Atomic Scattering from Bose-Einstein

Condensates

Dissertation

zur Erlangung des akademischen GradesDoctor rerum naturalium

(Dr. rer. nat.)

vorgelegt

der Fakultät Mathematik und Naturwissenschaftender Technischen Universität Dresden

vonIvo Häring

aus Antholing bei Münchengeboren am 23.3.1972 in Ebersberg

Max-Planck-Institut für Physik komplexer Systeme Dresden2003

——————————————————————–

Eingereicht am 5.9.2003

Erster Gutachter: Prof. Dr. Jan-Michael RostZweiter Gutachter: Prof. Dr. Rüdiger SchmidtDritter Gutachter: Prof. Dr. Michael Fleischhauer

Verteidigt am

To A. H., B. H., C. H., S. H. and U. H.

Contents

1 Introduction 11.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Connecting threads through the chapters . . . . . . . . . . . . . . 10

2 Hartree-Fock-Bogoliubov theory 122.1 Background to the Hartree-Fock-Bogoliubov theory . . . . . . . . 122.2 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Consequences of condensation . . . . . . . . . . . . . . . . . . . . 162.4 Diagonalisation of the grand canonical Hamiltonian . . . . . . . . 192.5 Approach with Heisenberg’s equation of motion . . . . . . . . . . 242.6 Approximations and generalisations . . . . . . . . . . . . . . . . . 252.7 Spherically symmetric trapping potential . . . . . . . . . . . . . . 27

2.7.1 Expansion in spherical harmonics at zero temperature . . . 272.7.2 Bound and free quasi-particle modes . . . . . . . . . . . . 292.7.3 An iterative scheme for finite temperatures . . . . . . . . . 32

3 Solution of the Gross-Pitaevskii equation 343.1 Wave function propagation . . . . . . . . . . . . . . . . . . . . . . 353.2 Sine transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Representation of functions and operators . . . . . . . . . . . . . 373.4 Imaginary time propagation . . . . . . . . . . . . . . . . . . . . . 383.5 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Solutions of the Hartree-Fock-Bogoliubov equations 414.1 Diagonalisation for bound states . . . . . . . . . . . . . . . . . . . 414.2 Propagation from two sides . . . . . . . . . . . . . . . . . . . . . 424.3 Potential scattering reference problem . . . . . . . . . . . . . . . . 44

4.3.1 Neglecting the hole quasi-particle mode . . . . . . . . . . . 454.3.2 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Examples for zero temperature neglecting non-condensed modes . 464.5 Thomas-Fermi regime and harmonic confinement . . . . . . . . . 56

4.5.1 Field operator and quasi-particle energies and modes . . . 56

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Contents

4.5.2 Can we go to finite temperatures? . . . . . . . . . . . . . . 57

4.6 Example for finite temperature . . . . . . . . . . . . . . . . . . . 57

5 Levinson theorem for Bogoliubov equations 60

5.1 Introduction and summary . . . . . . . . . . . . . . . . . . . . . . 60

5.2 The coupled equations . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.1 Two-channel scattering . . . . . . . . . . . . . . . . . . . . 65

5.3.2 Bogoliubov equations with negative chemical potential . . 67

5.3.3 Remarks about positive chemical potential . . . . . . . . . 70

5.4 The physical solutions . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 The regular solutions . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 The irregular solutions . . . . . . . . . . . . . . . . . . . . . . . . 74

5.7 The Jost matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.8 The Fredholm determinant of the physical solution . . . . . . . . 80

5.9 Zeros of the Fredholm determinant . . . . . . . . . . . . . . . . . 82

5.10 Contour integration . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.11 Analytical Thomas-Fermi model for a finite square-well trap . . . 90

5.11.1 Fredholm determinant of the Bogoliubov equations . . . . 90

5.11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.11.3 Jost function for the potential scattering model . . . . . . 93

5.11.4 Number of bound states for selected cases . . . . . . . . . 94

5.11.5 Non-standard zeros of the Fredholm determinant . . . . . 97

6 Single particle scattering 99

6.1 Basic approach to the Born approximation . . . . . . . . . . . . . 99

6.1.1 Hamiltonian for distinguishable atoms and its splittings . . 100

6.1.2 Description of the target . . . . . . . . . . . . . . . . . . . 101

6.1.3 Direct processes . . . . . . . . . . . . . . . . . . . . . . . . 103

6.1.4 Exchange processes . . . . . . . . . . . . . . . . . . . . . . 103

6.1.5 Scattering of distinguishable and identical particles . . . . 104

6.2 Field operator description . . . . . . . . . . . . . . . . . . . . . . 105

6.3 Elastic scattering from a spherically symmetric condensate at finitetemperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4 Comparison with analytical expressions . . . . . . . . . . . . . . . 109

6.4.1 Thomas-Fermi solution and generalised condensate radii . 110

6.4.2 Gaussian trial function . . . . . . . . . . . . . . . . . . . . 118

6.4.3 Exponential trial function . . . . . . . . . . . . . . . . . . 120

6.5 Analytical fit of the computed distributions . . . . . . . . . . . . 122

6.6 Beyond the first Born approximation . . . . . . . . . . . . . . . . 123

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7 General formalism for scattering of several particles 1287.1 One indistinguishable probe atom in the initial and final channel . 128

7.1.1 Full field operator . . . . . . . . . . . . . . . . . . . . . . . 1297.1.2 Bogoliubov field operator . . . . . . . . . . . . . . . . . . . 131

7.2 One probe atom in the initial channel and two atoms in the finalchannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2.1 Full field operator . . . . . . . . . . . . . . . . . . . . . . . 1337.2.2 Bogoliubov field operator . . . . . . . . . . . . . . . . . . . 135

7.3 Several atoms in the final channel . . . . . . . . . . . . . . . . . . 136

8 Scattering from vortex condensates 137

9 Conclusion 149

A Appendix 154A.1 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . 154A.2 Scaling properties of the Gross-Pitaevskii equation . . . . . . . . . 157A.3 Spherical Bessel, Neumann, Hankel and Ricatti functions . . . . . 160A.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162A.5 Three phase shift formulae . . . . . . . . . . . . . . . . . . . . . . 163A.6 Ortho-normal Jacobi polynomials . . . . . . . . . . . . . . . . . . 164A.7 Integration of oscillating functions . . . . . . . . . . . . . . . . . . 164

Bibliography 168

Thanks 189

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Contents

viii

1 Introduction

1.1 Outline

Griffin’s Hartree-Fock-Bogoliubov equations describing trapped ultra-cold weakly-interacting atomic gases at finite temperature close to and below the Bose-Einstein transition temperature are applied to elastic atomic scattering fromBose-Einstein condensates by using finite trapping potentials. For a sphericallysymmetric potential a partial wave analysis of the quasi-particle modes of theequations results in bound and free modes. In analogy with potential scatteringthe cross sections are formulated and finite-temperature effects are seen. Variousnumerical and analytical examples show that the phase of the free quasi-particlemode builds up to π times the number of pairs of bound quasi-particle modes.

To prove the observed Levinson relation the Bogoliubov equations are treatedas scattering equations and different kinds of matrix solutions are given. Theirsymmetry properties carry over to the Fredholm determinant of the Bogoliubovequations. The phase shift of each angular momentum is connected with thenumber of bound quasi-particle modes by a contour integration. This is a gener-alisation of Levinson’s theorem for potential scattering and quite different fromLevinson theorems of multi-channel scattering. Specific features of the Bogoliu-bov scattering equations such as complex energy and continuum bound states arediscussed.

In the first Born approximation elastic scattering of identical atoms probes theconfining potential and the density of the condensed and non-condensed (thermal)atoms in the trap. Neglecting at zero temperature the confinement and depletioneffects it directly probes the two-particle interaction potential. In the two limitsof small and large two-particle interactions a Gaussian and a Thomas-Fermi trialfunction describe the cross sections well when compared to the numerical distrib-utions obtained within the Gross-Pitaevskii theory. Generalised condensate radiiare extracted from the differential and total cross sections.

In a simple example an expansion of the Born series with respect to thenumber of particles in the condensate is given. The higher Born terms enter thedescription of certain inelastic processes which are not accessible within the firstBorn approximation. Two further applications of the first Born approximationare discussed: A formalism to describe fragmentation processes is given where one

1

1 Introduction

atom is in the initial and serveral atoms are in the final channel. The signaturesof a condensate with a vortex are found in its differential and total cross sections.

2

1.2 General introduction

Figure 1.1: Schematic representation of a fermionic system and of scattering frombosons.

1.2 General introduction

On the microscopic quantum level there are two types of particles: fermions andbosons. Fermions, in a way, rather abide normal expectations. For instance theshell structure of atoms is determined by the fermionic nature of the electrons.Pauli’s exclusion principle asserts that each quantum-mechanical state of theatom is occupied by only one electron. The shells fill up as we add electrons.As a result each electron needs a certain spatial volume. This coincides withpractical experience, e.g. if we fill a coffee cup with peas we anticipate it tooverflow soon. This is known to be true also for the case of fermions that do notexert forces on each other. In figure 1.1 we indicate this behaviour drawing singlefermions as open circles on the energy levels of a harmonic oscillator. If all levelshave to be filled the greatest energy is fixed by the number of fermions.

In contrast, if we had taken non-interacting bosons instead of fermions, wecould fill one energy level with whole sacks of them. We visualise this in ourschematic by drawing several filled circles right on the lowest energy level. Sucha counter-intuitive behaviour should have consequences even less expected.

The spin-statistics theorem states that particles with integer spins are bosonsand particles with half-integer spins are fermions. So let us consider a gas ofatoms with integer spins. Bose (1924) and Einstein (1924, 1925a,b) predictedthat a gas of non-interacting bosonic atoms will, below a certain finite tempera-ture, suddenly develop a macroscopic population of the lowest energy quantummechanical state. Since the predicted temperature necessary to reach this newstate was extremely low, it was an open question for a long time whether such astate of matter is observable at all.

Fritz London suggested in 1938 to explain with the ”peculiar condensationphenomenon of the ’Bose-Einstein’ gas” the λ-phenomenon of Helium II (4He),a phase transition of the specific heat at low temperature. It was soon arguedby Tisza (1938) that condensation is equally important to superfluidity whichgoverns a rich variety of experiments with liquid Helium II including frictionless

3

1 Introduction

flow, persistent current and wave propagation on liquid surfaces or the fountainphenomenon (Leggett, 1999). As it happens in a fluid, the predicted behaviour ismasked by strong interactions between the particles. The superfluid part of bulkhelium II can reach almost 100 per cent but the condensate fraction, that is thenumber of atoms which are in a macroscopic quantum state, is close to ten percent (e.g. Sokol (1995)).

Superconductivity was firstly connected to condensation by describing it ”asa kind of condensed state in momentum space” by London (1948). Onnes fromLeiden University Netherlands observed in 1911 that the electric resistance ofmercury disappears if it is cooled to about four Kelvin. In 1935 there werealready 16 elements known which exhibit superconductivity (McLennan et al.,1935). These are examples of low temperature superconductivity relevant totemperatures up to 23.3 K which is well understood by now (e.g. Bardeen et al.(1957); Tinkham (1996)). In 1986 a new class of superconductor materials basedon copper oxide ceramics with layered crystal structures was investigated whichled to the discovery of high temperature superconductivity (Bednorz and Müller,1986). The physics of high temperature superconductivity is a vivid and con-troversial field of today’s physics, both experimentally and theoretically. Thetemperature recently reached is 138 K (Dai et al., 1995).

A further example for Bose-Einstein condensation realised in semi-conductingcuprous oxide (Cu2O) is the condensation of a gas of excitons, each composed ofa bound electron-hole pair (Lin and Wolfe, 1993). Snoke and Baym (1995) giveexamples in the realm of nuclear and elementary particle physics.

The examples mentioned so far are realisations of condensation in rather com-plex systems. The role of Bose-Einstein condensation often is not very clearand the phenomenon difficult to identify unambigously. Indeed, for instancethe condensed gas of excitons just mentioned certainly has to be reinvestigated(O’Hara et al., 1999). Nevertheless the early theoretical attempts to describethese processes relied on substantial simplifications. Experimentalists realisedsince 1995 a remarkable set of systems exhibiting condensation that allow forthe direct application and test of fundamental paradigms of many-body theory.Microscopic quantum-mechanical theory can thus be compared with experiment.In a series of experiments with dilute gaseous vapours of stable atomic alkalispecies condensation was reported for two isotopes of rubidium, 87Rb and 85Rb(Anderson et al., 1995; Cornish et al., 2000), sodium, 23Na (Davis et al., 1995),lithium, 7Li (Bradley et al., 1995), potassium, 41K (Modugno et al., 2001), andrecently cesium 133Cs (Weber et al., 2003). The condensation of the light el-ements hydrogen, 1H (Fried et al., 1998), and metastable triplet helium, 4He∗

(Robert et al., 2001; Dos Santos et al., 2001), was also achieved. 4He∗ is in the23S1 electronic state 19.8 eV above the electronic ground state. The 4He isotopeof helium is an example for the achievement of condensation in the gaseous andin the liquid phase where 4He is in the electronic ground state.

Why did it take so long to experimentally verify Bose and Einstein’s prediction

4

1.2 General introduction

in the atomic realm? A heuristic approach to understand condensation in diluteatomic vapours is to compare the mean distance d of the atomic particles of massm with their thermal de-Broglie wave length λdB. The latter defines a length scaleat which quantum-mechanical effects start appearing from a given temperatureT ,

λdB =h√

2πmkBT. (1.1)

Intuitively we can expect the onset of condensation if there is more than oneparticle per cubic de-Broglie wave length. The particle waves overlap at thepoint of highest density in a generic situation if the following condition is met(Bagnato et al., 1987)

2.612 <λ3dBd3

. (1.2)

That is roughly three particles in the de-Broglie cube are sufficient. For a longtime it was an elusive goal to meet this stringent phase-space requirement fordilute atomic vapours. The first realisation was done by a hybrid approach ap-plying methods developed over decades (e.g. Cornell and Wieman (2001)). Atfirst laser cooling and trapping of the neutral atoms (Adams and Riis, 1997)increased their phase-space density by 15 orders of magnitude. Then the remain-ing five to six orders of magnitude were overcome by loading the sample intoa magnetic trap allowing for evaporative cooling (e.g. Masuhara et al. (1988);Burnett (1996); Ketterle et al. (1998)). The hottest atoms leave the trap andthe remaining atoms rethermalise, effectively cooling the vapour. The tempera-tures necessary to reach condensation are of the order of 0.5-2 µK, the numberof atoms in the condensate is of the order of 102 − 109 while the densities arebetween 1014 − 1015cm−3 which is about a hundred thousandth the density ofnormal air. The size of the condensed cloud is about 10-50 µm if it is approxi-mately spherical. A typical cigar-shaped condensate has a diameter of 15 µm anda length of 0.3 mm (Ketterle, 2001). Especially large condensates are achievedwith sodium and hydrogen.

What prevents the transition of the atoms to a liquid or solid phase at suchlow temperatures? At low density and thermal energies of the dilute vapoursthe binary elastic collisions, necessary for rethermalisation, are dominant whencompared to inelastic two- or three-body collisions which lead to spin exchange,excitations and molecule or cluster formation. This effectively results in traplosses (e.g. Parkins and Walls (1998)). So thermal equilibrium is achieved in ameta-stable state, since the chemical equilibrium requires liquefaction or solidifi-cation. The time scales of both processes are different and allow for experiments.One complete circle of loading into the magneto-optical trap and cooling to quan-tum degeneracy lasts of the order of seconds to minutes.

5

1 Introduction

In the fist experiments it was important to verify that condensation has oc-curred, e.g. by means of the expansion method (Holland and Cooper, 1996;Castin and Dum, 1996). After the break-through of realising an experimentalcondensate, the interest has shifted to more involved experiments which explorethe nature of this newly available matter. Such experimental achievements in-clude the first creation of vortices (Matthews et al., 1999; Williams and Holland,1999). Vortices are intimately connected to superfluidity. The superfluid flow isquantised. Condensates with and without a vortex are topologically distinct ob-jects. Solitons are localised waves which travel without attenuation and changeof shape over long distances. The dispersion is compensated by non-linear effects.An example are dark solitons, also called kink-states, which are well known inthe optical realm (e.g. Kivshar and Luther-Davies (1998)). These were producedby Burger et al. (1999) and Denschlag et al. (2000). They optically imprinteda phase pattern onto the condensate and let the condensate evolve in time. Anotch in the density profile and in addition a phase step across the soliton centrecharacterises the dark soliton. The expansion method was used to detect thesolitons. The phase imprinting method was originally proposed to create vortices(Dobrek et al., 1999). Recently, the first matter-wave bright solitons were ex-perimentally created in elongated traps (Khaykovich et al., 2002; Strecker et al.,2002). Such elongated traps effectively confine the condensate to one dimension.In such a situation the balance between the kinetic energy dispersion and theattractive two-particle interaction can lead to the formation of a density hump.A further achievement is the celebrated atom laser which allows the output ofcoherent matter waves out of the condensate (Mewes et al., 1997; Bloch et al.,1999). It is the non-linear matter wave analog of an intense monochromatic beamof coherent light. Theoretical descriptions of atom lasers are given by Ballaghet al. (1997), Japha et al. (1999), Choi et al. (2000), Schneider and Schenzle (1999)and Schneider and Schneider (2000). By means of optical loading of a condensateone could create a continously reloaded atom laser (Santos et al., 2000; Floegelet al., 2003). Two-component condensates were realised by Myatt et al. (1997)and Stenger et al. (1998). The experiments use hyperfine states of rubidium andsodium, respectively. How to treat multi-component condensates theoretically isdescribed for instance by Leggett (2001). The attainment of simultaneous quan-tum degeneracy in a mixed gas of bosons (7Li) and fermions (6Li) is reportedby Truscott et al. (2001). Gaseous fermionic 3He and bosonic 4He are furthercandidates for the study of quantum degenerate mixtures of bosons and fermi-ons. Despite their challenges (Presilla and Onofrio, 2003) such experiments mighthelp to understand strongly interacting condensed matter systems which exhibitsuperconductivity. Bose-Einstein condensates can serve as experimental modelsystems of solid state physics. The quantum phase transition from a superfluid toa Mott insulator is observed in a condensate with repulsive interactions, held ina three dimensional lattice potential (Greiner et al., 2002). Tuning the potentialdepth of the lattice the system reversibly changes between the two ground states.

6

1.2 General introduction

A point of fundamental interest is the investigation of the ultra-cold cloudby collisions. We subsume the interaction of radiation with the condensate, thescattering of electrons, atomic scattering of several atoms and the interaction ofcondensates with each other as collisional processes.

The absorption of resonant light was used from the very beginning to observeshadow images of the ballistically expanding rubidium cloud after the suddenswitching off of the trapping potential. The first sodium experiment used de-structive absorption imaging. In the first lithium experiment, absorption wasmeasured in situ. Near-resonant light in transmission measures the optical den-sity. The optical density corresponds to the column density of the atoms andmay be compared to mean-field predictions, revealing the shape of the condensedmode (Hau et al., 1998). The more flexible phase-contrast method is especiallysuitable for the small lithium condensate and measurements in situ (Bradleyet al., 1997b,a). Phase-contrast imaging results in the depletion of the conden-sate and phase diffusion (Leonhardt et al., 1999). Optical methods also allowa direct detection of vortices (Goldstein et al., 1998) and the condensate phase(Lewenstein and You, 1996b).

Lewenstein and You (1996a) and Parkins and Walls (1998) distinguish co-herent light scattering, which probes the density of cold atomic samples throughthe first oder correlation function, and incoherent light scattering, which probesdensity-density correlations, i.e. density fluctuations. In the first case, since thedensity changes if a condensate is present, quantum statistical effects enter onlyimplicitly via the form factor which is the Fourier transform of the density distri-bution. Incoherent light scattering offers an explicit examination of fluctuationsbecause the dynamic structure factor enters which is the Fourier transform ofan average of density fluctuation operators. Off-resonant light which can onlyinduce virtual transitions of the ground state atoms in the trap is determined bycorrelation functions of the Schrödinger field density (Javanainen, 1995; Csordaset al., 1996). As an example we refer to Andrews et al. (1996) who use disper-sive light scattering in experiments with sodium. It is an example of incoherentscattering.

An alternative to incoherent light scattering to probe the dynamic structurefactor is stimulated two-photon Bragg scattering where the momentum transferand the energy are preset (Stenger et al., 1999). The high momentum and en-ergy resolution were experimentally used to confirm that the coherence length isof the size of the condensate. Goldstein and Meystre (1998) proposed a schemethat combines localised ionisation of trapped atoms through light and simulta-neously detecting the atoms which allows the measurement of normally orderedcorrelation functions of the Schrödinger field.

Wang et al. (2001) treat the scattering of electrons by a condensate of alkali-metal atoms. They find that the elastic differential cross section is proportionalto the square of the number of condensed atoms. This dramatic enhancementof the scattering cross section is due to the coherence of the condensate. The

7

1 Introduction

Chinese group computes for the inelastic scattering process the stopping powerof the electron which determines the energy loss of the incident electron along itspath through the condensate.

Timmermans and Côté (1998) investigate the scattering of impurity atomswithin a homogeneous condensate. Their calculations model sympathetic coolingof fermionic alkali atoms or inert gases through a condensate. Chikkatur et al.(2000) create impurity atoms using a Raman transition from a trapped to anuntrapped hyperfine state of sodium. The momentum transfer from the lightfield to the untrapped atoms can be tuned allowing them to find the Landaucritical velocity of the condensate.

The present work is focused on atomic scattering from Bose-Einstein conden-sates. There are several theoretical papers dealing with this topic. Bijlsma andStoof (2000) consider the case where the entire scattering process takes place ina harmonic trapping potential. The probe atoms are identical to those that arecondensed. In a non-destructive way they probe the density and the phase ofthe condensate by propagating a wave packet. This is analogous to Andreev re-flection from a superconductor-normal-metal interface where electrons probe themagnitude and the phase of the superconducting order parameter. Besides thedip in the density there is a velocity field if a vortex is present. The latter ex-tends over the whole condensate. The evolution of the wave packet is influencedby the velocity field. The authors see similarities with the Aharonov-Bohm ef-fect which originally was noted in the description of the interaction of an electronwith a magnetic flux confined to a thin tube (Sonin, 1997). Wynveen et al. (2000)show that for incident atoms indistinguishable from those that are condensed afinite-size weakly-coupled Bose system may exhibit effective transparency. Suchtransparency effects were originally suggested for localised strongly-coupled liq-uid helium. This phenomenon is similar to the Ramsauer-Townsend effect ofpotential scattering (e.g. Schiff (1968)). They show that the elastic cross sectionwhich exhibits the effect dominates when compared to the inelastic cross section.Poulsen and Mølmer (2003) compute transmission and reflection coefficients ofone-dimensional condensates and find a negative time delay for the transmis-sion. Idziaszek et al. (1999) investigate the scattering of slow atoms from thecondensate. They show that elastic scattering is sensitive to the density of thecondensate whereas inelastic channels probe collective excitations. They derivethe scaling properties of the cross sections with respect to the total number ofcondensed atoms. Considering special cases of fragmentation, Kuklov and Svis-tunov (1999) propose to measure the one-particle density matrix with fast atoms.The method is sensitive to distances of the order of the interatomic spacing. Thusit could allow to measure short-range density correlations and the emergence ofcoherence. They investigate the cases of a single condensate containing a vortexand a split condensate characterised by some phase difference.

A closer look at the publications discussed in the last paragraph reveals thatwe may divide the theoretical works into two groups. The first three apply

8

1.2 General introduction

coupled Bogoliubov equations and the last two use the first Born approximationto describe the scattering process. These two approaches are also pursued inthe present work. We will see that the coupled-equation approach results fromallowing deviations from the macroscopically occupied mode which is governedby a generalised Gross-Pitaevskii-equation. The deviations describe scatteringparticles as well as thermal excitations. Comparing along the way the resultswith the works mentioned in the last paragraph we aim at precise formulationsand answers to the following questions:

• What is a consistent description of cross sections of the condensate and thenon-condensed cloud at finite temperatures?

• Which signatures result from the non-condensed cloud in the cross sections?

• Do the scattering cross sections give information about low-energy linearcollective excitations of the condensate cloud?

• How are the first Born approximation and the Bogoliubov approach relatedto each other?

• Which signatures of the cross sections within the first Born approximationare lost if various analytical trial functions of the condensate wave functionare used?

• How do higher terms of the Born series look like?

• Is there a consistent formulation of fragmentation processes?

• Is it possible to detect a vortex state in a simple scattering experiment?

There are a number of reviews connected with the condensation of cold atomicgases in traps by Dalfovo et al. (1999) who describe the mean-field approach,Leggett (2001) explains basic concepts with an emphasis on two-state systemswhile Stenholm (2002) gives a heuristic field theoretical approach. We mention,as well, the topical review on vortices by Fetter and Svidzinsky (2001). There aretwo recent monographs on Bose-Einstein condensation of dilute gases by Pethickand Smith (2002) and Pitaevskii and Stringari (2003). In this context we alsorefer to the two Nobel lectures by Cornell and Wieman (2001) and Ketterle (2001)which certainly are a good and stimulating starting point to get acquainted withthis new kind of soft condensed matter.

9

1 Introduction

1.3 Connecting threads through the chapters

We continue in chapter 2 with a theoretical framework connecting all chapters.The initial and final state of a scattering experiment can be described in terms ofthe objects of second quantisation introduced in this chapter. We show how theformalism of second quantisation is adapted in the presence of a macroscopicallyoccupied mode. In such a situation we can exactly diagonalise an approximateHamiltonian introducing the condensate wave function and quasi-particle excita-tions as deviations from the condensate. The condensate wave function is gov-erned by a generalised Gross-Pitaevskii equation. The quasi-particle modes aredetermined by a coupled set of Bogoliubov-type equations. They correspond tofirst order equations in the deviations. The deviations describe the non-condensedcloud at 0 ≤ T . All equations have to be solved self-consistently. For the caseof a spherically symmetric trapping potential and condensate we apply a partialwave analysis to the quasi-particle modes. We describe bound and scatteringexcitations.

The Gross-Pitaevskii equation determines the condensate wave function. Inchapter 3 we shortly describe how to obtain the lowest self-consistent solutionfor a given spherically symmetric trap. We rely on a proper representation ofthe kinetic energy operator and the radial dependent potentials to propagate thecondensate wave function in real and imaginary time. The matrix operators ofthe kinetic energy and the potential are also applied to the bound state quasi-particle solutions in chapter 4.

In chapter 4 the diagonalisation of the non-hermitian matrix of the Bogol-iubov equations is described leading to the quasi-particle solutions which arelocalised in the condensate region. From the propagation of the coupled set ofequations introduced in chapter 2 we obtain the non-localised quasi-particle solu-tions. Asymptotically away from the condensate we can extract the informationof the scattering process in terms of phase shifts which determine the differentcross sections. A potential scattering reference model facilitates the interpreta-tion. We give numerical examples for realistic setups at different temperatures.As in potential scattering we numerically observe that the phase shift counts thenumber of bound states of the Bogoliubov system. However, we have to specifycarefully what we regard as a bound state of our bosonic scattering system.

In chapter 5 a rigorous proof of a Levinson relation is given for the Bogoliubov-like equations, which relates the number of bound quasi-particle modes to thephase shift behaviour for all energies. We use the formalism of multi-channelscattering for a local coupling matrix. Solutions with different boundary condi-tions are introduced which allow us to go into the complex plane and to apply

10

1.3 Connecting threads through the chapters

symmetries of the system. In the final contour integration analytical propertiesof the Fredholm determinant are exploited. The scattering formalism exposesspecial features of our bosonic scattering system. As an exactly solvable examplewe present analytical solutions of the scattering system in the limit of a largeparticle number and square well confinement. By modifying the example we in-vestigate what happens if the condensate is not in the lowest lying self-consistentmode.

Starting in chapter 6 with a simple wave function approach we calculateelastic scattering for direct and exchange processes for single atoms in the firstBorn approximation. Switching to the formalism of second quantisation we findmuch more concise expressions for elastic and inelastic scattering. In the case ofelastic scattering of distinguishable atoms an attempt is made to go beyond thefirst Born approximation. Elastic scattering probes the density distribution of thecondensate wave function. For a spherically symmetric trap the wave functionand hence the density is determined by one single parameter. We ask for theinfluence of this parameter on the cross sections. We compare the numericalresults with various analytical trial functions for the condensate. Characteristiclengths of the condensate distribution are identified.

In chapter 7 we introduce a formalism capable of describing multiple particlescattering from the condensate. Specifically we give expressions that describefragmentation in the first Born approximation, e.g. one probe atom in the initialand two atoms in the final channel.

Chapter 8 treats the scattering from a vortex state in the first Born ap-proximation. A vortex is an example of a higher self-consistent solution of theGross-Pitaevskii equation and is not spherically symmetric.

11

2 Hartree-Fock-Bogoliubov theory

We consider two approaches to describe the scattering of atoms from the con-densate. One is based on the asymptotic studies of radial equations. The secondapproach needs an initial and final state description and an expression for thetransition operator between them. Largely we apply the first Born approximationto the transition operator. We use a combination of plane wave states and setsof occupation numbers to describe the initial and final scattering states. Withinthe Bogoliubov approach we see that the occupation numbers count quasi-particleexcitations with respect to the condensate which in this sense is our vacuum ofexcitations. The condensate wave function describes this vacuum and dominantlyenters into the description of the scattering process. An elegant way of writingthe various scattering matrix elements is to use the operator language of secondquantisation.

To proceed we need to know the space dependent condensate function and thequasi-particle modes and their respective energies. So we will see that ultimatelyboth approaches rely on a coupled set of equations relevant to the excitationsand a generalised Gross-Pitaevskii Equation for the ground state. They have tobe solved self-consistently.

2.1 Background to the Hartree-Fock-Bogoliubovtheory

A benchmark for the description of the condensate and the thermal cloud is givenby Griffin’s self-consistent quadratic mean field Hartree-Fock-Bogoliubov equa-tions (Griffin, 1996), first applied by Hutchinson et al. (1997). As Griffin pointsout, it is not a gapless theory in the sense of the Hohenberg-Martin classificationof approximations. Proukakis et al. (1998) propose two schemes to overcome thisrestriction by the replacement of the two-body T -matrix with the many-bodyT -matrix, effectively resulting in minor changes of the self-consistent scheme. Anapplication of such a so-called gapless Hartree-Fock-Bogoliubov theory is foundin (Hutchinson et al., 1998).

The Bogoliubov approach can be formulated in a particle-number-conservingway (Gardiner, 1997; Castin and Dum, 1998; Dziarmaga and Sacha, 2003). Suchformally precise theories allow to justify the standard approach in a quantitative

12

2.1 Background to the Hartree-Fock-Bogoliubov theory

way. The number conserving approach in combination with perturbation theoryby Morgan (2000) also confirms the standard Bogoliubov approximation.

Proukakis et al. (1998) and Proukakis (2001) pursue an approach involving thecoupled equation technique. In its advanced form it describes self-consistentlycondensate and non-condensate. Köhler and Burnett (2002) and Köhler et al.(2003) introduce a microscopic dynamics approach which does not depend on thecontact interaction approximation and can treat systems which are not in thermalequilibrium. Applications for this theory are atom-molecule oscillations in a Bose-Einstein condensate (Köhler et al., 2003). It is based on the Heisenberg equationof motion for operators and a cluster expansion for products of field operators(Fricke, 1996). The n-th order correlations appearing therein are termed non-commutative cumulants. The first- and second-order normal-ordered cumulantscan be identified with the condensate wave function, the pair function and theone-body density matrix of the non-condensed fraction, respectively.

The time-dependent density functional theory is an established tool for many-electron systems. Kim and Zubarev (2003) develop a Kohn-Sham like time-dependent theory for bosons in three- and quasi-two dimensional traps. A furtheralternative are quantum Monte Carlo calculations which suffer from no system-atic uncertainties other than the inter-atomic pseudo-potential approximation ofthe two-particle approximation (Krauth, 1996). For small particle numbers thedensity of the condensate fraction agrees well with the mean field prediction.

Certainly there are several inconsistencies known for the Hartree-Fock-Bogol-iubov variational formalism, which can be resolved with more rigorous approaches(Olshanii and Pricoupenko, 2002). Nevertheless we will use it as our tool todescribe the dilute Bose gas. Restrictions of the theory (Griffin, 1996) or minormodifications of the theory (Proukakis et al., 1998) agree well with experimentsand the inconsistencies are resolved.

The microscopic theories yield the condensate density and non-condensatedensity and possibly other mean fields like the anomalous average. One can de-rive stability estimates, specific heats and properties of various propagating soundwaves. These collective quantities which depend on the excitation spectrum as awhole may be compared to experiments. More accurate tests of the microscopictheories consist in the direct comparison of single excitation energies to experi-ments in harmonic traps (Stamper-Kurn et al., 1998; Chevy et al., 2002) and inoptical lattices (Fort et al., 2003).

Griffin’s Hartree-Fock-Bogoliubov equations include two-particle interactionsand neglect three-particle interactions. For instance Gammal et al. (2000) includethree body interactions which are important for loss processes. We assume thatthe condensate is static. Giorgini (2000) includes small amplitude oscillationsof the condensed and non-condensed cloud. Such oscillations allow to obtainthe trapping frequencies. We do not consider the non-equilibrium path of theformation of the condensate. The growth processes of the condensate out of thethermal cloud can be experimentally monitored in time (Köhl et al., 2002).

13

2 Hartree-Fock-Bogoliubov theory

2.2 Fock space

As indicated in the introduction to this chapter, we describe the initial and finalscattering states in the form of a tensor product of plane wave states and a setof occupation numbers. For example | k〉 | n0, n1, · · · 〉 corresponds to an atom ina pure momentum state plus a cloud of cold atoms possibly with a condensatepresent. Similarly the state | k1〉 | k2〉 | n0, n1, · · · 〉 consists of two atoms plus thecloud. A general scattering state is a superposition of this kind of basis states.

To make this ideas more explicit we introduce some notation mainly followingBlaizot and Ripka (1986). The Fock space is the direct sum of N -particle Hilbertspaces. Each N -particle Hilbert space consists of the tensor product of N single-particle Hilbert spaces. We can formally identify the space belonging to theinteger zero with the vacuum state | vac〉 which consists of a single state. Weintroduce single-particle states as eigenvectors of a single-particle HamiltonianH1 which form a complete orthonormal basis set,

H1 | k〉 = �k | k〉, 〈j | k〉 = δjk, 11 =∑

k

| k〉〈k | . (2.1)

Here j and k are a shorthand notation for a complete set of single-particle quan-tum numbers and may be multi-indexed. Sometimes we will write 〈r | k〉 = ϕk(r)for the single-particle functions.

A symmetric basis state appropriate to describe N bosons may be written asthe time-independent abstract state vector

| n0, n1, · · · 〉, N =∞∑

i=0

ni. (2.2)

where the integers 0 ≤ ni are the numbers of particles in the single-particlestates i. If we stick to the single-particle basis, an arbitrary state of N bosons isof the form

∑i Ci | n0i, n1i, · · · 〉 where the sum is over all basis state vectors with

N particles. Note that the occupation numbers start with n0. Subsequently wewill identify the particles with quantum number 0 as belonging to the condensate.Due to the symmetry of the bosons we require for the creation operators b̂†iand destruction operators b̂i of a bosonic state | i〉 that the elementary bosoniccommutation rules hold:

[b̂i, b̂†j] = b̂ib̂

†j − b̂†j b̂i = δij, [b̂i, b̂j] = [b̂†i , b̂†j] = 0. (2.3)

In addition we find the following properties

〈vac | vac〉 = 1, bi | vac〉 = 0, (2.4)b̂i | · · · , ni, · · · 〉 = (ni)

12 | · · · , ni − 1, · · · 〉,

b̂†i | · · · , ni, · · · 〉 = (ni + 1)12 | · · · , ni + 1, · · · 〉,

n̂i | · · · , ni, · · · 〉 = ni | · · · , ni, · · · 〉,

14

2.2 Fock space

where n̂i = b̂†i b̂i is the number operator.

Now we want to write a plane-wave state in terms of single-particle states.To do this we introduce the field operator. Quite generally we have for any twoorthonormal basis systems {| i〉} and {| j〉} that | i〉 = ∑j〈j | i〉 | j〉 which leadsto the operator relations b̂†i =

∑j〈j | i〉b̂†j. In the same spirit the creation and

destruction field operators of a boson at position r are linear combinations of thecreation and destruction operators, respectively,

Ψ̂†(r) =∑

k

〈k | r〉b̂†k, Ψ̂(r) =∑

k

〈r | k〉b̂k, (2.5)

Ψ̂†(r) | vac〉 =| r〉, Ψ̂(r) | vac〉 = 0.

We can now generate a plane wave function 〈r | k〉 = (2π)− 32 eikr with the help ofthe relations

b̂†k =∑

k

〈k | k〉b̂†k =∫〈r | k〉Ψ̂†(r)d3r, (2.6)

b̂†k | vac〉 =| k〉, b̂k | vac〉 = 0,

because we know how to write 〈r | and | k〉. Of course there is a similar expressionfor the annihilation operator of a plane wave state b̂k. For completeness and laterreference we introduce for a system which has translation invariance the fieldoperators

Ψ̂†(r) =∑

k

(2π)32

V12

〈k | r〉b̂†k, (2.7)

where V = LxLyLz ≡ L1L2L3 is the volume in which the system is enclosed.Our single-particle basis functions are in this case exp(ikr)/

√V and they are

orthonormal with respect to integration over the volume V . This kind of basisis especially suited to compute the kinetic energy. We can do so if we assumethe Born-von-Kármán periodic boundary conditions for the position dependentfunctions or operators F . They fulfill F (r +

∑3i=1 νiLiei) = F (r) for νi ∈

�. The

k-vectors are in this case elements of the discrete set K = {∑3i=1 2πνieiLi |νi ∈� },

where ei are the unit vectors in the Cartesian directions.

As an example we write the creation operator of a boson in the k mode in

three different ways, b̂†k =∫〈r | k〉Ψ̂†(r)d3r = (2π)

32

V12

∑k〈k | k〉b̂†k. A state where

all particles are in the 0-mode reads b̂†N0 /√N ! | vac〉 = |N,n1 = 0, n2 = 0, · · · 〉

and an example for several occupied states is b̂†N−30 b̂†21 b̂†2/√

(N − 3)!2!1! | vac〉= |N − 3, n1 = 2, n2 = 1, n3 = 0, · · · 〉. All introduced field operators obey as wellbosonic commutation relations, e.g. [Ψ̂(r), Ψ̂†(r′)] = δ(r− r′).

15

2 Hartree-Fock-Bogoliubov theory

2.3 Consequences of condensation

In the introduction we saw that the experimental temperatures for which thecondensation of alkali atoms becomes inevitably are of the order of a few 100 nK.A model for non-interacting bosons of mass m held in an anisotropic externalharmonic confining potential,

V (r) = m(ω2xx2 + ω2yy

2 + ω2zz2)/2, (2.8)

predicts in the large N limit for the critical temperature (e.g. Bagnato et al.(1987); Giorgini et al. (1996)),

T 0c ≈ 0.94�ωgakB

N13 , ωga = (ωxωyωz)

13 , (2.9)

where we introduced the geometrical average ωga of the trapping frequencies.Within this model one has the following relationship for the number of atoms inthe lowest lying single-particle mode N0 and the total number of atoms N in thetrap:

N0(T )

N= 1−

(T

T 0c

)3, T < T 0c . (2.10)

As the temperature is lowered we see that the condensate fraction becomes macro-scopic, 1 � N0 ≈ N . We say that a mode is macroscopically occupied if N0/Nis of the order of one. We speak of weak quantum depletion (e.g. Hohenbergand Martin (1965)) if N − N0 � N . We find a similar picture if we allow forweak two-particle interactions in the trap (Giorgini et al., 1996). Modificationsof the critical temperature in this situation are treated by Leggett (2003). Weexpress the idea of the macroscopic occupation of a single-particle mode moreprecisely following Leggett (2001). We consider a special correlation function forthe bosons. The one-particle reduced density matrix ρ(r, r′) of interacting bosonsis Hermitian with respect to the position coordinates. We see this if we note thatρ(r, r′) =

∫Ψ∗(r, r2, · · · , rN )Ψ(r′, r2, · · · , rN)d3r2 · · · d3rN . Here Ψ(r1, · · · , rN) is

the bosonic N -particle function. This allows us to write ρ in terms of orthonormalsingle-particle functions. We express the many-particle function of the bosons inthe trap in terms of this basis set as a Fock state and find

ρ(r, r′) = 〈n0, · · · |∞∑

i=0

n̂i〈r′ | i〉〈i | r〉 | n0, · · · 〉. (2.11)

Now we expect from the non-interacting model that one of the single-particlestates is macroscopically occupied. We give this unique state with macroscopicoccupation number the index zero and write n0 = N0 and 〈r | 0〉 for its single-particle mode. If we can identify exact one such single-particle state in the

16

2.3 Consequences of condensation

expression for ρ we say that Bose-Einstein condensation has occurred. Thisdefinition coincides with the definition given by Penrose and Onsager (1956) whogive as well alternative forms of the criterion.

We take advantage of the single macroscopically occupied mode by notingthat b̂†0b̂0 | N0, n1, · · · 〉 ≈ (N0 + 1) | N0, n1, · · · 〉 = b̂0b̂†0 | N0, n1, · · · 〉. We concludethat we can use b̂0 =

√N0 within Bogoliubov’s (1947) approximation for the field

operator

Ψ̂(r) = 〈r | 0〉b̂0 +∑

k 6=0〈r | k〉b̂k ≡ Ψ̂0(r) + Ψ̂′(r) (2.12)

≈ (N0)12 〈r | 0〉 +

∑′〈r | k〉b̂k ≡ Ψ0(r) + Ψ̂′(r).

We omitted the mode in which condensation occurred in the sum over all single-particle states in both lines of (2.12) which we indicated in the second line by theprimed sum. The operator Ψ̂′(r) describes deviations from the macroscopicallyoccupied mode or condensed mode which is described by the c-number 〈r | 0〉.The c-number Ψ0(r) represents the mean field of all N0 condensed bosons and isalso called the order parameter or macroscopic wave function (e.g. Ballagh et al.(1997); Giradeau and Wright (2000)). This splitting of the full bosonic field opera-tor will lead to substantial simplifications when diagonalising the grand canonicalHamiltonian. Castin and Dum (1998) exploit the existence of a macroscopicallypopulated state in a more concise way. They identify a small parameter andimplement an expansion procedure for a system with a well defined number ofparticles.

The occurence of condensation is sometimes defined via the existence of aso-called anomalous average 〈· · · | Ψ̂ | · · · 〉 of the field operator. With theBogoliubov field operator we immediately see, Ψ0(r) = 〈· · · | Ψ0(r) + Ψ̂′(r) | · · · 〉(Hohenberg and Martin, 1965; Bogoliubov, 1960)). If we take the Bogoliubovapproximated field operator it is obvious that this result is independent of thestate | · · · 〉 for which we take the average. If we do not use the Bogoliubovapproximation to the field operator within the average it is not clear at firstsight how a nonzero result is possible at all. The super-selection rule of totalparticle number conservation is violated if the average of a single field operatoris non-zero. In such a case the state | · · · 〉 must be a superposition of states withdifferent particle numbers. However, we defined that condensation has occured ifthere is a macroscopically occupied single-particle mode. Then the existence of anon-vanishing anomalous average follows if we accept (2.12). The existence of anon-vanishing anomalous average is referred to as ”spontaneously broken gaugesymmetry” (Goldstone, 1961; Anderson, 1966; Leggett, 1996). We as well do notdefine condensation via the concept of ”off-diagonal long-range order” (Yang,1962) which is more relevant for infinite homogeneous systems. We note thatthe adopted definition, of course, allows for off-diagonal long-range order withinthe condensate region and as well for phase coherence of the order parameter

17

2 Hartree-Fock-Bogoliubov theory

(Anderson, 1966). We consider these coherence phenomena as consequences ofthe condensation in a single-particle mode.

Next we observe the action of the deviation operator Ψ̂′ = Ψ̂ − Ψ0 on thestate | N0〉 ≡| N0, n1 = 0, · · · 〉. We find Ψ̂′ | N0〉 = Ψ0[| N0 − 1〉− | N0〉].However, in the exact case we have of course Ψ̂′ | N0〉 = 0. We conclude that theBogoliubov approximation implies that we set | N0〉 =| N0± 1〉 which enables usto use the exact relation. The operator nature of Ψ̂0 is lost in the Bogoliubovapproximation, [Ψ∗0,Ψ0] = 0. For the deviation operator we find the commutator(Fetter and Walecka, 1971)[Ψ̂′(r), Ψ̂′

†(r′)]

=∑′〈r | k〉〈k | r′〉 = δ(r− r′)− 〈r | 0〉〈0 | r′〉 ≡ δ(r− r′) (2.13)

We will now apply the Bogoliubov approximation to the second-quantisedHamiltonian

Ĥ =

∫Ψ̂†(r)H̃0Ψ̂(r)d

3r +1

2

∫ ∫Ψ̂†(r)Ψ̂†(r′)U(r, r′)Ψ̂(r′)Ψ̂(r)d3r′d3r. (2.14)

We introducec the single-particle operator of the kinetic energy of the particlesand the trapping potential, H̃0 = − � 22m∆r+V (r), and the two-particle interactionU(r, r′) which will be specified later. The number operator N̂ =

∫Ψ̂†(r)Ψ̂(r)d3r

produces the total number of particles when sandwiched between a symmetricbasis state. With Tolmachev (1960) we observe that the Hamiltonian and theparticle number operator do not commute when we use the approximated fieldoperator since all kinds of products of creation and destruction operators appeardue to the c-number replacement. If we had taken the full operators Ĥ andN̂ do have a common eigenbasis. As a remedy we add in a standard fashionthe condition of total particle number conservation ( e.g. Lewenstein and You(1996a); Lindner (1997)),

N = 〈n0, · · · | N̂ | n0, · · · 〉 = N0 + 〈n1, · · · |∫

Ψ̂′†(r)Ψ̂′(r)d3r | n1, · · · 〉, (2.15)

to the Hamiltonian Ĥ in the form of a Lagrange multiplier by defining the grand-canonical Hamiltonian

K̂ = Ĥ − µN̂ . (2.16)The deviation operator in the averaged term on the right of (2.15) acts only onstates with k 6= 0. The chemical potential is equivalent to the energy neededto add one particle to the condensate. The grand canonical Hamiltonian allowsfor a consistent treatment of the particle number conservation if the Bogoliubovapproximation is used. It describes the energy of the condensed mode and thecontributions of excitations in the vacuum. Equations (2.12) and (2.16) are to becombined for a correct description. For a subtle discussion of the grand canonicalHamiltonian for bosons see the bock by Fetter and Walecka (1971) on many-bodyphysics and the classical book by Noziéres and Pines (1990) on quantum liquids.

18

2.4 Diagonalisation of the grand canonical Hamiltonian

2.4 Diagonalisation of the grand canonicalHamiltonian

If we insert the Bogoliubov form of the field operator (2.12) into the grand-canonical Hamiltonian (2.16) we obtain terms from zeroth order up to forthorder in the deviation operator. The symmetry rule U(r, r′) = U(r′, r) allows usto write the terms of this Bogoliubov Hamiltonian which are linear and cubic inthe deviation operator in the form of an expression plus its Hermitian conjugate.In addition we can carry out one integration if we assume the following twoparticle interaction (Fermi, 1936; Weiner et al., 1999; Dalfovo et al., 1999)

U(r, r′) = 4π� 2aTT/m δ(r− r′) ≡ U0δ(r− r′), (2.17)

where aTT is the s-wave scattering length for binary scattering in vacuo betweentwo trapped atoms and m is the mass of the trapped atoms. We will laterintroduce the scattering length aTP which is relevant if the target and probe atomsare different. U0 is the two-body T -matrix in the low-energy limit (Bijlsma andStoof, 1997). Alternatively we can consider equation (2.17) as the modified firstterm of a pseudo-potential which correctly reproduces all scattering phase shiftsof the two interacting atoms (Huang and Yang, 1957; Huang and Tommasini,1996). The pseudo-potential depends on these phase shifts. In the case of ahard-sphere potential the first term of an expansion in the radius aTT of thehard-sphere is of the form U0δ(r− r′)∂rr. In lowest order perturbation theory forthis Fermi-Huang pseudopotential the operator ∂rr is equal to unity. The Fermi-Huang pseudopotential is embedded in a wider class of potentials (Olshanii andPricoupenko, 2002). Within this class (2.17) is the zero-energy two-body T -matrix in the limit of zero-range interaction.

The shape-independent contact interaction approximation is adequate for thelow-energy elastic collisions of the atoms in the cold dilute gas. We speak ofdilute if there are few atoms in a scattering length volume, equivalently if theso-called gas parameter |aTT |3n, where n is the average density, is much lessthan one. We assume that the energy of the atoms is too small for inelasticcollisions. Only s-wave scattering is relevant at ultra-low scattering energiessince the angular momentum barrier prevents the atoms to come close enoughto probe the inter-atomic potential for higher momenta. The existence of anelastic scattering regime with a scattering length which is not too small is crucialfor the rethermalisation process of the atoms. The numerical computation ofscattering lengths is involved because long tails of the interactions of the atomshave to be taken into account. A combination of experimental data and numericalcalculations is applied for actual computations of scattering lengths (van Abeelenand Verhaar, 1999).

An effectively attractive interaction leads to a negative scattering length, arepulsive interaction has a positive scattering length. In this context it is in-

19

2 Hartree-Fock-Bogoliubov theory

teresting to note that the scattering length can be tuned over a wide range inpresent experiments. The scattering length can be changed by variation of themagnetic field (Tiesinga et al., 1993; Inouye et al., 1998) or due to the influenceof nearly resonant light (Fedichev et al., 1996) or by applying radio-frequencyfields (Moerdijk et al., 1996). All these modifications are only pronounced in a socalled Feshbach-resonance (Feshbach, 1962), when a quasi-bound molecular statehas nearly zero energy and resonantly couples to the free state of the collidingatoms (Inouye et al., 1998). Modifications of the scattering length due to thescattering in a medium are considered by Stein et al. (1997) using a rank-oneseparable potential (Cho et al., 1993). The s-wave scattering cross section foridentical particles is given by σTT = 8πa2TT .

Now we proceed diagonalising the grand canonical Hamiltonian. We apply theself-consistent quadratic mean-field approximation (Griffin, 1996) for the termswhich are cubic and quartic in the deviation operator. The following factorisation

scheme based on Wick’s theorem is used (Giorgini, 2000) Ψ̂′†Ψ̂′Ψ̂′ = 2〈Ψ̂′†Ψ̂′〉Ψ̂′

+〈Ψ̂′Ψ̂′〉Ψ̂′† and Ψ̂′†Ψ̂′†Ψ̂′Ψ̂′ = 4〈Ψ̂′†Ψ̂′〉Ψ̂′†Ψ̂′ + 〈Ψ̂′Ψ̂′〉∗Ψ̂′Ψ̂′ + 〈Ψ̂′Ψ̂′〉Ψ̂′†Ψ̂′†. We”〈· · · 〉” mean taking expectation values in the sense of the averaged term on theright of (2.15). The quantity 〈Ψ̂′†Ψ̂′〉 is the non-condensed density and 〈Ψ̂′Ψ̂′〉is called anomalous average. If we use the single-particle basis of the densitymatrix without assuming the Bogoliubov splitting (2.12) the anomalous averagevanishes. The anomalous average can become non-zero if we switch to a quasi-particle basis. We will yet define this single-particle basis for the non-condensedparticles which we will use.

The contact interaction and factorisation approximations do not destroy thesymmery mentioned at the beginning of this section. We find

K̂ = K̂0 + K̂1 + K̂†1 + K̂2 (2.18)

where (Ψ0 and Ψ̂′ depend on r, H.c. is shorthand for Hermitian conjugate)

K̂0 =

∫Ψ∗0

[H̃0 − µ +

1

2U0|Ψ0|2

]Ψ0 d

3r, (2.19)

K̂2 =

∫Ψ̂′†LΨ̂′ +

[12U0 [Ψ

∗02 + 〈Ψ̂′Ψ̂′〉∗] Ψ̂′Ψ̂′ +H.c.

]d3r. (2.20)

We introduced for later convenience the Hermitian operator

L = H̃0 − µ + 2U0[|Ψ0|2 + 〈Ψ̂′†Ψ̂′〉]. (2.21)

The careful reader notices that the term which is linear in the deviation operatoris missing between (2.19) and (2.20). We require that the Hamiltonian has onlybilinear terms in the deviation operator. This is necessary to proceed with thediagonalisation. The term which is linear in the deviation operator is of the form

20

2.4 Diagonalisation of the grand canonical Hamiltonian

K̂1 =∫

Ψ̂′†[· · · ]d3r, hence we let the expression in the brackets vanish and reap

already at this stage an equation which describes the condensate. It is equivalentto a generalised Gross-Pitaevskii equation, namely

[H̃0 − µ + U0 [|Ψ0|2 + 2〈Ψ̂′

†Ψ̂′〉]

]Ψ0 + U0〈Ψ̂′Ψ̂′〉Ψ∗0 = 0. (2.22)

Without the non-condensed density and the anomalous average this is the wellknown simple form of the Gross-Pitaevskii equation, Ginsburg-Pitaevskii-Grossequation, or nonlinear Schrödinger equation, (Ginzburg and Pitaevskii, 1958;Pitaevskii, 1961; Gross, 1961, 1963; Lifshitz and Pitaevskii, 1996). It is interestingto note in this context that a non-linear equation of the same structure as thesimple Gross-Pitaevskii equation describes the modulation of deep water waves,so-called wave trains (Yuen and Lake, 1975). The factor of two in (2.22) is justas obtained from Hartree-Fock theory for bosons (Goldman et al., 1981).

To diagonalise (2.20) we introduce the space-dependent linear Bogoliubovtransformation (e.g. de Gennes (1989))

Ψ̂′(r) = Ψ̂′(r, 0), Ψ̂′(r, t) =∑

k

uk(r)e−iEkt/ � β̂k − v∗k(r)e+iEkt/ � β̂†k (2.23)

For later reference we give in (2.23) the time dependent deviation operator Ψ̂′(r, t)The annihilation and creation operators β̂k and β̂

†k of the new quasi-particle states

obey bosonic commutation rules as introduced in section 2.2. Note that the sumis over all quasi-particle modes. For later reference we introduced the time-dependent linear transformation Ψ̂′(r, t). It is shown later that for a certain ap-proximation the lowest quasi-particle state can be identified with the condensatemode. As Leggett (2001) points out, the Bogoliubov approximation is equiva-lent to assume a correlated wave function in contrast to the Hartree-Fock-typeansatz which is a product of single-particle wave functions. In particular the wavefunction which is actually allowed for in a particle-number-conserving version ofthe Bogoliubov approximation is of the shape Ψ(r1, · · · , rN ) ∝ S

∏i

2 Hartree-Fock-Bogoliubov theory

find

K̂2 = −1

2

∑

k

(E∗k + Ek)

∫|vk|2 d3r (2.25)

+1

2

∑

jk

β̂†j β̂k(E∗j + Ek)

∫u∗juk − v∗j vk d3r

−12

[∑

jk

β̂jβ̂kEj

∫ujvk − ukvj d3r +H.c.

].

We see that certain orthogonality conditions diagonalise K̂2 of (2.25) which isof zeroth order or quadratic in the quasi-particle operators. Which criteria aresufficient that the quasi-particle modes fulfil these conditions?

We multiply the first line of (2.24) with u∗ from the left side and use thefact that the operator L is Hermitian and the first line of (2.24) to modifythe first addendum. This gives the expression U0

∫[Ψ∗0

2 +〈Ψ̂′Ψ̂′〉∗] v∗kuj −[Ψ02+〈Ψ̂′Ψ̂′〉] u∗kvj d3r = (Ej − E∗k)

∫u∗kuj d

3r. We can derive three further indepen-dent equations of a similar form if we multiply the second, first and second lineof (2.24) by v∗, u and v, respectively. We add the first two and likewise the lasttwo,

(E∗j − Ek)∫u∗juk − v∗jvk d3r = 0, (2.26)

(Ej + Ek)

∫ujvk − ukvj d3r = 0. (2.27)

Now we successively introduce two criteria which lead to a positive definite Hamil-tonian that is bound from below. Then the residual diagonalisation scheme fol-lows as in the T = 0 case. Necessary are the properties of the standard Bogoliubovquasi-particle states. Later we discuss the properties of states which violate thecriteria and classify them with the help of the criteria which are necessary to bein the standard situation. The scattering process probes both kinds of states.

For the first criterion we utilise the following symmetry: If the pair of functions(uk, vk) is a solution of (2.24) with eigenvalue Ek then (v∗k, u

∗k) is a solution with

eigenvalue −Ek which is easily seen from (2.24). We define as the first criterionthat the pseudo-norm does not vanish,

∫|uk|2 − |vk|2 d3r 6= 0. (2.28)

This leads to the possibility of diagonal contributions to the Hamiltonian. Withthe symmetry we see that we can always choose a pair (uk, vk) which has a positivepseudo-norm. With (2.26) we conclude that the quasi-particle energies Ek mustbe real. If Ek = 0 all contributions from the corresponding quasi-particle mode to

22

2.4 Diagonalisation of the grand canonical Hamiltonian

the Hamiltonian vanish. In this sense we see that if a zero-energy mode exists itis automatically excluded from the final form of the Hamiltonian. We can restrictourselves to the case Ek 6= 0.

Now we have choosen quasi-particle modes with positive pseudo-norm. Look-ing at the contributions to K̂2 in (2.25) we see that the second criterion is thatthe quasi-particle energies are positive,

0 < Ek. (2.29)

If these two restrictive criteria are met we have pairs of quasi-particle modes withpositive quasi-particle energy and positive pseudo-norm.

In the standard situation we can conclude from (2.26)

∫u∗juk − v∗jvk d3r = δjk, (2.30)

provided the quasi-energies are not degenerate. We are in the situation 0 < Ek,hence from (2.27)

∫ujvk − ukvj d3r = 0. (2.31)

With the last two expressions, (2.30) and (2.31), and adding the restriction thatthe quasi-particle energies are real we complete the diagonalisation of (2.25),

K̂2 = −∑

k

Ek

∫|vk|2d3r +

∑

k

n̂kEk. (2.32)

For an infinite harmonic potential the first sum shifts the energy of the bosonsby a constant value and must be finite. Note that for a homogeneous conden-sate the first term is divergent. The term is due to quantum depletion at zerotemperature. To find the energy from the grand-canonical Hamiltonian the orderparameter and the quasi-particle modes must be known. The confined bosonsare now described by the order parameter and non-interacting quasi-particles. Afinal set of solutions simultaneously obeys (2.22) and (2.24). We note that thegeneralised Gross-Pitaevskii equation is connected to the Bogoliubov-like equa-tions via the two averages which we introduced to reduce the full Hamiltonianto a quadratic form in the deviation operator. For the bosonic quasi-particles〈n̂k〉 = [exp(βEk) − 1]−1 is the only non-zero average in the sense of the lastterm in (2.15) for one- or two-quasi-particle operators (e.g. Doerre et al. (1979))applying the standard definition 〈· · · 〉 = Tr[· · · exp(−βK̂)]/Tr[exp(−βK̂)] forthe ensemble average. In the last expression the temperature T enters throughT = 1/(kBβ), where kB is the Boltzmann constant. We encounter averages ofthe bosonic quasi-particle operators if we calculate the non-condensed density

23

2 Hartree-Fock-Bogoliubov theory

and the anomalous average and find, respectively,

〈Ψ̂′†Ψ̂′〉 =∑

k

[|vk|2 + 〈n̂k〉 [|uk|2 + |vk|2]

], (2.33)

〈Ψ̂′Ψ̂′〉 =∑

k

ukv∗k

[1 + 2〈n̂k〉

]. (2.34)

In the standard situation the description is now complete.Now we address ourselves to the situation where the two criteria are not met.

If we approximate the non-linear full Hamiltonian with the Bogoliubov approachabout an excited state we distinguish two cases. First we think of a non-zeropseudo-norm which we choose positive and simultaneously negative real energy.Such anomalous modes are energetic instabilities and allow for the creation of oneor more unstable quasi-particles thereby lowering the total energy. The excitedcondensate mode is unstable and decays in the presence of dissipation. Conden-sate candidates for such anomalous modes are condensates with vortices (Castinand Dum, 1998; Fetter and Svidzinsky, 2001) and condensates where dark soli-tons are present (Muryshev et al., 1999; Busch and Anglin, 2000; Dziarmaga andSacha, 2002).

The second possibility is a vanishing pseudo-norm. In contrast to the first casethe existence of a real energy is not an automatic conclusion. The energy may bepure imaginary or complex in this case. The complex modes indicate dynamicalinstabilities which are always triggered by quantum fluctuations (Garay et al.,2001). For a short period of time the linearised theory remains valid. Suchcomplex modes are discussed for vortex states by Aranson and Steinberg (1996),Pu et al. (1999) and Garćıa-Ripoll and Pérez-Garćıa (1999) and for dark solitonsby Fedichev et al. (1999), Feder et al. (2000), Anderson et al. (2001), Brand andReinhardt (2002) and Muryshev et al. (2002).

2.5 Approach with Heisenberg’s equation ofmotion

This derivation allows us to understand the quasi-particle excitation energies asbelonging to ”classical” linearised deviations from the static ”classical” Bose fieldwhich is governed by the time-dependent generalised Gross-Pitaevskii equation(2.22).

We start with the equation of motion

i�∂tΨ̂1(r, t) = −[Ĥ, Ψ̂1(r, t)] (2.35)

for the time-dependent field operator Ψ̂1(r, t) = eiĤt/ � Ψ̂(r, 0)e−iĤt/ � in the Heisen-berg picture, where Ĥ is the many-body Hamiltonian of (2.14). In a standardway we have the field operator equation by computing the commutator of the

24

2.6 Approximations and generalisations

right hand side (Schwabl, 1997). In the resulting equation we insert the ansatzΨ̂1(r, t) = e

−iµt/ �[Ψ0(r) + Ψ̂

′(r, t)]≡ e−iµt/ � Ψ̂(r, t) and find

i�∂tΨ̂(r

′, t) =[H̃0 − µ+

∫Ψ̂†(r′, t)U(r, r′)Ψ̂(r′, t)d3r′

]Ψ̂(r, t). (2.36)

Griffin (1996) showes how to derive the condensate equation (2.22) and an equa-tion for the deviation operator Ψ̂′(r, t) within the contact interaction and self-consistent mean-field approximation. For a non-stationary Bose condensate weobtain a time-dependent version of (2.22). The time-dependent version withoutthe averages is often sufficient to describe the dynamics of a condensate. If we use

in this equation the ansatz Ψ0(r, t) = e−iµt/ �[Ψ0(r) +u(r)e−iEkt/ � −v∗(r)e+iEkt/ �

]

which describes small classical oscillations with the frequencies ωk = Ek/�

of theorder parameter around its static value we find (2.24) without the averages bykeeping only terms linear in the u- and v-modes. If we view them in this perspec-tive the simplified version of (2.24) descirbes ”classical” deviations and frequen-cies of the Bose field. Note that the ”classical” field approach works without thequasi-particle operators and agrees with the diagonalisation approach.

Let us continue without dropping completely the operator nature of the fieldoperator. The approach allows one to proceed to a more elaborate Green’s func-tion formulation. Griffin uses various averages of products of the full field operatorΨ̂. Finally he subtracts the condensate equation from (2.36),

i�∂tΨ̂′(r, t) =

[H̃0 − µ

]Ψ̂′(r, t) (2.37)

+2U0〈Ψ̂†(r)Ψ̂(r)〉Ψ̂′(r, t) + U0〈Ψ̂(r)Ψ̂(r)〉Ψ̂′†(r, t).

We can convince ourselves that (2.37) is equivalent to the coupled equations(2.24) by replacing the deviation operator with the time-dependent linear trans-formation given in (2.23). We see that the quasi-particle modes must obey thecoupled equations.

2.6 Approximations and generalisations

Let us first streamline and summarise the results. Introducing the densities

n(r) ≡ 〈Ψ̂†Ψ̂〉 = |Ψ0(r)|2 + 〈Ψ̂′†Ψ̂′〉 ≡ nc(r) + nn(r), (2.38)

m(r) ≡ 〈Ψ̂Ψ̂〉 = Ψ20(r) + 〈Ψ̂′Ψ̂′〉 ≡ Ψ0(r)2 +mn(r). (2.39)

Both densities are made up of a condensate part and a non-condensed part. Aswe learn from (2.33) and (2.34) the latter has temperature dependent and inde-pendent contibutions. The temperature independent part of the non-condensed

25

2 Hartree-Fock-Bogoliubov theory

density at T = 0 is called quantum-depletion. Now we can rewrite the gener-alised Gross-Pitaevskii equation (2.22) and the Hartree-Fock-Bogoliubov equa-tions (2.24)) in the following compact form:

[H̃0 − µ+ U0 [nc(r) + 2nn(r)]

]Ψ0(r) + U0mn(r)Ψ

∗0(r) = 0, (2.40)

[H̃0 − µ+ 2U0n(r)

]uk(r)− U0m(r)vk(r) = Ekuk(r), (2.41)

[H̃0 − µ+ 2U0n(r)

]vk(r)− U0m∗(r)uk(r) = −Ekvk(r).

The system of equations consisting of the generalised Gross-Pitaevskki equa-tion (2.40) and the Bogoliubov equations (2.41) can be simplified and extendedin various ways (see also Griffin (1996)). We proceed from crude approximationsto more Daedalian schemes indicating for which purpose they are useful.

The first two are single-particle descriptions in mean-field approximation (Dal-fovo et al., 1999). In the case of harmonic confinement they become accurate forthe high-energy part of the quasi-particle spectrum which is numerically foundto dominate the thermodynamic behavior even at low temperature. The v-modeis completely neglected and we may distinguish the two schemes:

• vk = nn = mn = 0 in (2.40) and (2.41) – the single-particle Hamiltonianfor the excitations at T = 0 and

• vk = mn = 0 in (2.40) and (2.41) – this Hartree-Fock Hamiltonian describesnon-interacting bosons in a self-consistent mean field at finite temperaturewith an excitation spectrum valid near the critical temperature Tc of Bose-Einstein condensation (Goldman et al., 1981; Huse and Siggia, 1982).

The coupling between the modes is responsible for the collectivity of the solu-tions (Dalfovo et al., 1999). The first two schemes give a gapless single-particlespectrum Griffin (1996) and belong to the following two approximated Hartree-Fock-Bogoliubov equations:

• nn = mn = 0 – standard Bogoliubov approximation. It is valid at T = 0.The quantum depletion due to interactions is negelected. (Fetter, 1972;Fetter and Svidzinsky, 2001).

• mn = 0 – the Popov approximation in a finite temperature scheme (Popov,1987). This is a reasonable guess to describe both the high and low temper-ature regimes below the critical temperature for condensation, (Shi et al.,1994; Dalfovo et al., 1999). It is a gapless theory and it obeys Goldsone’stheorem (Virtanen et al., 2001a). Like in the standard Bogoliubov approx-imation we can make the phase explicit (Fetter, 1972) and show that thereexists a solution with u0 = v∗0 ∝ Ψ0 with E0 = 0. This can be unterstoodas a Goldstone mode resulting from the ”U(1) symmetry breaking” of theapproximation (Lewenstein and You, 1996a).

26

2.7 Spherically symmetric trapping potential

In the Popov approximation an equation similar to (2.27) may be derivedprovided the order parameter is real, Ψ0 = Ψ

∗0.

∫Ψ0uk −Ψ0vk d3r = 0. (2.42)

In this case we see with (2.24) that u0 = v0 = Ψ0 is a solution withE0 = 0. So under certain conditions we have the quasi-normalisationsor biorthonormality conditions (2.30), (2.31) and (2.42) for the Popov ap-proximation. In the sense of the pseudo-norm (2.30) the quasi-particleexcitations are orthogonal to the condensate. In (2.42) we can require that∫

Ψ0ukd3r =

∫Ψ0vk d

3r = 0. Then each quasi-particle mode is orthogonalto the condensate.

Finally we generalise the standard Bogoluibov equations to a situation with amore general two-particle interaction.

• nn = mn = 0 and in addition U(r, r) is assumed to be real and symmetricwith respect to the exchange of the coordinates – we obtain generalisednon-local Bogoliubov equations (Esry, 1997)

[H̃0 − µ +

∫U(r, r′)|Ψ0(r′)|2d3r′

]Ψ(r) = 0, (2.43)

L1[uk(r)]−∫U(r, r′)Ψ0(r)Ψ0(r

′)vk(r′)d3r′ = Ekuk(r), (2.44)

L∗1[vk(r)]−∫U(r, r′)Ψ∗0(r)Ψ

∗0(r′)uk(r

′)d3r′ = −Eνvk(r),

where the non-local operator L1f(r) =[H̃0 −µ+

∫U(r, r′)|Ψ0(r′)|2d3r′

]f(r)

+∫U(r, r′)f(r′)Ψ∗0(r

′)d3r′ Ψ0(r) is used. Köhler and Burnett (2002) ad-dress (2.43) as the stationary Gross-Pitaevskii equation in the Born ap-proximation.

2.7 Spherically symmetric trapping potential

2.7.1 Expansion in spherical harmonics at zero temperature

We solve the Hartree-Fock-Bogoliubov equations for a spherically symmetric con-fining potential, V (r) = V (r). In addition, we assume that the potential ap-proaches zero at infinity decaying faster than 1/r2. Therefore it only allows for afinite number of bound states. In addition there are free or continuum states. Tocover the finite-temperature regime we will first thoroughly describe the T = 0

27

2 Hartree-Fock-Bogoliubov theory

case. Then we show how to proceed to a self-consistent iteration scheme for finitetemperatures.

As a further restriction we choose the lowest spherically symmetric self-con-sistent solution. In this case the condensate wave function may be chosen real(Edwards and Burnett, 1995). We will nevertheless keep the distinction of thecondensate mode and its complex conjugate in our notation till further consid-erations force us to drop it. It will prove convenient to define a reduced radialcondensate wave function u, not to be confused with the quasi-particle modeuk, and a radial condensate density nc(r) which as an argument has the radialcoordinate r,

Ψ0(r) =

(N04π

)12 u(r)

r, |Ψ0(r)|2 =

nc(r)

4π. (2.45)

The reduced condensate wave function u is normalised to unity with respect toradial integration, whereas the condensate wave function Ψ0(r) =

√N0〈r | 0〉 is

normalisable to N0 with respect to three-dimensional integration.

We expand the quasi-particle u-mode and v-mode in spherical harmonics,

uk(r) =

∞∑

l=0

l∑

m=−l

u(k)lm (r)

rYlm(r

0). (2.46)

The expression for the v-mode is identical with u replaced by v. The superscriptlabels possibly existing further quantum numbers which are different from thelm-numbers of the spherical harmonics. See the appendix (A.1) for the definitionwhich we adopted for the spherical harmonics.

Let us now consider the standard Bogoliubov equations as introduced in sec-tion 2.6. We assume that a reduced radial solution u(r) of the stationary Gross-Pitaevskii equation (2.40) with mn(r) = nn(r) = 0,

[−

�2

2m

d2

d2r+ V (r) + U0

N04π

∣∣∣∣u(r)

r

∣∣∣∣2]u(r) = µu(r), (2.47)

is computed. All particles are in the condensate within this approximation. Wehave a negative chemical potential, µ = −|µ|. Now we insert the expansions(2.46) of the quasi-particle modes into (2.41) with mn(r) = nn(r) = 0. Wecompute the action of the kinetic energy operator taking advantage of (A.19).The orthogonality relations for the spherical harmonics (A.2) allow us to project

28

2.7 Spherically symmetric trapping potential

out one pair of modes (u, v). Finally multiplication with r entails[ �

2

2m

[− d

2

d2r+l(l + 1)

r2

]+ V (r) + 2U0

N04π

∣∣∣∣u(r)

r

∣∣∣∣2]u(k)lm (r) (2.48)

−U0N04π

[u(r)

r

]2v

(k)lm (r) = [µ+ E

(k)lm ]u

(k)lm (r),

[ �2

2m

[− d

2

d2r+l(l + 1)

r2

]+ V (r) + 2U0

N04π

∣∣∣∣u(r)

r

∣∣∣∣2]v

(k)lm (r)

−U0N04π

[u∗(r)

r

]2u(k)lm (r) = [µ−E

(k)lm ]v

(k)lm (r).

We observe that in the case of a real condensate wave function the confining po-tential V (r) and the radial condensate density nc(r) determine the spectrum ofthe quasi-particle modes. The equations are independent of the azimuthal quan-tum number m of the spherical harmonics. Therefore possible radial solutionsare (2l + 1) times degenerate.

It is convenient to introduce harmonic oscillator units. This corresponds to thefact that harmonic confinement at least at the bottom of the trap is a realisticassumption for present experiments. The length unit is aho =

√ �/(mω), the

energy unit�ω and the time unit 1/ω, see also the appendix A.2. We have

assumed that for small r the potential is of the form V (r) = const. + mω2r2/2.If the order parameter is real we find

[−1

2

d2

d2r+V (r)

�ω

+aTTaho

nc(r)

]u(r) =

µ�ωu(r), (2.49)

[− 1

2

d2

d2r+

1

2

l(l+ 1)

r2+V (r)

�ω

+aTTaho

2nc(r)

]u(k)lm (r) (2.50)

−aTTaho

nc(r)v(k)lm (r) =

µ + E(k)lm�ω

u(k)lm (r),[− 1

2

d2

d2r+

1

2

l(l+ 1)

r2+V (r)

�ω

+aTTaho

2nc(r)

]v

(k)lm (r)

−aTTaho

nc(r)u(k)lm (r) =

µ− E(k)lm�ω

v(k)lm (r).

2.7.2 Bound and free quasi-particle modes

We examine the behaviour of the reduced condensate wave function and thequasi-particle modes at zero and in the limit of large r. We distinguish twodifferent regimes of the quasi-particle modes due to their behaviour at large r.It depends on the constant effective energy composed of the chemical potential

29

2 Hartree-Fock-Bogoliubov theory

PSfrag replacements

bound

modes

Elnl

k21/2

k22/2

2

free

modes

Elk

k21/2

k22/2

3trapping potential

µ

r

1

Figure 2.1: Scheme to illustrate the different solutions of the Bogoliubov equations.The condensate is indicated at the bottom of the trapping potential. It islabeled by ”1”. For bound quasi-particle solutions (”2”) with energy Elnlwe find k21,2/2 = (µ ±Elnl)/(

�ω) < 0. For free quasi-particle modes (”3”)

we have 0 < k21/2 = (µ+ Elk)/(�ω) but k22/2 = (µ −Elk)/(

�ω) < 0. The

momentum k1 = k of the free solutions is introduced in (2.56).

µ = −|µ| and the positive quasi-particle energy E = |E| at the right hand sidesof equations (2.48).

From (2.47) we can choose the following boundary conditions for the realspherically symmetric reduced condensate wave function:

u(r)r→0−→ const. r, u(r) r→∞−→ const. exp

[√(−2µ)/( � ω) r

]. (2.51)

The wave function is regular at the origin and decays exponentially if r approachesinfinity.

We require the quasi-particle modes to be regular at the origin, independentof their asymptotic behaviour at infinity,

u(k)lm (r)r→0−→ const. rl+1, v(k)lm (r)

r→0−→ const. rl+1. (2.52)

If (µ + Ek)/(�ω) < 0 we have also (µ − Ek)/(

�ω) < 0. We find that the two

quasi-particle modes decay exponentially for large r, they obey the bound state

30

2.7 Spherically symmetric trapping potential

boundary conditions (Setty, 1998; Wynveen et al., 2000),

u(k)lm (r)r→∞−→ const. exp

[−√

(−2µ − 2E(k)lm )/(�ω) r

], (2.53)

v(k)lm (r)r→∞−→ const. exp

[−√

(−2µ + 2E(k)lm )/(�ω) r

].

The v quasi-particle modes decay faster than the u-modes for large r. Fora given l the equations (2.48) are satisfied only for certain discrete energies

0 ≤ E(k)lm /(�ω) ≤ |µ|/( � ω), which we label according to the number of nodes

nl of the u mode. This number is the same for the v mode. Since the equa-tions (2.48) are independent of m there are (2l + 1) degenerate radial reducedmodes for each angular momentum l. Thus we label the bound state energiesElmnl = Elnl , where l = 0, 1, · · · , lmax, m = −l,−l+ 1, · · · , l and for each l thereare nmaxl + 1 solutions, nl = 0, 1, · · · , nmaxl , or none. For a finite potential thereis only a finite number of bound states. The total number of bound states is∑lmax

l=0 (2l + 1)(nmaxl + 1).

If 0 < (µ + Ek)/(�ω) we have nevertheless (µ − Ek)/(

�ω) < 0. The v-mode

is localised in the condensate region while the u-mode extends over the wholespace. The quasi-particle states obey the free or continuum boundary conditions

u(k)lm (r)

r→∞−→ const. sin[√

(2µ+ 2E(k)lm )/(

�ω) r − lπ/2 + δ

E(k)lm

], (2.54)

v(k)lm

r→∞−→ const. exp[−√

(−2µ + 2Ek)(�ω) r

]. (2.55)

In the continuum case it is convenient to introduce the free momentum

k =√

(2µ + 2E)/(�ω). (2.56)

For each l and and free momentum k there is just one (2l+1) times degenerate freesolution corresponding to the quasi-particle energy Elmk = Elk, m = −l, · · · , l.Therefore we can write δ(k)lm = δl(k), m = −l, · · · , l, for the phase shifts. A nu-merically favourable boundary condition equivalent to (2.54) is of course available(see, e.g., Joachain(1975))

ulk(r)r→∞→ const. kr[jl(kr) cos δlk − nl(kr) sin δlk] ≡ u(∞)lk (δl(k), r), (2.57)

where the spherical Bessel and Neumann functions jl and nl are given by (A.40)and (A.42) in the appendix A.3. This boundary condition has the advantage thatit takes the repulsive angular barrier into account. It is superior for situationswhere the confining potential is almost zero but l(l+ 1)/r2 is not yet negligible.The boundary condition (2.54) is identical with (2.57) if the constants in theequations are equal.

We can label the solutions in the same way as the corresponding energies. Tobe as explicit as possible, we write

ulmnl(r) = Ylm(r0)ulnl(r)/r, (2.58)

ulmk(r) = Ylm(r0)ulk(r)/r.

31

2 Hartree-Fock-Bogoliubov theory

The same expression exists for the v-mode with u replaced by v. The functionsulnl and ulk may be chosen real. If the argument of the u-mode is the threedimensional vector r we have the full set of quantum numbers, lmnl in the boundstate case and lmk in the continuum case. If the argument of the u-mode is 0 < rthe quantum numbers are only lnl or lk.

Note that we resticed ourselves to the case of a positive pseudo-norm (2.30)and positive quasi-particle energy. The same boundary conditions hold if we con-sider the standard situation and exchange the role of the u- and the v-mode. Wehave then negative pseudo-norm and negative quasi-particle energy. In this situa-tion the u-mode is bound and the v-mode is free. We try to integrate the variouspossible quasi-particle energies, bound and free states and boundary conditionsof the standard situation in figure 2.1.

2.7.3 An iterative scheme for finite temperatures

Let us assume that we have solved the standard Bogoliubov equations. We areinterested in the stationary density distribution, restricting ourselves to the caseof bound states. We expect that these states describe the condensate shortly afterrethermalisation. We know the reduced radial solution u of the Gross-Pitaevskiiequation (2.47) and its chemical potential µ and a finite number of pairs of modes(ulmnl , vlmnl) corresponding to one energy Elmnl each.

The first iteration step is to compute the non-condensed density (2.33) andthe anomalous average (2.34). We insert the explicit expressions (2.58) for thebound quasi-particle states into the equations for the densities. Since the energiesare the same for all azimuthal quantum numbers m we may use (A.20) which isa special case of the addition theorem for spherical harmonics. In an identicalmanner as for the condensate we introduce radial densities,

nn(r) =lmax∑

l=0

2l + 1

4πr2

nmaxl∑

nl=0

[|vlnl(r)|2 +

|ulnl(r)|2 + |vlnl(r)|2eElnl/(kBT ) − 1

]≡ nn