Ultracold Bose gases - from the Gross-Pitaevskii
to the fractional quantum Hall regimebySatyan Gopal Bhongale
B.Tech., Indian Institute of Technology, Bombay, 1996

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2004

This thesis entitled:
Ultracold Bose gases - from the Gross-Pitaevskii to the fractional quantum Hall regime
written by Satyan Gopal Bhongale
has been approved for the Department of Physics

Murray Holland

Prof. John Bohn

Date

The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Satyan Gopal Bhongale, (Ph.D., Physics)
Ultracold Bose gases - from the Gross-Pitaevskii to the fractional quantum Hall regime
Thesis directed by Prof. Murray Holland

Ultra-cold Bose gases present an ideal environment for the study of many-body
physics. These systems can be prepared under various experimental conditions with
precise control. Techniques like Feshbach resonances allow us to dynamically tune theinter atomic interaction, from strongly attractive to a strongly repulsive one.
In the first part of the thesis, we study the weakly interacting Bose gas in connection with the dynamics of an atom laser. Here we propose a possible optical pumping
model for loading the reservoir of a continuous wave atom laser. The finite temperatureeffects like phase diffusion require a thorough understanding of the kinetic regime of thedilute Bose gas. In this respect, we develop a non-Markovian quantum kinetic theory
and thereby show the emergence of different time scales for correlation and subsequentrelaxation to an equilibrium states. Using numerical simulations, we also predict thedamping rates and frequencies of collective modes.
In the second part, we study the strongly correlated regime where the interactionenergy is greater than any other (single particle) energy scale of the problem. Here,in the presence of a Feshbach interaction we predict the generation of novel strongly
correlated paired states. Such states while similar to the one observed in a 5/2 fractional
quantum hall effect, are unique in symmetry to the Bose gas system.

Dedication

To my parents

v

Acknowledgements

I would first like to thank my advisor Prof. Murray Holland for his guidance andencouragement through out the course of this thesis. He has provided me with a researchatmosphere that enhanced my ability towards becoming an independent researcher. His
informal attitude, not only made him available to me when ever I was stuck with aresearch problem but also allowed me to express my thoughts without hesitation.
I am also extremely grateful to Prof. John Cooper who has been like my coadvisor. I cannot thank him enough for the countless discussion hours and invaluableadvice.
I would like to thank my colleagues, Josh Milstein, Jochen Wachter, Servaas
Kokkelmans, and Reinhold Walser for the incredible amount of help I received during
the course of this thesis.
Apart from those who have been directly connected with my thesis work, I would
like to mention a few people without whose support and encouragement I would not
be here writing this thesis. Firstly my parents to whom this thesis is dedicated. They
were a constant source of motivation without which I would not be the person I am
today. My father Dr. Gopal Bhongale who inspired me to think of the world. My
undergraduate advisor Prof. Bhanu Das who got me interested in the field of AMO
physics.
Lastly I would like to thank my best friend Patricia Rettig for all the support
and for standing by me in every situation possible.

vi

Contents

Chapter
1 Introduction

1

2 Loading a conitnuous wave atom laser by optical pumping techniques
Published in Physical Review A 62, 043604 (2000)

7

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

The Model

92.3

Effect of time dependent Ω and finite detuning on the transient spectrum 15

2.4

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3 Memory effects and conservation laws in the quantum kinetic evolution of a dilute
Bose gas
Published in Physical Review A 66, 043618 (2002)

20

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

3.2

Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.3

Conservation laws and Quasi-particle damping . . . . . . . . . . . . . .

28

3.4

Application to a dilute Bose gas in a spherical box trap . . . . . . . . .

31

3.5

Real time response to perturbation . . . . . . . . . . . . . . . . . . . . .

38

3.6

Conclusion

43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Bosonic fractional quantum Hall effect4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46
46

vii
4.2

Fractional quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . .

47

4.3

Electron system versus dilute Bose gas . . . . . . . . . . . . . . . . . . .

49

4.4

Landau levels and the many-body Laughlin wavefunction . . . . . . . .

50

4.5

Exact diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.6

Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.7

Conclusion

57. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Chern-Simons theory for a strongly correlated rotating Bose gas

59

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

5.2

Statistics in 2-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . .

60

5.3

Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.4

Anyons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5.5

Composite particle mean field theory . . . . . . . . . . . . . . . . . . . .

64

5.6

Chern-Simons Ginzburg-Landau theory . . . . . . . . . . . . . . . . . .

66

5.7

Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

6 Resonant formation of novel strongly correlated paired states in rotating Bose
gases
Published in Physical Review A 69,053603 (2004)

71

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

6.2

Inclusion of Feshbach resonance in the Chern-Simons theory . . . . . . .

73

6.3

Pairing in the strongly correlated ground state . . . . . . . . . . . . . .

76

6.4

Conclusion

80

7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

viii
Bibliography

84

Appendix
A Reference distribution90

B Matrix elements

91

C Anyon field operator

92

ix

TablesTable
4.1

Comparison between the electronic fractional quantum Hall system andthe rotating Bose gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

xFiguresFigure1.1

The total angular momentum of the stable ground state plotted as afunction of the rotational frequency for N = 6 atoms with two-bodyinteraction potential in units of harmonic oscillator quanta given by V =(1/N )δ(r i −rj ). (Figure obtained from N. K. Wilkins and J. M. F. Gunn,
Phys. Rev. Lett. 84, 6 (2000)) . . . . . . . . . . . . . . . . . . . . . . .

1.2

Flow chart of the chapters in this thesis. Figure also shows the openingof new possibilities for further study as a result of this work.

2.1

4. . . . . .

6Level scheme illustrating: the ground state |gi, excited state |ei, and trap
state |ti. A laser is used to couple states |gi and |ei with an intensity
characterized by the Rabi frequency Ω. Spontaneous emission of photons
occurs with rate Γe from |ei to |ti and rate Γt from |ti to |gi. . . . . . .

2.2

9Schematic diagram showing the proposed loading scheme for obtaining
a steady state BEC and CW atom laser. Atoms entering the laser field
are in the ground state. As the atoms pass through the laser field, thepopulation is continuously transferred to the trap state by a sequence ofcoherent pumping and spontaneous emission. . . . . . . . . . . . . . . .

2.3

10

Populations as a function of time; ground state |gi shown by a solid line,
excited state |ei shown by dashed and trap state |ti shown by dot-dashedline. The parameters used were Ω = 5, Γe = 1 and (a) Γt = 2, (b) Γt = 0.

12

xi
2.4

Comparison of the |ei → |ti transition spectrum using the steady statecalculation (solid) and the quantum trajectory method (dash-dot). . . .

2.5Ω as a function of time, t. The three different curves correspond todifferent time scales over which Ω reaches its maximum value. . . . . . .

2.6

17

Hartree-Fock energies as a function of particles number in the box groundstate, f11 with all other fij ’s equal to zero. . . . . . . . . . . . . . . . . .

3.2

16

Figure shows the effect of detuning on the transient spectrum for |ei → |titransition. Dashed for δ = 0, solid for δ = 1 and dot-dashed for δ = 2. .

3.1

16

Transient spectrum for |ei → |ti transition for the different scenarios forΩ(t) shown in Fig.2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

13

32

The scaled mean-field density, Uf (r, r)/ε0 , as a function of the scaled
radial distance, r/R, at a relatively hot temperature, β = 0.01 is shownfor three different values of particle numbers: N = 10 (solid curve), N =100 (dashed), and N = 500 (dot-dashed). Inset shows the corresponding
number density as a function of scaled radial distance. . . . . . . . . . .

3.3

34

The scaled mean-field density, Uf (r, r)/ε0 , as a function of the scaled
radial distance, r/R, at a relatively cold temperature, β = 0.5 is shown forthree different values of particle numbers N = 10 (solid curve), N = 100(dashed), and N = 500 (dot-dashed). Inset shows the corresponding
number density as a function of scaled radial distance. . . . . . . . . . .

3.4

35

Comparison between different damping functions, exp(−ητ ) (solid), exp(−ητ 2 )(dot-dashed), and 1/ cosh(ητ ) (dashed). Note that the hyperbolic secant
function asymptotes to an exponential form for large τ and a Gaussian
form for small τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

xii
3.5

Evolution of the diagonal elements fnn (n = 1, 2.., 5) (shown with curves
from bottom to top) toward a self-consistent steady-state solution starting from an approximate solution. Two different damping functions areused in the evaluation of the Γ’s. Dashed and solid curves correspond toexponential and hyperbolic secant respectively. . . . . . . . . . . . . . .

3.6

37

Linear behavior of θ(fnn ) as a function of εHF
n . Initial distribution shownwith diamonds. Final distribution shown with circles for the case of
exponential damping function, and squares for the case of hyperbolic
secant damping function. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

38

Absolute value of f eq , the self consistent steady-state solution for the second order kinetic equation. The single particle density matrix is plottedin the Hartree-Fock basis. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.8

39

Non-negative frequency eigenvalues scaled with respect to ε0 shown withcrosses for the Hartree-Fock equation and squares for the quantum-Boltzmann
equation. The dotted lines correspond to difference energies (εHF−i
εHFj )/ε0 . The modes labelled (a) and (b) – non-zero frequency and damping rate, (c) – zero mode, and (d) – zero frequency and non-zero damping
rate will be considered for further discussion. . . . . . . . . . . . . . . .

3.9

40

The perturbation λδfωs (bottom) is shown in a rotated frame such thatf eq is diagonal; the resulting oscillatory and damped behavior (top) of
the element f12 of f in the box basis is due to the perturbation. The left
and right figures correspond to the points marked (a) and (b) in Fig. 3.8
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.10 The perturbation λδfωs (bottom) is shown in a rotated frame such thatf eq is diagonal; the resulting damped behavior (top) of the element f12
of f in the box basis is due to the perturbation. The left and right figurescorrespond to the points marked (c) and (d) in Fig. 3.8, respectively. . .

42

xiii
3.11 The change in the total energy ∆E = E(f )−E(f eq ) as the system relaxes
to its new equilibrium. In the top figure, the solid and dashed lines correspond to cases (a) and (b), respectively. Similarly in the bottom figure,the solid and dashed lines correspond to cases (c) and (d), respectively.
4.1

The FQHE in ultra high-mobility modulation-doped GaAs/AlGaAs. Many
fractions are visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

43

48

Atomic density in the lowest Landau level: left - density in the lowest
Landau level with angular momentum quantum number l = 3 is sharply
peaked at radius r =pring of thickness ∆r =lh̄/mω, right - density is shown by a concentricph̄/4lmω in the x-y plane. . . . . . . . . . . . .

52

4.3

The atomic density in the 21 - Laughlin state for N = 5 atoms. . . . . . .

55

4.4

Results of exact diagonalization for a small system of N = 5 bosonicatoms in a rotating 2-dimensional harmonic trap. The stirring frequency
is chosen to be smaller than but very close to the trapping frequency,
1 − Ω/ω = 0.01. The parameter η is chosen such that the interactionenergy is very small compared to the Landau level spacing, 2h̄ω. Such a
choice allows us to use a truncated Hilbert space of lowest Landau levels.

5.1

56

Statistics associated with the exchange of two identical particles. The net
effect of making the interchange twice is to take one particle around another in a closed loop. In 3-dimensions the loop can be deformed withoutcrossing the other particle which is not possible in two dimensions.

5.2

. .

60

Trajectory of the vector R(t) in the parameter space. The vector R(t)represents a set of parameters that define the Hamiltonian at some timet. The parameters are changed adiabatically from their initial value attime ti such that the vector moves in a closed loop and comes back to its
original position at some final time tf. . . . . . . . . . . . . . . . . . .

62

xiv
5.3

Left figure: two-dimensional two-electron system. The electron is localized separately potential valleys V 1 and V 2. The valleys are separated
such that there is negligible overlap between the electronic wavefunctions.
Right figure: the same situation as on the left except now the electrons
are independently attached to magnetic flux φ and the flux moves alongwith the electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

63

Composite particle picture. A boson is transformed to a composite boson(fermion) by attaching an even(odd) number of magnetic flux quanta.
The extra fictitious flux cancels the real flux in the mean field approximation resulting in free composite bosons/fermions. . . . . . . . . . . .

5.5

65

A Feshbach resonance results when a closed channel potential possesses
a bound state in proximity to the scattering energy in an open channel
potential. The detuning of the bound state from the edge of the collisioncontinuum is denoted by ν0 .

6.1

. . . . . . . . . . . . . . . . . . . . . . . .

69

The Feshbach resonance pairing mechanism is illustrated by the above
Born-Oppenheimer curves. Pairs of composite atoms composed of single atoms and an associated number of quanta of angular momentum(represented by the arrows) approach each other within an open channel potential of background value U (x, x′ ). They may form a composite
molecule due to the presence of a closed channel bound state, at a detuning ν from the scattering continuum, which is coupled to the open
channel with strength g(x, x′ ). . . . . . . . . . . . . . . . . . . . . . . .

7.1

72

Origin of different possible fractions in the fermionic- and bosonic- fractional quantum Hall effect. The tuning of interactions via Feshbach resonance may result in generation of fractions, the underlying mechanism
could be compared to that of the even FQHE. . . . . . . . . . . . . . . .

82

Chapter 1

Introduction

Dilute atomic gases present an ideal environment for studying various aspects ofmany-body physics. These systems can be prepared with precise control and extreme
purity. Advancement in the techniques of trapping and cooling based on the detailed
understanding of atom-light interactions have allowed experimentalists to cool atomic
gases to ultra-low temperatures. This has opened new possibilities for the understanding
and investigation of many-body phenomena, for example the prediction of Bose-Einsteincondensation (BEC). BEC occurs in bosonic systems where the ground state acquires
a macroscopic population when cooled below a certain critical temperature Tc . While
first predicted by Bose and Einstein in 1924, it wasn’t until 1995 when it was confirmedin the form in which it was originally posed by experiments in a trapped dilute atomic
vapor of Rb87 , Na and Li7 [1, 2, 3]. Recently BEC has been achieved in dilute atomic
gases of H, He, K41 ,Cs and Yb, and has also been realized in micro-traps on chips [4]and also in two dimensional surface traps [5].
Since 1995, dilute atomic gases have generated tremendous theoretical and experimental interest and have led to the study of phenomena like atom lasers and vortices.
More recently with the advent of techniques based on Feshbach resonances, the interactions between atoms can now be dynamically tuned by changing the external magnetic
field. While interactions in condensed matter systems could be modified, for example,
by doping a semiconductor material, dynamic control is typically limited. In fact, in2some condensed matter systems such as those that give rise to the fractional quantum
Hall effect, there is virtually no control over interactions. Feshbach tuning allows theinter-atomic interaction to be dynamically varied from a repulsive to an attractive one.
The ability to tune interactions has allowed us to stretch the scope of possibilities tostudying BCS superfluidity [6, 7, 8, 9], guiding of matter waves [10] and molecule formation in a Fermi gas [11]. Thus dilute atomic gases have provided us with a new setting
to answer some of the questions and test ideas that have previously been inaccessible
in the domain of condensed matter physics.
From a theory point of view, dilute atomic gases are extremely challenging since
the theoretical framework required to describe them can be very different in different
regimes based on the strength of interaction, temperature scale of interest and densityof the atomic gas. For example, in a dilute Bose gas with weak repulsive interactions
at T = 0 (BEC), a description in terms of mean fields given by the Gross-Pitaevskii(GP) equation is sufficient to account for all the observable properties. However for
finite temperatures 0 < T < Tc , a quantum kinetic description that includes the effectof binary collisions may be necessary. In another situation, a dilute Bose gas at T = 0
can be configured to be strongly correlated making the mean field description invalid.
In this case, as we will see in the later part of this thesis, an effective field theory picture
in terms of composite particles is convenient to completely account for the correlations
and predict the physical observables.
Given the richness of this area of research, in this thesis we theoretically investigate some particular aspects of the trapped Bose gas system in two different regimesthe weak interaction regime and the strong interaction regime.
(1) Weak interaction regime:
In this regime we focus on the behavior and working of a continuous-wave (cw) atom
laser. An atom laser is essentially a device that produces an intense coherent beam ofatoms. Since a BEC is a coherent superposition of atoms in the ground state, one could

3
simply think of an atom laser as a beam of Bose condensed atoms. The working of an
atom laser can be described in complete analogy to the optical laser. As in the case ofan optical laser, if a threshold condition is satisfied, the atom laser will operate far fromequilibrium in a dynamical steady state. In this dynamic state, the macroscopicallyoccupied quantum state is continuously depleted by loss to the output field, and continuously replenished by pumping from an active medium. The threshold condition for an
atom laser is analogous to the critical point associated with the phase transition froma normal gas to a BEC. A version of a “pulsed” atom laser was demonstrated [27, 28]
soon after the first experimental realization of BEC in dilute alkali gases. Therefore
one of the objectives of this thesis is to model a possible pumping scheme that would
continuously replenish the reservoir allowing the BEC to be generated in a dynamical
steady state and hence a cw- atom laser.
In order to study the coherence and phase diffusion properties of any such device,
it is very important to understand the collisional regime of the dilute trapped Bose gas.
While binary collisions is the necessary mechanism for the evaporative cooling techniqueemployed in the realization of a BEC, it is also responsible for the loss of coherence ofthe condensed component due to collisions with the non-condensed atoms. In order tobetter understand the effect of collisions, we discuss the dilute Bose gas in the quantum
kinetic regime (T > Tc , absence of a condensed component). Here, we derive a nonMarkovian generalization to the quantum kinetic theory described by Walser et al. [13]for temperatures above the BEC temperature. We show that within this framework,
finite duration effects like quasiparticle damping arise naturally and lead to a systematic
Markov approximation from a non-Markovian Born theory. Using numerical simulation
we demonstrate the emergence of different time scales for correlation and relaxation as
well as predict damping rates and frequencies of collective modes [14]. This formulation
can be extended to the temperature regime, 0 < T < Tc , by including the condensed
component. Such a theory will be an important step towards a complete understanding

4of the cw- atom laser.

Figure 1.1: The total angular momentum of the stable ground state plotted as a functionof the rotational frequency for N = 6 atoms with two-body interaction potential in unitsof harmonic oscillator quanta given by V = (1/N )δ(ri − rj ). (Figure obtained from N.
K. Wilkins and J. M. F. Gunn, Phys. Rev. Lett. 84, 6 (2000))(2) Strong interaction regime:
This regime is defined by the situation where the interaction energy is large compared
to any other single particle energy scale of the system. A dilute Bose gas can be configured in this regime in various ways. One possibility is to tune the interactions by using
Feshbach resonance techniques. Another possibility is to introduce degeneracy in thesystem, for example by using an optical lattice trapping potential as was used in theexperiment involving a superfluid to a Mott transition [89]. In this thesis we consider
a two dimensional trapped Bose gas but instead introduce degeneracy by rotating the
trap. At a certain critical rotational frequency the system is predicted to enter thestrongly correlated regime where the many-body ground state is given by the bosonicvariant of the Laughlin wavefunction. This is shown in Fig. 1.1 where the total angularmomentum of the stable ground state is plotted as a function of the rotational frequency. The Laughlin state becomes stable only when the rotational frequency is close
to the trapping frequency. The Laughlin ground state is precisely the state responsible

5for the observation of the fractional quantum Hall effect in two dimensional electron
systems. Therefore in this thesis we will refer to this regime of the dilute Bose gas as
the fractional quantum Hall regime. Using the tunability of interaction made available
by Feshbach resonances, which as mentioned earlier is almost impossible to achieve inan electronic fractional quantum Hall system, we study this regime of the dilute Bosegas in the presence of Feshbach interaction. The importance of employing such resonantinteractions in the strongly correlated regime will be apparent from the possibility ofcreating novel states as well as the opening of new possibilities for answering questions
related to the many-body physics of fractional quantum Hall effect.
Overview:
The general flow of topics covered in this thesis is shown in the figure below. In Ch. 2,
we discuss a model based on an optical pumping scheme to continuously load the reservoir of a continuous wave atom laser. Here we show that despite loss mechanisms, it ispossible to generate BEC in a steady state that is crucial to the working of a continuous
wave atom laser. In Ch. 3, we develop a non-Markovian quantum kinetic theory andthereby derive a systematic Markov approximation that includes finite duration effects
such as quasiparticle damping. The remainder of the thesis is devoted to the strongly
correlated regime of the dilute Bose gas. In Ch. 4, we introduce the electronic fractional
quantum Hall effect and show how an analogous many-body state can be generated in atrapped dilute Bose gas system. In Ch. 5, we develop a field theory picture required todescribe this regime. Within this picture the strongly correlated regime can be thoughtof as a Bose condensed phase of composite bosons. The idea of composite particles andits connection to the existence of fractional statistics is also discussed. The importanceof such a description will be apparent as it will allow us to theoretically study thisregime in the presence of Feshbach interactions. The implications of introducing resonant interactions in this regime are discussed in Ch. 6. The main result of this chapter
is the prediction of a novel strongly correlated state unique to the trapped Bose system.

6

Figure 1.2: Flow chart of the chapters in this thesis. Figure also shows the opening of
new possibilities for further study as a result of this work.

Chapter 2

Loading a conitnuous wave atom laser by optical pumping techniques
Published in Physical Review A 62, 043604 (2000)

2.1

Introduction
An atom laser is a device which generates an atomic beam which is both intenseand coherent [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. It is typically defined by direct
analogy with the optical laser. As in the case of an optical laser, if a threshold condition is satisfied the atom laser will operate far from equilibrium in a dynamical steadystate [26]. In this regime, the macroscopically occupied quantum state is continuously
depleted by loss to the output field, and continuously replenished by pumping from anactive medium. The threshold condition for the atom laser is analogous to the criticalpoint associated with the phase transition from a normal gas to a Bose-Einstein condensate (BEC). A version of the “pulsed” atom laser [27, 28] was demonstrated soonafter the first experimental realizations of BEC in dilute alkali gases [1, 2, 3, 29].
The three basic components of an atom laser are: a cavity (or resonator), an
active medium (or reservoir), and an output coupler. The cavity is typically a threedimensional magnetic trap which for containment uses the force on the atomic magnetic
dipole via an inhomogeneous potential. The active medium may be composed of a saturated dilute thermal gas formed through evaporative cooling or other techniques. Saturated in this context implies that the energy distribution function of the atoms is suchthat thermodynamic relaxation of the gas through binary collisions would result in the8stimulated accumulation of atoms into the ground state. The output coupler can be implemented by either applying a short radio-frequency (RF) pulse to the condensate andthereby separating the condensate into a trapped component and an un-trapped component [27, 28, 30, 31], or alternatively by creating a situation in which the atoms tunnel
out through a potential barrier [32]. A recent demonstration of a quasi-continuous atom
laser used an optical Raman pulse to drive transitions between trapped and un-trappedmagnetic sublevels giving the output-coupled BEC fraction a non-zero momentum [33].
To date, no continuous wave (CW) atom laser has been demonstrated in which
there is replenishing of the reservoir in direct analogy with CW-optical laser. Replenishing the reservoir is required in steady state to compensate for the loss due to output
coupling as well as to compensate for various intrinsic loss mechanisms such as collisions with hot atoms from the background vapor and inelastic two-body and three-bodycollisions between trapped atoms [16, 25, 34].
In this chapter we propose a scheme for loading the thermal cloud into a magnetic
trap by optically pumping atoms from an external cold atomic beam source. Opticalpumping is necessary since the magnetic potential is conservative and therefore a dissipative process is required for confinement of the atoms to be achieved. When combinedwith evaporative cooling, our approach may allow both steady state BEC and the CW
atom laser to be demonstrated. The crucial point is that while one is trying to obtaina steady state BEC by injecting atoms into the magnetic trap, the loading mechanism
itself should not heat the system or destroy the delicate condensate component.
There have been related proposals to ours using single atom “dark states” forreaching low temperatures by purely optical means—even well below the photon recoil
limit [35, 36, 37]. As shown by Castin et al. in Ref. [38], the main hurdle in achieving
BEC in all these schemes is that of photon reabsorption. Each reabsorption can remove
one atom from the dark state and increment the energy of the cloud on average by
one recoil energy. We engineer the analogous effect in our scheme to be negligible by

9

e
Γe

Ω
Γt

t
g

Figure 2.1: Level scheme illustrating: the ground state |gi, excited state |ei, and trap
state |ti. A laser is used to couple states |gi and |ei with an intensity characterized by
the Rabi frequency Ω. Spontaneous emission of photons occurs with rate Γe from |ei to|ti and rate Γt from |ti to |gi.

configuring the frequency of the emitted photons and adjusting the spatial geometry of
the system to reduce the probability of reabsorption by the condensate.

2.2

The Model
We consider atoms whose internal degrees of freedom are described by a threelevel atomic system as shown in Fig. 2.1. Our description of the loading scheme ismotivated by the experimental setup in Ref. [39]. There the atoms were loaded froma background vapor and precooled in a magneto-optical trap (MOT). They were thenlaunched into the second MOT, optically pumped to the desired internal atomic hyperfine state, and transfered to a magnetically confining potential in which evaporative
cooling was implemented. Here we study the possibility for loading the thermal cloud
into the magnetic trap directly and continuously. This would allow the evaporative
cooling to be carried out in steady state, which is the situation considered in Ref. [26].
Our scheme is illustrated in Fig. 2.2. Atoms are injected into the trap in state |gi.
As the atoms enter the spatial region containing the thermal cloud they pass througha laser field and are coherently pumped to the excited state |ei from which they may
spontaneously emit a photon and end up in the trap state |ti.
As will be shown later the spontaneously emitted photons may be out of resonancewith the bare |ti → |ei transition and therefore reabsorption by atoms in the thermal
cloud is greatly reduced (the exception to this general statement is the negligible fraction

10

hν

Precooledatomic beam

Laser
Condensate
Ground state g
Excited state etTrap state

Thermal cloud

Figure 2.2: Schematic diagram showing the proposed loading scheme for obtaining a
steady state BEC and CW atom laser. Atoms entering the laser field are in the groundstate. As the atoms pass through the laser field, the population is continuously transferred to the trap state by a sequence of coherent pumping and spontaneous emission.

of thermal atoms in the localized spatial region of the focussed laser beam.) Using thisapproach atoms may be continuously loaded into the thermal cloud and be confinedby the trapping potential without prohibitive reabsorption of the spontaneous photons
emitted in the required dissipative process.
We use a master equation [40, 41] to study the evolution of this system. We define
the spontaneous emission rates between the excited state and trap state and between
the trap state and ground state as Γe and Γt respectively, and for simplicity neglect
spontaneous emission directly from |ei to |gi. Assuming that the laser is on resonancewith the atomic transition frequency (detuning δ = 0), the coherent interaction can bemodeled by the Hamiltonian in the interaction picture
H=

Ω
ge
ge
h̄(σ+
+ σ−
)
2

(2.1)

ge ge
where σ+, σ− are the raising and lowering operators respectively between the groundstate |gi and the excited state |ei. The master equation for the reduced density op-

11erator of the atom includes terms due to both the spontaneous emission and a part
corresponding to the coherent driving source. In the interaction picture this takes the
form
∂ρ
∂t

Ω ge
Γe
ge
et
et
et et
et et
= −i [σ++ σ−, ρ] + (2σ−ρσ+− σ+σ− ρ − ρσ+
σ− )
2
2
Γt
gt
gt
gt gt
gt gt
+ (2σ−ρσ+− σ+
σ− ρ − ρσ+
σ− ),
2

(2.2)

et , σ et are the raising and lowering operators respectively between the excitedwhere σ+−gt gtand trap state and σ+, σ− are the raising and lowering operators respectively between

the trap state and ground state. From the above master equation we obtain the coupled differential equations for the evolution of matrix elements of ρ with the constraint
Tr[ρ] = 1. The atomic population in the three levels given by the matrix elements
ρgg (t), ρee (t) and ρtt (t) are obtained for values of Γe = 1 and Ω = 5 and initial condition given by ρgg (0) = 1, ρee (0) = ρtt (0) = 0. We consider two different values for Γt ;
Γt = 2 and Γt = 0, which give qualitatively distinct dynamics. The zero value for Γtcorresponds to optically pumping the atom from the ground to trap state, and leads tocomplete transfer of population at long times. The non-zero value of Γt gives a steadystate solution for the level populations in which the atom cycles indefinitely, which will
be useful later for the calculation of the spectrum of the spontaneously emitted photons.
The steady state spectrum of spontaneously emitted photons on the |ei → |titransition is given by the Fourier transform of the two-time correlation functionetetK(t + τ, t) = hσ+(t + τ )σ−
(t)i,

(2.3)

the differential equation for which is obtained by combining the equation for ρtg and ρteabove along with the quantum regression theorem (QRT) [41].
The QRT states that for a set of operators Yi , which evolve in steady-state according toX∂
Gij hYj (τ )i,
hYi (τ )i =
∂τ
j

(2.4)

12

Population1Γe=1, Γt=2

0.5

0
0

5

10

15

20

15

20

Population1Γ =1, Γ =0
e

0.5

0
0

t

5

10

time (in units of 1/Γ )eFigure 2.3: Populations as a function of time; ground state |gi shown by a solid line,
excited state |ei shown by dashed and trap state |ti shown by dot-dashed line. The
parameters used were Ω = 5, Γe = 1 and (a) Γt = 2, (b) Γt = 0.

then two-time averages can be related to the one-time expectation values byX∂
Gij hYj (t + τ )Yl (t)i,
hYi (t + τ )Yl (t)i =
∂τ
j

(2.5)

where the Gij ’s are the same coefficients in both Eq. (2.4) and Eq. (2.5). For a general
non-steady-state problem in which transient dynamics must be considered, applicationof the QRT is typically more complicated. For simplicity our approach is to take afictitious non-zero value of Γt , and then consider the limit Γt → 0. The spectrum
defined asS(ω) = lim

Z tt→∞ −tdτ K(τ )e−iωτ(2.6)

is then given byS(ω) =

ρss
tt
πΩeff

where Ωeff =

 



p

i Γ4e + Ω2eff

Γt
Γe
4 + 2

2

Ω2 − Γ2e /4





Γe
Γt
4 + 2

+ ω + Ω2eff 2  +

ρss
tt
πΩeff



Ωeff
Γe
 −i 4 + 2



Γt
Γe
4 + 2

2




Γe
Γt
4 + 2

+ ω − Ω2eff
2  (2.7)

13
1.4
1.2
1

Ω/Γ =5

 S(ω)

2

e

0.8
0.6
0.4
0.2
0
−5

−4

−3

−2

−1

0

ω/Γ

1

2

3

4

5

e

Figure 2.4: Comparison of the |ei → |ti transition spectrum using the steady statecalculation (solid) and the quantum trajectory method (dash-dot).

In Fig. 2.4 we plot |S(ω)|2 shown by the solid curve for the limiting case of Γt → 0.
The spectrum is AC-Stark split into two Lorentzians separated by Ωeff . This spectrum
is commonly referred to as the Autler-Townes doublet [43, 44].
For comparison we now use the method described in Ref. [42] to provide an alternative calculation of the fluorescence spectrum, extending our treatment to include thepossibility for the time dependence of Ω. This involves the decomposition into “quantum
trajectories”, where the evolution is conditional on a certain record of dissipative events.
In the case of the evolution conditional on the spontaneous emission of no photons, thequantum trajectory |ψ(t)i, evolves according to∂|ψ(t)i1= Heff |ψ(t)i,
∂t
ih̄

(2.8)

where Heff is the non-Hermitian Hamiltonian defined by
Heff = H − ih̄Γe et et
σ σ .
2 + −

(2.9)

For the evolution conditional on the spontaneous emission of one photon, each quantum

14
trajectory |ψωj (t)i is labelled by the frequency of the photon ωj . If we consider a timeinterval t ∈ [0, τ ], then according to the Fourier sampling theorem a complete description
requires choosing each ωj from a discrete but infinite set of frequencies spaced (2π/τ )apart. The observed spectrum may then be defined asS(ω) = hψω (τ )|ψω (τ )i(2.10)1Γe etσ− |ψ(t)i + (Heff + h̄ωj )|ψωj (t)i.
τih̄

(2.11)

with (see Ref. [42])d|ψωj (t)i =dts

By solving the coupled Eqns. (2.8) and (2.11) we get|ψωj (t)i =

where









0
0
fωj (t)










(2.12)−1fωj (t) = qMN −1 (iωj − Γe /4)2 + Ω2eff /4Ωeff t −Γe t/4
M = sine(iωj − Γe /4)2Ωeff t −Γe t/4 Ωeff −iωj t
Ωeff
cos
e
e
+
−
2
2
2


q =

4
Ω

s



Γe
λ+ λ−
τ λ+ − λ −(2.13)The single-photon spectrum which is so obtained, |S(ω)|2 , is shown by dot-dashed curvein Fig. 2.4. The normalization constant N is introduced so that |fωj |2 → 1 as t → ∞.
We see a close agreement between the spectra calculated by the two methods, thequantum regression theorem and the quantum trajectory approach of solving conditional
dynamics. The crucial observation is the absence of a significant contribution in thezero-frequency region of the fluorescence spectrum where there would be strong photon
reabsorption by the thermal cloud. Virtually all of the spontaneous photons are off
resonance with the bare |ti → |ei transition which is the resonant frequency for all
thermal atoms out of the laser field focus.

15

2.3

Effect of time dependent Ω and finite detuning on the transient
spectrum
The assumed time independent nature of the interaction picture Hamiltonian (2.1)

may not be a realistic situation. In an actual experiment, as the atoms pass throughthe laser beam they would see a continuously changing intensity and hence a changingΩ. At the same time, existence of finite detuning δ (assumed previously to be zero forgege
simplicity), may result in the time dependence of the operators σ+
and σ−
given byge
ge ±iδtσ±(t) = σ±e(2.14)

Thus, in a more realistic situation the interaction picture Hamiltonian of Eq.(2.1) should
be replaced by the time dependent form
H(t) =Ω(t)ge iδtge −iδth̄(σ+e + σ−e)2(2.15)

Using this interaction Hamiltonian we now recalculate the AC-Stark split spectrum. Firstly we consider the δ = 0 case with the time dependence solely due to Ω(t).
Since the problem is a transient one in which population is completely transferred tothe trap state over a certain time scale, the variation of Ω over this time scale is crucial.
Also, from the calculations in the previous section it is clear that a large value of Ω isfavorable for our scheme. We therefore consider various time scales over which Ω reaches
its maximum value compared to the optical pumping time as shown in Fig. 2.5. The
corresponding spectrum shown in Fig. 2.6 is calculated using the method of conditional
dynamics discussed in the previous section. The calculation shows that the crucial requirement for our model i.e. that the emitted photons cannot be resonantly absorbedby the thermal cloud outside the laser focus,is only satisfied if the atom sees a field
increasing in intensity on a time scale which is short compared to the time it takes to
optically pump the population from |gi to |ti. For a particular average velocity of theatoms in the atom beam source, this gives the requirement for the spatial variation of

16
6
5

Ω

4
3
2
1
0
0

5

10

15

20

time( in units of 1/Γe)

Figure 2.5: Ω as a function of time, t. The three different curves correspond to different
time scales over which Ω reaches its maximum value.
3.5

1

3

0.8
0.6

ρtt

2.5

 S(ω)

2

0.4

2

0.2

1.5

0
0

10

20

time(1/Γ )
e

1
0.5
0
−5

−4

−3

−2

−1

0

ω/Γ

1

2

3

4

5

e

Figure 2.6: Transient spectrum for |ei → |ti transition for the different scenarios forΩ(t) shown in Fig.2.5.

the laser intensity associated with the beam waist. Alternatively, the laser field could
be pulsed on and off to achieve a sufficiently rapid rise time.

17
3

1
0.8

2.5
ρ

tt

0.6
0.4

 S(ω)

2

2

0.2

1.5

0
0

10

20

time(1/Γe)

1
0.5
0
−5

−4

−3

−2

−1

0

ω/Γ

1

2

3

4

5

e

Figure 2.7: Figure shows the effect of detuning on the transient spectrum for |ei → |titransition. Dashed for δ = 0, solid for δ = 1 and dot-dashed for δ = 2.

Now we consider the effect of finite detuning, δ, in addition to the time dependence of Ω. For the case of constant Ω, a finite δ has the effect of transforming Ω to√
Ω2 + δ2 [44]. Therefore, even if Ω depends on time, it is intuitive to expect a similareffect resulting in an increased AC-Stark splitting. This is illustrated in Fig. 2.7 where
the spectrum is calculated for the case of Ω(t) shown by dot-dashed curve of Fig. 2.5
for various values of δ. Thus a large detuning may seem to be favorable for our purpose
since the spontaneous photons are emitted further away from resonance. On the other
hand, large detuning means slower optical pumping rate and hence a wider laser focus
will be required for the same Ω(t) as illustrated in the inset of Fig. 2.7. Therefore, it
is important to achieve an optimum set of parameters that would allow a maximum
possible splitting depending on the average atomic speed and line width of the atomic
levels under consideration.

18

2.4

Conclusion
The AC-Stark splitting of the spectrum indicates that this scheme may potentiallybe used to provide a method for continuous loading of the thermal cloud in an atom
laser. In the case of constant Ω the spectrum of the spontaneously emitted photons
consists of two peaks separated by Ωeff . Even for the realistic case of time varying Ω,large splitting between the two peaks can be achieved provided that the rise time of
the laser field intensity is sufficiently small. The splitting can be further enhanced byintroducing a finite detuning.
Large splitting of the spectrum is essential to avoid heating due to inelastic multiple scattering of photons. Usually heating in an optical pumping scheme is causedby photon reabsorption and re-emission. Each reabsorption can remove one atom fromthe tr ap state and increment the energy of the cloud on average by one recoil energy.
This effect is significantly reduced in our model since the spontaneously emitted photons are off-resonance with the |ti → |ei transition and cannot be reabsorbed by atoms
outside the laser focus. Therefore the heating of the thermal component due to photon
reabsorption may be suppressed. One can also reduce the cross section for photon reabsorption by designing the shape of the trap, for example a cigar shape or disc shape,such that there is a limited solid angle for photon reabsorption. Another possibility
is to align the optical pumping laser and the geometry of the trap to use the intrinsic
dipole radiation pattern to reduce spontaneous emission into unfavorable directions.
Our model requires that the atoms be in the trap state before they are out of thelaser field, which may place a limit on the maximum speed of atoms in the atomic beamsource. However, the faster the incoming atoms, the smaller the rise time of Ω which aspreviously discussed is required to avoid the emission of photons at zero-frequency. The
optimum speed of incoming atoms is therefore a balance between these two requirements.
Since our goal is to obtain BEC in a steady state and not just BEC, one possibility to19obtain sufficiently slow speeds is to use a condensate prepared in another trap as asource instead of just laser precooled atoms formed from a magneto-optical-trap.
For a typical alkali atom, the Zeeman level diagram is composed of many hyperfine
levels. Even though our three level system is a simplification of a real situation, mostof the omitted processes, such as imperfect optical pumping to the trapping state, willlead to additional loss of atoms and modify the effective incident flux. This will impact
the possibility for achieving a sufficiently high pumping rate of cold atoms to allow theCW atom laser and steady state BEC to be achieved, as discussed in Ref. [26].

Chapter 3

Memory effects and conservation laws in the quantum kinetic
evolution of a dilute Bose gas
Published in Physical Review A 66, 043618 (2002)

3.1

Introduction
In the mean-field approximation, a Bose-condensed phase is well described by the

Gross-Pitaevskii (GP) equation [45]. Examples of collective phenomena that arise at themean-field level include the formation of vortex states [46, 47, 48, 49, 50] and collective
excitations [51, 52, 53]. In this context, the GP equation is often sufficient to describe
the dynamics at T = 0. However, at finite temperatures, 0 < T < Tc , it is important
to include effects due to the presence of a thermal component. These thermal, noncondensed atoms interact mutually and with the condensed atoms via binary collisions.
In fact, the collisional dynamics is the microscopic mechanism for evaporative coolingand it leads to BEC. Finite temperature effects are also responsible for phenomena suchas phase diffusion and damping of collective excitations. In order to describe theseeffects, the inclusion of collisions due to the thermal component becomes essential. A
generalized GP equation, for example the Hartree-Fock-Bogoliubov approach, includes
the effects of collisions only indirectly through their energy shifts and hence is valid onlyat very low temperature (collisionless regime).
In the other limit, T > Tc , the condensate component is absent and the thermal
component completely determines the dynamics. This limit is well described by the21quantum-Boltzmann (QB) equation. Thus, a non-equilibrium kinetic theory, whichinterpolates between the GP and the QB limits corresponding to T = 0 and T > Tc ,respectively, is necessary to provide a such a complete description.
Currently, there exist a number of kinetic theories including those based on quantum stochastics [54, 55], the Fokker-Planck equation [56], generalized single-time masterequations [57, 58, 59], a semiclassical hydrodynamic approach [60, 61] and a Green’s
function approach [62, 63, 64]. However, obtaining explicit solutions for the nonequilibrium dynamics has remained a challenge. In particular, the condensate growth dynamics
is still a very active area of research [65, 66]. More recently, Monte Carlo simulations
have provided an alternative approach to the solution of non-equilibrium master equations [67, 68, 69, 70]. In particular, Jackson and Zaremba [70] have shown good agreement with experimental observations of decay rates and frequencies. However Monte
Carlo approaches suffer from the simulation noise which can lead to a spread in thevalue of quantities that should be exactly conserved. Also the theory as formulated can
only be applied to situations in which the anomalous pair correlations are not important(Popov approximation). Under certain conditions, typically when the interactions arestrong, anomalous densities can play a significant role. This was the case for example in
the recent observation of Ramsey type oscillations [71] and their subsequent theoreticalexplanation [72].
Energy conservation in theories with Boltzmann type collision integrals is typically enforced by requiring exact energy conservation in each individual collision event.
From a fundamental perspective this is unsatisfactory as it would be preferable if energy
conservation would arise intrinsically from the theoretical formulation. The essential
point is that such an approach would allow off the energy shell collision events so longas energy is conserved overall. In practice this requires including effects arising fromthe finite duration of a collision and quasiparticle damping.
Another issue for consideration is the question of Markovian vs. non-Markovian

22
dynamics in a inhomogeneous system. Even though, for a dilute Bose gas, memory
effects can be neglected under the principle of rapid attenuation of correlations, nonMarkovian behavior is intricately related to the conservation properties [73, 74, 75, 76,77]. For example, the Markovian-Boltzmann type kinetic equations derived by Walser
et. al. [13] conserves energy only to first order [78]. Previous attempts to address
this issue have been limited to the discussion of systems with translational symmetry[79, 80].
In this chapter, we generalize the kinetic theory derived in [13] in order to address
the problems discussed above. Presently, we limit our goal to the derivation of a systematic Markov approximation from the non-Markovian Born theory by including finite
duration effects and quasiparticle damping. Using the short and long time behaviorof the two time Green’s function, we obtain a damping function that gives improved
energy conservation to second order in the perturbation parameter.
We apply this theory to a simple model of an inhomogeneous Bose gas confinedin a spherical box above Tc with discrete basis states. The usual Born-Markov theory
gives a δ-function energy conservation which for any interacting system will result in
only exchange collisions. Therefore introducing finite duration effects facilitates thecalculation by relaxing the energy conservation condition. The simplicity of this modelallows us to study the real time evolution of the system from some initial state to a final
state of equilibrium. We perform a linear response calculation to study the stability anddamping rates for the steady-state solution. A calculation on similar lines with a more
realistic model of a harmonic trapping potential and non-vanishing condensate (α) andpair-correlation (m̃) components will ultimately make comparison with experimental
observations of the frequency and damping rates of collective excitations possible.
This chapter is organized as follows. In Sec. 3.2 we derive a non-Markovian
generalization to the kinetic theory of Ref. [13] using the prescription of a nonequilibrium
statistical operator method [81]. Both quasiparticle damping and damping arising from23
the finite time of collision events are discussed in Sec. 3.3. This has implications forthe underlying symmetries of the theory and their associated conserved quantities. In
Sec. 3.4, we apply this generalized kinetic theory to a simple model of an inhomogeneous
dilute Bose gas confined in a spherical box and obtain a self consistent steady-statesolution to the second-order kinetic theory. Finally in Sec. 3.5, we study the responseof the system to a small perturbation. This allows us to determine the frequencies anddamping rates of collective modes.

3.2

Kinetic equations
We start with the many-body Hamiltonian for a weakly interacting Bose gas givenbyĤ = Ĥ (0) + Ĥ (1) ,(3.1)

where Ĥ (0) is the single particle Hamiltonian that is defined asĤ (0) = ε12 â†1 â2(3.2)

using the implicit summation convention for repeated indices. The two-body energyĤ (1) is given byĤ (1) = φ1234 â†1 â†2 â3 â4 .

(3.3)

The bosonic operators â1 and â†1 annihilate and create a particle in a single particle state|1i, respectively. The abbreviated notation |1i represents a state specified by a complete
set of quantum numbers for both the motional and electronic degrees of freedom.
We assume that the particles are confined by an external trapping potential Vext .
Thus, the matrix elements of the single particle Hamiltonian are given by
ε12 = h1|p̂2
+ Vext (x̂)|2i,
2m

(3.4)

where m is the mass. The binary interaction is mediated by a short range repulsive

24
potential Vbin . This gives the symmetrized (S) matrix elements
φ1234 = (S)h1| ⊗ h2|Vbin (x̂ ⊗ I − I ⊗ x̂)|3i ⊗ |4i.

(3.5)

In the low energy limit Vbin can be approximated by a contact potential with the matrix
elements given by
1234

φ

≈

V0
2

Z ∞

−∞

h1|xih2|xihx|3ihx|4i d3 x.

(3.6)

The interaction strength V0 , is related to the scattering length as by V0 = 4πh̄2 as /m.
Here, we use the well-known nonequilibrium statistical operator method [81, 82] toobtain an explicitly non-Markovian version of the kinetic theory [13]. In this approach,the nonequilibrium state of a weakly interacting quantum gas is specified by a set of
single time master variables. For our system the most important master variable is the
single particle density matrix f (t),nof12 (t) = hâ†2 â1 i = Tr â†2 â1 σ(t) ,(3.7)

where h. . .i = Tr{. . . σ(t)} and σ(t) is the statistical many-body density operator. We
focus our studies on the temperature regime above and in close proximity to the criticaltemperature for BEC. For this reason, we do not consider either symmetry breaking
fields, hâ1 i, or the anomalous fluctuations, hâ1 â2 i. We therefore define {γ̂0 = I, γ̂k =â†2 â1 |k ∈ {(1, 2)}} as our complete set of relevant operators. The expectation values ofthese operators, γk (t) = hγ̂k i, are the only quantities that will appear in the final kinetic
equations.
The time evolution of the nonequilibrium statistical operator σ(t) is described by
the Liouville equation with an extra source term on the right hand side:id
σ(t) + [Ĥ, σ(t)] = −η σ(t) − σ (0) (t) .
dt
h̄

(3.8)

Such a source term breaks the time reversal symmetry of the Liouville equation andrepresents a convenient way to incorporate the irreversible character of macroscopic

25
processes. We will see later that this procedure leads to finite duration of collisioneffects and quasiparticle damping. The relevant distribution σ (0) (t) given by
n

(0)

o

σ (0) (t) = σ{γ(t)} = exp γ̂k Υk (t) ,(3.9)

where Υk (t) are the Lagrange multipliers, represents a special solution that maximizes
the information entropy S ′ = −Tr{σ ′ log(σ ′ )} for the given averages γk (t). Furthermore,at some initial instance t = t0 in the remote past, we can assume that σ(t0 ) corresponds
to its noninteracting value and therefore
σ(t0 ) = σ (0) (t0 ).

(3.10)The Lagrange multipliers Υk (t) are calculated from the self-consistency condition
γk (t) = Tr{γ̂k σ (0) (t)} = Tr{γ̂k σ(t)}.

(3.11)

This essentially enforces the Chapman-Enskog condition [83] for the restricted set of
relevant operators at all times.
From the Liouville equation (3.8) one can easily establish the basic equations of
motion for the average values,
o
o
d
i n
i n
γk (t) = Tr [Ĥ (0) , γ̂k ] σ(t) + Tr [Ĥ (1) , γ̂k ] σ(t) .
dt
h̄
h̄

(3.12)

The form of Ĥ (0) enables us to express the first trace on the right hand side of the above
equation in terms of the averages γk (t). The second trace plays the role of the “collision”
term, the evaluation of which requires us to seek an integral solution of the Liouvilleequation. But, before we proceed, it is instructive to repartition the total Hamiltonian
Eq. (3.1) into single particle and two-particle contributions,ˆ (0) (t) + H̄ˆ (1) (t)Ĥ = H̄
=

h

i

h

i

Ĥ (0) + Q̂(t) + Ĥ (1) − Q̂(t) .

(3.13)

26
This modification anticipates self-energy shifts
Q̂(t) = Q12 (t)â†1 â2 ,(3.14)

which will inevitably arise in the course of the calculation.
An integral solution for σ(t) can then be obtained easily from the Liouville equation (3.8), by using the single particle time evolution operator Û (0) (t, t1 ), the boundary
condition Eq. (3.9), and an additional partial integration. Thus, one finds
σ(t) = σ (0) (t) −+Z t

dt1 e−η(t−t1 ) Û (0) (t, t1 ) ×t0i ˆ (1)[H̄ (t1 ), σ(t1 )] Û (0)† (t, t1 ).
h̄




d (0)
i ˆ (0)σ (t1 ) + [H̄(t1 ), σ (0) (t1 )]dt1
h̄
(3.15)

ˆ (0) (t) depends on time through the Q̂(t), the time evolution
Since the Hamiltonian H̄
operator Û (0) is in general a time-ordered exponent
Û

(0)

i
(t, t0 ) = T̂ exp −
h̄


Z t
t0



ˆ (0) (t ) .
dt1 H̄
1(3.16)

To establish the time derivative dσ (0) (t1 )/dt1 in Eq. (3.15), we recall from Eq. (3.9)
that the relevant operator depends only implicitly on time through the averages γk (t1 ).
Moreover by exploiting the transformation properties of a quantum Gaussian (see Appendix A), one finds thatd (0)σ (t) =dtdγk (t) ∂γk σ (0) (t)
dt


(1)
i
i ˆ (0)(0)ˆ= − [H̄ (t), σ (t)] + Tr [H̄ (t), γ̂k ] σ(t) ∂γk σ (0) (t). (3.17)h̄h̄Using these relations we eliminate the time derivative of σ (0) (t1 ) in Eq. (3.15), to obtain
the integral form of the statistical operator
σ(t) = σ (0) (t) −
ˆ
+ [H̄

(1)

i
h̄

Z t
t0





ˆ
dt1 e−η(t−t1 ) Û (0) (t, t1 ) × Tr [H̄(t1 ), σ(t1 )] Û (0)† (t, t1 ).

(1)(t1 ), γ̂k ] σ(t1 ) ∂γk σ (0) (t1 )(3.18)

27
Since we are only interested in a weakly interacting gas, we seek a power series expansion
in the interaction strength
σ(t) = σ (0) (t) + σ (1) (t) + ...
σ (1) (t) = −

i
h̄

ˆ
+ [H̄

Z t

(1)

t0

(3.19)(1)ˆdt1 e−η(t−t1 ) Û (0) (t, t1 ) × Tr [H̄(t1 ), γ̂k ] σ (0) (t1 ) ∂γk σ (0) (t1 )(t1 ), σ (0) (t1 )] Û (0)† (t, t1 ).

(3.20)

With this explicit expression for the statistical operator, the evaluation of the equationof motion (3.12) is straight forward and one obtains the quantum-Boltzmann equationdf (t) = L[f ] + L[f ]† .
dt

(3.21)

Here, Wick’s theorem has been used to express the higher order averages in terms of
the single particle ones. The kinetic operator L consists of a reversible Hartree-Fock(HF) part LHF and a collisional quantum-Boltzmann contribution LQB ,L[f ] = LHF [f ] + LQB [f ],i
LHF [f ] = − HHF (t)f (t),h̄
LQB [f ] = Γf f (1+f )(1+f ) − Γ(1+f )(1+f )f f .
HHF = ε + 2 Uf is the Hartree-Fock Hamiltonian, with Uf14 = 2 φ1234 f32 the self energy,
while Γ’s are the collision integrals given by
Γ15
ABCD

=

1
h̄

Z t
t0



′′ ′′ ′′ ′′dt1 e−η(t−t1 ) φ1234 φ1 2 3 4

× K1′′ 1′ (t, t1 )K2′′ 2′ (t, t1 )K3†′′ 3′ (t, t1 )K4†′′ 4′ (t, t1 )

(3.22)

(3.23)

× A31′ (t1 )B42′ (t1 )C4′ 2 (t1 )D3′ 5 (t1 ) .
The propagators K(t, t0 ) are given byK(t, t0 ) = T exp −

i
h̄

Z t
t0

dt1 HHF (t1 ) .

Therefore, unlike the collision terms of Ref. [13], the Γ’s defined above depend on thepast history of the system. Thus Eq. (3.21) represents a non-Markovian generalizationof the kinetic equation previously derived in Ref. [13].

28

3.3

Conservation laws and Quasi-particle damping
The conserved quantities for a closed isolated system are the total energy E andthe total number N . These quantities then represent the constants of motion for the fullkinetic equation. The total number operator can be represented as a linear combinationof the relevant operators,N̂ = â†1 â1 ,(3.24)and therefore the functional N (f ), representing the total number is given by
N (f ) = Tr{f }.

(3.25)The kinetic equation for N (f ) can then be written asnodN (f ) = Tr L[f ] + L[f ]†
dt
n

o

n

o

= Tr LHF [f ] + LHF [f ]† + Tr LQB [f ] + LQB [f ]† .

(3.26)

The first order term on the right-hand side involves the trace of a commutator and istrivially equal to zero. The Γ’s associated with the second order terms have the following
property due to the symmetries of φ,nonTr Γf f (1+f )(1+f ) = Tr Γ(1+f )(1+f )f fo∗.

(3.27)As a result, the second order contribution in Eq. (3.26) can also be shown to be zero.
Hence the total number N is a constant of motion. The important point here is that the
total number conservation is a result of the symmetries of φ and does not dependent on
the non-Markovian nature of the collision integral. Therefore a Markov approximation
would leave this conservation law unchanged.
While the number conservation is a natural consequence of the self-consistency
condition, the total energy conservation is not obvious as the Hamiltonian Ĥ cannot
be represented as a linear combination of the relevant operators. We start with writing

29
the total energy functional E(f ) as a perturbative expansion in φ,E(f ) = Tr{Ĥ (0) σ (0) }hi

Tr{Ĥ (1) σ (0) } + Tr{Ĥ (0) σ (1) }

+

h

i

Tr{Ĥ (1) σ (1) } + Tr{Ĥ (0) σ (2) } + ...

+

io
i nh
= Tr {(ε + Uf )f } + Tr Γf f (1+f )(1+f ) − Γ(1+f )(1+f )f f + ..
2

(3.28)(3.29)

Using the self-consistency condition Eq. (3.11) for the relevant operators, one can showthat the third and the fifth trace terms in the right-hand side of the above expression
drop out. The kinetic equation for the energy functional E(f ) can then be written asnodE(f ) = Tr εf˙ + Uf f˙ + Uf˙f
dt
 h
i
∂
i
Tr
+
Γf f (1+f )(1+f ) − Γ(1+f )(1+f )f f .
2∂t(3.30)

Again, we use the symmetry properties of φ, to write the simplified equationodη nE(f ) = −i Tr Γf f (1+f )(1+f ) − Γ(1+f )(1+f )f f .
dt
2(3.31)
One can now see that the energy is conserved only in the η → 0 limit. In this limit
the kinetic equation (3.21) represents the Born approximation. A finite value of η
could then be thought of as resulting in additional terms that are beyond the Born
approximation. Such terms model the duration of collision effects and quasi-particle
damping. In principle if this effect is treated self-consistently, η will be a time dependentfunction at least of order φ. This means that the rate of change of E given by Eq. (3.31)
is of the order φ3 .
Also, if one is only interested in times grater than the correlation time τcor , thefiniteness of η allows us to extend the lower limit of the collision integral to −∞. Now
we can approximate the f (t′ ) in the non-Markovian expression of Γ by its instantaneous

30
value f (t) to obtain the Markov form
(m)15
ΓABCD

=

1
h̄

Z t

−∞



′′ ′′ ′′ ′′dt1 e−η(t−t1 ) φ1234 φ1 2 3 4

× K1′′ 1′ (t, t1 )K2′′ 2′ (t, t1 )K3†′′ 3′ (t, t1 )K4†′′ 4′ (t, t1 )
× A31′ (t)B42′ (t)C4′ 2 (t)D3′ 5 (t)(3.32)

One can verify that the above Markov form results in an energy conservation, up
to the most significant order given by

o
n
d
(m)
(m)
E(f ) = Tr HHF LQB [f ] + LQB [f ]† .
dt

(3.33)

If we compare this with the expression for the correlation energy Ecor in Refs. [73, 74],
the right hand side of Eq. (3.33) is exactly −∂Ecor /∂t. This is not surprising andcan be understood more intuitively by writing the collision integral as a sum of twocontributions: correlation and collisionΓ = Γcor + Γcol =Z 0...dt′ +−∞Z t...dt′ .

(3.34)

0

For a finite η, the Γcor contribution decays to zero as e−ηt . Therefore the decaying
correlation energy Ecor associated with this part shows up in the rate of change of thetotal energy E. In the Born approximation Γcor is constant because η → 0.
The exponential damping results in a widened delta function and therefore therate of change of E(f ) Eq. (3.33) can be shown to be of order ηΓ(m) . Thus, by including terms beyond the Born approximation, we have obtained a collision integral thatis Markovian and still conserves energy up to φ2 order. Now if we assume that theequilibration time is of the order 1/Γ(m) , the total change in energy ∆E is therefore∆E = E(f eq ) − E(f in )∼ η Γ(m) ×



1
Γ(m)



= η,

where f in and f eq are the initial and equilibrium distributions, respectively.

(3.35)

31
The importance of the damping term becomes more obvious when one attempts tosolve the kinetic equation numerically. One no longer has to worry about the η → 0 limit
in the Born-Markov approximation, which for a finite system with discrete levels canresult in the nonphysical situation of only exchange collisions and hence no equilibration.

3.4

Application to a dilute Bose gas in a spherical box trap
In the previous section, we introduced the general methods and concepts to de-scribe a weakly interacting Bose gas under nonequilibrium conditions. We will now
apply these to a simple model of a typical 87 Rb experiment, as realized by many laboratories around the world, for example Refs. [1, 51, 84]. The physical parameters areusually quoted in the natural units for a harmonic oscillator trapping potential, i. e.,
the angular frequency ω = 2π 200 Hz, the atomic mass m87 = 86.9092 amu, the groundstate size aH = [h̄/(ω m87 )]1/2 = 763 nm, and the s-wave scattering length as = 5.82 nm.
However, in the present chapter, we do not pursue the usual harmonic confinement, but rather explore the properties of a radial box as a particle trap. This choice ismotivated by previous studies of the self-consistent Hartree-Fock single-particle states[78]. As soon as repulsive mean-field potentials are added to the bare harmonic trapping
potentials, the corresponding eigenstates widen in size and look remarkably close to theeigenstates of a box, provided the spatial extensions of the box is chosen appropriately.
In particular, we pick a box of radius R = 1000 aS = 5.82 µm.
Thus, our model is represented by a spherical trap with box potential given by
Vext (r) =




 0,

r<R

(3.36)

 ∞, r ≥ R,

The eigenfunctions are a product of spherical Bessel functions j(l) and spherical harmonics Y(lm) :ψ(nlm) (r, θ, φ) =




 N


 0,

(nl) j(l) (r k(nl) ) Y(lm) (θ, φ),r<Rr ≥ R.

(3.37)

32
30

25

εHF
(ε0)
n

20

15

10

5

0
0

200

400

600

800

1000

f11

Figure 3.1: Hartree-Fock energies as a function of particles number in the box groundstate, f11 with all other fij ’s equal to zero.

A normalization constant is represented by N(nl) . The eigenenergies are given in termsof the wave-vectors k(nl) , which can be obtained from the n-th nodes of the sphericalBessel-functions of angular momentum l,ε(nl) = (R k(nl) /π)2 ε0 .

(3.38)

Here, ε0 represents the ground state energy and defines the energy scale of the problem
ε0 =h̄2 π 2.
2 mR2(3.39)All the physical parameters are scaled with respect to this energy unit ε0 and the radiusof the box R.
For simplicity, we assume all the atoms to be in the l = 0 state initially. One
therefore needs to consider only the l = 0 manifold and get for the normalization
constant N(n0) = n(π/(2R3 ))1/2 and radial wave-vector R k(n0) = nπ. If the cloud isrelatively cold, then most of the population resides in the lowest few energy states.
Therefore, we can also limit the number of radial modes 1 ≤ n ≤ nmax . For the present

33
case, we take nmax = 5. Obviously, all these simplifications reduce the number ofdegrees of freedom significantly and thus we are able to study certain aspects of thenonequilibrium dynamics of the trapped Bose gas in great detail.
With the above definitions, the bare single particle box Hamiltonian is given byεpq = δpq q 2 (noimplicitsumover q).

(3.40)The interaction part Ĥ (1) involves the matrix elements of the interaction potential
defined by Eq. (3.6)
φpqrs =

4aS
π

Z π

sin (p x) sin (q x) sin (r x) sin (s x)

0

dx
,
x2
(3.41)

which in general have to be computed numerically. Interestingly, in the case of a spherical box this integral can be evaluated analytically and simplified to a finite sum of sine
integrals and cosine functions (see Appendix B for details).
The nonequilibrium state of the above system is represented by the single particle distribution function f with time dependence given by Eq. (3.21). Neglecting thesecond order collision terms, the first order evolution is governed by the Hartree-Fock
Hamiltonian HHF and is given bydf = LHF [f ] + LHF [f ]† .
dt

(3.42)The energy eigenstates of the interacting system are therefore shifted from the bare box
states due to the self energy effect. These shifts can be significant depending on the
total particle number. This is clear from Fig. 3.1 where we plot the eigenenergies as afunction of f11 , the total particle number in the box ground state (with all other fij ’sequal to zero).
Now note that the time dependent Hartree-Fock equation (3.42) for the density
matrix, f , is nonlinear and hence we seek a self consistent solution such thatf=Xi

P (εi )|εi ihεi |,(3.43)

34
where HHF |εi i = εi |εi i, and for a Bose-Einstein distribution P (ε) is given by
P (ε) =

1.
exp((ε − µ)/kB T ) − 1(3.44)For a given total particle number, N , and temperature, β = 1/kB T , a self-consistentchemical potential, µ(β, N ), and hence a self-consistent Bose-Einstein distribution is
obtained. For example, let us consider three different total particle numbers N =10, 100, 500 at two different temperatures, β = 1/kB T = 0.01, 0.5, corresponding to hot
and cold clouds, respectively. In Figs. 3.2 and 3.3, we plot the self-consistent solution
and the self-energy density in the position space representation. We see that the selfenergy density is proportional to the number density only near the center of the trap
and drops off faster with increasing radius. This can be attributed to the restricted
number of basis states used in our calculation and effectively gives a finite range to thetwo body potential Vbin .
500

4000
3000

f(r,r)

400
2000
1000

Uf(r,r)/ε0

300

0
0

0.2

200

0.4

0.6

0.8

1

r/R

100

0
0

0.2

0.4

0.6

0.8

1

r/R

Figure 3.2: The scaled mean-field density, Uf (r, r)/ε0 , as a function of the scaled radial
distance, r/R, at a relatively hot temperature, β = 0.01 is shown for three different
values of particle numbers: N = 10 (solid curve), N = 100 (dashed), and N = 500(dot-dashed). Inset shows the corresponding number density as a function of scaled
radial distance.

35
2000
300

1000
500

200

0
0

f

U (r,r)/ε0

f(r,r)

1500

0.2

0.4

0.6

0.8

1

r/R
100

0
0

0.2

0.4

0.6

0.8

1

r/R

Figure 3.3: The scaled mean-field density, Uf (r, r)/ε0 , as a function of the scaled radial
distance, r/R, at a relatively cold temperature, β = 0.5 is shown for three different
values of particle numbers N = 10 (solid curve), N = 100 (dashed), and N = 500 (dotdashed). Inset shows the corresponding number density as a function of scaled radial
distance.

Up to first order, the f equation is totally reversible. The inclusion of the second
order terms (collisions) break the reversibility, and therefore represents a relaxation of
the system from some initial state to a final equilibrium state. Here we will be using theMarkov form (3.32) for the collision integral. We take the Hartree-Fock self-consistentstate for the initial condition.
In the previous section, we interpreted the function exp(−ητ ) to account for
duration of collision effects and quasi-particle damping. But the exponential form wasoriginally introduced to break the time reversal symmetry and it has the correct longtime behavior. However, an exponential damping will result in a Lorenzian line shapefor the final equilibrium distribution. Due to the long-reaching wings of the Lorenzian
curve, in the Markov limit, off-the-energy shell collisions get weighted strongly. To seek
an improved damping function that will have the correct short and long time behavior,

36
we use the equivalence of kinetic theories based on the Green’s function approach [63]and the non-equilibrium statistical operator method [81] as shown in Ref. [85]. The
behavior of the retarded Green’s function g(t, t1 ) for very large and very small timescales is given byg(t, t1 ) ∼ e−η(t−t1 )t − t1 ≫ τcor e−η(t−t1 )2(3.45)t − t1 ≪ τcor .

Therefore, behavior over the intermediate time scale will be best represented by aninterpolating function. This is also true for the damping function. From Fig. 3.4 we seethat the function
F(t, t1 ) = 1/ cosh(η(t − t1 )),(3.46)

has exactly this behavior and therefore represents a better choice than the exponential
form.
With either choice of the damping function, for a particular value of the parameter1

damping function

0.8

0.6

0.4

0.2

0
0

0.5

1

1.5

2

2.5

3

3.5

4

τ (h/ε0)Figure 3.4: Comparison between different damping functions, exp(−ητ ) (solid),exp(−ητ 2 ) (dot-dashed), and 1/ cosh(ητ ) (dashed). Note that the hyperbolic secant
function asymptotes to an exponential form for large τ and a Gaussian form for small
τ.

37
3

10

2

f

nn

10

1

10

0

10

0

50

100

150

200

t (h/ε )0

Figure 3.5: Evolution of the diagonal elements fnn (n = 1, 2.., 5) (shown with curves
from bottom to top) toward a self-consistent steady-state solution starting from anapproximate solution. Two different damping functions are used in the evaluation ofthe Γ’s. Dashed and solid curves correspond to exponential and hyperbolic secant
respectively.

η, a time propagation results in a self-consistent steady-state solution, f eq . Figure
3.5 shows such a time evolution for N = 500 particles with an initial temperaturecorresponding to β = 0.01. We have chose η to be of the order Re[Γ] ≈ 2.3. In principle
η should be obtained self-consistently at every time step. So long as one considers large
enough number of basis states, the final result is reasonable insensitive to the value of ηas long as η ≥ Re[Γ]. As mentioned earlier we see that the exponential damping function
results in significant transfer of population to the excited states. This effect is less withthe hyperbolic secant damping function. We get different steady-state solutions because
different damping functions correspond to different initial correlations. Also a plot ofθ(fnn ) ≡ ln(1/fnn + 1) vs the Hartree-Fock energies εHFn , shown in Fig. 3.6 shows thatthe f eq is very close to a Bose-Einstein distribution. The slope, which represents theself-consistent value of β, shows that the change in temperature is far greater for the

38

nn

θ(f )

0.4

0.2

0
0

5

10

HF

15

εn (ε0)

20

25

30

Figure 3.6: Linear behavior of θ(fnn ) as a function of εHF
n . Initial distribution shownwith diamonds. Final distribution shown with circles for the case of exponential damping
function, and squares for the case of hyperbolic secant damping function.

exponential damping function compared to the hyperbolic secant case, because off-theenergy shell effects are larger as explained previously.
Thus we have obtained a self-consistent steady-state solution to the second orderkinetic equation. We emphasize here that the steady-state solution is a result of the realtime non-equilibrium evolution of the system. Deriving such an equilibrium solution,
whose absolute value is plotted in Fig. 3.7, is a prerequisite step to the study of collectivemodes and damping rates of a dilute gas.

3.5

Real time response to perturbation
The properties of the equilibrium solution f eq exhibit the expected characteristics

of a Bose-Einstein distribution. In order to verify the stability of this solution and to
study the damping rates of the collective excitations, we will now examine the real-time
response of the system to a perturbation. First, we will outline the linear response

39

500
400

|f|

300
200
100
0
1
2
5

3

4
3

4

m

5

2
1

n

Figure 3.7: Absolute value of f eq , the self consistent steady-state solution for the second
order kinetic equation. The single particle density matrix is plotted in the Hartree-Fockbasis.

theory and discuss the structure of the modes, their frequencies, and the life-time of theexcitations. Subsequently, we will use these modes to initially prepare the system andto evolve the full nonlinear quantum kinetic equation towards equilibrium.
One of the fundamental properties of the quantum kinetic equation (3.21)
df (t) = L[f ] + L[f ]† ,dt

(3.47)

is its Hermitian structure. Thus, if we prepare a physical state initially, it will remain
Hermitian with f (t) = f (t)† , indefinitely. We will now consider a weak perturbation ofan equilibrium state,f (t) = f eq + δf (t),(3.48)and calculate the first order response of the system. As usual, we want to assume that
we can decompose a general perturbation into fundamental damped and/or oscillatory
eigenmodes of the system. Therefore, such a specific perturbation can be parameterized
as

40
30

25

Re[hω/2π]/ε0

20

15

10

5

(b)
(a)

0

(c)

(d)

−5
−1

−0.5

0

0.5

Im[hω/2π]/ε0

Figure 3.8: Non-negative frequency eigenvalues scaled with respect to ε0 shown withcrosses for the Hartree-Fock equation and squares for the quantum-Boltzmann equation.
− εHF
The dotted lines correspond to difference energies (εHF
j )/ε0 . The modes labelledi(a) and (b) – non-zero frequency and damping rate, (c) – zero mode, and (d) – zerofrequency and non-zero damping rate will be considered for further discussion.

(+)

where δfωδf (t) = e−i ωt δfω(+) + h.c.,(3.49)

δfω(+) = δfω(c) + i δfω(s) ,(3.50)

denotes a positive frequency amplitude. It turns out to be useful to de-compose it into quadrature components
δfω(c)=

δfω(s) =1†
δf (+) + δfω(+) ,
2  ω

1
†
δfω(+) − δfω(+) ,2i
(3.51)(3.52)and evaluate the kinetic operator L[f ] only for such Hermitian arguments. From a
Taylor series expansion of the kinetic operator around the equilibrium distribution, one
obtains finally the linear response equations for the fundamental modes(−iω) δfω(+) = L(1) [δfω(c) ] + i L(1) [δfω(s) ] + L(1) [δfω(c) ]† + i L(1) [δfω(s) ]† , (3.53)

41
−48.3

−48.4

f12

−48.42−48.44

−48.5
0

5

10

15

20

−48.460

5

10

t (h/ε )20t (h/ε )

0

0

0.1

|δ f|

0.1

|δ f|

15

0.05
0
1

2

m

0.05
0
1

3

4

5

1

2

4

3

n

5

2

m

3

4

5

1

2

3

4

5

n

Figure 3.9: The perturbation λδfωs (bottom) is shown in a rotated frame such that f eq
is diagonal; the resulting oscillatory and damped behavior (top) of the element f12 off in the box basis is due to the perturbation. The left and right figures correspond to
the points marked (a) and (b) in Fig. 3.8 respectively.

where we have defined a linear response operator through an appropriate centered difference limit

L(1) [δf ] =L[f eq + λδf ] − L[f eq − λδf ]
.
λ→0
2λ
lim

(3.54)

We solve Eq. (3.53) as an eigenvalue problem. In general, the eigenvalues arecomplex, with frequency and damping rate given by the real and the imaginary parts
respectively. The eigenvalues appear as complex conjugate pairs. The eigenmodes
corresponding to non-zero eigenvalues are Hermitian conjugates of each other and aretraceless with normalization given byTr{δfω(+) δfω(+) † } = 1.

(3.55)The physical linear response mode is given by the quadrature components δfωs and δfωc .
There also exists a zero mode that has non vanishing trace. The damping rates are all42−48.43−48.4−48.435f12−48.44−48.445−48.45
−48.5
0

5

10

15

20

−48.4550

5

10t (h/ε0)

20

0.2

|δ f|

|δ f|

0.2
0.1
0
1

15

t (h/ε0)

2

0.1
0
1

3

m

4

5

1

2

3

4

5

n

2

3

m

4

5

1

2

3

4

5

n

Figure 3.10: The perturbation λδfωs (bottom) is shown in a rotated frame such that f eq
is diagonal; the resulting damped behavior (top) of the element f12 of f in the box basis
is due to the perturbation. The left and right figures correspond to the points marked(c) and (d) in Fig. 3.8, respectively.

negative, thus confirming the stability of the collective modes. In Fig. 3.8, we plot thepositive frequency eigenvalues. The dotted lines correspond to the difference frequenciesof the Hamiltonian HHF .
It is interesting to see how these different modes evolve in real time. For this we
use the equilibrium distribution obtained in the previous section and perturb it with
one of the quadrature components
f → f eq + λδfω(s) ,(3.56)

where λ = 0.2 determines the smallness of the perturbation. In particular we willconsider the modes labeled by (a), (b), (c) and (d) in Fig. 3.8.
The real-time response is shown by plotting the off-diagonal matrix element f12of the single particle density matrix in the box basis as shown in Figs. 3.9 and 3.10. In
Fig. 3.11, we plot the change in the total energy ∆E = E(f ) − E(f eq ) as a function

43
0.04

∆ E (ε0)

0.02
0

−0.02
−0.04
−0.06
0

5

10

15

20

10

15

20

t (h/ε0)0.8∆ E (ε0)

0.6
0.4
0.2
0
0

5

t (h/ε0)Figure 3.11: The change in the total energy ∆E = E(f ) − E(f eq ) as the system relaxes
to its new equilibrium. In the top figure, the solid and dashed lines correspond to cases(a) and (b), respectively. Similarly in the bottom figure, the solid and dashed lines
correspond to cases (c) and (d), respectively.

of time. In cases (a) and (b) we see that the ∆E oscillates about zero and eventually
goes to zero. This is expected because such a perturbation tends to create coherences,resulting in an energy change by the amount of Ecoh (coherence energy) that would
eventually decay down to zero. Similar damped behavior is observed for case (d). Suchan oscillatory damped behavior of the total energy could be attributed to the Markov
approximation in the collision integral Eq. (3.32). On the other hand, perturbations of(s)

the kind (c) increase the total particle number by the amount δN = Tr{λδfω } andhence result in a finite change in total energy.

3.6

Conclusion
A non-Markovian version of the quantum kinetic theory is derived using the pre-scription of a nonequilibrium statistical operator method as outlined in Ref. [81]. This
theory is shown to conserve energy in the η → 0 limit. Inclusion of quasi-particle

44
damping and duration of collision effects results in a description beyond the Born approximation with energy conservation to φ2 order even in the Markov limit. To obtain
collision integrals that involve quasi-particle damping and duration of collision effects
and conserves energy precisely, one will have to calculate the T -matrix in the full collision operator keeping terms of all orders in the interaction.
We applied the generalized second order kinetic theory to the nonhomogeneous
dilute Bose gas confined in a spherical box to numerically study the full non-equilibrium
evolution of the system towards equilibrium. The self-consistent distribution f eq thus
obtained is very close to the Bose-Einstein distribution as shown in Fig. 3.6. We also
observe a significant Hartree-Fock self energy shift which depends on the single particle
distribution function f . The form of the damping function is important in determining
the line shape. Particularly, the function with a 1/ cosh type of behavior is found to beappropriate and gives improved energy conservation due to smaller initial correlation
effect.
The importance of such a real-time calculation is apparent from the full realtime response calculation, where we have calculated the damping rates and frequencies.
These damping rates correspond to a shorter time scale compared to the equilibration
time scale, which depend on rates in and out of the various levels.
This simple model of a spherical trap can be easily extended to a more realistic
situation of a harmonic trap. As in Refs. [13, 86, 87], the condensed component canbe easily included by introducing a symmetry broken mean-field, αi = hâi i, as one of
the relevant observables and Hartree-Fock-Bogoliubov quasi-particle excitations. Even
though this extension of the kinetic theory discussed in this chapter may seem simple,
the actual calculations are complicated and involved due to the presence of anomalous
fluctuations. Also the theory will need to be renormalized in order to ensure a gapless
spectrum. Such a calculation will allow us to make experimentally verifiable predictionsof damping rates of collective excitations. One can also explore the possibility of in-45cluding a time dependent potential or an external force term to selectively excite one
or more of the collective modes.

Chapter 4

Bosonic fractional quantum Hall effect4.1

Introduction
With the advancements in methods of trapping and cooling atomic gases, a whole

new frontier has opened up for the study of many body physics. Due to the extreme
control, purity and detailed theoretical understanding of interactions of dilute atomic
systems, these systems provide a perfect scenario to test many of the ideas previously
limited to condensed matter systems. Since the achievement of Bose-Einstein condensation in weakly interacting atomic gases [1, 2, 3], innumerable papers have been publishedon the equilibrium properties and dynamical behavior of dilute Bose gases at temperatures above and below the Bose-Einstein condensation temperature, Tc . While many
new phenomena have been studied in the regime of weak interactions, the other limit
in which the atoms can be strongly correlated has remained unexplored. Strong correlations between particles have shown to produce fascinating effects like the fractional
quantum Hall effect (FQHE) in condensed matter systems.
The regime of strong correlations in dilute atomic gases can be achieved by two
ways. One possibility is to tune the interactions such that the interaction energy is
greater that any other (single particle) energy scale of the problem, for example by using
Feshbach resonance [88]. The other possibility is to make the interactions dominate bycreating degeneracies. This could be achieved for example by using a periodic trapping
potential. Under the conditions that the tunneling amplitude between wells is small

47
compared to the typical interaction energy, the system has been predicted to undergo
a transition from a superfluid to a strongly correlated Mott state [89]. Another way ofachieving degeneracies is to rotate the harmonic trap that confines the dilute Bose gas.
At rotational frequencies close to the trapping frequency, the system is predicted to enter
the fractional quantum Hall regime where the ground state is given by the variational
Laughlin wavefunction [90]. This is precisely what we will focus in the remaining of thisthesis.
In this chapter, we will discuss the fractional quantum Hall effect (FQHE) in brief.
Then we will see how a dilute Bose gas can be prepared under conditions identical tothose required for observing the electronic FQHE. Having met the requirements, withsimple quantum mechanical arguments we will show that this bosonic system exhibits
single and many-particle states identical to that of the electronic FQHE system.

4.2

Fractional quantum Hall effect
The quantization of the Hall effect discovered by von Klitzing, Dorda, and Pep-

per [91] in 1980 is a remarkable macroscopic quantum phenomenon which occurs in
two-dimensional electron systems at low temperature and strong perpendicular magnetic field. Under these conditions the Hall-conductivity exhibits plateaus at integral
multiples of a fundamental constant of nature e2 /h. This is the integer quantum Hall
effect (IQHE). The striking result is the accuracy of the quantization irrespective of the
impurities or the geometric details of the two-dimensional system. Each plateau is associated with a deep minimum in the diagonal resistivity indicating a dissipationless flowof current. This was followed by the discovery of fractional quantum Hall effect (FQHE)
by Tsui, Stormer, and Gossard [92] in higher mobility samples where the plateaus occur
at fractional multiples of e2 /h as shown in Fig. 4.1. Despite the apparent similarity ofexperimental results, the physical mechanism responsible for the IQHE and the FQHE
are quite different. In the former the random impurity potential plays an important role

48

Figure 4.1: The FQHE in ultra high-mobility modulation-doped GaAs/AlGaAs. Many
fractions are visible.

while in the latter it is the collective effect arising from the coulombic electron-electron
repulsion. Also note that the fractional values originally observed as shown in Fig. 4.1
were all odd denominator fractions. While this is in fact related to the fermionic natureof electrons, more recently [93, 94] even fractions like 52 have also been observed. Occurance of such even fractions is rare, and their existence can be attributed to a Cooper
instability [95]; nevertheless, it is again a collective phenomenon due to the interaction
between electrons.
Regardless of the nature of the fractions, the many body physics that is important

49
to the current discussion is that the fractional quantum Hall state is a liquid state
exhibiting a gap in the excitation spectrum. This has been very well understood bymethods based on exact diagonalization and trial wavefunctions. The variational groundstate proposed by Laughlin [90] has been very successful in explaining the FQHE andpredicts the existence of quasi-particles with fractional charge. In the forthcoming
sections we show that identical many body effect can be predicted for a rotating Bosegas. In fact a bosonic variant of the many body Laughlin wavefunction can be written
down to describe the ground state of the system in this strongly correlated regime.

4.3

Electron system versus dilute Bose gas
While at first glance a rotating Bose gas may seem very different from the frac-tional quantum Hall system, there exists an interesting symmetry with respect to themeasurable physical quantities between these systems. This is summarized in the tablebelow where we list the necessary conditions and physical quantities for a rotating Bosegas corresponding to those which define the electronic fractional quantum Hall system.

1

2
3
4

5

Electronic FQHE system
2-dimensional at interface between semiconductors

Repulsive coulombic interaction
Vector potential A
Magnetic field B and magneticR
flux φ, B = ∇ × A; φ = B ·n̂dSLandau levels
fermionic-Laughlin state

Rotating Bose gas
effective 2-dimensional due tostrong confinement along thez direction compared to the x
and y directions
Repulsive two body potential
Velocity field u
Vorticity Ω and circulation
Θ,RΩ = ∇ × u; Θ = Ω · n̂dS
Landau levels
bosonic-Laughlin state

Table 4.1: Comparison between the electronic fractional quantum Hall system and therotating Bose gas.

50
As mentioned previously, the FQHE occurs in a two-dimensional electron systemin the presence of a strong perpendicular magnetic field. Identical conditions can beachieved for a dilute Bose gas by having the confinement in the z direction strong
compared to the x and y directions while the vorticity Ω of the rotating Bose gas
plays the role of the B field. The many body effects are generated by the repulsive
two-body inter-atomic potential which replaces the coulombic repulsion for electrons.
Thus, apart from the intrinsic statistics and charge of the particles, it is theoreticallypossible to configure a dilute Bose gas to meet the requirements necessary for observing
a FQHE. Therefore in principle one could predict a strongly correlated ground state forthe rotating Bose gas.
To begin with we will first address the last row of the above table by showingthat the Landau levels describe the single particle spectrum of a rotating Bose gas incomplete analogy to the electronic case. By including inter-atomic repulsive interactions
we will also show that a bosonic variant of the Laughlin wavefunction can be written
down as the variational many body ground state of the rotating Bose gas.

4.4

Landau levels and the many-body Laughlin wavefunction
We start with writing the Hamiltonian for N interacting bosonic atoms confined

to a harmonic potential in the rotating frame. Also, we assume the confinement along
the axis of rotation to be strong enough that we can neglect the excitations in that
direction and consider the system to be essentially two dimensional:
H=

N
X

j=1

( 2

pjx
p2jy
mω 2 x2jmω 2 yj2+++− ΩLjz
2m 2m
2
2

)

+

X
i<j

V (ri − rj ).

(4.1)

Here, m is the mass of the atoms, the angular momentum operator is given by Lz =xpy − px y, and Ω is the rotating frequency. The repulsive two body potential V depends
only on the relative coordinate of the individual atoms. Now if we make the following

51
identification
Ajx =

mω
yj ;
e

Ajy = −

mω
xj ,
e

(4.2)

the above Hamiltonian can be cast in a form very similar to the fractional quantum
Hall Hamiltonian
H=

N 
X
1

j=1

2m

(pjx + eAjx )2 +NXX
1(pjy + eAjy )2 +(ω−Ω)
V (ri −rj ). (4.3)
Ljz +
2m
i<j
j=1



To obtain the single particle spectrum we will first consider the non-interacting case,i.e. V = 0. The Hamiltonian can then be rewritten in a more convenient form
H=

N
X
1
2h̄ω + h̄(ω − Ω)(b†j bj − a†j aj ),aj aj +
2
j=1

N 
X
†

j=1



(4.4)

where the operators a and b are defined in terms of the covariant momentum (Px , Py ) =1
1
Py , y − 2mωPx )(px + eAx , py + eAy ) and the guiding center coordinate (X, Y ) = (x + 2mω

aj =

r

1
(Pjx + iPjy );
4h̄mω

a†j =

r

1
(Pjx − iPjy )
4h̄mω

(4.5)

s

b†j =

s

h̄
(Xj + iYi )
4mω

(4.6)

and
bj =

h̄
(Xj − iYj );4mω

obeying the following commutation relations[ai , a†j ] = [bi , b†j ] = δij ;(4.7)[ai , aj ] = [ai , bj ] = [a†i , bj ] = [bi , bj ] = 0.

(4.8)

The operators a and a† annihilate and create a quanta in the Fock space, while theoperators b and b† are the usual angular momentum ladder operators. Now one caneasily see that in the absence of interactions, at a certain critical rotational(stirring)frequency Ω = ω, the Hamiltonian depends only on the a’s and a† ’s. The corresponding
single particle energy states are the Landau levels given by|nj , lj i =s1(a† )nj (b†j )lj +nj |0i
nj !(lj + nj )! j

(4.9)

52

Figure 4.2: Atomic density in the lowest Landau level: left - density in the lowest
Landaup level with angular momentum quantum number l = 3 is sharply peakedpat radiusr = lh̄/mω, right - density is shown by a concentric ring of thickness ∆r = h̄/4lmω
in the x-y plane.

with energy eigenvalues Enj = (nj + 12 )2h̄ω and angular momentum Lj = lj h̄, where|0i is the Fock vacuum. Since the angular momentum operator commutes with the
Hamiltonian, each Landau level is degenerate with respect to the angular momentum
quantum number l. Now using the action of the operators a and b on the Fock states,
the configuration space wavefunction of the zero angular momentum lowest Landau
state can be easily shown to be given byψ00 (x) =rmω|x|2
mω
exp −
πh̄
2h̄

!

.

(4.10)

Therefore using Eq.(4.9), the generic Landau wavefunction can then be written as
ψln (x) = hx|n, li =s1(a† )n (b† )l+n ψ00 (x).
n!(l + n)!

(4.11)

The atomic density in the lowest Landau level given by |ψl0 (x)|2 is sharply peaked atradius r =plh̄/mω as depicted in Fig. 4.2. The lowest Landau states are therefore

represented by concentric rings labeled by the angular momentum quantum number l(Fig. 4.2). Thus the single particle spectrum for the rotating Bose gas shows Landau

53level structure identical to that of the electrons in a strong perpenticular magnetic field.
Now we include interactions in order to calculate the many-body ground state.
We start by writing the Hamiltonian (4.3) as
H = HLandau + HL + V(4.12)

Here HLandau is the quantum Hall single particle Hamiltonian whose single particle
states are the Landau levels separated by 2h̄ω. The Hamiltonian HL = (ω − Ω)Lz is
proportional to the z component of the total angular momentum Lz =PN

j=1 Ljz , andV is the interaction term. From now on we would like to consider the limit in which
the energy scale characterizing the Hamiltonian HLandau and V is much greater thanthe one corresponding to HL . This is possible when the stirring frequency is very close
to the harmonic trapping frequency. Also we will consider only contact interaction sothat V can be written as
V =η

N
X
i<j

δ(xi − xj ).

(4.13)In this limit, the ground states and elementary excitations of the system will lie in thesubspace of common zero energy eigenstates of HLandau and V . Thus the many body
wavefunction Ψ(z) must lie within the subspace generated by the tensor product of thelowest Landau level single particle states|zk |2exp −Ψ({z}) = P(z1 , ..., zN )
2
k
Y

!

,

(4.14)

where P(z) is a polynomial in each of the scaled complex atomic coordinates
zj =

r

mω
(xj + iyj ).
h̄(4.15)

Now if Ψ(z) is also an eigenstate of V , then due to the delta function nature of V , thepolynomial P(z) has to satisfy the form
P({z}) = Q({z})

Yi<j(zi − zj )2(4.16)

54
Here, the “2” in the exponent, is due to the even symmetry of bosons under exchange.
We can now diagonalize HL within the truncated Hilbert space of wavefunctions of
the form specified by (4.14) and (4.16). At the same time when P(z) is a homogeneouspolynomial in z, the state (4.14) is an eigenstate of HL with eigenvalue EM = h̄(ω−Ω)M ,where M is the homogeneous degree of P(z). Now since the ground state is the statewith the lowest angular momentum, we conclude that Q(z) = 1 and the ground state
wavefunction can be written as|zk |2Ψ({z}) = (zi − zj )
exp −
2
i<j
k
Y

2

Y

!

.

(4.17)

This is the essentially the bosonic variant of the Laughlin wave function [96, 97] originallyproposed for fractional quantum Hall electrons. The most generic Laughlin wavefunction
is represented by|zk |2exp −Ψq ({z}) = (zi − zj )q
2
i<j
k
Y

Y

!

(4.18)

where 1q represents the fraction in the FQHE. Therefore as one can immediately notice
that, q has to be odd(even) for the case of electrons(bosons) to satisfy the Fermi(Bose)
symmetry. Physically the fraction 1q , represents the number of electrons(bosons) per
flux-quanta(vortex) and is called the filling fraction. Thus the Laughlin wave function(4.17) for the rotating Bose gas represents a 21 bosonic FQHE. The atomic density profilein the 21 -Laughlin wavefunction is shown in Fig. 4.3.

4.5

Exact diagonalization
To confirm the validity of the arguments presented in the previous section wenumerically diagonalize the Hamiltonian (4.3) for a small system of five bosonic atoms.
As before we assume hard core interaction potential represented by a delta function.
Also in order to limit the basis states to the lowest Landau levels, the interaction
parameter η is chosen such that the interaction energy is small compared to the Landaulevel spacing, 2h̄ω. We choose the stirring frequency to be smaller than but very close to55

Figure 4.3: The atomic density in the 12 - Laughlin state for N = 5 atoms.

the trapping frequency, 1 − Ω/ω = 0.01. Such a choice allows us to see the effect of the
HL piece of the Hamiltonian while satisfying the assumptions of the previous section.
In Fig. 4.4 we plot the eigenenergies as a function of the total angular momentum Ltot .
We notice that there is a branch of states well separated from the rest of the spectrum.
These states are polynomial states of the form given by (4.14) and (4.16). The groundstate is the state with angular momentum Ltot = N (N − 1) = 20 and is the Laughlin
state corresponding to the bosonic 12 FQHE.

4.6

Quasiparticles
The Laughlin wavefunction (4.17) describes a bound state of atoms with evennumber of zeros or vortices. Now if the area of the system is slightly increased, thenumber of vortices increases, resulting in a change of the filling fraction from 12 . Now
consider the case where we introduce one extra vortex. The new wavefunction can thenbe written asΨ2+ε (z) =Yizj Ψ2 (z)(4.19)

56
7.5

Energy [hω/(2π)]

7

6.5

6

5.5

5
14

16

18

20

22

24

26

Ltot [h/(2π)]
Figure 4.4: Results of exact diagonalization for a small system of N = 5 bosonic atoms ina rotating 2-dimensional harmonic trap. The stirring frequency is chosen to be smaller
than but very close to the trapping frequency, 1 − Ω/ω = 0.01. The parameter η ischosen such that the interaction energy is very small compared to the Landau levelspacing, 2h̄ω. Such a choice allows us to use a truncated Hilbert space of lowest Landaulevels.

and possesses a zero at the origin. Physically, the above operation shifts the atomic
distribution corresponding to (l1 , l2 , ..lN ) to the one corresponding to (l1 +1, l2 +1, ..lN +1). This can be verified trivially by expanding the above wavefunction and will be left
to the reader. From the experimental point of view, such a situation can be realizedby shining a laser at z = 0 as proposed by Paredes et al. in [97]. More generally we
could shine the laser at some point z0 (localized within an area ∼ h̄/(mω)) in whichcase the Laughlin wavefunction will be multiplied byj (zj − z0 ) and hence reducingQ

the atomic density at that point to zero. Such an isolated zero acts like a quasi-holeexcitation with fractional statistics. This is very easy to see: consider a hypotheticalsituation in which we introduce two zeros at the same point z0 , and then pin an extra
atom to that point. Such an operation would return the ground state to the 21 Laughlin
wavefunction of the previous section except now for (N + 1) bosons. Thus two quasi-

57
holes are annihilated by an atom and hence the quasi-hole corresponds to a particle
with 21 statistics. The rotating Bose gas can also be predicted to be incompressible.
This is because compressing or expanding the state would require removal or injectionof an atom requiring a finite amount of energy. In essence, it is not possible to induce
an infinitesimal change in the system by an infinitesimal change in the pressure.

4.7

Conclusion
In this chapter we showed that the rotating Bose gas under certain conditions

represents a strongly correlated system in complete analogy to the electronic FQHE
system. Here the ground state is given by the bosonic variational 21 - Laughlin wavefunction. The 21 - quasi-hole excitations can be created by shining a laser at some specificpoint.
While theoretical understanding of the occurence of such a strongly correlated
state is quite clear, experimental realization seems difficult. The main difficulty is dueto the extreme low temperature required to access this state. Typically the temperature
required is such that [97]
1 η
kT
≪ , .
h̄ω
N N

(4.20)The remainder of this thesis will be devoted to precisely address this issue. We will beconsidering an alternative approach, where along with creating degeneracies by rotating
the trap, we increase the interaction energy by applying Feshbach resonance. The
motivation for this approach can be understood from the numerical results of Fig. 4.4
and the nature of the Laughlin wavefunction. The strongly correlated Laughlin statecorresponds to the lowest energy state in the separated branch of states. This is because,
the Laughlin wavefunction is annihilated by the delta function interaction potential.
Therefore, except for the states in this branch, eigenenergies of all the other states havea contribution corresponding to the interaction energy. Therefore, naively, increasing

58
the interaction energy would increase the gap between the separated branch and thecontinuum resulting in an increased temperature for observing the ground state.
However, in implementing such an approach, an alternative formulation of therotating Bose gas is required. This is because, while the physics of such resonances iswell formulated in a mean field picture, mean field theories fail to describe the strongly
correlated regime. The order parameter describing for example the Laughlin state is nolonger given by the expectation value of the atomic field operator. This difficulty canbe resolved in a gauge transformed Chern-Simons approach [98, 99]. Here the problem
can be formulated in terms of mean fields of composite particles. In the next chapter,
we discuss the basic ideas behind this effective field theory picture. Then continuingalong the same lines, in Ch. 6 we predict novel strongly correlated states in the presenceof Feshbach resonance.

Chapter 5

Chern-Simons theory for a strongly correlated rotating Bose gas

5.1

Introduction
The quasi-hole excitations in the bosonic Laughlin state are 21 bosons as argued

in the last chapter. Therefore a field theory model appropriate for the Bose gas in thestrongly correlated Laughlin regime should exhibit vortex solutions corresponding tosuch quasi-holes. Initial ideas of a field theory appropriate to describe such a strongly
correlated state were put forth by Girvin and MacDonald [100] in relation to the electronic FQHE. A related model was developed by Zhang, Hansson, and Kivelson (ZHK)[98] directly from the microscopic Hamiltonian. While the starting points were different,
both these theories involved coupling of a scalar field to a gauge potential (a0 , a) witha Chern-Simons action (or topological mass term). To understand the necessity of sucha gauge term and its relevance to the strongly correlated Bose system that we wish to
study, we first need to understand the origin of fractional statistics in a two-dimensional
space.
We begin by discussing the origin of anyons, particles with statistics intermediate
between Fermi and Bose statistics in two-dimensional space. In fact it is possible tochange the statistics of particles in two-dimensional space, for example by attachingfictitious magnetic flux in the case of electrons. The ZHK theory uses precisely thistransformation property associated with the gauge term of the Chern-Simons theory
to describe the strongly correlated fractional quantum Hall state as a Bose condensed

60

Figure 5.1: Statistics associated with the exchange of two identical particles. The net
effect of making the interchange twice is to take one particle around another in a closed
loop. In 3-dimensions the loop can be deformed without crossing the other particle
which is not possible in two dimensions.

state of bosonized electrons (composite bosons). Within this view point, we show thatthe atomic Laughlin state of previous chapter can be described as a Bose condensed
state of composite bosons.

5.2

Statistics in 2-dimensions
In 3-dimensional space there are only two types of particles, bosons and fermions.

They are distinguished by the following properties: (a) intrinsic spin of bosons is sh̄; s =0, 1, 2.. while that of fermions is sh̄; s = 12 , 32 , ..., (b) when two identical particles areexchanged, the wavefunction changes its overall sign for fermions but not for bosons,and (c) any number of bosons can occupy a single quantum state, but a single statecannot be occupied by more than one fermion. The non-Abelian algebra,[Si , Sj ] = ih̄εijk Sk for i, j, k = x, y, z

(5.1)

61
is responsible for the quantization of spin in 3-dimensions. To the contrary, fractional
spin is possible in a 2-dimensional space because there is only one way of rotation, whichis around the z axis by embedding the 2-dimensional space into the 3-dimensional space
and identifying it with the xy plane. Here the spin operator is Sz with an arbitrary
eigenvalue since the rotational group is Abelian. Now for the exchange statistics, as
shown in Fig. 5.1, if we move one particle around the other in a closed loop, in 3dimensions, the loop can be continuously deformed to a point which in 2-dimensional
space is not possible without crossing the second particle. Hence in 2-dimensional space,
the exchange statistics depends on the loop contrary to 3-dimensional space, where itis an intrinsic property of the particles. Thus particles in 2-dimensions can act likefermions or bosons under exchange depending on the geometry of exchange. Lastly, thePauli exclusion principle can be interpolated from fermions to anyons (particles in two
dimensions with fractional spin) by introducing a statistical interaction g [101] definedbyg=dN − dN +∆N
,
∆N

(5.2)

where N is the number of particles and dN is the dimension of the one-particle Hilbert
space obtained by holding the coordinates of N − 1 particles fixed. We have g = 0 for
1
bosons, g = 1 for fermions and g = m for the composite particles in the mfractional

quantum Hall state.

5.3

Berry phase
Consider a system in which the Hamiltonian depends on some external parameter

represented by a vector R. Also assume that the Hamiltonian has a non-degenerate
eigenstate ΨR
H(R)ΨR = ER ΨR

(5.3)

62

C

R(t)
O
ti
tf

Figure 5.2: Trajectory of the vector R(t) in the parameter space. The vector R(t) represents a set of parameters that define the Hamiltonian at some time t. The parameters
are changed adiabatically from their initial value at time ti such that the vector moves
in a closed loop and comes back to its original position at some final time tf.

Now we start at some initial time ti and change the parameters such that the vector
R moves in a closed loop C in the parameter space as shown in Fig. 5.2 and returns to
its initial value at some final time tf , R(tf ) = R(ti ). If the change is sufficiently slow
the state changes continuously and remains an eigenstate at all times. Since we haveassumed the state to be non-degenerate, the state at time tf should be the same as thatat ti except for a multiplicative phase factor. In fact, the states are related byiΨR(tf ) (tf ) = exp −
h̄


Z tf
ti

′

′



dt E(R(t )) + iγ(C) ΨR(ti ) (ti ).

(5.4)

The first part of the phase factor is the dynamical phase and is due to the eigenenergyof the state. The second part depends on the contour C and is independent of tf − ti
and is referred to as the geometric phase. This phase can be expressed asγ(C) = iIChΨR(t) |∇R(t) ΨR(t) i · dR(t)(5.5)

and is called the Berry phase [102]. A typical example is the Aharonov-Bohm (AB)phase [103], which appears when an electron moves in a magnetic field. The AB phaseis given by γ(C) = eBS/h̄, where BS is the flux passing through the contour C. In thenext section we will see how this extra geometric phase is associated with the idea of63

Figure 5.3: Left figure: two-dimensional two-electron system. The electron is localized
separately potential valleys V 1 and V 2. The valleys are separated such that there is
negligible overlap between the electronic wavefunctions. Right figure: the same situation
as on the left except now the electrons are independently attached to magnetic flux φand the flux moves along with the electrons.

anyons, the particles with statistics intermediate between Fermi and Bose statistics.

5.4

Anyons
One can change the statistics of the particles in a two-dimensional system basedon the fact that the AB phase arises in the presence of magnetic flux. For example,
consider a two-electron system, in which each of the electrons are localized in separate
potential valleys at R1 and R2 as shown in Fig. 5.3. We assume that the valleys are
separated so that the overlap of the wavefunctions is negligible. The two electron wave

64
function can then be written in terms of their bound state wavefunctions ΨR (r)1ΨR1 ,R2 (r1 , r2 ) = √ [ΨR1 (r1 )ΨR2 (r2 ) − ΨR2 (r1 )ΨR1 (r2 )]
2

(5.6)

Now if we exchange the position of the potential wells adiabatically, the Hamiltonian
returns to its original form, but due to the Fermi statistics of the electrons, the total
wave function acquires an extra minus sign. Next we consider the situation in which
each of the electrons is attached to an infinitesimally thin magnetic flux φ as shown in
the right figure of Fig. 5.3 and also assume that the flux moves along with the electrons.
Again we exchange the position of the potential wells as before, but now there is an
extra contribution eφ/(2h̄) to the phase due to the flux attachment. We can choose theflux to be such that the additional phase cancels the minus sign due to Fermi statistics.
Thus the electrons behave like bosons when a certain amount of flux is attached tothem. Since the choice of φ is arbitrary, it is possible to make particles with statistics
intermediate between that of Fermi and Bose or anyons.

5.5

Composite particle mean field theory
Having introduced the key concepts behind exchange statistics in two-dimensions,

we now show how these ideas can be combined into building a field theory applicablein the strongly correlated regime of the rotating Bose gas described by the bosonicLaughlin wavefunction. We start by defining a unit quantum of flux by φ0 = h/|e|.
Then bosons and fermions can be transformed into fermions and bosons respectively byattaching an odd number of flux quanta to each particle. On the other hand the statistics remains unchanged by attaching an even number of flux quanta. This is shown in
Fig. 5.4. Thus electrons in a 1q ; q = odd FQHE, can be considered as composite bosons
with extra qφ0 fictitious flux in the opposite direction. The mean field of the fictitious
flux cancels with the real flux, and the system can be considered as a system of interacting composite bosons in a zero magnetic field. Thus in the mean field picture, the65

Figure 5.4: Composite particle picture. A boson is transformed to a composite boson(fermion) by attaching an even(odd) number of magnetic flux quanta. The extra
fictitious flux cancels the real flux in the mean field approximation resulting in freecomposite bosons/fermions.

interaction experienced by the composite bosons is just the repulsive coulomb interaction. At the same time we know that a repulsive bosonic system in two dimensions
undergoes Bose condensation at zero temperature. Hence in this picture, an odd fractional quantum Hall state can be thought of as a Bose condensed phase of composite
bosons.
The same argument holds for our case of bosonic 21 FQHE in rotating Bose gases.
Since q = 2 = even, the rotating Bose system can be reduced to a system of interacting
composite bosons with zero rotational frequency. The composite bosons still experience
the repulsive two-body potential; therefore, the strongly correlated bosonic Laughlin
state can be considered as a Bose condensed state of composite bosons.

66

5.6

Chern-Simons Ginzburg-Landau theory
The starting point of this theory is the second quantized Hamiltonian for atoms(bosons)in the rotating frame. Since we have already made the mapping between the vorticity
and magnetic field, we write the second quantized Hamiltonian in terms of the vector
potential A instead of rotating frequency Ω. Also here we specialize to the case ofΩ = ω, so that HL = 0 in (4.3). ThusĤ =Zψ̂ † (r)11(−iD)2 ψ̂(r)d2 r +
2m
2


Z Z

ψ̂ † (r)ψ̂ † (r′ )V (r − r′ )ψ̂(r)ψ̂(r′ )d2 rd2 r ′(5.7)

where Dµ = ∂µ + ieAµ and ψ̂(r) =P∞

LL
l=0 al ψl (r) where al is the annihilation operator

for a particle in the lth single particle lowest landau state. As discussed in the previous
section, we would like to work in a composite boson picture. The composite boson field
operator ϕ̂ is defined by (see App. C)
ϕ̂(r) = e−J(r) ψ̂(r)(5.8)

where the operator J is given byJ(r) = iqZd2 r ′ ρa (r′ )θ(z − z ′ )(5.9)

In the composite picture the second quantized Hamiltonian retains the same form asgiven above with the operators ψ̂ and ψ̂ † replaced by the corresponding composite boson
operators ϕ̂ and ϕ̂† and D replaced by D,
Dµ = Dµ + ieaµ (r)(5.10)

where aµ is the fictitious Chern-Simons gauge field. The Chern-Simons field is an auxiliary field subject to certain constraints rather than a dynamical field obeying Maxwell’sequation. In order to derive the constraint equation and the stationary solution it isconvenient to use the Lagrangian formulation. In the semi-classical approximation,

67
the Lagrangian density corresponding to the full second quantized Hamiltonian written
above is given by
L = La + Lcs(5.11)

where
La = ϕ∗ (r)(iD0 )ϕ(r) +−1
2

Lcs = −

Z

1 ∗ϕ (r)D 2 ϕ(r)2mϕ∗ (r)ϕ∗ (r′ )V (r − r′ )ϕ(r)ϕ(r′ )d2 r ′e2 µνλε aµ (r)∂ν aλ (r)4qπ(5.12)(5.13)

This Lagrangian has the usual form except for the last term called the Chern-Simons
term. Here µ is the chemical potential and εµνλ is the complete antisymmetric tensor
with the convention ε012 = 1. Therefore the action is given byS=Z

Ld2 rdt(5.14)In the rest of the thesis we consider the interaction potentials to be local V (r − r′ ) =V δ(r − r′ ). The ground state is the state that minimizes the action S. This implies thatthe action be invariant under small fluctuations of ϕ. But first we consider the variationof the action with respect to the zeroth component of the gauge field, ∂S/∂a0 = 0, which
results in the constraint equation∇ × a|z = −2qπ ∗(ϕ ϕ),e(5.15)

which is essentially the Chern-Simons condition. The extremum condition ∂S/∂ϕ∗ = 0
results in the following classical equation of motion for the field ϕ(iD0 + µ)ϕ(r) +1 2
D ϕ(r) − ϕ∗ (r)V ϕ(r)ϕ(r) = 0.
2m(5.16)The most simple solution to the above equation is the one where the Bose field is uniform
in space and time ϕ =√ρ̄. Also since the system is homogeneous, to avoid infinite mean

68
field contribution we shift the zero of energy to this mean field energy. Now we see thatthe uniform solution is possible only if
A+a = 0and a0 = 0(5.17)

This implies that B = (2qπ/e)ρ̄ or the filling fraction is ν = 1/q. Thus a uniform
solution can exist only when a Laughlin wavefunction can be written down. Thus abosonic Laughlin state of a rotating Bose gas can be thought of as the Bose condensed
phase of composite bosons formed from atoms attached to an even number of vortices.

5.7

Feshbach resonance
From the discussion of the Chern-Simons field theory of the previous section, it isclear that, within this picture, Feshbach interaction can be systematically introduced totune the inter-atomic interaction. In a typical Feshbach resonance illustrated in Fig. 5.5,
the properties of the collision of two atoms are controlled through their resonant coupling
to a bound state in a closed channel Born-Oppenheimer potential. By adjusting an
external magnetic field, the scattering length can be tuned to have any value. The
field dependence of the scattering length is characterized by the detuning ν and obeys
a dispersive profile given bya(ν) = abg

κ
1−
2ν





,

(5.18)

where κ is the resonance width, and abg is the background scattering length. In fact, all
the scattering properties of a Feshbach resonance system are completely characterizedby just three parametersU0 =

4πh̄2 abg
,
m

g=

p

κU0 ,

and ν.

(5.19)

Physically, U0 represents the energy shift per unit density on the single particle eigenvalues due to the background scattering processes, while g, which has dimensions of energy
per square-root density, represents the coupling to the Feshbach resonance connecting
the open and closed channel potentials.

Energy69

Closed Channel

νOpen Channel

Internuclear Separation
Figure 5.5: A Feshbach resonance results when a closed channel potential possesses abound state in proximity to the scattering energy in an open channel potential. The
detuning of the bound state from the edge of the collision continuum is denoted by ν0 .

The second quantized Hamiltonian for a dilute Bose gas with binary interactions
is given by
H=

Z

3

d x ψa† (x)Ha (x)ψa (x) +Z

d3 xd3 x′ ψa† (x)ψa† (x′ )U (x, x′ )ψa (x′ )ψa (x)(5.20)

where Ha (x) is the single particle Hamiltonian, U (x, x′ ) is the binary interaction potential, and ψa (x) is the atomic field operator. In cold quantum gases, where the atoms
collide at very low energy, we are only interested in the behavior of the scattering abouta small energy range above zero. There exist many potentials which replicate the low
energy scattering behavior of the true potential; therefore, it is convenient to carry out
the calculation with the simplest one, the most convenient choice being to take theinteraction potential as a delta-function pseudo potential when possible.
For a Feshbach resonance this choice of pseudo potential is generally not available
since the energy dependence of the scattering implies that a minimal treatment mustat least contain a spread of wave-numbers which is equivalent to the requirement of a

70
nonlocal potential. Since the solution of a nonlocal field theory is inconvenient, we takean alternative but equivalent approach. We include in the theory an auxiliary molecularfield operator ψm (x) which obeys Bose statistics and describes the collision between
atoms in terms of two elementary components: the background collisions between atoms
in the absence of the resonance interactions and the conversion of atom pairs intomolecular states. This allows us to construct a local field theory with the property thatwhen the auxiliary field is integrated out, an effective Hamiltonian of the form given
in Eq. (5.20) is recovered with a potential U (x, x′ ) = U (|x − x′ |) which generates theform of the two-body T -matrix predicted by Feshbach resonance theory [104]. The local
Hamiltonian which generates this scattering behavior:
H =


h̄2 2
∇x + Va (x) − µa ψa (x)
2m
Z

 h̄2
†
∇2x + Vm (x) − µm ψm (x)+ d3 x ψm
(x) −
4m
Z
U
3
†
+
d x ψa (x)ψa† (x)ψa (x)ψa (x)
2
Z
g
†
+
d3 x ψm(x)ψa (x)ψa (x)2Zd3 x ψa† (x) −(5.21)

has the intuitive structure of resonant atom-molecule coupling. Here Va,m are the external potentials and µa,m are the chemical potentials. The subscripts a, m represent theatomic and molecular contributions, respectively. The Feshbach resonance is controlled
by the magnetic field which is incorporated into the theory by the detuning ν = µm −2µa
between the atomic and molecular fields. The Hamiltonian in equation (5.21) contains
the three parameters U0 , g, and ν which account for the complete scattering propertiesof the Feshbach resonance.
Thus in order to introduce Feshbach interaction within the framework discussed
earlier in this chapter, we will have to define composite bosons for the molecules exactly
in the same way as we did for the atoms, except that the composite molecular bosons
will be attached to twice the number of flux quantaas that for the atoms. All theseideas will be put to work in the next chapter.

Chapter 6

Resonant formation of novel strongly correlated paired states inrotating Bose gases
Published in Physical Review A 69,053603 (2004)

6.1

Introduction
Recently, there has been considerable interest from both the experimental andtheoretical perspective in the behavior of rotating Bose gases confined to an effective twodimensional space [105, 106, 107, 96, 97, 108]. One goal has been to understand and tocreate strongly correlated states, such as the Laughlin state generated by the fractional
quantum Hall effect (FQHE), within ultracold atomic gases. A major challenge torealizing such a state is the need to reach extremely low temperatures in order toresolve the lowest Laughlin state from the first excited state. The size of this energy gap,
however, is directly related to the strength of interatomic interactions, so by increasing
the interactions it would seem natural that one could increase the gap, making thesystem more accessible to experiment.
Atomic systems can now be created which allow the microscopic interactions to be
dynamically tuned [109, 110]. Feshbach resonances have proven very successful at doing
this, allowing one to tune the interactions by adjusting the resonant detuning ν (forexample, by varying a magnetic field), and would seem an excellent tool for increasing
the gap within a FQHE system. To account for the full effects of the resonance, however,
we cannot simply scale the mean-field energy but must incorporate the entire resonant

72

Figure 6.1: The Feshbach resonance pairing mechanism is illustrated by the above
Born-Oppenheimer curves. Pairs of composite atoms composed of single atoms andan associated number of quanta of angular momentum (represented by the arrows)
approach each other within an open channel potential of background value U (x, x′ ).
They may form a composite molecule due to the presence of a closed channel boundstate, at a detuning ν from the scattering continuum, which is coupled to the open
channel with strength g(x, x′ ).

structure into our model. This means that we must include the process of molecularformation generated by the introduction of a bound state within the open channel ofscattering states [111, 112, 113].
By introducing a bound state, however, we not only modify the relative interaction
strength, but we also introduce a physical mechanism for generating pair correlations
between particles (see Fig. 6.1). In the context of two-dimensional condensed mattersystems, such a mechanism, although arising from a very different source [114], can havea significant effect on the ground state properties. Before we can study the resonant
behavior of the gap, we must first understand the effect of resonant interactions on the
ground state wavefunction. We will find that as we approach the Feshbach resonancethe Laughlin state transforms into a unique, strongly correlated state.

73

6.2

Inclusion of Feshbach resonance in the Chern-Simons theory
We begin by writing down an effective Hamiltonian, in second quantized form, fora resonant gas of Bosons of mass m rotating in two dimensions with stirring frequencyΩ approaching the trapping frequency ω, i.e. ω − Ω → 0+ :

Ĥ =
+

Z

1
2

d2 x ψ̂a† (x)
Z

Z



−1
(∇ − iA(x))2 ψ̂a (x)
2m


(6.1)

d2 x′ d2 x ψ̂a† (x)ψ̂a† (x′ )U (x, x′ )ψ̂a (x′ )ψ̂a (x)−1(∇ − 2iA(x))2 + ν ψ̂m (x)
+
4m


Z
Z
′
1
† x+x
2 ′
2
′
′
+
d x d x ψ̂m ()g(x, x )ψ̂a (x)ψ̂a (x ) + H.c. .
2
2
Z

2

†
d x ψ̂m(x)

† (x), ψ̂
Here ψ̂a,ma,m (x) are the creation and destruction operators for atoms and molecules

which satisfy the commutation relations [ψ̂1 (x), ψ̂2† (x′ )] = δ(3) (x, x′ )δ1,2 , where 1, 2 ∈{a, m}. We define the two-dimensional vector potential A(x) = (mωy, −mωx), U (x, x′ )is the two-particle background scattering potential, g(x, x′ ) is the resonant coupling between the open and closed channel potentials, and ν is the detuning of the open channel
continuum from the level of the bound state in the closed channel.
Recently, it has been noted that there is a direct mapping between the Hamiltonian for a rotating two-dimensional gas, the first two terms of Eq. (6.1), and that of the
FQHE [96]. The analog of the magnetic field is realized by the angular rotation and the
Coulomb interaction is replaced by the two-particle scattering. Therefore, if we were toneglect resonant effects we would expect the many-body ground state to be describedby the Laughlin wavefunction [90]

ΨL (x1 , x2 , ...xN ) =Yi<j(zi − zj )2

Y
k





exp −|zk |2 /2 ,(6.2)

where the products run over the indices i, j, k = (1, 2, ...N ) at position z = x + iy
for N particles. We will see that the inclusion of the resonant terms in Eq. (6.1) can

74
significantly modify the form of Eq. (6.2) due to the growth of two-particle correlations.
We approach this problem by an extension of the Chern-Simons theory [115] whichallows us to develop a mean-field theory for the rotating system that has removed thecomplications of the associated rotation. This is done by constructing a composite
particle composed of the original particle and an artificially attached number of flux
quanta. The composite particle is designed so that the attached flux quanta cancel
the total rotation of the original system leaving a system of non-rotating, interacting
composite particles. For the resonant system, the composite particles can formally beobtained by the following transformation:

ϕ̂a,m (x) = exp



−i q

Z

d2 x′ θ(x − x′ )× [ρa (x′ ) + 2ρm (x′ )] ψ̂a,m (x),(6.3)

where ρa (x′ ) and ρm (x′ ) are the atomic and molecular spatial densities, respectively,and θ(x − x′ ) is the topological phase. To guarantee that the resulting composite particle is a Boson, with no loss of generality we set q = 2.
In the composite picture, we replace the ψ̂ operators in Eq. (6.1) with the corresponding composite operators ϕ̂ and introduce the statistical Chern-Simons field a(x)through the gauge transformation

A(x) → A(x) + a(x)(6.4)

These modifications generate the Hamiltonian formulation of our composite atom/moleculetheory. The composite picture can be shown to be equivalent to the single particle picture of Eq. (6.1).
We will now shift to a functional representation of the composite atom/molecule
system to clarify the resonant modifications of the Chern-Simons theory and then return

75
to the Hamiltonian formulation to derive the ground state wavefunction. Defining theaction within one temporal and two spatial dimensions

S=

Z

3

d x

X

σ=a,m

ϕ∗σ (x)i∂0 ϕσ (x) −Z

dx0 Ĥ,(6.5)

we generate a Chern-Simons term which couples to the statistical vector field aµ (x)Z

SCS = − d3 x

1 µνλǫ aµ (x)∂ν aλ (x),
8π

(6.6)

where the indices of µ, ν, and λ run over the three dimensions (0, 1, 2) and the summation
convention over repeated indices is invoked. We have also introduced the antisymmetrictensor ǫ012 = 1. To simplify the following calculations we assume contact interactions of
the form U (x, x′ ) = U δ(x, x′ ) and g(x, x′ ) = gδ(x, x′ ). Any complications arising fromthis replacement of the true potentials with contact potentials should be remedied as
explained in Ref. [104]. We next perform the lowest order variation of the action.
Varying with respect to the zeroth component of the gauge field, ∂S/∂a0 = 0,
reproduces the Chern-Simons condition∇ × a(x)|z = −4π |ϕa (x)|2 + 2|ϕm (x)|2 .

(6.7)

Equation (6.7) is a statement of Gauss’s law for the statistical gauge field associating
an even number of rotational flux quanta with each particle. This relation is simply arestatement of our choice of quasiparticle.
Since we will be interested in the ground state properties of the atom/molecule
system, let us assume that the fields ϕa (x) and ϕm (x) are uniform. By minimizing theaction with respect to the atomic and molecular fields, i.e. ∂S/∂ϕa = 0, ∂S/∂ϕm = 0,
we generate the following constraint equation for the molecules:

ϕm =

gϕ2a
.
2[ν + |A + a|2 /m]

(6.8)

76
Equation (6.8) allows us to eliminate the molecular field from the theory and arrive at
a self-consistent relationship for the gauge field2|A + a| =!

g2
U+2m|ϕa |2 .
4(ν + |A + a|2 /m)(6.9)

Equation (6.9) is the usual result relating the gauge field to the background density
only now it is dependent upon the detuning from the resonance.

6.3

Pairing in the strongly correlated ground state
We now switch back to the Hamiltonian form of our theory to derive the groundstate wavefunction. After Fourier transforming the composite form of Eq. (6.1) by substitution of the field operators ϕ̂a (x) =ik·x and ϕ̂ (x) = P b̂ eik·x , we followmk k
k âk ePthe usual Hartree-Fock-Bogolubov (HFB) approach to construct a quadratic Hamiltonian which accounts for the lowest order pairing. As before, we assume contact interac-tions and now make the additional assumption that we may neglect the excited modesof the molecular field keeping only the lowest condensed mode b̂k = b0 . The resulting
Hamiltonian for the composite system can be written in the form

H = H0 +X

A†k Mk Ak .

(6.10)

k6=0

H 0 is composed of all terms of less then quadratic order in the operator âk , we define acolumn vector Ak = (âk , â†−k ), and Mk is the self-energy matrix. For our purposes weneed only concern ourselves with the structure of the second term in Eq. (6.10). Herethe self-energy matrix is expressed as Ek

Mk = 


∆∗



∆ 

E−k




with the additional definitions for the diagonal and off-diagonal terms(6.11)

77

Ek = Ek0 + U (|ϕa |2 + n)

(6.12)∆ = U (ϕ2a + p) + gϕm .

(6.13)

Equations (6.12) and (6.13) are expressed in terms of the pairing-field p =the normal-field n =PPk′ hâk′ â−k′ i,†0
k′ hâk′ âk′ i, and Ek is the effective kinetic term which contains

the contribution from the gauge field A(x).
Eq. (6.10) can be rewritten in terms of the quasiparticles

1†α̂k = p
(E
+
ω
)â
+
∆â
k
k
−k
k
(Ek + ωk )2 + |∆|2


1
†
(E
+
ω
)â
+
∆
â
α̂†−k = pkk−kk
(Ek + ωk )2 + |∆|2

(6.14)(6.15)

which result in the diagonal Hamiltonian

H = H0 +Xωk α̂†k α̂k ,(6.16)

k6=0

where H0 contains the ground state contribution to the energy and the excitations are
given by the spectrum of frequencies ωk =q

Ek2 − |∆|2 .

Since there are no quasiparticles present in the ground state |gsi , which is what
one would expect from an interacting bosonic system at T = 0, the ground state mustsatisfy the condition α̂k |gsi = 0. Substitution of Eqs. (6.14) and (6.15) for the quasiparticle operators result in the relation(Ek + ωk )âk |gsi = −∆â†−k |gsi.

(6.17)

Because âk and â†k are canonically conjugate variables there is no loss of generality
in making the replacement âk → ∂/∂â†k [116]. This converts Eq. (6.17) into a simple
differential equation for the ground state with the solution

78|gsi = exp

"

X
k

#

−∆
â† â† |0i.
Ek + ωk −k k(6.18)

To derive the many-body wavefunction we must now move from second to first quantization. The relationship which links the second quantized ground state |gsi with thefirst quantized wavefunction ΨCB can be written for an even number of noncondensed
particles 2N asΨCB (x1 , x2 , ...x2N ) = h0|ϕ̂a (x2N )...ϕ̂a (x2 )ϕ̂a (x1 )|gsi,(6.19)

where it should be noted that ΨCB is the full-many body wavefunction for the composite
Bose particles. If we are able to assume that Ek + ωk ≫ ∆ , an assumption which
will remain valid as long as we are not too close to resonance, we may truncate thepower expansion of the exponent in Eq. (6.18). This results in the composite boson
wavefunctionΨCB (x1 , x2 , ...x2N ) = S(ψ12 ψ34 ...ψ(2N −1)2N )(6.20)

comprised of a symmetrized product S of paired wavefunctions

ψij =

−∆
eik·(xi −xj ) .
E
+
ω
k
k
k

X

(6.21)

If we were dealing with a system of fermions, equation (6.20) would be antisymmetrized
and would result in a Pfaffian wavefunction [6, 117]. Here, because of the statistics of theparticles, we generate a bosonic analogue to this result. The many-body wavefunction
for the bare particles can now be extracted from the composite wavefunction [118]resulting inΨM B = ΨCB (x1 , x2 ..., x2N ) × ΨL (x1 , x2 ..., x2N )(6.22)

79
which is a product of the composite particle wavefunction of Eq. (6.20) and the Laughlin
wavefunction of Eq. (6.2).
Equation (6.22) is the final result for the ground state wavefunction of the resonant rotating Bose system. This result has important consequences for the generationof the FQHE within a resonant atomic gas. It would imply that an increase of interparticle interactions by a resonant tuning of the interactions simultaneously generates anincrease in 2-particle correlations between composite particles resulting in a modification to the ground state wavefunction. As is clear from the form of Eq. (6.20), for large
detuning from the resonance, corresponding to small pairing and molecular field, themany-body wavefunction reduces to the Laughlin wavefunction. As one moves nearer tothe resonance, however, the off-diagonal part of the self-energy matrix, ∆, grows. This
results in an increasing modification of the many-body wavefunction from the Laughlin
wavefunction. The ability to tune a Feshbach resonance therefore allows for the direct
study of this crossover from a Laughlin wavefunction to a paired wavefunction.
A similar, yet distinct, paired wavefunction as in Eq. (6.22) was found for electronic FQHE systems [114, 117, 119]. This has been used to explain the previously
unresolved even denominator filling fractions which result from a pairing instability,
such as the observed incompressibility of the 5/2 filling. In this case, a straightforward
generalization of the Laughlin wavefunction would result in a symmetric wavefunction,
violating the asymmetry of the fermions. However, the generation of an antisymmetric Pfaffian wavefunction which multiplies the generalized Laughlin state allows theoverall ground state to be correctly antisymmetrized. For the bosonic system we havetreated, the overall wavefunction must remain symmetric, so the corresponding paired
wavefunction is symmetric in comparison to the antisymmetric Pfaffian wavefunction.

80

6.4

Conclusion
In conclusion, because of the extreme diluteness of trapped atomic gases, ifstrongly correlated effects such as the FQHE are to be observed in these experiments,
the interatomic interactions will most likely need to be resonantly enhanced. Feshbachresonances are a convenient method for increasing the interactions, but the molecular
processes involved in such a resonant system force one to account for the effects ofatomic pairing. The growth of pair correlations among the composite particles leads toa modification of the expected ground state for a rapidly rotating Bose gas. The new
ground state wavefunction, which is generated by the Feshbach resonance, exists as astrongly correlated state unique to trapped Bose gases. The tunability of the resonanceopens the possibility for the direct study of the crossover transition between the paired
state and the Laughlin state.
These results have important implications for the production of the FQHE within
atomic gases since the use of a Feshbach resonance results in a state quite different froma rapidly rotating gas where the interactions are quantified by only a large scattering
length. For instance, many of the observeable properties of the gas may be modified
such as the density profile for both atoms and molecules and the nature of collective
excitations. It should also be noted that the crossover transition we have discussed isonly a part of a much more general crossover theory made accessible by the tunabilityof a Feshbach resonance. Although the methods presented here are invalid close to theresonance, one could imagine extending these ideas to describe the resonant system as
one passes from a gas of interacting rotating atoms, through the resonance, to a systemof tightly bound, rotating molecules.

Chapter 7

Summary

To summarize, in this thesis we have obtained some interesting results for the
dilute Bose gas in different regimes.
In the first part, we studied the weak interaction regime. Here we discussed theworking of an atom laser and also proposed an all optical method for continuously
loading the thermal component (reservoir) of a cw- atom laser. We showed that despite
the presence of spontaneously emitted photons, it is possible to configure the trappedsystem such that the probability of photon re-absorption is small and hence negligible
heating of the atomic cloud.
Another essential component is the continuous evaporative cooling mechanism.
This would have the effect of pumping thermal atoms into the condensate mode tocompensate for the loss due to output coupling as well to compensate for other intrinsic
loss mechanisms. Therefore, a proper understanding of a cw- atom laser would requirea model satisfying all three basic requirements– loading, evaporative cooling and output
coupling. Such a model will then allow us to calculate the phase diffusion propertiesof the cw- atom laser. Our discussion of the quantum kinetic regime in the second
chapter would be very crucial to understanding the collisional contribution to the phasediffusion and hence the line width of the cw- atom laser.
In the second part of this thesis, we showed that under certain conditions a dilute
Bose gas can enter the strongly correlated regime. In fact in this regime the Bose sys-

82

Figure 7.1: Origin of different possible fractions in the fermionic- and bosonic- fractional
quantum Hall effect. The tuning of interactions via Feshbach resonance may result ingeneration of fractions, the underlying mechanism could be compared to that of theeven FQHE.

tem showed fractional quantization quite analogous to that in the FQHE. The relevant
fraction was found to be 12 with the corresponding ground state given by the bosonic
Laughlin wavefunction. Field theory method based on the Chern-Simons gauge wasfound to be extremely useful in treating the Feshbach mechanism for tuning the interatomic interactions in this regime. While controlling interactions is virtually impossible
in the condensed matter fractional quantum Hall systems, our study of the dilute Bose83
gas in this regime has opened a whole new set of possibilities for investigating many
body effects previously inaccessible in condensed matter systems. The most interesting example would be the existence of even fractions in electronic FQHE. In fact the
only observed fraction is 25 and its origin is still an active area of research. From thisperspective, a rotating Bose gas with tunable interactions would be very useful. As
depicted in the flow chart of Fig. 7.1, an identical mechanism in the Bose case would
result in odd bosonic FQHE. Therefore, observing such strongly correlated phases with
odd fractional statistics may help enhance our understanding of the even FQHE.

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90

Appendix A

Reference distribution(0)

The reference distribution σ{f } of Eq. (3.9) is parameterized through its expectation values in Eq. (3.11). From the structure of this quantum Gaussian operator, it
follows that
(0)

(0)

σ{K(t,t0 ) f K† (t,t0 )} = Û † (t, t0 ) σ{f } Û (t, t0 ).

(A.1)

In the above, Û represents the single particle propagator of Eq. (3.16) acting in manyparticle Fock-space and K is the corresponding single particle Hilbert-space propagatorof Eq. (3.23). This condition implies that∂γk σ (0) Tr{[Ĥ(0) , γ̂k ]} = −[Ĥ(0) , σ (0) ]and was used to obtain Eq. (3.17).

(A.2)

91

Appendix B

Matrix elements

The matrix element of Eq. (3.41) can be evaluated easily by expanding the sinfunctions into co- and counter-propagating complex exponents and by an additional
partial integration. This results in eight separate terms, i.e.,φijkl /aS =

4
π

Z π

sin (i x) sin (j x) sin (k x) sin (l x)

0

dx
,
x2

= F (i + j + k − l) + F (i + j − k + l)
+ F (i − j + k + l) + F (i − j − k − l)
− F (i + j − k − l) − F (i − j + k − l)
− F (i − j − k + l) − F (i + j + k + l),

(B.1)

where
1
[cos (nπ) + nπ Si(nπ)],2π 2Z zsin(t)
Si(z) =dt.
t0

F (n) =

(B.2)
(B.3)

An asymptotic expansion of the sine integral leads to the following approximation thatis correct at the 1% level, i.e.,
F (0) =F (n > 0) ≈

1
,
2π 2
sin (nπ) 2 cos (nπ)
1 π
[ |nπ| −
+
].
2
2π 2
nπ
(nπ)2

(B.4)

Appendix C

Anyon field operator

The anyon field operator φ̂an (t, x) is represented byφ̂an (t, x) = eiαξ̂(t,x) ψ̂b (t, x)(C.1)

in terms of the boson operator ψ̂b (t, x) with the phase fieldξ̂(t, x) = αZd2 yθ(x − y)ρ(t, x).

(C.2)

where ρ(t, x) = ψ̂b† (t, x)ψ̂b (t, x). The Chern-Simons field is expressed asαh̄ak (x) =eZd2 y∂k θ(x − y)ρ(y).

(C.3)

We now examine how the exchange phase arises theoretically. We start with the commutation relation[ξ̂(y), ψ̂b (x)] =Z

d2 zθ(x − y)[ρ(z), ψ̂b (x)] = −θ(y − x)ψ̂b (x).

(C.4)

Now, we use the Hausdorff formula
1
eA Be−A = B + [A, B] + [A, [A, B]] + ...,
2

(C.5)

ˆ
by setting A = iαξ(y)and B = ψ̂b (x) we find thate−iαξ̂(y) ψ̂b (x)eiαξ̂(y) = eiαθ(y−x) ψ̂b (x).

(C.6)

93
With this formula we getφ̂an (x)φ̂an (y) = eiαθ(y−x) eiαξ̂(x) eiαξ̂(y) ψ̂b (x)ψ̂b (y)(C.7)φ̂an (y)φ̂an (x) = eiαθ(x−y) eiαξ̂(y) eiαξ̂(x) ψ̂b (y)ψ̂b (x).

(C.8)

Now using ψ̂b (x)ψ̂b (y) = ψ̂b (y)ψ̂b (x) and θ(x − y) = θ(y − x) + π, we compare (C.7)
and (C.8) to find thatφ̂an (y)φ̂an (x) = eiαπ φ̂an (x)φ̂an (y)This is the operator version of the interchange of two anyons.

(C.9)