent real-valued function θ(ω), we can model a standard thin layer spectral filterwith transmissivity function τ(ω) = cos(θ(ω)) and reflectivity function ρ(ω) = sin(θ(ω)). Theassociated Bogoliubov transformations read

a†(ω) =τ(ω) c†(ω) + ρ(ω) d†(ω) (7.8)

b†(ω) = −ρ(ω) c†(ω) + τ(ω) d†(ω) . (7.9)

Transmissivity and reflectivity obey the usual constraint to ensure energy conservation andpositive energy states at the beam-splitter:

|τ(ω)|2 + |ρ(ω)|2 = 1 (7.10)

Note that unlike spectral mode functions, τ and ρ do not have to be square-integrable functions.The associated unitary operator UBS describing this filter operation will transform an incomingbroadband state of light such that the broadband mode

A†j =

∫dω ξj(ω) a†(ω) (7.11)

will be substituted with

A†j →∫

dω τ(ω) ξj(ω) c†(ω) +

∫dω ρ(ω) ξj(ω) d†(ω) (7.12)

Therefore the spectrum of the transmitted part (in mode c) of a beam will go from initially|ξj(ω)|2 to |τ(ω) ξj(ω)|2. If the spectral functions {ξj} constitute an orthonormal basis, we caninfer two facts: Firstly, every spectral broadband mode ξj is partly transmitted, and secondly, thetransmitted mode functions are not generally orthogonal any more∫

dω |τ(ω)|2 ξ∗i (ω) ξj(ω) 6= δij , (7.13)

so that a new set of spectral basis functions has to be found for the transmitted beam[18]. Fromthis we see that a spectral filter cannot be used to select one broadband mode from a beam of light,while discarding all orthogonal modes.

Mathematically, such an operation must test orthogonality between input mode ξj(ω) and filterfunction φ(ω). Also, the transformation must describe the spectral output mode ξj(ω), so interms of spectral functions we can write down the transformation for the transmitted part of thebeam as

ξj(ω′)→ ξj

(ω′) ∫

dωφ∗(ω) ξj(ω) (7.14)

This filter will transmit light in the spectral mode φ, and reflect light in any orthogonal mode. Sofor the special case where the filter function coincides with an element of the input spectral basis

7.2 Broadband mode SFG 87

φ(ω) = ξi(ω), in terms of broadband mode operators we find the simple operator transformationrule

A†j →∫

dω ξj(ω) c†(ω) = C†j if j = i (7.15)

A†j →∫

dω ξj(ω) d†(ω) = D†j if j 6= i (7.16)

Formally, we have now defined a mode-selective process that targets exactly one broadband modefrom a basis set, disregarding all others. To realize a true broadband mode filter, we must ensurethat the transmitted mode Ci can be physically separated from the reflected modes Dj .

In order to physically implement the mode-selective filter, we ponder sum frequency generation(SFG) and identify the ’transmitted’ and ’reflected’ modes of our as yet purely theoretical filter withthe frequency-converted and the unconverted parts respectively of an optical beam undergoing anSFG process. The frequency gap between them allows us to easily separate them with a dichroicmirror.

7.2 Broadband mode SFG

To demonstrate that a SFG process can be used to generate a mode transformation of a simpleform according to Eq. 7.16, we first express its Hamiltonian operator in terms of broadband modes.For a bright classical gating pulse, the effective Hamiltonian of SFG that up-converts a photon inmode ’a’ to mode ’c’ is given by

H = θ

∫∫dωi dωo f(ωi, ωo) a(ωi)c

†(ωo) + h. c. (7.17)

Here we introduced the coupling constant θ ∝ χ(2)√P with χ(2) denoting the second order

nonlinear polarization tensor element of the SFG process and P the SFG pump pulse power.The SFG transfer function f(ωi, ωo) = α(ωo − ωi)× Φ(ωo, ωi) maps the input frequencies ωi tothe sum frequencies ωo, where α is the spectral amplitude of the gating pulse and Φ the phasematching distribution of the SFG process.

In parametric down-conversion (PDC), the Schmidt decomposition of the joint spectral ampli-tude of the generated photon pairs reveals their broadband mode structure. Applying the sameapproach to SFG[105] to decompose the spectral transfer function we find

f(ωi, ωo) =∑j

κj ξj(ωi) ψj(ωo). (7.18)

The decomposition is well-defined and yields two correlated sets of orthonormal spectral ampli-tude functions {ξj(ω)} and {ψj(ω)} and the real Schmidt coefficients κj which satisfy the relation∑

j κ2j = 1. If the gating pulse has the form of a Hermite function uj (ω) ∝ e

(ω−ω0)2

2σ2 Hj

(ω−ω0σ

)with Hj the Hermite polynomials, the basis functions of both sets are in good approximationHermite functions as well. In the Schmidt-decomposed form, the transfer function describes amapping between pairs of broadband modes ξj(ω)→ ψj(ω).

By defining broadband mode operators Aj =∫dω ξj(ω) a(ω) and Cj =

∫dω ψj(ω) c(ω)

corresponding to the Schmidt bases, the effective Hamiltonian from Eq. 7.17 can be rewritten as

88 7 Quantum pulse manipulation

H = θ∑j

κj

(AjC

†j + A†jCj

), (7.19)

An optical beam splitter has a Hamiltonian of the form HBS = θ a c† + h. c.; so with respectto broadband modes, SFG can be formally interpreted as a set of beam splitters, independentlyoperating on one pair of broadband modes each, such that Aj → cos(θj)Aj + ı sin(θj)Cj . Theeffective coupling constant θj = θ · κj ∝

√P takes the role of the beam splitter angle. Its

transmission probability – the probability to find a photon in the up-converted mode Cj if itinitially has been in mode Aj – is ηj = sin2(θj).

Figure 7.2: (A1-A3) SFG transfer function f(ωi, ωo) with (A1) and without (A2,A3) frequencycorrelations. (B1-B3) Coefficients κj for the first four Schmidt mode pairs of thetransfer functions. (C1-C3) SFG efficiencies Aj → Cj for the first four Schmidt modesagainst gating power dependent SFG coupling constant θ

In Fig. 7.2 A1-C1, we illustrate an example for a non-engineered SFG process, as commonlyfound in pulsed SFG experiments: The transfer function f(ωi, ωo)(Fig. 7.2 A1) exhibits spectralcorrelations, causing more than one non-zero Schmidt coefficient (Fig. 7.2 B1). This leads tothe simultaneous conversion of multiple modes Aj at once with non-zero coupling constantsθj ∝

√P for any given gating pulse power P (Fig. 7.2 C1). Hence a SFG process in general is not

mode-selective.

7.3 Spectral engineering and the Quantum Pulse Gate

However, spectral engineering can make SFG mode-selective by eliminating its spectral corre-lations so that the frequency of an up-converted photon gives no information about its originalfrequency. Now, Schmidt decomposition yields one predominant parameter κj ≈ 1 with all othersclose to zero and a separable transfer function f(ωi, ωo) ≈ κjξj(ωi)ψj(ωo). Also, now the fullcoupling θj ≈ θ is exploited, allowing for relatively weak gating beams for unit conversion effi-ciency. We achieve this by choosing a SFG process with an already correlation-free phasematching

7.4 Critical group velocity matching and QPG mode-switching 89

function Φ. If the phasematching bandwidth is narrow compared to gating pulse width, spectralcorrelations are negligible (Fig. 7.2 A2-A3), and we can approximate a separable transfer function(Fig. 7.2 B2-B3). The effective SFG Hamiltonian is now formally a beam splitter Hamiltonian

HQPG = θAjC†0 + h. c. (7.20)

meaning that only mode Aj is accepted for conversion. Because of the horizontal phasematching,the target mode is always the Gaussian pulse C0.

This process implements the QPG, with the bright input pulse used as gate pulse to select aspecific broadband mode. By tuning the central wavelength and spectral distribution of the gatingpulse, we can control the selected broadband mode’s shape, width and central wavelength. Wecompare the effect of different gating pulse forms: Gating with mode u0 (i. e. a Gaussian spectrum,Fig. 7.2 A2-C2) selects input mode A0, gating with mode u1 (Fig. 7.2 A3-C3) selects A1 from theinput pulse for frequency up-conversion.

Pure heralded single photons are a crucial resource in many quantum optical applications,but the widely used PDC photon pair sources emit mixed heralded photons in general. We nowconsider the application of the QPG to “purify” those photons. In type-II PDC, a pump photondecays inside a χ(2) -nonlinear medium into one horizontally polarized signal and one verticallypolarized idler photon. For a collinear type-II PDC source pumped by ultrafast pulses the generaleffective Hamiltonian in terms of broadband modes reads

HPDC = ζ∑j

cj

(˜A†jB

†j +

˜AjBj

). (7.21)

Using such a photon pair source for the preparation of heralded single photons, one finds thatthose are usually not in pure, but spectrally mixed states[134], and thus of limited usefulness formost quantum optical applications. We feed the signal photon (containing all broadband modes˜Aj) from the PDC source into the QPG which is mode-matched such that ˜

A0 = A0. We notethat for heralding pure single photons or pure Fock states[106], mode-matching is not necessaryand an engineered SFG process according to Eq. 7.20 is sufficient. In that case however, theresulting pulse shape is a coherent superposition of all input modes. Here, only the 0th mode isselected, and the higher modes do not interact with the QPG because the according beam splittertransformations yield the identity Aj → Aj for j > 0. We choose the gating pulse power suchthat θ0 = π

2 for optimal conversion efficiency. Combining the PDC source with a subsequent QPG

results transforms the PDC Hamiltonian as HPDC → H′ = e−ıHQPGHPDCeıHQPG , and we obtain

H′ = ıζc0B†0C†0 + ζ∞∑k=1

cj˜A†jB

†j + h. c. (7.22)

Since mode C0 is centered at the sum frequency of input and gating pulse, it can be split off easilyinto a separate beam path with a dichroic mirror. Conditioning on single photon events on thepath of C0 provides us with pure heralded single photons in mode B0. Fig. 7.3 illustrates thisscheme: A photon detection event heralds a pure single photon pulse in broadband mode u1. Thisprocess can be cascaded to successively pick off several modes Aj from the input beam.

7.4 Critical group velocity matching and QPG mode-switching

Group velocity matching (GVM) for SPDC (see section 3.2) is subtly different from SFG, albeit thatdifference exists in terminology only. The three involved waves swap roles: What was the pump

90 7 Quantum pulse manipulation

Figure 7.3: A QPG application: Generating pure heralded broadband single photons in differentmodes from a PDC source of multi-mode photon pairs

beam in the former case, is now considered the output beam in the latter, while signal or idleract as pump and input beams. Consequently, to fulfill the GVM condition we now are lookingfor a SFG with the output wave’s group velocity vo between pump and input velocities, vp and virespectively, so either

vp ≤ vo ≤ vi (7.23)

orvi ≤ vo ≤ vp (7.24)

has to hold.While it would be possible to find an uncorrelated case of ultrafast upconversion with regular

GVM, just as we did to realize the uncorrelated bi-photon spectrum for our two-mode-squeezersource in section 6, choosing critical GVM comes with a great advantage for the QPM: The abilityto switch, during operation, between orthogonal incoming modes.

Consider the general form of the SFG mapping function:

f(ωi, ωo) = α(ωo − ωi)× Φ(ωo, ωi) . (7.25)

Critical GVM is distinguished by a locally horizontal or vertical phasematching function. Wechoose input and output group velocities to be equal ( 1

k′o(ωo)= 1

k′i(ωi)), so that in a (ωo, ωo) graph

of Φ phasematching is horizontal. After linear expansion of Φ around the central input and outputfrequencies ωi and ωo we find

Φ(ωo, ωi) = e− (ωo−ωo)2

2σpm . (7.26)

The pump beam we assume to be emitted by a pulse shaper such that has the form of Hermitemode j:

α(ωo − ωi) = uj

(νo − νiσ

)=

1√√π2jj!σ

Hj

(νo − νiσ

)e−

(νo−νi)2

2σ2 (7.27)

f(ωi, ωo) ≈ e− ν2

o2σpm uj

(νo − νiσ

)(7.28)

7.4 Critical group velocity matching and QPG mode-switching 91

Figure 7.4: Critically group velocity matched SFG phase matching pumped with a Gaussianspectrum u0(ωp)

Figure 7.5: Critically group velocity matched SFG phase matching pumped with a Hermite modespectrum u1(ωp)

For a phasematching width much smaller than pump width σpm

σ � 1, we can see from Fig.7.4 that the mapping function’s contour plot is wide and thin; in the regions where f(ωi, ωo) issignificantly different from zero, the detuning of the output frequency νo is also much smallerthan the input detuning νi, so that the approximation uj

(νo−νiσ

)≈ uj

(−νiσ

)is justified. It follows

that the mapping function is approximately separable:

f(ωi, ωo) ≈ e− ν2

o2σpm uj

(−νiσ

). (7.29)

If one changes the spectral mode of the pump beam from u0 to u1, the resulting two-photonspectrum is still close to separability, owing to the horizontal, narrow phasematching function(c. f. 7.5). The phasematching width is inversely proportional to the SFG interaction length L,so that the above case can in principle be reached in experiment by increasing the length of thenonlinear crystal or waveguide. This can however be impractical, or bring unwanted side effectsdue to dispersion, absorption or production imperfections.

As we have pointed out in section 2.6.7, the SPDC joint spectral amplitude is connected to theeffective mode numberK of the process via the visibility V = 1

K of a theoretical two-source HOMexperiment with low photon pair flux. The relationship between K and f carries over to SFG,even though the experiment does not:

1

K=

∫dω1

∫dω2

∫dω3

∫dω4 f

∗(ω1, ω2) f∗(ω3, ω2) f(ω1, ω4) f(ω3, ω4) (7.30)

92 7 Quantum pulse manipulation

To gain greater insight into how the ratio between phasematching width and pump width r =σpm

σimpacts separability, we plot K versus ratio r for the pump in the first four Hermite modes inFig. 7.6(left). As long as r is in the order of 1

10 or smaller, the effective mode number stays belowK = 1.035. For r > 1

10 however, the effective mode number rises approximately linear: K ∝ r.

1

1.02

1.04

1.06

1.08

1.1

0 0.05 0.1 0.15 0.2 0.25

Eff

ect

ive m

od

e n

um

ber

Kj

Ratio r

K0(r)K1(r)K2(r)K3(r)

K3(0.1)=1.035

0.98

0.985

0.99

0.995

1

0 0.05 0.1 0.15 0.2 0.25M

inim

um

Sch

mid

t num

ber

κ0,m

in,j

Ratio r

κ0,min,0(r)κ0,min,1(r)κ0,min,2(r)κ0,min,3(r)

κ0,min,3(0.1)=0.991

Figure 7.6: Left: Effective mode number Kj of a critically group velocity matched SFG processpumped with Hermite mode uj over ratio r =

σpm

σ . Right: Lower boundary for the firstSchmidt coefficient of the same process.

We can also make an estimate for the Schmidt coefficient κ0 of upconversion of the strongestSchmidt mode. For a fixed κ0 the worst-case effective mode number is reached when there isexactly one other mode, i. e. κ1 > 0 and κj = 0 for j ≥ 2. From 1

K =∑

j |κj |4 = |κ0|4 + |κ1|4 ≥

11.035 and |κ0|2 + |κ1|2 = 1 we calculate

|κ0|4 + (1− |κ0|2)2 =1

K

2 |κ0|4 − 2 |κ0|2 + 1− 1

K= 0

|κ0|2 =1

2±√

1

2K− 1

4

(7.31)

From the two possible solutions for |κ0|2 we choose the greater one, since we had assumed j = 0to be the strongest mode, therefore |κ0|2 > |κ1|2 = 1− |κ0|2 and consequently |κ0|2 > 1

2 . So forK ≥ 1.035 we find as lower boundary for the probability of the first Schmidt mode

κ0 ≥

√1

2+

√1

2K− 1

4= 0.991. (7.32)

with the results depending on the ratio r plotted in Fig. 7.6 (right).In Eq. 7.32 we have established that for Hermitian pump modes j ≤ 3 the strongest mode

of the Schmidt decomposition of the SFG mapping function f has a Schmidt coefficient κ0 thatis still almost at unity, so that we can consider the SFG process almost single-mode. The formof the first Schmidt mode in the limit of a vanishing pump-to-phasematch width ration r → 0

7.5 Experimental feasibility 93

is exactly that of the pump spectrum, only the central frequency is now that of the input beaminstead; this is the central idea of the QPG, and also of the closely related, but independentlydeveloped pulse shaping method in [71]. For small values of r, we expect small variance from thisspectral form as well, but instead of analyzing this in the idealized case of an exactly horizontal,Gaussian-shaped phase-matching function, we present numerical calculations for more realisticexperimental parameters in the next section.

7.5 Experimental feasibility

Finally we give realistic parameters to show the feasibility of an experimental implementation ofthe QPG, and analyze the mode selection performance. For the SFG process we use a periodicallypoled LiNbO3 (PPLN) waveguide with an area of 8× 5µm2, a length of L=50 mm, a Λ = 4.2µmperiodic poling period and at 175◦C to achieve phasematching for SFG of an input pulse at1550 nm to 557 nm. It is gated by coherent laser pulses at 870 nm with 2 ps pulse length or aspectrum with 0.635 nm FWHM to ensure a transfer function separability. The uncorrelated,separable transfer functions in Fig. 7.2 (A2-A3) are calculated from these parameters, using gatingpulses with u0 and u1 as spectral amplitude, respectively.

Figure 7.7: Overlap between input pulse mode ul and QPG Schmidt mode ξj for mode-matched(left) and non-mode-matched (right) case.

In Fig. 7.7 we illustrate the switching capabilities of our QPG, as well as the impact of modematching. For the given material parameters, we employ gating pulses with pulse form u0 to u10,determine the Schmidt decomposition of the resulting transfer function f(ωi, ωo), and plot theoverlap of the predominant Schmidt function ξj (with κj ≈ 1) with an Hermitian input modeul from an incident light pulse. On the left, gating and input pulse have equal frequency FWHM,which is essential for good mode matching. Now, by switching the order j of the gating mode (andwithout changing the physical parameters of the QPG), we select with high fidelity only the input

mode j to be converted. For j ≤ 10, the overlap∣∣∣∫ dωu∗j (ω) ξj(ω)

∣∣∣2 exceeds 99%, and the overlap

for all other input modes combined therefore is less than 1%: Only a negligible fraction of modesother than the selected input mode are converted.

In contrast, Fig. 7.7 (right) has no mode matching, the gating pulse FWHM is twice that of theinput pulse. Multiple strong overlaps between SFG Schmidt modes ξj and input modes ul appear:

94 7 Quantum pulse manipulation

A wide range of modes is converted for any given input spectrum. The checkerboard patternreflects the fact that only Hermite modes of the same parity overlap, and the SFG Schmidt modesare in good approximation Hermite modes.

7.6 The Quantum Pulse Shaper

The “switchability” of the QPG process hinges on the locally horizontal phasematching function ofthe underlying SFG; geometrically speaking the spectral shape of the gating beam determines thespectral shape of the QPG input mode, while the output mode follows a perpendicular cut throughthe phasematching function, leading to a sinc profile, or a Gaussian profile in the approximation.In short, the input function is variable, while the output function is fixed.

Any three-wave-mixing process with a vertical phasematching function turns this around, as isillustrated in Fig. 7.8: We find a mode-selective process with a fixed input mode and a variableoutput mode that follows the spectral shape of the bright pump beam. This is the Quantum PulseShaper (QPS)[21], an technique to directly imprint an arbitrary pulse form on a single photon,rather than indirectly as in the scheme in Fig. 7.3. The Hamiltonian again has the structure of abroadband mode beamsplitter:

HQPS = θA0C†j + h. c. (7.33)

The difference to the QPG Hamiltonian is that now the output mode Cj rather than the inputmode is switchable, so that the pump spectrum will be repeated by the SFG output mode.

Figure 7.8: Critically group velocity matched SFG with a horizontal phase matching function,pumped with a Hermite mode u1(ωp)

7.7 Time ordering and strongly coupled three-wave-mixing

Our model of three-wave-mixing processes up until now has implicitly assumed that multiplephoton pairs in the case of SPDC or multiple conversion events in the case of SFG are independentfrom each other. However, this is not the case in general, and for processes with a strong couplingconstant ζ this self-interaction will influence its modal structure. We can understand the underly-ing reasons by considering the time evolution operator of an initially general quantum mechanicalprocess described by the Hamiltonian formalism.

The unitary time evolution operator U(t, t0) associated with a physical process described byHamiltonian H(t) propagates an arbitrary state at time t0 to the state at time t:

U(t, t0) |Ψ(t0)〉 = |Ψ(t)〉 . (7.34)

7.7 Time ordering and strongly coupled three-wave-mixing 95

For SPDC, the time difference t− t0 corresponds to the travel time of the pump pulse through thenonlinear material of length L such that t− t0 = L

vpwhere vp is the pump beam’s group velocity.

Up to this point, we have assumed the operator to be defined by

UTaylor(t, t0) = e−ı∫ tt0

dt′H(t′) (7.35)

which can readily be developed into a Taylor series:

UTaylor(t, t0) = 1− ı∫ t

t0

dt′H(t′)− 1

2

[∫ t

t0

dt′H(t′)]2

+ ... (7.36)

However, this definition implicitly assumes that the Hamiltonian operator at different timescommutes, but in general we have [

H(t1) , H(t2)]6= 0, (7.37)

and for three wave mixing we can easily verify that this is the case by calculating this commutatorfor H(t) = abc†e±ı(ωt) + h. c..

We can understand the physical impact of this fact for SPDC: The photons’ bosonic charactermakes the emission of a photon pair into two already populated mode more probable than into avacuum mode. Therefore, there is a difference between the first and the second produced photonpair in a PDC process. But the second order term of Eq. 7.36 implies that the creation of bothphoton pairs is completely independent, as it can be expressed as the non-time-dependent productof two commuting (indeed identical) Hamiltonian operators. We can now see that this is not thecase here, and that Eq. 7.36 is a good approximation only if we truncate after the first order term,i. e. in the weakly coupled regime where with high probability only one photon pair is produced.The same argument can be made for SFG, and likewise the simplified Taylor approach is applicableto weak coupling θ � 1.

The exact form of the time-ordered time evolution operator can be derived by considering thetime-dependent Schrödinger equation for an arbitrary quantum state |Ψ(t)〉[142]:

H(t) |Ψ(t)〉 = ı∂

∂t|Ψ(t)〉 (7.38)

When we substitute Eq. 7.34 into 7.38, we find

H(t) U(t, t0) |Ψ(t0)〉 = ı∂

∂tU(t, t0) |Ψ(t0)〉 . (7.39)

We can consider this as a Schrödinger equation for the time evolution operator – rather than thestate it is generating – and write

ı∂

∂tU(t, t0) = H(t) U(t, t0) . (7.40)

Eq. 7.40 is sometimes referred to as the Tomonaga-Schwinger equation. Transformed into anintegral equation through straight-forward integration it reads

U(t, t0) = 1− ı∫ t

t0

dt1H(t1) U(t1, t0) (7.41)

96 7 Quantum pulse manipulation

Recursive self-substitution results in the Dyson series representation of the time evolution opera-tor:

U(t, t0) = 1− ı∫ t

t0

dt1H(t1)

−∫ t

t0

dt1H(t1)

∫ t1

t0

dt2H(t2)

+ı

∫ t

t0

dt1H(t1)

∫ t1

t0

dt2H(t2)

∫ t2

t0

dt3H(t3) + ...

(7.42)

In each summand term, we see that the nested integrals set the constraint t > t1 > t2 > ..., andthe time-dependences appear in this descending order; therefore, the series is time-ordered. Thecommutation relation Eq. 7.37 forbids trivially switching time-dependences out-of-order. If theHamiltonians at different times do commute, then this series is equal to the Taylor series in Eq.7.35.

In section 2.6.6, we observed that with the help of the Schmidt decomposition of the jointspectral amplitude, one can express the broadband squeezing operator S as a tensor productof broadband two-mode squeezing operators. The same applies to the non-time-ordered timeevolution operator UTaylor of SFG

UTaylor(t, t0) = e−ıH = e−i(∑

j θκjAjC†j+θ

∗κ∗j A†jCj)

=⊗j

e−i(θκjAjC

†j+θ

∗c∗j A†jCj)

(7.43)

where H is once again the effective SFG Hamiltonian. But while this is a very good approximationfor weakly coupled SFG with θ � 1, in general it cannot be applied to the exact evolutionoperator U(t, t0), since it implicitly assumes that photon sum frequency conversion events areindependent from each other. We can understand this from the definition of U in Eq. 7.42: Startingpoint for the Schmidt decomposition is the mapping function f(ωi, ωo) from the effective SFGHamiltonian H, which is the integral of the interaction Hamiltonian over the interaction time.The operator U features H only in the first non-constant term, all higher terms are time-orderednested integrals over the interaction Hamiltonian Hint(t) that represent the interaction betweenmultiple conversion events. Therefore in the general, non-perturbative case the operator U cannotbe decomposed to single-mode operators according to Eq. 7.43, and consequently its broadbandmode basis must differ from the Schmidt modes of the perturbative case.

To gain insight into the mode structure of SFG (or any three wave mixing process) with a classicalpump beam for arbitrary coupling strengths, we consider the general input-output relations orBogoliubov transformations for the mode operators a and c. Since it is known that every timeevolution U(t, t0) of a bi-linear Hamiltonian generates linear Bogoliubov transformations, wewrite as ansatz[76, 87, 31]:

aout(ω) = U(t, t0) a(ω) U†(t, t0) =

∫dω′ Ca

(ω, ω′

)a(ω′)

+

∫dω′ Sa

(ω, ω′

)c(ω′)

cout(ω) = U(t, t0) c(ω) U†(t, t0) =

∫dω′ Cc

(ω, ω′

)c(ω′)

+

∫dω′ Sc

(ω, ω′

)a(ω′) (7.44)

The operator transformations are governed by the bivariate integral kernels Ca, Sa, Cc and Sc.According to the Bloch-Messiah theorem for bosons[20, 141, 87], there exist four ultrafast spectral

7.7 Time ordering and strongly coupled three-wave-mixing 97

mode bases A′j , C′j , A

outj and Cout

j that decompose the integral kernels such that 7.44 can bere-written as

Aoutj = cos(κj) A′j + ısin(κj) C′j

Coutj = cos(κj) C′j − ısin(κj) A′j .

(7.45)

We note that the input mode sets {A′j} and {Aoutj } in general differ from each other as well as

from the input Schmidt mode set {Aj} that arises from the decomposition of the perturbativeapproximation for the time evolution UTaylor. The form of the Bloch-Messiah decomposition andthus the spectral shape of the modes now depends on the coupling constant θ. We performed adecomposition of the QPG’s SFG process for the experimental parameters in section 7.5 and θ0 = π

2 ,by solving numerically for the integral kernels and applying a singular value decomposition tothem.

Figure 7.9: QPG input mode (left) and output mode (right) solution for the perturbative (blue) andrigorous, time-ordered (red) case.

Figure 7.10: Mode occupation coefficients κj for the perturbative (red) and time-ordered (blue)case.

In Fig. 7.9, we see the impact of time ordering on the mode shapes of the QPG: The inputmode is slightly wider than the perturbative solution predicts, but still of a Gaussian shape. In

98 7 Quantum pulse manipulation

experiment this can be easily compensated for by using a spectrally narrower pump beam. Theoutput mode’s sinc shape is still visible but slightly “washed out” in the rigorous solution. Thecomparison of mode occupation coefficients κj in Fig. 7.10 reveals that for strong coupling, theoccupation probability of the ground mode drops to |κ9|2 ≈ 90% while the probabilities of thehigher modes increase. We conclude that taking into account time ordering for strongly coupled,approximately single mode SFG decreases the “single-modedness” and increases the effectivemode number K, lowering the working fidelity of the QPG.

7.8 Conclusion

In conclusion, we have introduced the concept of the QPG, a flexible device to split well-definedbroadband modes from a light pulse based on spectrally engineered SFG. The selected mode canbe switched by shaping the gating pulse spectrum and converted with high efficiency. Further,we have given a realistic set of experimental parameters for a QPG realized in a PPLN waveguideand demonstrated the high flexibility of the QPG achieved through shaping the gating pulse form.We proposed as an initial application the preparation of pure heralded single photons from anarbitrary type II PDC source. We investigated the effects of time-ordering in the strong couplingregime and found that for a QPG approaching unit conversion efficiency we have to take intoaccount coupling-strength dependent changes in spectral shape of the input and output modes aswell as working fidelity.

8Conclusion and outlook

In the course of this thesis, we have investigated the means to create and manipulate ultrafastquantum pulses with χ(2) -nonlinear optical processes.

We have designed and implemented the first waveguide source of ultrafast two-mode squeezedvacuum states in the telecom wavelength regime based on type II SPDC[42]. To minimize the spec-tral correlations between signal and idler beams, we took advantage of the dispersion propertiesof the nonlinear KTP crystal used in the experiments. After setting up the basic experiment anddetecting photon pairs, we built a single photon pair fiber spectrometer to map the joint spectrumof our source, demonstrated our control over the form of the spectral correlations, and found theoptimal spectral pump width for spectrally uncorrelated beams at 1.95 nm. We then characterizedthe squeezed vacuum output state with the help of a measurement of the second order correlationfunction g(2) to ensure state separability on the single photon pair level. The measured value isg(2) = 1.80, and after background analysis we find a corrected value of g(2) = 1.95, correspondingto an effective Schmidt mode number K = 1

1−g(2) = 1.05, meaning we generate a state close to

a two-mode squeezed vacuum with g(2) = 2 and K = 1. For moderate pump pulse energiesof 75 pJ, our source produces on average 2.8 photon pairs per pump pulse with a conversionefficiency of 6.0 × 10−8. This is equivalent to a two-mode squeezing of 11.2 dB. In terms ofmode number and efficiency, our source constitutes a considerable improvement over previousseparable photon pair/two-mode squeezed vacuum experiments both in χ(2) -nonlinear bulkcrystal materials and χ(3)-nonlinear optical fibers.

The quantum light pulses at telecom wavelengths are well suited for transmission in wide areafiber communication networks. Thanks to its implementation in a waveguide chip of 10 mmlength, this compact source is ideal for the generation of high-photon number single modequantum pulses or alternatively high-purity single photon pulses in integrated optics experiments.For an even higher level of integration one could implement be the separation of the pump beamfrom the output state with a waveguide-integrated Bragg grating, and pig-tail the waveguide withpolarization-maintaining fibers aligned to the SPDC input and output polarization directions.

The first priority in improving the waveguide source itself in the future should be the suppressionof uncorrelated background photons created through coupling to radiation modes. This can beachieved with a steeper refractive index step between waveguide and surrounding, by using

100 8 Conclusion and outlook

different dopants or different concentration of dopants to define the waveguides, using ridgewaveguide structures, or using a different waveguide material than KTP altogether.

The next logical step towards a CV quantum repeater to counter the decoherence threateningthe security of long distance quantum communication is a photon subtraction experiment alongthe lines of [98] to improve, by increasing the available two-mode squeezing, the teleportationfidelity of a CV light state.

With the fiber spectrometer[8] we created a useful tool to characterize the spectra of single-or few-photon signals. With no moving parts involved, it is easy to set up and very robust, andalready has seen adoption in the field of quantum optics[23, 53].

We also have introduced the quantum pulse gate[41], a flexible filter device sensitive for broad-band modes or, equivalently, pulse forms based on spectrally engineered SFG in a PPLN waveguide.Given an arbitrary input state of light and a bright, coherent gating pulse, it converts one broad-band mode with high fidelity to one well-defined broadband mode in another wavelength regimeand lets transmit all broadband modes orthogonal to the selected one. The selected mode isdetermined by the gating pulse shape, and the converted mode is constant. A broadband-modeselection cannot be achieved with standard spectral filters since they transmit part of everybroadband mode. Subsequently, the converted mode can be conveniently split off by a dichroicmirror.

While this new concept still awaits experimental demonstration, and the modal distortionsthrough self-interaction at high powers need to be fully investigated, already several quantumoptical applications are conceivable: The implementation of a source of pure heralded photonswith the ability to control the output broadband mode of the photon, the de-multiplexing ofmultiple quantum information channels in orthogonal broadband modes, or the extraction of atwo mode squeezed vacuum state from a multimode squeezer. Also it has spawned the quantumpulse shaper[21]: By reversing the quantum pulse gate’s working principle, it is possible to create athree-wave-mixing process that allows to directly shape a quantum light pulse into an arbitrarypulse form at another wavelength regime.

Acknowledgments

First and foremost I would like to thank my supervisor Prof. Christine Silberhorn for providingme with the opportunity to work within the IQO group as both a diploma and a PhD student atthe MPL Erlangen on such a fascinating topic, while always having the freedom and support topursue my own ideas.

I would also like to thank my collaborators within the group: Andreas Christ, who first as adiploma student supported me in the lab, and from whose theoretical work as a PhD student thisthesis has greatly benefited. Benjamin Brecht, who worked out the experimental parameters for thequantum pulse gate. Malte Avenhaus, who together with me implemented the fiber spectrometer,especially (but far from only) for taking care of the computer hardware side of things. PeterJ. Mosley, whose experience and practical help in the lab was invaluable in realizing the SPDCsource. And finally Thomas Lauckner, whose thesis on modal dispersion in waveguides helped tolaunch several new research projects within the group. My thanks go also to our intern PatrickBronner and student helpers Peter Vogt, Philip Weber and Thomas Dirmeier, who decided tostay on and further develop my project within the QIV group at the MPL. I want to thank myother colleagues, present and past, for making our group a place I enjoyed working at. HendrikColdenstrodt-Ronge showed me the ropes when I had never seen a quantum optics lab from theinside. I thank Christoph Söller, Kaisa Laiho, and Andreas Schreiber for being always ready tohelp, discuss physics or play badminton, and also Peter Rohde, Katiúsca Cassemiro, WolfgangMauerer and Felix Just.

Many people outside the group were also involved in making this thesis a success. Prof. BerhardSchmauss from engineering department at Erlangen University and Prof. Georgy Onishchukovfrom the MPL generously provided us with the DFC coils for our spectrometer. My special thanksgo to the QIV group at MPL, to name just a few, to Christoph Marquardt, Christoffer Wittmann andJosef Fürst, for loaning us any lab equipment we happened to need in a hurry, and also to PhilipHölzer, Sebastian Stark, and the Russel Division in general, for lending us their fiber cleavers andoptical spectrum analyzers on a regular basis. A thank you goes also to Chris Poulton, on whosemode solver code the work of Thomas Lauckner was originally based. Prof. Jan-Peter Meyn was agreat source of knowledge for all things phase-matching, and supported our first steps towardslithium niobate technology by providing PPLN-samples, which were fabricated by Birgit Stiller.

I would like to thank Marga Schwender, Tina Schwender, Sabine König, Ulrike Bauer-Buzzoni,Margit Dollinger, Carolin Haßler, Nadine Danders, Manfred Eberler, Michael Zeller, BenjaminKlier, Bernard Thoman and Robert Gall for support on many occasions, be it an urgent order, aguest who needed rooming, a networking problem, impossibly heavy equipment to install, or aflooded lab, and generally for keeping the institute running.

I also thank the IQO group Mk. II at the University of Paderborn for welcoming me as a guestduring the last stages of my work, and I especially want to thank Irmgard Zimmermann for herhelp in finding a flat.

Finally, I would like to thank my parents for their ongoing love and support that made thisthesis possible.

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