University of trento

facUlty of MatheMatical, Physical and natUral sciences

GradUate ProGraM in theoretical and coMPUtational Physics

FINAL THESIS

Mixtures of atomic Bose gases: ground state and dynamics

thesis advisor candidate

dr. alessio recati alberto sartori

Prof. sandro strinGari

ACADEMIC YEAR 2011-2012

CONTENTS

Contents i

List of Figures iv

Introduction 1

Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 Ultracold gases: cooling and trapping 5

1.1 Phase space density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Role of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Optical forces on neutral atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Trapping techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Dipole force optical traps . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Magnetic traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Cooling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.1 Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.2 Sub-Doppler cooling: the Sisyphus mechanism . . . . . . . . . . . . 10

1.5.3 Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.4 Sympathetic cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Single particle in a harmonic potential 15

2.1 Instantaneous ladder operators . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Time evolution of impurity’s width . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Mixtures of classical gases 21

3.1 Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

i

Contents

3.2 The method of averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Coupled Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Gases in an isotropic harmonic trap . . . . . . . . . . . . . . . . . . . 25

3.3.2 One-dimensional collisional integrals . . . . . . . . . . . . . . . . . 26

3.3.3 Three-dimensional collisional integrals . . . . . . . . . . . . . . . . 32

3.4 Collisionless and hydrodynamic regime . . . . . . . . . . . . . . . . . . . . . 33

3.5 Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Bose-Einstein condensates 39

4.1 Key concepts in BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 One body density matrix and long-range order . . . . . . . . . . . . 39

4.1.2 Order parameter and symmetry breaking . . . . . . . . . . . . . . . 40

4.2 Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Ideal Bose gas in the box . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.2 Ideal Bose gas in the harmonic trap . . . . . . . . . . . . . . . . . . . 42

4.3 Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1 Thomas-Fermi approximation . . . . . . . . . . . . . . . . . . . . . . 44

4.3.2 Hydrodynamic formulation . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 Dynamics of a condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Elementary excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.2 Collective modes in harmonic traps . . . . . . . . . . . . . . . . . . . 51

4.5 One-dimensional BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Mixtures of Bose-Einstein condensates 57

5.1 Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.1 Mixtures in homogeneous case . . . . . . . . . . . . . . . . . . . . . 60

5.2.2 Harmonic trapped mixtures . . . . . . . . . . . . . . . . . . . . . . . 63

6 Numerical results 67

6.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1.1 Computational units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1.2 Collecting the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

ii

Contents

Conclusion 79

Directions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Appendices

A Useful formulas in calculating collision integrals 83

A.1 Momentum gained by a particle in a collision . . . . . . . . . . . . . . . . . . 83

A.2 Kinetic energy gained by a particle in a collision . . . . . . . . . . . . . . . . 83

A.3 Centre of mass kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

B Collective excitations of an ultracold atoms gas 85

Bibliography 87

iii

LIST OF FIGURES

1 Images of the velocity distribution of rubidium atoms at JILA . . . . . . . . . . 3

1.1 Model of an atomic transition in terms of a two-level system . . . . . . . . . . . 7

1.2 One dimensional Sisyphus cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Maximum oscillation amplitude without interaction: comparison between

classical and quantum behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Time evolution of potassium width for no interspecies interaction . . . . . . . 19

2.3 Experimental data for the oscillation of the widths at η= 0 . . . . . . . . . . . . 20

2.4 Oscillations of the width of potassium as a function of time and temperature . 20

3.1 Evolution of normalized functions ∆i (t ) in the collisionless regime . . . . . . . 33

3.2 Evolution of normalized functions ∆i (t ) in the intermediate regime . . . . . . . 34

3.3 Evolution of normalized functions ∆i (t ) in the hydrodynamic regime . . . . . 35

3.4 Experimental data point for the oscillations of the K impurities axial width for

different interaction strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Theoretical oscillations of rubidium and potassium for different values of TK

and σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Theoretical oscillations of rubidium and potassium for different values of σ . . 38

4.1 Excitation spectrum of a homogeneous Bose gas . . . . . . . . . . . . . . . . . . 50

4.2 Excitation frequencies of a condensate in an isotropic harmonic trap . . . . . . 53

4.3 Diagram of state for a trapped 1D Bose gas . . . . . . . . . . . . . . . . . . . . . 55

5.1 Speeds of sound for a mixture of two BEC. . . . . . . . . . . . . . . . . . . . . . . 62

6.1 Numerical breathing oscillations of the mixture . . . . . . . . . . . . . . . . . . 71

6.2 Numerical CM oscillations of the mixture . . . . . . . . . . . . . . . . . . . . . . 72

iv

List of Figures

6.3 Breathing oscillations of the condensate 2 for different values of χ. . . . . . . . 73

6.4 Ground state density profiles of the two component at phase separation and

mixed phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.5 Variation of the breathing frequency with g22 . . . . . . . . . . . . . . . . . . . . 74

6.6 Varying of the CM frequency with g22 . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.7 Evolution in real time of the two condensates in a phase separated regime . . . 76

6.8 Breathing oscillation for different value of N2/N1 . . . . . . . . . . . . . . . . . . 77

v

INTRODUCTION

Every object in the universe, being it the most fundamental particle or the widest cluster

of galaxies, can be classified on the base of its spin: if it is integer the object is said to be a

boson while, if it is semi-integer, the object is said to be a fermion. At the energy scale we

can experience in our life bosons and fermions behave in the same way but, when the

energy (hence the temperature) is sufficiently low, matter unveils its wave-like nature

and the quantum mechanics formalism replaces classical mechanics. In this framework

the spin classification becomes of special importance: particles with integer spin follow

Bose-Einstein statistics while particles with semi-integer spin follow Fermi-Dirac statis-

tics, which implies the Pauli exclusion principle. This thesis is devoted to the study of

some properties of the bosonic part of matter in the vicinity of the absolute zero temper-

ature and, in particular, it deals with experiments with ultracold pure bosonic mixtures

of 41K and 87Rb.

In 1920, the Indian physicist Satyendra Nath Bose wrote a paper [6] on the quantum

statistics of light quanta (photons), in which he argued that Maxwell-Boltzmann statis-

tics might not be the right statistics for microscopic particles, where fluctuations due to

the uncertainty principle will be significant. In his derivation of the right statistics for

a photon gas he assumed that the number of quantum states available to a single pho-

ton of fixed energy is given by the number of phase space cells of volume h3, using only

Einsteins light quantum hypothesis and the laws of statistical physics and obtaining a

version of Planck’s law at the end.

However, European physics journals refused to publish his results, as it was their

contention that Bose was mistaken. Discouraged, he sent his work to Albert Einstein

in 1924, who immediately recognised its significance and arranged its publication in the

Zeitschrift für Physik after he had translated it to German.

Einstein extended Bose’s work to an ideal gas of massive particles, publishing his re-

sults in a first article [12] some months later. In a follow-up article [13], he noted that a

1

Introduction

system of particles satisfying both the Bose statistics and the conservation of the number

of particles should undergo a before unknown phase transition at some critical temper-

ature. Below this critical temperature a macroscopic fraction of all particles “condense”

into one single state of the system, the quantum mechanical ground state. As these con-

densed particles do not contribute to the entropy of the system anymore, Einstein in-

terpreted this phenomenon as a phase transition. While the properties of normal Bose

gases were extensively studied in the decade after Einstein’s 1925 paper, nothing much

happened concerning BEC until 1938. Apparently a major reason was because George

Uhlenbeck criticized Einsteins prediction of a “phase transition” by arguing it would not

occur in a finite system. Second-order phase transitions were not understood yet and

Uhlenbecks criticism was generally accepted.

Although Einsteins prediction was for an ideal Bose gas, that is a non-interacting

gas satisfying the Bose statistics, F. London suggested in 1938 to explain the observa-

tion of superfluidity in liquid 4He, which is a strongly interacting gas, as a manifestation

of a Bose-Einstein condensation. Remarkably, this is still the basis for our present un-

derstanding of superfluidity and superconductivity. In 1995, the existence of BEC was

proven experimentally in helium from measurements of momentum distributions and

in semiconductors, where para-excitons were found to condense.

Also in 1995, pure BEC was observed in systems very different from 4He, namely di-

lute alkali gases [10, 2]. Such gases were confined in a magnetic trap and cooled down to

extremely low temperatures of the order of fractions of microkelvins. The first evidence

for the condensation emerged from time-of-flight measurements. The atoms were left

to expand by switching off the confining trap and then imaged with optical methods. A

sharp peak in the velocity distribution was observed below a certain critical tempera-

ture (figure 1), providing a clear signature for BEC. Over the last years these systems have

been the subject of a research explosion, both experimentally and theoretically. Many

different fields of physics like atomic collision, quantum optics, condensed matter or

even astrophysics contributed ideas and problems to these specific systems displaying

the attractiveness of BEC for researchers. As has been concisely formulated by J. R. An-

glin and W. Ketterle in 2002, “Our field is now at a historic turning point, in which we

are moving from studying physics in order to learn about atom cooling to studying cold

atoms in order to learn about physics.”

The most attractive feature of such a system is that it can be completely described

theoretically due to the weakly interacting nature, and thus it represents a testing ground

for studying more general physical phenomena, such as superfluidity and coherence.

2

Introduction

FIGURE 1: Images of the velocity distribution of rubidium atoms taken from Cornell, 1996 [9].The left frame corresponds to a gas at a temperature just above condensation; the centre frameto a gas just after the appearance of the condensate; in the right frame, after further evaporation,the gas becomes a sample of nearly pure condensate.

Indeed, since Bose-Einstein condensation is the macroscopic occupation of the lowest

available quantum state, the system is described by a single wavefunction. Bose-Einstein

condensate is thus phase-coherent as an optical laser field and it shows all the features

related to superfluidity, as experimentally observed. As a consequence of this, quantum

effects of a single atom which are essentially invisible may be spectacularly amplified up

to a macroscopic level.

Nowadays, after more than fifteen years of experience, we are able to cool small sam-

ples of dilute gases near the absolute zero by means of several different techniques and

we are also able to manipulate their properties by magnetic fields as well as laser light or

low-frequency electromagnetic radiations. The ability to completely control almost all

the parameters of such systems has been the key feature of the success of the work on

cold atoms. Indeed ultracold atoms have proven to be a powerful tool to experimentally

reproduce and simulate a wide range of physical phenomena that have been studied

theoretically but, often, not realized in other fields of physics. Among them the most

notable examples being solid state physics (adding an optical lattice), few-body physics

and quantum computation.

Moreover the above scenario can be enormously enriched when two gases of dis-

3

Introduction

tinguishable atoms are brought near or below the quantum degeneracy threshold. The

first theoretical study of such a system has be done by Ho and Shenoy [17] in 1996 while

the first experimental realization of a binary mixture of Bose-Einstein condensates is re-

ported by Hall et al. in [16]. Also the study of the boson-boson 41K-87Rb mixture has al-

lowed the realization of a binary Bose-Einstein condensate of two different species [24].

Such a mixture will be particularly important in this work.

Outline of this thesis

In this work we will study the problem of a mixture of two ultracold bosonic gases deter-

mining its ground state and focusing on its dynamics, passing from high temperatures

where they behave as classical gas to very low temperature where the Bose-Einstein con-

densation theory must be used.

Some chapters are of purely introductory nature; here we give the necessary infor-

mation and references to understand the next chapters. In chapter 1 we expose the main

experimental methods used to reach such low temperatures, the techniques to trap such

ultracold gases and the physical mechanism at their bases. In chapter 4 we explain what

is a Bose-Einstein condensate in more details, we introduce the Gross-Pitaevskii equa-

tion and its hydrodynamic formulation, we explain what changes if we consider a BEC in

a uniform potential or in a harmonic one and what change reducing the dimensions to

only 1D.

As a background of all others chapters there is an experiment by Catani et al. [8]

where it is studied the breathing oscillation of a potassium impurity in 1D tube of rubid-

ium atoms. In every chapter the exposition is general but at their ends the results derived

are applied to this experiment. In chapter 2 we will see how to describe the oscillation

of a particle in a harmonic potential after the sudden change of trapping frequency. In

chapter 3 we will study the mixture at high temperatures where both gases can be con-

sidered to behave as classical gases. We use here the coupled Boltzmann equations and

the method of averages. In chapter 5 we consider the two gases as BECs and we describe

the ground state and dynamics of the mixture obtaining an equation for the oscillations

of one component under the assumption that its density is much smaller than the other.

Finally, in chapter 6, we verify this result with some numerical simulation for the one

dimensional case.

4

CH

AP

TE

R

1ULTRACOLD GASES:

COOLING AND TRAPPING

1.1 Phase space density

A central concept of quantum mechanics is the so called wave-particle duality which

postulates that all particles exhibit both wave and particle properties. As a reflection of

this concept, to every gas of massive particles in equilibrium at temperature T is associ-

ated a wavelength, the thermal de Broglie wavelength, defined as

λT =√

2πħ2

mkB T.

We can take the average interparticle spacing in the gas to be approximately (1/n)1/3

where n is the density of the gas. When the thermal de Broglie wavelength is much

smaller than the interparticle separation the gas can be considered as a classical gas. On

the other hand, when the thermal de Broglie wavelength is larger than the interparticle

separation, quantum effects will dominate and the gas must be treated as a Fermi gas or

a Bose gas, depending on the nature of the particles.

Therefore, in order to have huge quantum effects, we have to increase the de Broglie

wavelength λT decreasing the temperature and/or decrease the interparticle separation

5

Ultracold gases: cooling and trapping

increasing the density of the gas n. It is then useful to introduce a quantity called phase

space density ρ(r,p, t ) and defined as the probability that a single particle is in a region

dr around r and has momentum dp around p at time t . For a gas of cold atoms it is

convenient to choose the elementary volume for ρ(r,p, t ) to be ħ3 so the phase space

density becomes the dimensionless quantity

ρ = nλ3T.

It is then this quantity that has to be increased in order to reach the degeneracy regime.

This choice is particularly appropriate also for BEC. In fact for a homogeneous ideal Bose

gas at the critical temperature we have

ρ = nλ3Tc

= g3/2(1) = 2.612

showing that for ρ¿ 1 we have T À Tc and for ρÀ 1 we have T ¿ Tc.

The Liouville theorem requires that ρ cannot be increased by use of conservative

forces. For instance, by increasing the strength of the trapping potential of particles in a

trap one can increase the density of the atoms but at the same time the compression of

the sample results in a temperature increase leaving the phase space density unchanged.

It is then necessary to use a force that is not conservative, such a velocity dependent one.

1.2 Role of collisions

Apart for the problem of cooling the gas, we should note that all interacting atomic sys-

tems, with the exception of helium, undergo a phase transition to the solid phase at low

enough temperature. The BEC regime can be achieved only as a metastable state with

thermodynamic equilibrium corresponding to the crystal phase and only if some criteria

are satisfied:

• The density of the gas should be so low that three-body collisions (responsible for

the solidification) are rare. This implies that one should work at extremely low

temperatures, of order of µK. Two-body collisions instead play a crucial role not

only for ensuring kinetic equilibrium but are also at the origin of sizeable interac-

tion effects which characterize in a unique way the behaviour of a Bose-Einstein

condensate.

• The gas should be kept far from any material wall where the interaction with other

atoms would favour the formation of molecules. So the gas should be confined in

magnetic and/or optical trap in regime of ultrahigh vacuum.

6

1.3 Optical forces on neutral atoms

FIGURE 1.1: Modelling of an atomic transition in terms of a two-level system.

1.3 Optical forces on neutral atoms

To classify the basic physical processes which are used for manipulating atoms by light,

it is useful to distinguish two large categories of effects: dissipative (or absorptive) effects

on the one hand, reactive (or dispersive) effects on the other hand.

Let us consider a light beam with frequency ωL propagating through a gas of atoms

with resonance frequencyωA. The incident photons can be absorbed and scattered in all

directions. The corresponding attenuation of the light beam is maximum at resonance.

It is described by the imaginary part of the index of refraction which varies with ωL −ωA

as a Lorentz absorption curve. We will call such an effect a dissipative (or absorptive)

effect. The speed of propagation of light is also modified. The corresponding dispersion

is described by the real part n of the index of refraction with n−1 that varies withωL−ωA

as a Lorentz dispersion curve. We will call such an effect a reactive (or dispersive) effect.

Dissipative and reactive effects also appear for atoms, as a result of their interaction

with photons. Consider for simplicity an atom sitting in a point r, which can be modelled

by a two-level system involving a ground state g and an excited state e, with a radiative

lifetime Γ−1. For relatively low laser intensities, these dissipative and reactive effects can

be understood as a broadening and a shift of the atomic ground state. The broadening

Γ′ is the rate at which photons are scattered from the incident beam by the atom. The

shift ħ∆′ is the energy displacement of the ground state, as a result of virtual absorp-

tions and stimulated emissions of photons by the atom within the light beam mode; it is

called light-shift or AC Stark shift. The expressions for Γ′ and ∆′ as functions of the local

intensity I (r) of the light beam, the saturation intensity IS and the detuning ∆=ωL −ωA

7

Ultracold gases: cooling and trapping

are

Γ′(r) = Γ I (r)

2IS

1

1+ I /IS +4∆2/Γ2 ,

∆′(r) =∆ I (r)

2IS

1

1+ I /IS +4∆2/Γ2 .

These expressions are valid as long as Γ′ and ∆′ are respectively small compared with Γ

and∆. For large detuning (|∆|ÀΓ), Γ varies as 1/∆2 and becomes negligible compared to

∆ which varies as 1/∆. On the other hand, for small detuning, (|∆|¿Γ), Γ is much larger

than ∆. In the high intensity limit these expression are no longer valid and the atomic

scattering rate saturates to Γ/2.

These two type of effects correspond to two type of radiative forces. Dissipative forces

are associate with the transfer of linear momentum from the incident light beam to the

atom in resonant scattering process and are proportional to Γ′. Being ħk the mean mo-

mentum transferred to the atom in an absorption-spontaneous emission cycle (k is the

laser wave vector), the radiation pressure force saturates to ħkΓ/2. The corresponding

acceleration communicated to an atom of mass m is equal to amax = ħkΓ/2m = vR/2τ,

where vR = ħk/m is the recoil velocity of the atom and τ = 1/Γ is the lifetime of the ex-

cited state.

Dispersive forces can be interpreted in terms of position dependent light shift ħ∆′(r)

due to a spatially varying intensity. It plays the role of a potential energy giving rise to a

force F =−ħ∇∆′(r).

1.4 Trapping techniques

In order to confine any object it is necessary to exchange kinetic energy for potential

energy in the trapping field and, in neutral atom traps, the potential energy must be

stored as internal atomic energy. Thus, practical traps for ground-state neutral atoms are

necessarily very shallow compared with thermal energy because the energy-level shifts

that result from convenient size fields are typically considerably smaller than kBT for

T = 1 K. Neutral atom trapping therefore depends on substantial cooling of a thermal

atomic sample, and is often connected with the cooling process.

1.4.1 Dipole force optical traps

The simplest imaginable optical traps are dipole ones. They consist of a single, strongly

focused Gaussian laser beam whose intensity at the focus varies transversely with r .

8

1.5 Cooling techniques

When the detuning is negative (ωL < ωA), light shifts are negative. If the laser beam is

focused, the focal zone where the intensity is maximum appears as a minimum of po-

tential energy, forming a potential well where sufficiently cold atoms can be trapped. If

the detuning is positive, light shifts are positive and can thus be used to produce poten-

tial barriers.

1.4.2 Magnetic traps

Alkali atoms have an unpaired electron resulting in magnetic moments µm of the order

of the Bohr magneton. So they strongly interact with an external magnetic field with an

interaction energy given by E =−µm ·B. Quantum mechanically the energy levels of such

an atom are given by E(mF) = gµBmFB . In order to have an atom trap we need a local

minimum of the magnetic potential energy E(mF). So, for g mF > 0 (weak field seeking

states), this requires a local magnetic field minimum. Strong field seeking states (g mF <0) can not instead be trapped by static magnetic fields because Maxwell’s equations do

not allow a magnetic field maximum in free space. Several different magnetic traps with

varying geometries have been studied in detail in literature giving rise to a large number

of trapping potentials [4].

1.5 Cooling techniques

In order to create a Bose-Einstein condensate in a dilute gas, atoms must be cooled and

compressed in a trap until thermal de Broglie wavelength becomes of the order of the

spacing between atoms. In doing so pre-cooling is a fundamental prerequisite for trap-

ping because conservative atom traps can only confine neutral atoms with a maximum

energy of one kelvin at best. Pre-cooling is done by laser cooling and the final cooling by

evaporation. Table 1.1 shows how these techniques together can reduce the temperature

of atoms by a factor of a billion and increase the phase space density of a factor of 1020.

1.5.1 Doppler cooling

Doppler cooling is based on a Doppler induced imbalance between two opposite radia-

tion pressure forces. Restricting to one dimension, we take two counterpropagating laser

waves with the same weak intensity and the same frequency slightly detuned to the red

(ωL < ωA). For an atom at rest this results in two equal and opposite force and the net

force is then zero. For a moving atom the apparent frequencies of the two laser waves

9

Ultracold gases: cooling and trapping

Temperature Density Phase space densityOven 500 K 1014 cm−3 10−13

Laser cooling 50 µK 1011 cm−3 10−6

Evaporative cooling 500 nK 1014 cm−3 1BEC 107

TABLE 1.1: Multi-stage cooling to BEC in the MIT experiment [10]. Through a combination ofoptical and evaporative cooling the temperature of the gas is reduced by a factor of 109, whilethe density at the BEC transition is similar to the initial density in the atomic oven. In each stepshown the ground state population increases by about 106. BEC can be regarded as free cooling asit increases the quantum occupancy by another factor of about a milion without any extra effort.Table taken from [19].

are Doppler shifted. The counterpropagating one is closer to the resonance and exerts a

stronger radiation pressure than the copropagating one, thus leading to a net force op-

posite to the atomic velocity v and proportional to v for small velocities, F = −αv . By

using three pairs of counterpropagating laser waves along three orthogonal directions,

one can damp the atomic velocity in a very short time, on the order of a few microsec-

onds, achieving what is called an optical molasses.

Doppler friction is necessarily accompanied by spontaneous emission of photons

that can communicate to the atoms a random recoil momentum ħk responsible for a

momentum diffusion. This interplay between friction and diffusion leads to a steady

state with an equilibrium temperature always larger than the Doppler limit kBTD =ħΓ/2.

For alkali atoms this temperature is on the order of 100 µK.

1.5.2 Sub-Doppler cooling: the Sisyphus mechanism

Most atoms, in particular alkali atoms, have a Zeeman structure in the ground state and

both differential light shifts and optical pumping transitions exist for the various Zee-

man sublevels of the ground state. Furthermore the laser polarization can, in general,

vary in space so that light shifts and optical pumping rates are position dependent. This

two effects lead to a very efficient cooling mechanism which explains the sub-Doppler

temperatures reached in optical molasses.

We take as an example the system of figure 1.2. Two counterpropagating laser waves

with same frequency, same intensity and orthogonal linear polarization generate a field

whose polarization changes from σ+ to σ− every λ/4. Let us consider an atomic ground

state with angular momentum 1/2 and two Zeeman sublevels with mg =±1/2. Because

of the different light shifts the Zeeman degeneracy in zero magnetic field is removed

10

1.5 Cooling techniques

FIGURE 1.2: One dimensional Sisyphus cooling. The laser configuration is formed by two coun-terpropagating waves along the z-axis with orthogonal linear polarizations. The polarization ofthe resulting field is spatially modulated with a period λ/2. For an atom with two ground Zeemansublevels mg =±1/2, the spatial modulation of the laser polarization results in correlated spatialmodulations of the light shifts of these two sublevels and of the optical pumping rates betweenthem. Because of these correlations, a moving atom runs up potential hills more frequently thandown.

giving the energy diagram shown in figure. In addition there are also real absorptions

of photons by the atom followed by spontaneous emissions which give rise to optical

pumping transfer between two levels: from mg = −1/2 to mg = +1/2 for a σ+ polariza-

tion and vice versa for a σ− polarization.

These two effects are due to the same cause and they occur with the same periodλ/2.

So, with the proper sign of the detuning, optical pumping always transfer atoms from the

higher Zeeman sublevel to the lower one. In this way an atom moving along the z-axis

runs up potential hills more frequently than down. When it climbs a potential hill its

kinetic energy is transformed in potential energy which is then dissipated by light since

the spontaneously emitted photon has an energy higher than the absorbed laser photon.

When the energy of the atom becomes smaller than U0 (depth of the optical potential)

the atom remains trapped in the potential wells.

Therefore Sisyphus cooling leads to temperatures TSis such that kBTSis 'U0 with U0

proportional to I /∆. We can not however decrease indefinitely the laser intensity be-

cause the recoil due to the spontaneously emitted photon increases the kinetic energy

of an atom by an amount on the order of the recoil energy, ER = ħ2k2/2m. When U0 be-

11

Ultracold gases: cooling and trapping

comes of the order of ER the Sisyphus cooling no longer works. For rubidium atoms this

limit temperature is of the order of a few microKelvins.

1.5.3 Evaporative cooling

Evaporative cooling is done by continuously removing the high-energy tail of the ther-

mal distribution of the atoms from the trap. The evaporated atoms carry away more

than the average energy, which means that the temperature of the remaining atoms de-

creases. The high energy tail must be constantly repopulated by collisions, thus main-

taining thermal equilibrium and sustaining the cooling process. So, the essential con-

dition for evaporative cooling is a long lifetime of the atomic sample compared to the

collisional thermalization time. Trapped atom clouds are extremely dilute (about ten or-

ders of magnitude less dense than a solid or a liquid) and collisional thermalization can

take seconds.

In rf-induced evaporation, the rf radiation flips the atomic spin. As a result, the at-

tractive trapping force turns into a repulsive force and expels the atoms from the trap.

This scheme is energy selective because the resonance frequency is proportional to the

magnetic field, and therefore to the potential energy of the atoms. In the case of tran-

sitions between magnetic sublevels mF, the resonance condition for the magnetic field

strength B is |g |µBB =ħωrf. Since the trapping potential is given by mFgµB[B(r )−B(0)],

only atoms which have a total energy E > ħ|mF|(ωrf −ω0) will evaporate (ω0 is the rf fre-

quency which induces spin flips at the bottom of the trap).

Another method to get evaporation consists in lowering the depth of the trap thereby

allowing the atoms with energies higher than the depth to escape. But rf induced evap-

oration has several advantages over other evaporation methods. First of all, the evap-

oration process can be completely separated from the design of the magnetic trapping

potential. Furthermore, atoms evaporate from the whole surface where the rf resonance

condition is fulfilled. This makes the evaporation three-dimensional in velocity space,

and therefore very efficient.

1.5.4 Sympathetic cooling

When we have a mixtures of two different gases we can use the so-called sympathetic

cooling technique. It consists in cooling one of the two species (for instance by evapora-

tive cooling) and then letting the second one cool by elastic collisions with the evapora-

tively cooled one. [25]

12

1.5 Cooling techniques

In order for this method to work we must have overlap between the two clouds and

high interspecies collision rate permitting thermalization. Advantages are small loss

rates for the second gas and high efficiency also in cooling atoms for which evaporative

cooling works bad. For example it would be very difficult to cool fermionic atoms into a

highly degenerate regime using normal evaporative cooling because of the requirement

of a large number of elastic collisions per trap lifetime, combined with the vanishing elas-

tic collision rate in low-temperature spin-polarized fermionic gases. However this can be

done using bosonic atoms as a working fluid to sympathetically cooling. This technique

also allows to cool to BEC the many species for which inelastic processes make it impos-

sible to obtain high enough densities for conventional evaporative cooling.

13

CH

AP

TE

R

2SINGLE PARTICLE IN A HARMONIC

POTENTIAL: SUDDEN CHANGE OF

TRAPPING FREQUENCY

In this chapter we will study the very particular problem of the breathing oscillations of

a particle in a harmonic potential after a sudden change of its trapping frequency from

an initial to a final value. We will restrict to the 1D case not only for simplicity but also

to apply the results to the experiment cited in the introduction [8]. As written in this

article, the initial trapping frequency corresponds to ω0/(2π) = 1 kHz (the species selec-

tive dipole potential frequency compressing the particles at the centre of the tubes) and

the final one to ω2/(2π) = 87 Hz (the residual trapping frequency of the optical potential

along x-direction of the tubes).

This will be done using the formalism of instantaneous ladder operators.

2.1 Instantaneous ladder operators

In such a system the impurity is described by the time-dependent Hamiltonian

H(t ) = p2

2m2+ 1

2m2ω(t )2x2

15

Single particle in a harmonic potential

which can be easily diagonalized to the form H = ħω(t )a(t )†a(t ) by introducing the in-

stantaneous ladder operators (see for example [30] for a recent application in the context

of cold gases)

a(t ) = 1p2ħ

(√m2ω(t )x + i

1pm2ω(t )

p

),

a(t )† = 1p2ħ

(√m2ω(t )x − i

1pm2ω(t )

p

).

Calculating their time evolution we get a mixing between a and a† and, once the func-

tional form ofω(t ) is fixed, it is easy to use it for calculating whatever observable we want.

In the case of our interest, namely the case of a sudden change of frequency, the ladder

operators simply become the following [21]

a(t ) = 1

2

[(√ω2

ω0+

√ω0

ω2

)a(0)+

(√ω2

ω0−

√ω0

ω2

)a(0)†

]e−iω2t , (2.1)

a(t )† = 1

2

[(√ω2

ω0+

√ω0

ω2

)a(0)† +

(√ω2

ω0−

√ω0

ω2

)a(0)

]eiω2t . (2.2)

2.2 Time evolution of impurity’s width

We now use these operators in order to get the width of the impurity. For doing so we

have to evaluate the expectation value of the square of the position operator which, writ-

ten in terms of creation and annihilation operators, is

x(t ) =√

ħ2mω2

[a(t )† +a(t )

].

Let us start by taking the case with T = 0 where we can consider the impurity occu-

pying only the ground state of the harmonic oscillator that in the following will be de-

noted by |0⟩. For brevity we rewrite equation (2.1) as a(t ) =(αa0 +βa†

0

)e−iω2t and (2.2)

as a(t )† =(αa†

0 +βa0

)eiω2t so in calculating

⟨0∣∣x2(t )

∣∣0⟩

we have to evaluate the expec-

tation values of a0a0, a†0a†

0, a†0a0 and a0a†

0. Clearly the first three vanish so we are left

with

⟨0∣∣x2(t )

∣∣0⟩= ħ

2mω2

∣∣αe−iω2t +βeiω2t∣∣2

⟨0∣∣∣a0a†

0

∣∣∣0⟩

= ħ4mω2

(ω2

ω0+ ω0

ω2

)[1+

(ω2

2 −ω20

ω22 +ω2

0

)cos(2ω2t )

]. (2.3)

16

2.2 Time evolution of impurity’s width

Some observation are now required. First of all we notice that the oscillation frequency

is exactly 2ω2 as measured in the experiment, moreover the value of the width at t = 0 is

x2(t = 0) =ħ/2mω0 as expected, i.e. the quantum expectation value of the square radius

in the presence of an harmonic potential with frequency ω0.

Another interesting quantity is the maximum amplitude of the oscillation that we

reach for cos(2ω2t ) =−1 because of his negative coefficient. With this substitution equa-

tion (2.3) becomes

x2max =

ħ2mω2

ω0

ω2À ħ

2mω2

and it shows that the oscillations of the impurity can be much greater than the mean

value in trap.

If the temperature is not zero the impurity is no longer in a pure state but in a mixed

state, i.e. a statistical mixture of pure states distributed according to the Gibbs measure.

Therefore we have to take into account the occupation of excited levels introducing the

density operator

ρ = 1

Z

∑n

e−βεn |n⟩⟨n| with Z =∑n

e−βεn ,

where the basis |n⟩ is formed by the eigenvectors of the harmonic oscillator, and we must

calculate the trace of the operator ρx2 in order to obtain the evolution of the width. Call-

ing C = ∣∣αe−iω2t +βeiω2t∣∣2

we get

⟨x2⟩ (t ) = Tr

[ρx2]=∑

m

⟨m

∣∣ρx2∣∣m

⟩= ħ

2mω2

C

Z

∑n

e−βεn∑m

⟨m |n ⟩︸ ︷︷ ︸δmn

⟨n

∣∣∣(a0a†0 +a†

0a0

)∣∣∣m⟩

= ħ2mω2

C

Z

∑n

e−βεn (2n +1).

At this point we substitute the explicit expression for the energy of a harmonic oscilla-

tor with frequency ω0 in one dimension, εn = ħω0(n +1/2), and we use the well-known

formulas∑

n e−An = (1−e−A)−1 and∑

n ne−An = e−A(1−e−A)−2 to express

⟨x2⟩ (β, t ) = ħ

2mω2C

(1+ 2

eβħω0 −1

)= ħ

2mω2

(ω2

ω0+ ω0

ω2

)[1+

(ω2

2 −ω20

ω22 +ω2

0

)cos(2ω2t )

](1+ 2

eβħω0 −1

). (2.4)

As we can see, this equation is totally identical to the one at T = 0 (2.3) except for the last

piece which takes into account finite temperature effects. In fact, letting T go to zero,

17

Single particle in a harmonic potential

ħω0 = kB T

0 50 100 150 200 250 300 350 400 4500

2

4

6

8

10

12

14

16

18

T (nK)

p x2

(µm

)

FIGURE 2.1: Maximum oscillation amplitude of impurity and surrounding gas without interspe-cies interaction as a function of temperature. The blue line represents the full behaviour, i.e. eq.(2.4) with cosine taking the value −1. The violet and orange ones represent respectively the im-purity and surrounding gas maximum classical values, the second of equations (2.5) with m1ω

21

and m2ω22. The coloured regions show the declared experimental situation where T = 350(50) nK.

i.e. ħω0 À kB T or βħω0 →∞,(eβħω0 −1

)−1vanishes and we are left with equation (2.3)

where only the ground state is occupied.

In the opposite case of “high” temperature, i.e. ħω0 ¿ kB T or βħω0 → 0 (for ω0 =1 kHz as in the experiment we have ħω0/kB ' 48 nK), the last piece becomes 2kB T /ħω0.

Also in this regime we analyse the behaviour of the maximum amplitude and initial value

of the oscillation obtaining

x2(t = 0) = kB T

mω2ω0

ω2

ω0= kB T

mω20

and x2max =

kB T

mω2ω0

ω0

ω2= kB T

mω22

. (2.5)

By equipartition theorem it follows that 12 mω2

⟨x2

⟩ = 12 kB T and this is exactly what we

got. We start at t = 0 having the mean value in the narrow trap and then it evolves reach-

ing the mean value in the broadest one as the maximum of oscillation.

In conclusion we observe that the quantum behaviour differs from the classical one

only at very low temperature (kB T < ħω0, see figure 2.1) where quantum fluctuations

raise the value of maximum amplitude which classically would tend to zero. At higher

18

2.3 Comparison with experiment

0 2 4 6 8 10 12 14 160

5

10

15

20

25

t (ms)

p x2

(µm

)

FIGURE 2.2: Time evolution of potassium width for no interspecies interaction. The red linerepresents the case in which TK = 460 nK and the black one is for TK = 830 nK. It is also indicated

the classical value of rubidium width√

kB T /mRbω2Rb with TRb = 460 nK.

temperature the two predictions become undistinguishable but they remain above the

rubidium one because of the different parameter of two species (ω and m).

2.3 Comparison with experiment

We used the formalism of instantaneous ladder operators applied to the problem of har-

monic oscillator with a sudden frequency change in order to get an equation that de-

scribes the oscillation of the width of a particle in this potential. Now we want to com-

pare this results with those obtained from experimental data [8].

In figure 2.3 are shown the results of the experiment. The first interesting feature is

that the frequency, within the error bars, is exactly 2ω2 so there is no shift. We also ob-

serve that, even in the absence of interspecies interactions, a residual damping occurs,

likely due to intraspecies collisions in tubes with more than one K atom (impurity), to

the anharmonicity of the trapping potential and to the fact that the the interspecies in-

teraction is zero within a certain value. Finally the maximum amplitude reached by the

19

Single particle in a harmonic potential

FIGURE 2.3: Experimental data for the oscillation of the width of potassium and rubidium atη= 0. Image is taken from [8].

FIGURE 2.4: Oscillations of the width of potassium as a function of time and temperature (seeequation 2.4) with a sudden change of trap frequency, from ω0/(2π) = 1 kHz to ω2/(2π) = 87 Hz.Values of other parameters are taken from [8].

potassium is larger than the one for rubidium (surrounding gas).

We will try to explain all this properties of the oscillations in view of equation (2.4).

First of all we notice that a rubidium width of about 17 µm corresponds to a temperature

TRb = 460 nK that in the next we will take as reference temperature. Assuming the potas-

sium initially thermalized with rubidium, we obtain a maximum amplitude just above

equilibrium rubidium value (figure 2.2, blue line) in contrast with what observed in the

experiment (figure 2.3). The only way to reproduce the experimental data is to take an

initial TK of about 830 nK (figure 2.2, violet line) that is almost double of rubidium one.

In any case, with no interspecies interactions and with the assumption of having only

one potassium particle for tube, no damping is expected.

We finally report in figure 2.4 the full behaviour of oscillation as a function of tem-

perature and time.

20

CH

AP

TE

R

3MIXTURES OF CLASSICAL GASES

When the temperature of a Bose gas is notably larger than the critical temperature for

Bose-Einstein condensation, the mean-field effects become less important and the col-

lisional terms can no longer be ignored. At this point the dynamical behavior of a dilute

gas is well described by the Boltzmann equation.

3.1 Boltzmann equation

We consider a dilute gas of N particles of mass m in an external potential V (r). We as-

sume that the temperature is sufficiently high and the density sufficiently low so that the

particles can be described as localized wave packets whose extensions are small com-

pared to the average interparticle separation. This is to say that the de Broglie wavelength

λT of a particle has to be much smaller than the average interparticle separation:

n1/3λT = n1/3

√2πħ2

mkB T¿ 1 (3.1)

The particles interact with each other through collisions whose nature is specified by a

given differential scattering cross section dσ/dΩ. In addition we consider a dilute gas so

that only binary collisions can happen.

Since we are interested in the collective motion of the system, we do not care about

the dynamics of every single particle but instead we want to derive the equation of mo-

21

Mixtures of classical gases

tion for the distribution function of the gas f (r,v, t ), namely the Boltzmann equation:(∂

∂t+v ·∇r + F

m·∇v

)f (r,v, t ) =

(∂ f

∂t

)coll

The right hand side of this equation takes into account all the changes in the distribu-

tion function coming from the collisions between particles and, for this reason, is called

collisional integral, Icoll[ f ]. Using the assumption of molecular chaos, that is that the

momenta of two particles in the volume element d3r are uncorrelated so that the proba-

bility of finding them simultaneously is the product of finding each alone, we can rewrite

the Boltzmann equation in this way:(∂

∂t+v1 ·∇r + F

m·∇v1

)f (v1) =

∫d3v2d2Ω|v1 −v2|dσ

dΩ

[f (v′1) f (v′2)− f (v1) f (v2)

](3.2)

where v1,v2 are the velocities of the particles before the collision and v′1, v′2 the velocities

of the particles after the collision.

From this equation it is easy to derive the equilibrium distribution of the gas f0(r,v)

solving ∂ f0(r,v)/∂t = 0. This is the well known Maxwell-Boltzmann distribution:

f0(r,v) =(

m

2πkB T

)3/2

e−m(v2

x+v2y +v2

z )

2kB T e− V (r)

kB T

3.2 The method of averages

Starting from equation (3.2) we can derive equations for the average of a general dynam-

ical quantity χ(r,v), where the average is taken in both position and velocity space and,

at every time t , the distribution function is normalized to particles number N :⟨χ⟩ = 1

N

∫d3r d3v f (r,v, t )χ(r,v)

We apply ∂/∂t on both sides of this definition and we use (3.2) to express ∂ f (r,v)/∂t

obtaining

∂⟨χ⟩

∂t= 1

N

∫d3r d3v

∂ f (r,v, t )

∂tχ(r,v)

= 1

N

∫d3r d3v

[−v ·∇r f (r,v, t )− F

m·∇v f (r,v, t )+ Icoll[ f ]

]χ(r,v)

= 1

N

∫d3r d3v

f (r,v, t )

[v ·∇rχ(r,v)+ F

m·∇vχ(r,v)

]+ Icoll[ f ]χ(r,v)

= ⟨

v ·∇rχ⟩+⟨

F

m·∇vχ

⟩+⟨

Icollχ⟩

(3.3)

22

3.2 The method of averages

In the second step we integrated by part killing the surface term and in the last one, with

a little notation abuse, we indicated by⟨

Icollχ⟩

the integral over space and velocity of the

product Icoll[ f ]χ(r,v). Now we want to find a more useful expression for this last piece.

We immediately notice that∫d3v1χ(v1)Icoll[ f ] =

∫d3v1d3v2d2Ωχ(v1)|v1 −v2|dσ

dΩ

[f ′

1 f ′2 − f1 f2

]=

∫d3v1d3v2d2Ωχ(v2)|v1 −v2|dσ

dΩ

[f ′

1 f ′2 − f1 f2

](3.4)

because the integral is symmetric by exchange of v1 and v2 in binary collisions. Moreover,

observing that d3v1d3v2 = d3v ′1d3v ′

2 and that [(dσ/dΩ)dΩ] = [(dσ/dΩ)dΩ]′, we obtain∫d3v1χ(v1)Icoll[ f ] =

∫d3v ′

1d3v ′2χ(v′1)|v′1 −v′2|

[dσ

dΩdΩ

]′ [f1 f2 − f ′

1 f ′2

]=−

∫d3v1d3v2χ(v′1)|v1 −v2|

[dσ

dΩdΩ

][f ′

1 f ′2 − f1 f2

](3.5)

=−∫

d3v1d3v2χ(v′2)|v1 −v2|[

dσ

dΩdΩ

][f ′

1 f ′2 − f1 f2

](3.6)

Putting together equations (3.4), (3.5) and (3.6) we finally obtain

4⟨

Icollχ⟩ = 4

N

∫d3r d3v1χ(v1)Icoll[ f ] = 1

N

∫d3r d3v1[χ1 +χ2 −χ′1 −χ′2]Icoll[ f ]

This formula is very useful because it shows that the collisional contribution⟨

Icollχ⟩

is

equal to zero if χ corresponds to a dynamic quantity conserved during the elastic col-

lision, i. e. if χ1 +χ2 = χ′1 +χ′2. This happens if χ can be written in the form χ(r,v) =a(r)+b(r) ·v+ c(r)v2. In fact, in this case, by conservation of number of atoms, momen-

tum and energy we get immediately the equality. But this is true only if we have collisions

between particles with same mass otherwise the factors in front of v and v2 are different.

If we want to study the time evolution of the average of a general dynamic quantity

χ(r,v) we have to use equation (3.3). But the term proportional to ∇rχ and ∇vχ can put

other dynamic quantities into the equation. The method of averages [15] consists in

using equation (3.3) also for every new quantity obtained in this way, until one gets a

closed set of differential equations ready to be solved.

3.2.1 Some examples

To better understand how to apply this method practically, we start analysing the mono-

pole mode (breathing mode) for a dilute gas in a harmonic isotropic trapping potential

Vtrap(x, y, z) = 1

2m

(ω2

x x2 +ω2y y2 +ω2

z z2)= 1

2mω2

0r 2.

23

Mixtures of classical gases

We are interested in the time evolution of the square radius of the cloud so we apply

equation (3.3) to χ(r) = r2 finding

d⟨

r2⟩

dt= 2⟨r ·v⟩ . (3.7)

In order to obtain a closed set of equations we also need to evaluate the time derivative

of ⟨r ·v⟩ and⟨

v2⟩

:

d⟨r ·v⟩dt

= ⟨v2⟩−ω2

0

⟨r2⟩ , (3.8)

d⟨

v2⟩

dt=−2ω2

0 ⟨r ·v⟩ . (3.9)

The collisional term does not contribute to the above equations because all the consid-

ered dynamic quantities are conserved during the collision. So there is no damping for

the breathing mode of a classical gas in a harmonic isotropic trap.

To obtain the frequency of oscillation of the breathing mode, we take the second

derivative with respect to time of equation (3.7) and we use equations (3.8) and (3.9)

getting

d2⟨

r2⟩

dt 2 =−4ω20

⟨r2⟩ .

By looking for solution of this equation evolving in time as e iωt we immediately find the

result ω = 2ω0 holding for all collisional regimes from the collisionless to the hydrody-

namical one.

It is interesting to compare this result with the one of a Bose-Einstein condensed gas

at temperature T = 0. In fact in the latter case the monopole oscillation is still undamped

but the frequency is ω=p5ω0 as we can see by equation (4.22).

3.3 Coupled Boltzmann equations

Up to now we considered only one gas and its oscillations: we have one distribution

function, we have no damping in the breathing mode and we do not have to calculate

any collisional integral. We want now to add some complication considering the case of

a two-component gas with intraspecies and interspecies interactions. Moreover, since

the basic mechanism described by the Boltzmann equation is binary collisions, two-

component systems exhibit all the complexities of n-component systems.

24

3.3 Coupled Boltzmann equations

In this case the dynamics of the gas is described by a set of two coupled Boltzmann

equations for each phase-space distribution function fi (r,vi , t ):(∂

∂t+vi ·∇r + Fi

mi·∇vi

)f (r,vi , t ) = Ii i [ fi ]+ Ii j [ fi , f j ], i 6= j . (3.10)

Obviously, this time, two collisional integrals appear on the right hand side of equa-

tion (3.10) as the result of the possibility for fi to change through both self-collisions (i−i

collisions) or cross collisions (i − j collisions). The expression for the new collisional in-

tegral is

Ii j[

fi , f j]= ∫

dΩd3vB |vB −vA|dσi j

dΩ

[fi (r,v′A, t ) f j (r,v′B, t )− fi (r,vA, t ) f j (r,vB, t )

], (3.11)

where i , j = 1,2 represent the two species and the particles involved in the collision are

labelled by A and B .

In the next we simplify the problem by an appropriate change of variables introduc-

ing the centre of mass velocity v0, the relative velocity vr ,v0 = mAvA +mBvB

mA +mB,

vr = vA −vB,(3.12)

the total mass and the reduced mass, respectively M = mA +mB and µ = mAmB/M . By

energy and momentum conservation it is easy to see that v′0 = v0 and |v′r | = |vr |, relations

that will be useful in proceeding.

Whit a small change, equation (3.3) for a dynamical quantity χ(r,vi ) now reads as

∂⟨χ⟩

i

∂t−⟨

vi ·∇rχ⟩

i −⟨

Fi

mi·∇viχ

⟩i= ⟨

χIi j⟩

i +⟨χIi i

⟩i . (3.13)

Also in this case we have⟨χIi i

⟩i = 0 if χ is in the form χ(r,vi ) = a(r)+b(r) ·vi +c(r)v2

i but

it is no longer true for the cross term⟨χIi j

⟩i . With this equation we can derive relations

governing the time evolution of quantities we are interested in and, using the method of

averages, we can solve them.

3.3.1 Gases in an isotropic harmonic trap

For the sake of simplicity we consider now the thermalization of an atomic mixture con-

fined in a harmonic and isotropic trap (different for each species):

Ui (x, y, z) = 1

2miω

2i

(x2 + y2 + z2) .

25

Mixtures of classical gases

The force acting on each atom is then Fi = −∇Ui = −miω2i

(x, y, z

) = −miω2i r. Starting

from the square radius, we apply the method of averages for both species using equation

(3.13), obtaining six coupled equations

d⟨

r2⟩

i

dt= 2⟨r ·vi ⟩i , (3.14)

d⟨r ·vi ⟩i

dt= ⟨

v2i

⟩i −ω2

i

⟨r2⟩

i +⟨

r ·vi Ii j⟩

i , (3.15)

d⟨

v2i

⟩i

dt=−2ω2

i ⟨r ·vi ⟩i +⟨

v2i Ii j

⟩i . (3.16)

The momentum and kinetic energy equations for the individual species (3.15 and 3.16)

include a contribution from the interspecies collision term, expressing the fact that are

the total momentum and total kinetic energy of the system that are collision invariants

rather than the individual species momentum and kinetic energy. These dissipative con-

tributions to the time evolution equations do not appear in the equation for the square

radius (3.14) since the collision is local and the number of atoms for each species is con-

served.

As we can easily see, in the absence of interspecies interactions these equations be-

come (3.7), (3.8) and (3.9) and they describe the undamped monopole mode for the two

species independently. In fact the collisional integral Ii i do not contribute because all

the dynamic quantities considered are collision invariant for intraspecies collisions.

So, all we need to do is to calculate the collisional integrals and then, analytically or

numerically, to solve the equations.

3.3.2 One-dimensional collisional integrals

We start considering the one-dimensional case not only because it is easier to deal with,

but also because it is more similar to the experimental setting.

The main differences from the three-dimensional case are in conservation laws that

become v ′0 = v0 and v ′

r = ±vr , in removing the integration on solid angle and in chang-

ing dσ(vr ,Ω)/dΩ to σ(|vr |). In this case the cross section is entirely determined by the

relative momentum of colliding particles, pr =µvr .

In order to calculate the collisional integrals we have to make an ansatz on the form

of the phase-space distribution function fi (r, vi , t ) of each species. We choose a Gaus-

sian ansatz with the inclusion of a small factor taking into account the space-velocity

26

3.3 Coupled Boltzmann equations

correlations:

fi (r, vi , t ) =Ni exp

(−mi v2

i +miω2i r 2

2kB Ti

)exp

(ηi mi r vi

)≈Ni exp

(−mi v2

i +miω2i r 2

2kB Ti

)(1+ηi mi r vi

), (3.17)

where Ni = miωi Ni /2πkB Ti is the normalization factor and Ti and ηi are the only time-

dependent variables. Such an ansatz is inspired by the exact solution for the phase-space

distribution function for the monopole mode, and provides a natural generalization of

the local equilibrium distribution. In [15] the Gaussian ansatz was shown to be accurate

for one species in investigating the damping of the quadrupole oscillation and in general

the transition between the hydrodynamic and the collisionless regimes of this mode.

Calculation of square velocity collisional integral

We begin from⟨

v2i Ii j

⟩i

(the calculation for⟨

r vi Ii j⟩

i will be analogous) introducing two

new quantities Σ± =Σ1 ±Σ2, where

Σi ≡⟨

Ni mi v2i

2Ii j

⟩i

.

The global conservation of kinetic energy leads immediately to Σ+ = 0 so we manipulate

the expression forΣ− using equations (3.11), (3.3) and symmetry properties of collisional

integrals, obtaining

Σ− =⟨

N1m1v21

2I12

⟩1

−⟨

N2m2v22

2I21

⟩2

= 1

N1

∫dr dv1

N1m1v21

2I12 − 1

N2

∫dr dv2

N2m2v22

2I21

=∫

dr dv1dv2|v1 − v2|(

m1v21 −m2v2

2

2

)σ(|v1 − v2|)Ξ1′2′

12

whereΞ1′2′12 = f1(r, v ′

1, t ) f2(r, v ′2, t )− f1(r, v1, t ) f2(r, v2, t ) andΞ1′2′

12 =−Ξ121′2′ . We now change

variables using transformations (3.12) whose Jacobian is one. So we get

Σ− =∫

dr dv0dvr |vr |A(v0, vr )σ(|vr |)Ξ1′2′12 (3.18)

where A(v0, vr ) ≡ m1v21 −m2v2

2

2= v2

0

2(m1 −m2)+2µv0vr +

µ2v2r

2

(1

m1− 1

m2

)

27

Mixtures of classical gases

which implies that A(v0, vr )− A(v0, v ′r ) = 2µv0(vr − v ′

r ) by conservation law on vr .

At this point we use a little trick summing on both sides of equation (3.18) the quan-

tity ∫dr dv ′

0dv ′r |v ′

r |A(v ′0, v ′

r )σ(|v ′r |)Ξ12

1′2′

which is exactly Σ− (we have only exchange all the v with the v ′). Noticing that dv ′0dv ′

r =dv0dvr and, because of conservation of centre of mass velocity and modulus of relative

velocity during the collision, |v ′r |A(v ′

0, v ′r )σ(|v ′

r |) = |vr |A(v0, v ′r )σ(|vr |), we obtain

Σ− = 1

2

∫dr dv0dvr |vr |

[A(v0, vr )− A(v0, v ′

r )]σ(|vr |)Ξ1′2′

12

=µ

∫dr dv0dvr |vr |

[v0(vr − v ′

r )]σ(|vr |)Ξ1′2′

12 . (3.19)

A minus sign appeared in order to exchange subscripts and superscripts of Ξ.

We have now to expand the quantity Ξ1′2′12 around the final equilibrium temperature

of the two gases T f . We do not consider the terms proportional to ηi because they are

odd functions of r and in equation (3.19), when integrated over all space, they give zero.

Terms proportional to η2i are instead omitted because they give contributions of higher

order. So we expand the initial temperatures of two gases in this way

Ti = T f + (Ti −T f ) =(1+ Ti −T f

T f

)T f = (1+∆Ti )T f ,

with ∆Ti being small variations.

Recalling our Gaussian ansatz for phase-space distribution functions (3.17), we see

that the temperature appears only in the denominator of the exponent. Therefore we use

the well-know expansion (1+x)α ≈ 1+αx for x ¿ 1, obtaining

exp

(− gi

Ti

)= exp

[− gi

T f (1+∆Ti )

]≈ exp

[− gi

T f(1−∆Ti )

]= exp

(− gi

T f

)exp

(gi

T f∆Ti

),

where gi = (mi v2i +miω

2i r 2)/2kB . In this way, using the conservation of kinetic energy

that implies g1 + g2 = g ′1 + g ′

2, we can expand Ξ1′2′12 to get

Ξ1′2′12 ≈N1N2 exp

(−g1 + g2

T f

)[exp

(g ′

1

T f∆T1

)exp

(g ′

2

T f∆T2

)−exp

(g1

T f∆T1

)exp

(g2

T f∆T2

)].

Now, being∆Ti ¿ 1, we can expand again the exponential neglecting terms proportional

to ∆T 2i or to ∆Ti∆T j , obtaining

Ξ1′2′12 ≈N1N2 exp

(−g1 + g2

T f

)[(1+ g ′

1

T f∆T1

)(1+ g ′

2

T f∆T2

)−

(1+ g1

T f∆T1

)(1+ g2

T f∆T2

)]=N1N2 exp

(−g1 + g2

T f

)[∆T1

T f

(g ′

1 − g1)− ∆T2

T f

(g2 − g ′

2

)].

28

3.3 Coupled Boltzmann equations

But g ′1 − g1 = (m1v ′

12 −m1v2

1)/2kB = (m2v22 −m2v ′

22)/2kB = g2 − g ′

2 for conservation of

kinetic energy and thus we can write

Ξ1′2′12 ≈N1N2 exp

(−g1 + g2

T f

)(g ′

1 − g1) ∆T1 −∆T2

T f

The expression we obtained is still written as a function of vi and v ′i , to transform

it into a function of v0, vr and v ′r we use expressions (A.3) and (A.2). We also note that

∆T1 −∆T2 = (T1 −T2)/T f , so

Ξ1′2′12 ≈−N1N2

(µ

kB T f

)(T1 −T2

T f

)[v0(vr − v ′

r )]

exp

(−M v2

0 +µv2r +m1ω

21r 2 +m2ω

22r 2

2kB T f

).

We can now insert this expression into equation (3.19) but, before doing this, we remem-

ber that, from conservation law on vr , we have v ′r = ±vr so vr − v ′

r becomes 2vr . With

this in mind we finally obtain

Σ− ≈−N1N2

(µ2

kB T f

)(T1 −T2

T f

)∫ +∞

−∞dr exp

(−m1ω

21r 2 +m2ω

22r 2

2kB T f

)

×∫

dv0dvr |vr |[v0(vr − v ′

r )]2σ(|vr |)exp

(−M v2

0 +µv2r

2kB T f

)(3.20)

=−N1N2

(µ2

kB T f

)(T1 −T2

T f

)∫ +∞

−∞dr exp

(−m1ω

21r 2 +m2ω

22r 2

2kB T f

)

×4∫ +∞

−∞dv0v2

0 exp

(− M v2

0

2kB T f

)∫ +∞

−∞dvr |vr |v2

rσ(|vr |)exp

(− µv2

r

2kB T f

). (3.21)

The first two integrals are standard Gaussian integrals and can easily be calculated, for

calculating the third one we have to know the expression ofσ(|vr |), at the moment we in-

dicate this last integral with ⟨σ⟩vr . This quantity has the dimensions of a square velocity

being the 1D cross section σ(|vr |) dimensionless.

As can be seen in equation (3.16), we would like to express the collisional integral in

the form⟨

v2i

⟩/τ with τ having the dimension of a time. For doing so we recall the fact

that the two gases are separately thermalized, therefore we can express their tempera-

tures using the equipartition theorem or calculating the square velocity mean value over

distribution fi , obtaining

T1 −T2

T f= m1

⟨v2

1

⟩1 −m2

⟨v2

2

⟩2

kB T f.

29

Mixtures of classical gases

We only need to solve the systemΣ1 +Σ2 = N1m1

2

⟨v2

1 I12⟩

1 +N2m2

2

⟨v2

2 I12⟩

2 = 0

Σ1 −Σ2 = N1m1

2

⟨v2

1 I12⟩

1 −N2m2

2

⟨v2

2 I12⟩

2 =Σ−

that gives as solutions

⟨v2

1 I12⟩

1 =Σ−

N1m1=−m1

⟨v2

1

⟩1 −m2

⟨v2

2

⟩2

N1m1τ

⟨v2

2 I12⟩

2 =− Σ−

N2m2=−m2

⟨v2

2

⟩2 −m1

⟨v2

1

⟩1

N2m2τ.

The factor τ can be obtained evaluating the integrals in equation (3.21) and substituting

the expressions for the normalizing constants N1 and N2:

1

τ= 2N1N2

π

µ3(ω1ω2)1/2

M(kB T f )2

(Mω1ω2

m1ω21 +m2ω

22

)1/2

⟨σ⟩vr . (3.22)

Calculation of space-velocity collisional integral

The calculation of space-velocity collisional integral⟨

r vi Ii j⟩

i is very similar to that just

done. So in this section we only sketch many of the steps of the calculation being them

easily recovered from last section.

We start here introducing the quantity

Λi =⟨

mi Ni r vi Ii j⟩

i

and noticing that, having introduced Λ± =Λ1 ±Λ2, the global conservation of momen-

tum ensuresΛ+ = 0. We evaluateΛ− and, after changing variables to v0 and vr , we get

Λ− =∫

dr dv0dvr |vr |B(v0, vr )σ(|vr |)Ξ1′2′12

where B(v0, vr ) ≡ (m1v1 −m2v2)r = [v0(m1 −m2)+2µvr

]r

Also in this case B(v0, vr )−B(v0, v ′r ) = 2µr (vr −v ′

r ) so we use the same trick of last section

summing on both sides the expression forΛ− with v ′ instead of v obtaining

Λ− =µ

∫dr dv0dvr |vr |

[r (vr − v ′

r )]σ(|vr |)Ξ1′2′

12 .

We have now to expand Ξ1′2′12 . The first non vanishing contributions are obtained by

setting T1 = T2 = T f in the ansatz (3.17) and keeping the linear terms in ηi . We do not

30

3.3 Coupled Boltzmann equations

expand around final temperature T f because in this way from the exponential would

come down terms proportional to r 2 that, multiplied by r and integrated over all space

give zero. Conservation of kinetic energy allows us to collect exponential in Ξ1′2′12 so we

remain with

Λ− ∝ [(1+η1m1r v ′

1)(1+η2m2r v ′2)− (1+η1m1r v1)(1+η2m2r v2)

]∝ [

η1m1(v ′1 − v1)−η2m2(v2 − v ′

2)]

r +o(η2)

∝ m1(v ′

1 − v1)(η1 −η2

)r

where in the last step we used the conservation of momentum during the collision to

express m1(v ′1 − v1) = m2(v2 − v ′

2).

Finally, using equations (A.3) and (A.1) we obtain

Λ− ≈−N1N2(η1 −η2

)µ2

∫ +∞

−∞dv0 exp

(− M v2

0

2kB T f

)

×∫

dr dvr |vr |[r (vr − v ′

r )]2σ(|vr |)exp

(−m1ω

21r 2 +m2ω

22r 2 +µv2

r

2kB T f

)

=−N1N2(η1 −η2

)µ2

∫ +∞

−∞dr r 2 exp

(−m1ω

21r 2 +m2ω

22r 2

2kB T f

)

×4∫ +∞

−∞dv0 exp

(− M v2

0

2kB T f

)∫ +∞

−∞dvr |vr |v2

rσ(|vr |)exp

(− µv2

r

2kB T f

). (3.23)

We would like to express the collisional integral in the form ⟨r vi ⟩ /τ with τ having

the dimensions of a time. For doing so we calculate the mean value of the quantity r vi

according to the phase-space distribution function fi getting

⟨r vi ⟩i =1

Ni

∫dr dvi r vi fi (r, vi ) = (kB T f )2ηi

miω2i

,

(kB T f

)2 (η1 −η2

)= m1ω21 ⟨r v1⟩1 −m2ω

22 ⟨r v2⟩2 .

We only have to solve the systemΛ1 +Λ2 = N1m1 ⟨r v1I12⟩1 +N2m2 ⟨r v2I12⟩2 = 0

Λ1 −Λ2 = N1m1 ⟨r v1I12⟩1 −N2m2 ⟨r v2I12⟩2 =Λ−

that gives as solutions

⟨r v1I12⟩1 =Λ−

2N1m1=−m1ω

21 ⟨r v1⟩1 −m2ω

22 ⟨r v2⟩2

N1m1ω1ω2τ

31

Mixtures of classical gases

⟨r v2I12⟩2 =− Λ−

2N2m2=−m2ω

22 ⟨r v2⟩2 −m1ω

21 ⟨r v1⟩1

N2m2ω1ω2τ.

The factor τ can be obtained evaluating the integrals in equation (3.23) and substituting

the expressions for the normalizing constants N1 and N2:

1

τ= N1N2

π

µ3(ω1ω2)3/2

(m1ω21 +m2ω

22)(kB T f )2

(Mω1ω2

m1ω21 +m2ω

22

)1/2

⟨σ⟩vr . (3.24)

3.3.3 Three-dimensional collisional integrals

In the three-dimensional case the calculation is very similar, at least initially, except

for some details like normalization factors and integration measures. From equation

(3.20) things change a little bit. We must be especially careful in manipulating the term[v0 ·

(vr −v′r

)]2 dσ/dΩ. This term in fact contains a dependence on the solid angle be-

tween vr and v′r that can not be simply integrated out. Indicating with ϑ the azimuthal

angle between the relative velocities before and after the collision and remembering that

conservation laws imply vr = v ′r , we can express the modulus of vector vr −v′r as

∣∣vr −v′r∣∣= 2vr sin

(ϑ

2

)= 2vr

√1−cosϑ

2.

There is not dependence on polar angle ϕ so integral (3.20) becomes

Σ− ≈−N1N2

(µ2

kB T f

)(T1 −T2

T f

)∫d3r exp

(−m1ω

21r 2 +m2ω

22r 2

2kB T f

)

×2∫

d3v0v20 exp

(− M v2

0

2kB T f

)∫d3vr v3

r σ(vr )exp

(− µv2

r

2kB T f

),

where σ(vr ) = 2π∫ π

0sinϑdϑ (1−cosϑ)

dσ

dΩ.

We now change variable on the last integral setting vr = cx and c =√

2kB T f /µ (in this

way, being c a velocity, the new variable x is dimensionless) so the integral over vr reads

4πc6∫ ∞

0dxσ(cx)x5e−x2 ≡ 4πc6 ⟨⟨σ⟩⟩ ,

with ⟨⟨σ⟩⟩ a thermal averaged cross section which clearly has the dimensions of an area.

At this point we can substitute the normalization factors Ni = m3i ω

3i Ni /(2πkB T f )3

and solve all integrals to get the final result. However we must pay attention in substitut-

ing T1−T2 because now the mean value of square velocities is⟨

v2i

⟩i= 3kB Ti /mi . Finally

32

3.4 Collisionless and hydrodynamic regime

0 5 10 15 20 25 30 35 40

0

0.2

0.4

0.6

0.8

1

ωt

∆i(

t)

FIGURE 3.1: Evolution of normalized functions ∆i (t ) as a function of ωt in the collisionlessregimeωτ0 = 10. They represent the difference in the mean value of the square radius (blue line),of the square velocity (orange line) and of the space-velocity correlations (red line).

we obtain (with the same notation of the one-dimensional case):

1

τ3D= 8N1N2

3π2

µ2(ω1ω2)3/2

M(kB T f )

(Mω1ω2

m1ω21 +m2ω

22

)3/2

⟨⟨σ⟩⟩ .

We repeat all this also in calculating space-velocity collisional integral getting

1

τ3D= 4N1N2

3π2

µ2(ω1ω2)5/2

(m1ω21 +m2ω

22)(kB T f )

(Mω1ω2

m1ω21 +m2ω

22

)3/2

⟨⟨σ⟩⟩ .

3.4 Collisionless and hydrodynamic regime

We now apply the formalism just developed to a specific example in three dimensions

with N1 = N2 = N , m1 = m2 = m, ω1 =ω2 =ω and keeping a constant cross section σ. In

this way ⟨⟨σ⟩⟩ become simply σ and

1

τ3D= N 2ω3m

3π2kB T fσ,

1

τ3D= N 2ω3m

6π2kB T fσ= 1

2τ3D.

33

Mixtures of classical gases

0 2 4 6 8 10 12

−0.2

0

0.2

0.4

0.6

0.8

1

ωt

∆i(

t)

FIGURE 3.2: Evolution of normalized functions ∆i (t ) as a function of ωt in the intermediateregime ωτ0 = 2. They represent the difference in the mean value of the square radius (blue line),of the square velocity (orange line) and of the space-velocity correlations (red line).

In order to simplify the problem it is convenient to introduce three new quantities

∆1(t ) = ⟨r2⟩

1 (t )−⟨r2⟩

2 (t ),

∆2(t ) = ⟨r ·v1⟩1 (t )−⟨r ·v2⟩2 (t ),

∆3(t ) = ⟨v2

1

⟩1 (t )−⟨

v22

⟩2 (t ).

With these definitions we can subtract each other every pair of equations (3.14), (3.15)

and (3.16) to obtain a new system of only three coupled equation

d∆1(t )

dt= 2∆2(t )

d∆2(t )

dt=∆3(t )−ω2∆1(t )− ∆2(t )

Nτ3D

d∆3(t )

dt=−2ω2∆2(t )− 2∆3(t )

Nτ3D

where we used the fact that τ3D = 2τ3D. In this way we can obtain analytical solution

whose collisional regime is fully characterized through the dimensionless parameterωτ0

where τ0 = Nτ3D.

34

3.5 Comparison with experiment

0 1 2 3 4 5 6 7 8

−0.2

0

0.2

0.4

0.6

0.8

1

ωt

∆i(

t)

FIGURE 3.3: Evolution of normalized functions ∆i (t ) as a function of ωt in the hydrodynamicregimeωτ0 = 0.4. They represent the difference in the mean value of the square radius (blue line),of the square velocity (orange line) and of the space-velocity correlations (red line).

All the solutions are functions of sinh[√

1−4(ωτ0)2ωt]

and cosh[√

1−4(ωτ0)2ωt]

so the decay exhibits oscillations for ωτ0 > 0.5. In the collisionless regime (ωτ0 À 1, see

figure 3.1) the time needed for thermalization is very long but it is the same for square

velocity and square radius, the decay is exponential and the space-velocity correlations

play a minor role. In the intermediate regime (ωτ0 ∼ 1, see figure 3.2) the space-velocity

correlations play instead a crucial role in the dynamics and the decay is not exponential

at all. Finally, in the hydrodynamic regime (ωτ0 ¿ 1, see figure 3.3) we have two different

time scales: a rapid one for the relaxation in velocity space and a slow one for the relax-

ation of spatial width. This is the classical behaviour of hydrodynamics with a short time

needed to reach local equilibrium and a longer time to reach the global equilibrium.

3.5 Comparison with experiment

We want now to use the equations derived for the one-dimensional case in order to re-

produce the experimental results described in [8] and shown in figure 3.4. We can not

use the quantities ∆i because the parameters for the two gases are different so we have

35

Mixtures of classical gases

FIGURE 3.4: Experimental data point and fitting curves for the oscillations of the K impuritiesaxial width (circles) for different interaction strengths, η = gKRb/gRb. Triangles represent the Rbaxial size. Image is taken from [8].

to solve the full system of six coupled equations.

We start considering a constant cross sectionσ; doing the integral in ⟨σ⟩vr (see equa-

tion 3.21), we obtain ⟨σ⟩vr = [4(kB T f )2/µ2]σ and, substituting it in equations for τ (3.22)

and τ (3.24), we get all we need to solve the problem. The only parameters we can change

in order to reproduce the data are the initial temperature of potassium TK and the cross

section σ.

This time the calculation can not be done analytically but it is not a so complicated

task doing it numerically. Once we set up the system of equations we easily get the time

evolutions of our six dynamical quantities⟨

x2Rb

⟩,⟨

x2K

⟩,⟨

v2Rb

⟩,⟨

v2K

⟩, ⟨xvRb⟩ and ⟨xvK⟩.

Analysing them we see that the parameter TK determines the amplitude of the first oscil-

lation of potassium, the higher the temperature the higher the peak, and the parameter

σ determines the damping of the oscillations, the higher σ the higher the damping. So,

letting TRb = 460 nK as noticed in section 2.3 and moving only this two quantities, we

tried to get the graphs as similar as possible to the experimental ones (see figure 3.5).

The first thing we notice is about the oscillation frequency. We know that for a classi-

cal gas in a harmonic trap of frequency ω this has to be exactly 2ω but we expect that for

two gases with interspecies interactions things may change. So we fit our functions with

a damped cosine oscillation with linear baseline

x(t ) = x0 +βt − Ae−γωt cos[ω(t − t0)]

in order to obtain ω. In all the four cases we find frequencies ranging from 173.7 Hz to

174.1 Hz which are compatible with 2ωK/(2π) = 174 Hz. So this point is in agreement

with experimental results that, within the error bars, do not show any evidence of fre-

quency shift. We compare with experimental fit also the slope β and the friction coef-

ficient γ finding a perfect agreement for the first (all our point within error bars of the

36

3.5 Comparison with experiment

0 2 4 6 8 10 12 14 16 18

6

8

10

12

14

16

18

20

22

24

t (ms)

p x2

(µm

)

FIGURE 3.5: Theoretical oscillations of rubidium and potassium for different values of TK andσ.Circles represent the rubidium mean value while solid lines the potassium evolution, all graphsare for TRb = 460 nK. The blue one is obtained for TK = 880 nK and σ = 0.002, the red one forTK = 670 nK and σ= 0.0045, the orange one for TK = 480 nK and σ= 0.005 and the black one forTK = 310 nK and σ= 0.005.

experimental ones) and only a partial agreement for the second [8].

The main difference is anyway in the parameters that describe the oscillations. In the

experiment, the evolution from the first to the last graph of figure 3.4 is only guided by

the raising of interaction strength, η = 0,1,4,30. With our theoretical model instead, we

have also (and mainly) to change the initial temperature of potassium. If we change only

the cross section σ we obtain a reduction of oscillations amplitude and thermalization

time but the maximum amplitude never get over the rubidium width (see figure 3.6).

3.5.1 Initial conditions

One last comment must be done about the initial conditions we have to impose in order

to solve the system of differential equations. The more straightforward thing to do is to

assume a totally classical behaviour for both gases and use as initial conditions the mean

values of x2, v2 and xv according to their phase-space distribution fi , equation (3.17), at

0th-order in η and∆Ti . Clearly in f2 the frequency isω0/(2π) = 1 kHz, the SSDP frequency

37

Mixtures of classical gases

0 2 4 6 8 10 12 14 16 18 20

6

8

10

12

14

16

18

20

t (ms)

p x2

(µm

)

FIGURE 3.6: Theoretical oscillations of rubidium and potassium for different values ofσ. Circlesrepresent the rubidium mean value while solid lines the potassium evolution, all graphs are forTK = TRb = 460 nK. The blue curve is obtained for σ= 0.002, the red one for σ= 0.005, the orangeone for σ= 0.01 and the black one for σ= 0.02.

compressing K at he centre of Rb tubes. Thus we obtain

x21(0) = kB T1

m1ω21

, x22(0) = kB T2

m2ω20

, v2i (0) = kB Ti

miand xvi (0) = 0.

As we have seen in chapter 2 the quantum fluctuations can rise the amplitude of os-

cillation over the classical value so an improvement to initial conditions can be to use

equation (2.4) at t = 0 for x22 and its analogous for v2

2 . But as we have already observed,

the temperature is too high to give appreciable differences from the classical initial con-

ditions, at such temperatures the difference between the two values are approximately

of 0.1%.

38

CH

AP

TE

R

4BOSE-EINSTEIN CONDENSATES

Bose-Einstein condensation is based on the indistinguishability and wave nature of par-

ticles, which are both basic concept of quantum mechanics. Defining Bose-Einstein con-

densation in one sentence, we can say it is the occupation of the lowest quantum state

of the external potential by a large fraction of bosons forming the system. Particles are

assumed to be bosons because only the Bose-Einstein statistics allows an arbitrary large

number of particles to occupy one single quantum state.

4.1 Key concepts in BEC

The key concepts underlying this phenomenon are long-range order, breaking of gauge

symmetry and the presence of an order parameter associated with the phase transition.

We now explain these concepts in more details.

4.1.1 One body density matrix and long-range order

We start introducing an important correlation function playing a central role in the theo-

retical description of Bose-condensed systems. This is the one-body density matrix which

is defined as the amplitude of the process where one particle is removed from position r′

39

Bose-Einstein condensates

and the same state is recovered by replacing the particle at position r:

n(1)(r,r′) =⟨Ψ†(r)Ψ(r′)

⟩.

The normalization of one body density matrix is defined by its diagonal part with the

condition N = ∫drn(r) = ∫

drn(1)(r,r).

Let us consider the case of a uniform and isotropic system of N particles in a volume

V . If we take the thermodynamic limit (N ,V →∞ with n kept fixed) the one body density

matrix depends only on the modulus of the relative variable s = r− r′ and we can write it

as the Fourier transform of its equivalent in the momentum space:

n(1)(s) = 1

V

∫dpn(p)e−ip·s/ħ. (4.1)

If the momentum distribution has a smooth behaviour at low momenta, as normally is,

the one body density matrix vanishes for s →∞. If we have condensation there is instead

a macroscopic occupation of the state with p = 0 that leads to a singular behaviour of

momentum distribution:

n(p) = N0δ(p)+ n(p).

Inserting this distribution into equation (4.1) we find that the one body density matrix

does not vanish at large distances but approaches a finite value, n(1)(s)s→∞ → n0.

The existence of off-diagonal long-range order can be explained saying that there

exists a finite probability that the particle is extracted from the condensate and replaced

in the condensate returning back to the same state. The amplitude of this process is

independent from the distance and the one body density matrix remains finite even in

the limit s →∞.

4.1.2 Order parameter and symmetry breaking

If we find the eigenvectors ϕi (r) (with eigenvalues ni ) of the one body density matrix

n(1)(r,r′), we can use them as a natural basis of orthonormalized single-particle wave

functions and write the field operator in the form

Ψ(r) =∑iϕi (r)ai =ϕ0(r)a0 +

∑i 6=0

ϕi (r)ai .

This is the starting point for the introduction of the Bogoliubov approximation that con-

sists in replacing the operators a0 and a†0 with the c-number

pN0. This is equivalent

to ignoring the noncommutativity of this operators and we can see it is a good approx-

imation for BEC if we observe that the commutator of a0 and a†0 is equal to 1 while the

40

4.2 Critical temperature

operators themselves are of orderp

N0 À 1. So we treat the macroscopic component

ϕ0a0 of the field Ψ(r) as a classical field and the sum over i 6= 0 as the field describing the

non-condensed part which is negligible at very low temperatures.

The function Ψ0(r) = pN0ϕ0(r) is called the wave function of the condensate and

plays the role of an order parameter in fact it vanishes above the critical temperature.

Being a complex quantity it can be written as

Ψ0(r) = |Ψ0(r)|eiS(r)

where the modulus determines the contribution of the condensate to the diagonal den-

sity and the phase S(r) plays an important role in characterizing the coherence and su-

perfluid phenomena.

The order parameter is defined up to a constant phase factor: if we multiply this

function for the numerical factor eiα any of the physical properties of the system changes.

This reflects the gauge symmetry to which all physical equations of the problem obey,

choosing a precise order parameter corresponds to a breaking of the gauge symmetry.

4.2 Critical temperature

Working in the grand canonical ensemble it is easy to obtain the total number of particles

as a sum of average occupation numbers

N =∑i

ni =∑

i

1

exp[β(εi −µ)]−1.

This result implies that the chemical potential µ must fulfil the condition µ < ε0 (where

ε0 is the lowest eigenvalue of the single particle Hamiltonian) in order to have no nega-

tive occupation numbers. When µ→ ε0 the occupation number of the ε0 state becomes

increasingly large and this is actually the mechanism at the origin of Bose-Einstein con-

densation. Indicating the total number of particles as N = N0 +NT we define the critical

temperature at which Bose-Einstein condensation occurs as

NT (Tc ,µ= ε0) = N . (4.2)

4.2.1 Ideal Bose gas in the box

In this case the single-particle Hamiltonian has the simple form

H = p2

2m

41

Bose-Einstein condensates

with plane waves solutions (using cyclic boundary conditions) of the form

ϕp = 1pV

eip·r/ħ.

The lowest eigenvalue has zero energy (ε0 = 0) so the chemical potential must always be

negative. Passing from sum to integral and using the transformation p2 = 2mkB T x we

get the number of atoms out of the condensate as

NT = V

λ3T

g3/2(eβµ)

where λT is the thermal wavelength (3.1) and g3/2 is a special case of the class of Bose

function:

gp (z) = 1

Γ(p)

∫ ∞

0dxxp−1 1

z−1ex −1=

∞∑l=1

z l

l p .

So, applying the definition of critical temperature (4.2), we finally obtain

kB Tc = 2πħ2

m

(n

g3/2(1)

)2/3

where g3/2(1) = 2.612.

Attention must be paid in distinguishing between the microscopic criterion

kB Tc ¿ ε1 −ε0 = h2

2mV 2/3

for the system to be in the ground state and the weaker condition NT ¿ N which only

requires T ¿ Tc . Bose-Einstein condensation for large system can take place also in the

case h2/(2mV 2/3) ¿ kB T ¿ kB Tc with macroscopic values of T. The use of macroscopic

values of T is actually crucial in order to treat the thermal component as a continuum of

excitations and to replace the sum over the single particle states with an integral.

4.2.2 Ideal Bose gas in the harmonic trap

As we have seen in chapter 1, the experimental realization of BEC has been achieved in

atomic gases trapped with potentials whose shape is, in most cases, well approximated

by a harmonic shape. So we now give the properties of a gas trapped in a potential of the

type

Vext(r) = 1

2mω2

x x2 + 1

2mω2

y y2 + 1

2mω2

z z2

42

4.3 Gross-Pitaevskii equation

where the eigenvalues of the single-particle Hamiltonian are given by the analytic ex-

pression

εnx ,ny ,nz =(nx + 1

2

)ħωx +

(ny + 1

2

)ħωy +

(nz + 1

2

)ħωz .

Recalling the definition of NT and of Tc (equation 4.2), setting µ = ε0 and replacing the

sum with an integral we finally get the critical temperature for the trapped gas

kBTc =ħωho

(N

ζ(3)

)1/3

= 0.94ħωhoN 1/3. (4.3)

In passing from sum to integral we used the so called semiclassical approximation where

the temperature of the gas is assumed to be much larger than the spacing between single-

particle levels, kBT Àħωk . The comparison between this expression and equation (4.3)

shows that the BEC regime is compatible with this approximation only if N À 1. There-

fore we can introduce the thermodynamic limit letting N tend to infinity andωho tend to

zero with the product Nω3ho being kept constant. In this way the semiclassical approx-

imation is automatically satisfied and the critical temperature (4.3) has always a well-

defined value.

The same result can be obtained without using the semiclassical approximation but

doing explicitly the sum as explained in [20] by W. Ketterle and N. J. van Druten. They

obtain an equation for N that must be resolved numerically but that can be approxi-

mated for kBT À ħω and expanded in kBT /ħω at various order. This is equivalent to

approximate better and better the density of states in calculating the integral of N −N0.

4.3 Gross-Pitaevskii equation

As we have seen in section 4.1.2 the Bogoliubov approximation permits the introduction

of the classical function Ψ0(r, t ). In order to establish the equation governing this field

we recall that the field operator Ψ(r, t ), in the Heisenberg representation, fulfils the exact

equation

iħ ∂

∂tΨ(r, t ) = [

Ψ(r, t ), H]= [

−ħ2∇2

2m+Vext(r, t )+

∫Ψ†(r′, t )V (r′− r)Ψ(r′, t )dr′

]Ψ(r, t ).

In order to replace Ψ(r, t ) withΨ0(r, t ) we have to use an effective soft potential Veff that

reproduces the same low energy scattering properties of the bare potential V . So, as-

suming that the functionΨ0(r, t ) varies slowly on distances of the order of the range of

interatomic force, we finally obtain the time dependent Gross-Pitaevskii equation

iħ ∂

∂tΨ0(r, t ) =

(−ħ2∇2

2m+Vext(r, t )+ g |Ψ0(r, t )|2

)Ψ0(r, t ) (4.4)

43

Bose-Einstein condensates

where g = ∫Veff(r)dr can be express in terms of s-wave scattering length a as

g = 4πħ2a

m.

Equation (4.4) was derived independently by Gross [14] and Pitaevskii [29] in 1961 and is

the main theoretical tool for investigating non uniform dilute Bose gases at low temper-

atures. It has the typical form of a mean field equation where the order parameter must

be calculated self-consistently. In the following for brevity we will indicate Ψ0(r, t ) only

withΨ(r, t ).

Let we now consider a stationary, zero-temperature condensate described by the

Gross-Pitaevskii equation (4.4). For such a system the condensate density is time invari-

ant and so the wave function evolution will consist solely of a spatially uniform phase

rotation due to the condensate’s energy. Hence we can separate the time-dependent

condensate wave function using the usual ansatz

Ψ(r, t ) =Ψ(r)e−iµt/ħ

where Ψ(r) is the now time-independent condensate wave function and µ is a char-

acteristic energy of the condensate that we will discuss below. Applying this ansatz to

the Gross-Pitaevskii equation (4.4) we straightforwardly arrive at the time-independent

Gross-Pitaevskii equation(−ħ2∇2

2m+Vext(r)+ 4πħ2a

m|Ψ(r)|2

)Ψ(r) =µΨ(r), (4.5)

To ensure consistency between these two equations (4.4 and 4.5), µmust be the con-

densate chemical potential. In fact the wave function phase factor reflects the fact that

microscopicallyΨ is equal to the matrix element of the annihilation operator Ψ between

the ground state with N particles and that with N −1 particles,

Ψ(r, t ) = ⟨N −1

∣∣Ψ(r)∣∣N

⟩∝ exp[−i(EN −EN−1)t/ħ] .

For large N the difference in ground state energies with N and N −1 particles is equal to

∂E/∂N which is exactly the chemical potential.

4.3.1 Thomas-Fermi approximation

The Gross-Pitaevskii equation (4.5) should be solved numerically except in some impor-

tant limits. If we are in presence of a repulsive interaction (g > 0) the gas will expand

44

4.3 Gross-Pitaevskii equation

and the size of the cloud will increase. Eventually, if the effect of the interaction is very

significant, the width of the gas will become so large and the density profile so smooth

that we can ignore the kinetic energy term in the Gross-Pitaevskii equation. This limit is

called the Thomas-Fermi limit and it is characterized by the analytical solution for the

order parameter

ΨTF(r) =√

nTF(r) =√

1

g

[µTF −Vext(r)

].

The chemical potential µTF is obtained imposing the normalization of nTF at N and it

takes the value

µTF = ħωho

2

(15N a

aho

)2/5

(4.6)

where aho =√ħ/(mωho) is the oscillator length associated with the geometrical average

of the three trapping frequencies.

We now introduce a crucial quantity for the theory characterizing the interacting na-

ture of the system, namely the healing length

ξ=√

ħ2

2mg n. (4.7)

It is the shortest distance over which the wave function can change and is obtained

equating the kinetic term to the energy scale of the system given by the chemical po-

tential. Typically ξ is much smaller than the dimensions of the system and smoothing of

the wave function only occurs in a thin boundary layer so these surface effects give only

small corrections to results calculated using the Thomas-Fermi approximation.

The condition of validity for the Thomas-Fermi approximation can be investigated

rescaling the Gross-Pitaevskii equation (4.5) with spherical harmonic trapping of fre-

quency ωho using aho and ħωho as units of length and energy:[−∇2 + r 2 +8π

(N a

aho

)Ψ2(r)

]Ψ(r) = 2µΨ(r)

where Ψ = N−1/2a−3/2ho Ψ0. So, in order to ignore the kinetic term we must have that the

so called Thomas-Fermi parameter N a/aho is much higher than 1.

The density profile takes the form of an inverted parabola which vanishes at the clas-

sical turning point where Vext(Rk ) = µTF. In case of isotropic trapping these radii reduce

to

R = aho

(15N a

aho

)1/5

(4.8)

45

Bose-Einstein condensates

and, if we evaluate the healing length at the centre of the trap where nTF(0) = µ/g , we

obtain for the ratio of ξ and Rξ

R=

(15N a

aho

)−2/5

showing that, as previously said, in the Thomas-Fermi limit the healing length becomes

much smaller than the size of the condensate.

4.3.2 Hydrodynamic formulation

In order to better understand the physical content of the Gross-Pitaevskii equation (4.4)

we now reformulate it as a pair of hydrodynamic equations. These are equations for the

density of the condensate, which is given by |Ψ|2, and for the gradient of its phase, which,

as we will see, is proportional to the local velocity of the condensate.

If we multiply the equation (4.4) by Ψ∗(r, t ) and we subtract the complex conjugate

of the resulting equation we obtain

∂|Ψ|2∂t

+∇ ·[ ħ

2mi

(Ψ∗∇Ψ−Ψ∇Ψ∗)]= 0,

that has the form of a continuity equation for the particle density n = |Ψ|2 and it may be

written as ∂n/∂t +∇ · (nv) = 0 defining the velocity of the condensate as

v = ħ2mi

Ψ∗∇Ψ−Ψ∇Ψ∗

|Ψ|2 . (4.9)

Simpler equations can be obtained writingΨ in terms of its phase and its amplitude,

Ψ = pneiφ. We indicate the amplitude with

pn just because its square modulus coin-

cides with the one of Ψ. Inserting this relation into equation (4.9) we get the expression

for the velocity as a function of the phase of the order parameter

v = ħm∇φ.

So we can conclude that the motion of the condensate corresponds to potential flow,

since the velocity is the gradient of a scalar quantity, which is referred to as the velocity

potential. Provided that φ is not singular, we can immediately conclude that the motion

of the condensate must be irrotational, that is

∇×v = ħm∇×∇φ= 0.

The possible motions of a condensate are thus much more restricted than those of a

classical fluid.

46

4.3 Gross-Pitaevskii equation

To obtain the equations of motion for the density and the velocity we must insert

the expression of Ψ in this new variables into equation (4.4) and then separate real and

imaginary parts. We first notice that

−∇2Ψ= [−∇2pn + (∇φ)2pn − i(∇2φ)p

n −2i∇φ ·∇pn

]eiφ,

so the equation for the imaginary part becomes

ħ∂p

n

∂t=− ħ2

2m

[(∇2φ)

pn +2∇φ ·∇p

n]

1

2p

n

∂n

∂t=−1

2

[pn∇ ·v+ 1p

nv ·∇n

]∂n

∂t=−∇ · (nv) , (4.10)

while the one for real part becomes

−ħpn∂φ

∂t= (

Vext + g n)p

n − ħ2

2m

[∇2pn − (∇φ)2pn]

m∂

∂t

( ħmφ

)=−(

Vext + g n)+ ħ2

2mp

n∇2pn − 1

2mv2

m∂v

∂t=−∇

(− ħ2

2mp

n∇2pn + 1

2mv2 +Vext + g n

)m∂v

∂t=−∇

(µ+ 1

2mv2

). (4.11)

The first one is nothing else than the continuity equation expressed in the new variables.

The second one can be rewritten also using the pressure of the gas p. In fact, at zero

temperature, changes in the chemical potential for a bulk system are related to changes

in the pressure by the Gibbs-Duhem relation dp = ndµ. Furthermore the terms g n is the

expression for the chemical potential of a uniform Bose gas omitting the contributions

from the external potential so its gradient can be recast in the form ∇p/n. So equation

(4.11) can be finally written as

∂v

∂t=− 1

mn∇p −∇

(v2

2

)+ 1

m∇

( ħ2

2mp

n∇2pn

)− 1

m∇Vext. (4.12)

Equations (4.10) and (4.12) are very similar to the hydrodynamic equations for a per-

fect fluid, (4.10) is exactly the same while the analogue of equation (4.12) is the Euler

equation∂v

∂t−v× (∇×v) =− 1

mn∇p −∇

(v2

2

)− 1

m∇Vext. (4.13)

47

Bose-Einstein condensates

There are two differences between equations (4.12) and (4.13). The first is that the Euler

equation contains the term v× (∇× v). However, since the velocity field of the super-

fluid corresponds to potential flow, ∇× v = 0 and this term would not contribute. The

only difference between the two equations for potential flow is therefore the third term

on the right hand side of equation (4.12), which is referred to as the quantum pressure

term. This describes forces due to spatial variations in the magnitude of the wave func-

tion for the condensed state. Like the term ∇v2/2, its origin is the kinetic energy term

ħ2|∇Ψ|2/2m = mnv2/2+ħ2(∇p

n)2/2m in the energy density, but the two contribu-

tions correspond to different physical effects: the first is the kinetic energy of motion of

particles, while the latter corresponds to zero-point motion, which does not give rise to

particle currents. If the spatial scale of variations of the condensate wave function is l ,

the pressure term in equation (4.12) is of order g n/ml , while the quantum pressure term

is of order ħ2/m2l 3. Thus the quantum pressure term dominates the usual pressure term

if spatial variations of the density occur on length scales l less than or on the order of the

coherence length ξ∼ħ/(mg n)1/2, and it becomes less important on larger length scales.

So far we specified the motion of the condensate in terms of local density and local

velocity. The reason for this is that the only degrees of freedom are those of the wave

function of the condensate: its magnitude and its phase. Ordinary liquid and gases have

instead many more degrees of freedom and, as a consequence, it is in general necessary

to employ a microscopic description. However, if collisions between particles are suffi-

ciently frequent that thermodynamic equilibrium is established locally, a hydrodynamic

description is possible. The state of the fluid may then be completely specified by lo-

cal density, local velocity and local temperature; at zero temperature the temperature

becomes an irrelevant variable and the description is just as for a condensate.

4.4 Dynamics of a condensate

The time-dependent behaviour of BoseEinstein condensed clouds, such as collective

modes, is an important source of information about the physical nature of the conden-

sate. In addition, the spectrum of elementary excitations of the condensate is an essen-

tial ingredient in calculations of thermodynamic properties.

In order to study the elementary excitations of a Bose-Einstein condensate one can

adopt the microscopic approach or the hydrodynamic one. We here choose the second

because of its clarity and simplicity and because it would be the base to study the more

complicated case of the mixture. In analysing the collective modes we focus on the three

48

4.4 Dynamics of a condensate

dimensional case of a condensate in a spherical harmonic trap. Dispersion law similar to

the one that we will find can be obtained also for deformed traps or in lower dimension

(see for instance [22] for the one dimensional case).

4.4.1 Elementary excitations

In order to find the properties of elementary excitations of such a system we can consider

small deviation of the state of the gas from the equilibrium and we can find the periodic

solution to the time-dependent Gross-Pitaevskii equation. Equivalently we can use the

hydrodynamic formulation writing the density as n = neq +δn (where neq is the equilib-

rium density and ∂t neq = 0) and linearizing equations (4.10) and (4.11) by treating the

velocity v and the departure from the equilibrium density δn as small quantities. In this

way we find

∂δn

∂t=−∇ · (neqv

)m∂v

∂t=−∇δµ

where δµ is obtained by linearizing the expression for µ defined in equation (4.11). Tak-

ing the time derivative of the first one and eliminating the velocity by means of the sec-

ond one we finally obtain the equation of motion for δn:

m∂2δn

∂t 2 =∇ · (neq∇δµ)

(4.14)

This equation describes the excitations of a Bose gas in an arbitrary external potential

which is contained in the δµ term.

As a first example of application of this formalism we investigate the spectrum of a

homogeneous gas where the external potential Vext is constant. In this case the density

n of the gas is the same everywhere and it may therefore be taken outside the spatial

derivatives. From now on we indicate neq simply with n. Looking for plane wave solu-

tions proportional to exp(iq · r− iωt ) we find that

δµ=(

g + ħ2q2

4mn

)δn

and the equation of motion becomes

mω2δn =(

g nq2 + ħ2q4

4m

)δn (4.15)

49

Bose-Einstein condensates

0 0.5 1 1.5 2 2.50

1

2

3

4

5

ħq/mc

ħω/g

n

FIGURE 4.1: Excitation spectrum of a homogeneous Bose gas (full line) plotted as a function ofthe wave number expressed as the dimensionless variable ħq/mc. The expansion (4.17) for highwave number is shown as a dashed line.

Non-vanishing solutions of equation (4.15) are possible only if the frequency satisfies the

relation

ħω=±εq =√

2g nε0q + (ε0

q )2 (4.16)

where ε0q is the free-particle energy ħ2q2/2m. At small q the second term in equation

(4.16) can be neglect, εq becomes a linear function of q

εq ' cħq

and the spectrum is sound-like with the velocity c equal to√

g n/m. This result, first de-

rived by Bogoliubov from microscopic theory [5], agrees with the expression of the sound

velocity calculated from the hydrodynamic theory where c2 = dp/dρ = (n/m)(dµ/dn).

So the repulsive interaction has turned the energy spectrum at long wavelengths, which

is quadratic in q for free particles, into a linear one. This provides a key to superfluid

behaviour and it was one of the triumphs of Bogoliubov’s calculation.

At short wavelengths the leading contributions to the spectrum are

εq ' ε0q + g n. (4.17)

50

4.4 Dynamics of a condensate

This is the free-particle spectrum plus a mean-field contribution. The transition between

the linear spectrum and the quadratic one occurs when the kinetic energy, ħ2q2/2m,

becomes large compared with the potential energy of a particle ∼ g n, or in other words

the quantum pressure term dominates the usual pressure term. This occurs at a wave

number ∼ (2mg n)1/2/ħ, which is the inverse of the coherence length ξ (see equation 4.7).

The coherence length is related to the sound velocity by ξ = ħ/p

2mc. On length scales

longer than ξ, atoms move collectively, while on shorter length scales, they behave as

free particles.

4.4.2 Collective modes in harmonic traps

If we now switch on an external harmonic potential things change very much. We can

no more expand the solution as plane wave and the dispersion law assumes a totally

different form.

We analyse here the case of a Bose-Einstein condensate in a spherical potential of

frequency ωho. In order to simplify the formulas as much as possible we want to rewrite

equation (4.14) in rescaled variables τ and r. We also assume Thomas-Fermi approxima-

tion so we can neglect the quantum pressure term in µ and we can write equation (4.14)

as∂2δn

∂t 2 =∇x ·[

neq

m∇x

(∂µ

∂n

∣∣∣∣n=neq

δn

)]= 0.

As a unit of time we use the inverse of the trap frequency and as a unit of length the

Thomas-Fermi radius that, using equation (4.6) and (4.8), can be written as

R2 = 2µ

mω2ho

.

So, defining τ=ωhot and r = x/R, we get

∂2δn

∂τ2 − 1

2∇r ·

[neq

µ∇r

(µ′(neq)δn

)]= 0 (4.18)

where neq = neq(|r|) = neq(r ) and µ′ = µ′(r ) because we are in presence of an isotropic

trapping potential. At this point, using the spherical symmetry of the problem, we write

the general solution with an angular dependence given by the spherical harmonics and

a time dependence given by exp(−iωt ) = exp(−iωτ/ωho)

δn(r,τ) = δn(r,ϑ,ϕ,τ) =Cr l F (r )Yl ,m(ϑ,ϕ)e−iωτ/ωho .

51

Bose-Einstein condensates

Inserting this solution into equation (4.18), writing the divergence and the gradient in

spherical coordinates and using the properties of the spherical harmonics we get

−(ω

ωho

)2

r l F (r ) = 1

2µ

∂

∂r

[neq(r )

∂

∂r

(µ′(r )r l F (r )

)]+ 1

r

neq(r )

µ

∂

∂r

(µ′(r )r l F (r )

)+

− r l

2r 2

neq(r )µ′(r )

µl (l +1)F (r ). (4.19)

We have now to insert the explicit dependence on r of µ′ and neq. In general one has µ=µ−Vext = Anγ

eq but for bosons it is very easy because γ= 1 and A = g . In appendix B we do

the calculation for general γ and A but here we restrict to the boson case. Inserting the

spherical harmonic potential and using the rescaled variable r we get neq(r ) =µ/g (1−r 2)

and µ′(r ) = g . We insert these expressions in equation (4.19) obtaining

r(1− r 2)F ′′(r )+2

(l +1)︸ ︷︷ ︸A

− (l +2)︸ ︷︷ ︸B

r 2

F ′(r )+ r

[2

(ω

ωho

)2

−2l

]︸ ︷︷ ︸

C

F (r ) = 0. (4.20)

The function F (r ) can be expanded as a polynomial in r so

F (r ) =∞∑

n=0cnr n

F ′(r ) =∞∑

n=1cnnr n−1 =

∞∑n=0

cn+1(n +1)r n

F ′′(r ) =∞∑

n=1cn+1n(n +1)r n−1 =

∞∑n=0

cn+2(n +1)(n +2)r n

Equation (4.20) is therefore nothing else than a polynomial in r with coefficients that

depend on n, A, B , C or n, l ,ω,ωho. For this to be equal to zero every of these coefficients

must be zero by themselves. So, considering the general coefficient of r n+1 and switching

indices where necessary, we obtain a condition on the cn ’s

cn+2 = cnn(n −1)+2Bn −C

(n +2)(n +1+2A). (4.21)

Dealing with bosons we want the function describing the density oscillation to be even.

But r l Yl ,m(ϑ,ϕ) is always an even function and so must be F (r ). We substitute then n

with 2n in the previous formula. In order to have a polynomial of degree 2n the coeffi-

cient c2n+2 must be zero, this implies the dispersion relation in this way:

0 = 2n(2n −1)+4Bn −C

= 2n(2n −1)+4(l +2)n −2

(ω

ωho

)2

+2l

52

4.4 Dynamics of a condensate

1

2

3

4

5

6l = 0 l = 1 l = 2 l = 3 l = 4

ω/ω

ho

FIGURE 4.2: Excitation frequencies of a condensate in an isotropic harmonic trap. Full lines rep-resent the case in which interparticle interactions are switched on (equation 4.22) while dashedlines represent the case in which there are not interparticle interactions (equation 4.23).

=⇒(ω

ωho

)2

= 2n2 +2nl +3n + l . (4.22)

This result was first given in [34] and clearly holds in the limit N a/aho À 1.

It is interesting to compare this dispersion law with that of a harmonic oscillator

model without interparticle interactions

ω=ωho(2n + l ). (4.23)

We first notice that in the dipole case (n = 0 and l = 1) both the predictions coincide

with the harmonic oscillator frequency ωho. This follows from the fact that the lowest

dipole mode corresponds to the oscillation of the centre of mass and is unaffected by the

interatomic forces.

For a fixed value of N the accuracy of the prediction (4.22) is expected to decrease

with the increasing of n and l . In fact the high energy states are associated with rapid

variations of the density space and consequently the kinetic energy contribution can not

be neglected any longer.

53

Bose-Einstein condensates

4.5 One-dimensional BEC

If we let ωx,y ≡ ω0 À ωz ≡ ω become higher and higher we can essentially freeze the

radial motion of particles (at sufficiently low temperature). In this way we can obtain a

quasi-1D gas whose behaviour is kinematically purely 1D and the only memories of three

dimensions are in the value of interparticle interaction which now depends on the radial

confinement.

The question if such a system shows a phase transition is a long standing problem.

In 1969 Yang and Yang [35] have proved the analyticity of thermodynamical functions at

any finite temperature T , which indicates the absence of phase transition. However at

low T the properties of a 1D Bose gas are different from classical high-T properties. In

the regime of a weakly interacting gas the density fluctuations are suppressed whereas

the long-wave fluctuations of the phase lead to an exponential decay of the one parti-

cle density matrix at finite T and to a power-law decay at T = 0, indicating that Bose-

Einstein condensation is absent at any T . This is confirmed by the Bogoliubov k−2 the-

orem as shown in [23] and in [28] for T = 0. Surely the decrease of temperature leads to

a continuous transformation of correlation properties from classical ideal gas to interac-

tion/statistic dominated gas.

Furthermore Ketterle and van Druten [20] find a BEC-like behaviour of a trapped

1D ideal gas and established that at temperatures kB T < Nħωz /ln2N the population of

ground state rapidly grows becoming macroscopic.

Finally Petrov et al. [27] obtained the diagram of state of this system. They identify

three regimes of quantum degeneracy at T ¿ Td, where KB Td ≈ Nħω is the degeneracy

temperature.

• At sufficiently large interaction and for N much smaller than a characteristic value

N∗ we have a Tonks gas characterized by a Fermi-gas density profile.

• Well below Td and for weak interactions density fluctuations are suppressed but

the phase still fluctuates and we have a quasicondensate with Thomas-Fermi pro-

file.

• At very low T also the phase fluctuations are suppressed because of the finite size

of the system and we get a true condensate with Thomas-Fermi profile.

To have a weakly interacting gas we must require that the healing length ξ is much

54

4.5 One-dimensional BEC

101 102

101

102

103

Classical Gas

Tonks gas

Quasicondensate

True Condensate

T/ħω

N

FIGURE 4.3: Diagram of state for a trapped 1D Bose gas. The black dot represent the positionof experiment [8] from which are also taken the values of α and N∗. The dashed line representthe transition temperature obtained by Ketterle and van Druten [20]. For the line separating thequasicondensate from the true condensate we used the better estimate (T /ħω) = (32N /9N∗)1/3.

larger than the mean interparticle separation 1/n, so we introduce a new parameter

γ= 1

(nξ)2 = mg

ħ2n, (4.24)

with γ¿ 1 in the weakly interacting case. Moreover, being the particle trapped, we in-

troduce the dimensionless quantity α= mg aho/ħ2 wich provides a relation between the

interaction strength and the trapping frequency.

At T = 0 we have a true condensate (the phase fluctuation are of order γ1/2) with a

Thomas-Fermi profile. The chemical potential is given by µ/ħω ∼ (Nα)2/3 so for α À1 we are always in Thomas-Fermi regime but in this case equation (4.24) requires that

N À N∗ = α2. Otherwise, if α ¿ 1, equation (4.24) is satisfied for all N but we have

Thomas-Fermi regime only if N À α−1 and a simply Gaussian profile in the opposite

case.

In order to characterize the transition from quasi to true condensate we have to study

55

Bose-Einstein condensates

the phase fluctuation. We find that at a temperature

Tph = Tdħωµ

the mean square phase fluctuation are of order 1 so we have a true condensate only for

T ¿ Tph ≈ ħω(4N /α2)1/3. In figure 4.3 is shown the diagram of state with parameters

given by the experiment described in [8].

56

CH

AP

TE

R

5MIXTURES OF BOSE-EINSTEIN

CONDENSATES

When the temperature of the gases is sufficiently low quantum effects become very im-

portant and the system must be studied with the Bose-Einstein formalism. In the next

we assume temperature T = 0 and we study the ground state of our mixture and its exci-

tations.

5.1 Ground state

In order to find the ground state wave functions of the mixture (i. e. the densities of the

two components) we start from the time independent Gross-Pitaevskii equation (4.5).

But in this case we have two coupled equations describing the two components where

the coupling is given by the interaction between component 1 and component 2. To

describe this interaction we use the same method used to describe the intraspecies in-

teraction and we add the terms g12|Ψ1|2 and g12|Ψ2|2 to our equations in this way:[−ħ2∇2

2m1+ 1

2m1ω

21r 2 + g11 |Ψ1(r)|2 + g12 |Ψ2(r)|2

]Ψ1(r) =µ1Ψ1(r),[

−ħ2∇2

2m2+ 1

2m2ω

22r 2 + g12 |Ψ1(r)|2 + g22 |Ψ2(r)|2

]Ψ2(r) =µ2Ψ2(r).

57

Mixtures of Bose-Einstein condensates

We have already inserted harmonic trapping potentials for the the two components and

clearly g12 = 2πħ2a12/µ where µ is the reduced mass.

In such a situation the ground state can be either a uniform mixture of the two com-

ponents, or phase separated. The condition for the miscibility can be easily established

investigating the energies of the two configurations in the homogeneous case:

Eunif =g11

2

N 21

V+ g22

2

N 22

V+ g12

N1N2

V(5.1)

Esepar = g11

2

N 21

V1+ g22

2

N 22

V2(5.2)

where V1 and V2 are the volumes occupied by the two separated phases and V1 +V2 =V .

To ensure the mechanical equilibrium between the two phases they must satisfy the con-

dition ∂Esepar/∂V1 = ∂Esepar/∂V2 that implies the relationship g11(N1/V1)2 = g22(N2/V2)2.

Using this last expression we can rewrite the phase separation energy (equation 5.2) in a

more suitable form:

Esepar = g11

2

N 21

V+ g22

2

N 22

V+p

g11g22N1N2

V. (5.3)

So, comparing equation (5.1) with equation (5.3), we see that in order to have phase sep-

aration (Esepar < Eunif) we must require

g12 >pg11g22. (5.4)

These considerations can be easily transferred to the case of a trapped mixture using the

local density approximation.

We now assume to have a large number of particles of both the components, N1 À 1

and N2 À 1, and we use Thomas-Fermi approximation (see section 4.3.1) in order to

neglect the kinetic terms. In this way the system simplifies very much and the densities

n1 = |Ψ1|2 and n2 = |Ψ2|2 can be find explicitly to be

n1(r ) = g22

∆

[µ1 − g12

g22µ2 − 1

2m1

(ω2

1 −g12

g22

m2

m1ω2

2

)r 2

]n2(r ) = g11

∆

[µ2 − g12

g11µ1 − 1

2m2

(ω2

2 −g12

g11

m1

m2ω2

1

)r 2

]where ∆ = g11g22 − g 2

12 and the chemical potentials must be determined imposing the

normalization of the densities at the particles number.

We now study in more details the case in which one components (1) is much bigger

than the other (2), i. e. n1(r ) À n2(r ) at each r in which n2(r ) > 0. In this case, remaining

58

5.1 Ground state

in the Thomas-Fermi regime, our Gross-Pitaevskii equations simplify to:1

2m1ω

21r 2 + g11n1(r )+g12n2(r ) =µ1,

1

2m2ω

22r 2 + g12n1(r )+ g22n2(r ) =µ2.

(5.5)

So, for the first component the density reduces to the standard Thomas-Fermi inverted

parabola for a single condensates with radius R21 = 2µ1/m1ω

21:

n1(r ) = 1

g11

(µ1 − 1

2m1ω

21r 2

)Θ(R1 − r ) (5.6)

where we have used the Heaviside step functionΘ(x)

Θ(x) =1 if x > 0

0 if x < 0

For the second one things don’t change so much in fact, substituting equation (5.6) into

the second of equations (5.5), we obtain

n2(r ) = 1

g22

[µ2 − g12

g11µ1 − 1

2m2

(ω2

2 −g12

g11

m1

m2ω2

1

)r 2

]Θ(R2 − r )

= 1

g22

(µ2 − 1

2m2ω

22r 2

)Θ(R2 − r ) (5.7)

that has the same form of equation (5.6) with R22 = 2µ2/m2ω

22. Clearly if g12 = 0 we get

also for the component 2 the standard Thomas-Fermi inverted parabola.

We have now to impose the normalization conditions in order to get the expression

for µ1 and µ2. The method is the same for both species and gives

N = 4π

g

∫ R

0dr r 2

(µ− 1

2mω2r 2

)⇒ µ= ħω

2

(15N a

aho

)2/5

where aho = pħ/mω. For the component 2 µ and ω must be replaced by their corre-

sponding µ2 and ω2 so, after the replacement of µ1 in µ2, we obtain the expression for µ2

as

µ2 = ħω2

2

(15N2a2

aho

)2/5

+ g12

g11

ħω1

2

(15N1a1

aho

)2/5

where clearly now aho =√ħ/m2ω2.

For a more detailed description of the topology of the ground state of two interacting

Bose-Einstein condensates see for example [31].

59

Mixtures of Bose-Einstein condensates

5.2 Dynamics

We now apply the methods of section 4.4 (hydrodynamic formulation and collective

modes in trap) to the mixture of two BECs.

5.2.1 Mixtures in homogeneous case

Let us start by considering a mixture of two Bose Einstein condensates in a homogeneous

external potential. In this case the two coupled time independent Gross-Pitaevskii equa-

tions that have to be satisfied are(−ħ2∇2

2m1+ g11|Ψ1(r)|2 + g12|Ψ2(r)|2

)Ψ1(r) =µ1Ψ1(r),(

−ħ2∇2

2m2+ g12|Ψ1(r)|2 + g22|Ψ2(r)|2

)Ψ2(r) =µ2Ψ2(r).

Here g11 and g22 are expressed as functions of the intraspecies scattering length a1 and

a2 while g12 is expressed as a function of the interspecies scattering length a12. In this

situation the equilibrium densities of the gases are the same everywhere (as their wave

functions) so the kinetic terms can be eliminated and, solving the remaining system in

neq1 = |Ψ1|2 and neq2 = |Ψ2|2, we finally get their values

neq1 = 1

∆

(g22µ1 − g12µ2

)neq2 = 1

∆

(g11µ2 − g12µ1

)where ∆ = g11g22 − g 2

12. Recalling the condition of phase separation, equation (5.4), we

can see that the parameter ∆ describes this phenomenon: if ∆ > 0 we have a uniform

mixture, if ∆< 0 we have phase separation.

It is now easy to see that the chemical potential µ defined in equation (4.11) simply

becomes

µ1 = g11n1 + g12n2 − ħ2

2m1p

n1∇2pn1,

µ2 = g12n1 + g22n2 − ħ2

2m2p

n2∇2pn2.

We can now apply the same method of the previous section: we linearize the hydrody-

namic equations around the solution with velocity v = 0 and density neq and we seek for

60

5.2 Dynamics

plane wave solutions proportional to exp(iq · r− iωt ) obtaining as variations of µ (from

now on we indicate neq simply with n)

δµ1 =(

g11 + ħ2q2

4m1n1

)δn1 + g12δn2,

δµ2 =(

g22 + ħ2q2

4m2n2

)δn2 + g12δn1.

Now we can insert these expressions into the equations of motion for the gases 1 and 2

(equation 4.14) to finally obtain the system:(

g11n1q2 + ħ2q4

4m1−m1ω

2)δn1 +

(g12n1q2)δn2 = 0

(g12n2q2)δn1 +

(g22n2q2 + ħ2q4

4m2−m2ω

2)δn2 = 0

For this system to be solvable the determinant of the matrix formed by the coefficient

of δn1 and δn2 must be equal zero. We assume for simplicity that m1 = m2 = m and,

imposing the condition on the determinant, we get the two dispersion relation

ħω± = ħq

2

√√√√√ħ2q2

m2 +2c21 +2c2

2 ±2

√√√√c41 + c4

2 +2c21c2

2

(2g 2

12

g11g22−1

)(5.8)

where c21 = g11n1/m and c2

2 = g22n2/m are the speeds of sound of the two condensates

separately.

Some remarks are necessary on this last result. First of all we observe what happens

if we switch off the interaction between the two gases (g12 = 0). In this case the internal

square root gives simply√

c41 + c4

2 −2c21c2

2 = c21 − c2

2 and we get

ħω1,2 =√

ħ4q4

4m2 +ħ2q2c21,2 =

√(ε0

q

)2 +2g11,22n1,2ε0q ,

that is the normal Bogoliubov dispersion relations of the two condensates separately (see

equation 4.16) that has already been studied in the last section.

Another interesting case is when the value of g12 is the one for which we have phase

separation, g 212 = g11g22. The internal root gives now c2

1 + c22 and we obtain two very

different dispersion relations. If we take the minus sign, the c’s cancel each other and we

remain with a free-particle dispersion law. On the other side, if we take the plus sign the

c’s sum each other giving a Bogoliubov-like dispersion law

ħω+ =√(

ε0q

)2 +2(g11n1 + g22n2

)ε0

q =√

ħ4q4

4m2 +ħ2q2c2,

61

Mixtures of Bose-Einstein condensates

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

g 212/g11g22

c 1,2

FIGURE 5.1: Speeds of sound for a mixture of two BEC as a function of their interactions (equa-tion 5.9). Values of velocities c1 and c2 when there is no interspecies interaction are arbitrarilychosen to be 1 and 1.5.

where the speed of sound is now c2 = c21 + c2

2 .

In conclusion the dispersion relation of equation (5.8) can be rewritten as a normal

Bogoliubov dispersion law with the speed of sound c replaced by

c2 = c21 + c2

2

2± 1

2

√√√√c41 + c4

2 +2c21c2

2

(2g 2

12

g11g22−1

). (5.9)

This two dispersion laws represent two different modes in which the two components of

the mixture move in phase (equation 5.8 with the plus sign) or counterphase (equation

5.8 with the minus sign). These are respectively the hard and the soft mode of the system.

As we have already seen, the sound velocity of the soft one goes to zero at the phase

separation. This is because increasing g12 the energy of phase separated configuration

decreases until it reaches the energy value of the ground state.

In figure 5.1 we show how these two velocities change varying g 212/g11g22, a parame-

ter that is equal 1 at phase separation. The distinction in hard and soft modes is evident

and we can observe how the soft mode goes to zero at the phase separation.

62

5.2 Dynamics

For completeness we briefly analyse also the case in which masses are different. With

exactly the same method we obtain the dispersion relation

ħω± =ħq

√√√√√ħ2q2(m21 +m2

2)

8m21m2

2

+ c21 + c2

2

2±

√√√√c21c2

2

g 212

g11g22+

[ħ2q2(m21 −m2

2)

8m21m2

2

+ c22 − c2

1

2

]2

.

The non interacting case is easy to analyse also in this situation in fact, putting g12 = 0

we immediately get

ħω1,2 =√√√√ ħ4q4

4m21,2

+ħ2q2c21,2 =

√(ε

0q1,2

)2 +2g11,22n1,2ε0q1,2,

has we expected.

At phase separation things become more complicated. We are no longer able to put

the dispersion relation in a Bogoliubov form without considering a q-dependent speed

of sound. The behaviour of the mixture is in any case approximately the same. At low q

we get the same plot of figure 5.1 but to obtain it we have to plot ħω±/qlow exploiting the

linear behaviour of the dispersion law to get c (qlow is an arbitrarily chosen sufficiently

small value of q). Clearly, the bigger is the difference between the masses the more im-

portant becomes the q-dependence of the speed of sound and more different from the

standard Bogoliubov dispersion becomes our dispersion relation.

5.2.2 Harmonic trapped mixtures

In analysing trapped mixtures we focus our attention on the case in which the number

of atoms of both species permits us to assume Thomas-Fermi approximation and the

density of one component is much bigger than the one of the other. So, as we have seen

in the first pages of this chapter, the densities assume the form of equation (5.6) and (5.7):

n1(r ) = 1

g11

(µ1 − 1

2m1ω

21r 2

),

n2(r ) = 1

g22

[µ2 − g12

g11µ1 − 1

2m2

(ω2

2 −g12

g11

m1

m2ω2

1

)r 2

]= 1

g22

(µ2 − 1

2m2ω

22r 2

).

The equation of motion (4.14) can be rewritten in a simpler way if we neglect the quan-

tum pressure term in µ

∂2δni

∂t 2 −∇ ·(

gi i ni

mi∇δni

)= 0

63

Mixtures of Bose-Einstein condensates

where we can define a position dependent speed of sound as

c2i (r ) = gi i ni (r )

mi.

The difference from the homogeneous case is just in this dependence from r . The density

ni (r ) in fact, can be no more taken out of the divergence and its variation δni can not be

written as a plane wave. We remain so with the system∂2δn1

∂t 2 −∇ · (c21(r )∇δn1

)= 0,

∂2δn2

∂t 2 −∇ · (c22(r )∇δn2

)= g12

g11∇ · (c2

2(r )∇δn1)

.

We first observe that the equation for δn1 is exactly the same as the one for one conden-

sate alone and so its dispersion law is (4.22) with ωho substituted by ω1. For the second

one we must do some other consideration. First of all, being the condensate 1 much big-

ger than the condensate 2, it is much more difficult to excite it and so we can consider

only small energy (small frequency) excitations. Furthermore its low frequency excita-

tions, being related to the spatial part of the dispersion law by the relation (4.22), will

produce a spatial variation of the density very small in the region in which also the con-

densate 2 is present so we get ||∇δn2|| À ||∇δn1||. This clearly happens also for the

trivial solution δn1 = 0. In this situation we can neglect the right hand side of the equa-

tion for δn2 obtaining also for it a dispersion law of the form of equation (4.22) but with

ω2ho = ω2

2 =ω22

(1− g12

g11

m1ω21

m2ω22

). (5.10)

We must stress that this relation is valid for all the modes of the condensate 2 provided

that the excited modes of the condensate 1 are enough low frequency.

This result was already present in the equation for the density of the component 2

(5.7) where we notice that the presence of the component 1 has the only effect of chang-

ing the value of the chemical potential and changing the trapping frequency to the value

ω2.

A very interesting thing to notice is that, in this case, the phase separation is not

related with the frequency going to zero as opposed to what happened for the soft mode

in the homogeneous case. We have in fact phase separation for g 212 > g11g22 but here

g22 does not appear at all. Therefore, depending on the value of g22 and of the other

parameters, the phase separation can happen before or after the frequency goes to zero.

This because we are assuming that n1 À n2 and so the particles of the component 2 does

64

5.2 Dynamics

not see each other and their mutual interaction (g22) does not enter in the problem, they

behave as single impurities surrounded by the condensate 1.

Equation (5.10) can be used to precisely measure the s-wave interspecies scattering

length a12 of a two component Bose-Einstein condensate. Having access to the oscil-

lation frequency is in fact easier than directly measure the scattering length and, using

this equation, one can in principle obtain it without too much efforts. This is what is

done in a recently article of Egorov et al. [11] published during the drafting of this work.

They showed that the three dimensional dynamics of a component with a small relative

population in a cigar-shaped trap can be described by a one dimensional Schroedinger

equation for an effective harmonic oscillator. In addition they found that the frequency

of the collective oscillation is determined by the parameter a12/a11 and, fitting numerical

simulation results with experimental ones, they obtained a value of a12 for a particular

mixture in a reasonable agreement with theoretical prediction.

65

CH

AP

TE

R

6NUMERICAL RESULTS

In this chapter we finally want to confirm our theoretical modification of the single com-

ponent condensate dispersion law (5.10) with some numerical simulations. We will use

a modified version of the Fourth-order Runge-Kutta in the Interaction Picture (RK4IP)

algorithm developed by Ballagh et al. [3] (or see appendix A of [26]) to propagate the

stochastic differential equations obtained with the truncated Wigner method [32]. Since

the purely numerical method is not the central part of this thesis, we only sketch its de-

scription and we prefer to focus on the results obtained. For a more detailed analysis

please see [26, 3, 33].

In the next we will study a one-dimensional system of two interacting Bose-Einstein

condensates in the case N1 À N2. We will analyse in particular the breathing and centre

of mass oscillations of the component 2 immersed in the component 1.

6.1 Algorithm

The truncated Wigner method is a particular method that belongs to a general class of

classical field methods. The essence of these methods is as follows. One expresses the

appropriate density operator for the system on a coherent state basis using a classical

quasiprobability distribution function. Assuming that such a distribution function ex-

ists, it can be shown that the action of any quantum-mechanical operator on the den-

67

Numerical results

sity operator is equivalent to the action of a classical operator on the quasiprobability

distribution function. In this way the von Neumann or master equation for the density

operator can be exactly mapped onto a corresponding partial differential equation gov-

erning the evolution of the quasiprobability function. Although simulations of this clas-

sical equation of motion are possible for simple systems, for realistic situations where

the number of modes is measured in the millions the necessary computational resources

greatly exceed anything currently available. Under certain conditions, the distributional

evolution can be transformed into a set of stochastic differential equations (SDEs) de-

scribing the evolution of a classical field, being the numerical equivalent of a single ex-

perimental realisation and giving the method its name. By assembling a number of these

trajectories large enough to adequately span the phase space, one can reconstruct the

quasiprobability function, and hence the full quantum-mechanical system including the

quantum expectation values.

In treatments of Bose-condensed systems, one often uses as quasiprobability func-

tion the Wigner function giving the name at the truncated Wigner method.

Using as a wave-functions basis the plane-wave basis, one can write the stochastic

differential equation for a mode in the form of a Fourier series. In addition, also the

wave-function, written in plane-wave basis is in the form of a Fourier series. So, one can

use some very efficient algorithms for processing these transforms, collectively known

as Fast Fourier Transforms (FFTs) [1].

6.1.1 Computational units

In order to ensure maximum numerical accuracy it is customary in computational treat-

ments of physical system to express the relevant equations in dimensionless computa-

tional units. In this chapter we choose as natural units of length and time

x0 =√

ħ2mω1

t0 = 1

ω1

where ω1 is the trapping frequency of the bigger component (N1 À N2). From these two

relations we easily get the units of energy, ε0 =ħω1, and the dimensionless wave-function

Ψ(x, t ) = Ψ(x, t )x−1/20

where the tilde is used to indicate quantities that are expressed in terms of our compu-

tational units.

68

6.1 Algorithm

Inserting these expressions into the two coupled time-dependent Gross-Pitaevskii

equations we geti∂Ψ1

∂t=

[−∇2 + 1

4x2 + g11

∣∣Ψ1∣∣2 + g12

∣∣Ψ2∣∣2

]Ψ1

i∂Ψ2

∂t=

[−m1

m2∇2 + 1

4

m2ω22

m1ω21

x2 + g12∣∣Ψ1

∣∣2 + g22∣∣Ψ2

∣∣2

]Ψ2

where the dimensionless coupling constants are

gi = gi

ε0x0= gi

ħω1

√2m1ω1

ħ .

6.1.2 Collecting the results

The oscillation frequencies are obtained in two step. First we have to find the ground

state of our mixture because clearly we don’t know an analytical form of the wave func-

tions of the two component. This is done running the program in imaginary time letting

all the high energy component of the initial guess wave-function die. At each step we

have to normalize the wave functions to the number of particles because the evolution

in imaginary time does not preserve the norm. If we want to study the breathing oscilla-

tions in this first part we have two traps centred in zero with the one of component 2 very

narrow compared with the other one. If instead we are interested in the CM oscillations

the trap of the second component is in addition centred in a point different from zero.

After we got the wave-functions, these are used as input of the program in real time

that evolves it for the number of time steps desired. At each time step values of mean

radii and root mean square radii,

⟨xi ⟩ =∫ ∞

0dx |Ψi (x)|2 x⟨

x2i

⟩ = ∫ ∞

0dx |Ψi (x)|2 x2

are calculated and saved (all express in computational units). In this second part the

conformation of the traps change before the evolution starts. If we want to study the

breathing oscillation the narrow trap of the component 2 is broadened to become com-

parable with the first component one. If instead we are interested in the CM oscillations

the trap of the second condensate is centred and broadened.

69

Numerical results

g11 : g12 : g22 m1/m2 (ω1/ω2)2 ωteo ωnum ωnoint

1 : 0.388 : 1.076 1.00 1.00 1.35 1.39 1.731 : 0.778 : 1.076 1.00 1.00 0.82 0.85 1.731 : 0.648 : 1.076 1.00 1.00 1.03 1.06 1.731 : 0.648 : 1.076 1.00 0.50 2.01 2.03 2.451 : 0.648 : 1.076 1.23 1.00 0.77 0.79 1.731 : 0.648 : 1.076 1.50 0.70 1.17 1.19 2.07

TABLE 6.1: Comparison between theoretical and computational frequencies for the breath-ing mode with N1 = 200 and N2 = 10. Parameters for the condensate 1 are: g11 = 3.848,m1 = 86.91 a.m.u., ω1/(2π) = 200 Hz. The trap frequency of the condensate 2 during the imagi-nary time evolution is ω2/(2π) = 500 Hz.

g11 : g12 : g22 m1/m2 (ω1/ω2)2 ωteo ωnum ωnoint

1 : 0.388 : 1.076 1.00 1.00 0.78 0.74 1.001 : 0.778 : 1.076 1.00 1.00 0.47 0.48 1.001 : 0.648 : 1.076 1.00 1.00 0.59 0.57 1.001 : 0.648 : 1.076 0.90 1.00 0.65 0.62 1.001 : 0.648 : 1.076 1.00 0.60 1.01 1.00 1.291 : 0.648 : 1.076 0.50 0.60 1.16 1.16 1.29

TABLE 6.2: Comparison between theoretical and computational frequencies for the CM modewith N1 = 200 and N2 = 10. Parameters for the condensate 1 are: g11 = 3.848, m1 = 86.91 a.m.u.,ω1/(2π) = 200 Hz. The trap frequency of the condensate 2 during the imaginary time evolution isω2/(2π) = 500 Hz.

6.2 Results

We restrict our analysis to the one dimensional case in order to have reasonable compu-

tational time. In this case the oscillations for a single Bose gas are [22]

ω2 =ω2trap

k

2

[2+γ(k −1)

]whereγdistinguishes between Thomas-Fermi (γ= 1/2), 1D mean-field (γ= 1) and Tonks-

Girardeau (γ = 2) regimes and k indicates the mode considered, CM motion (k = 1),

breathing (k = 2). So in cases of our interest we get ω = ωtrap for the CM mode and

ω=p3ωtrap for the breathing one.

Equations (5.10) in computational units becomes

ω= ω

ω1= ω2

ω1

√√√√1− g12

g11

m1ω21

m2ω22

× 1 CM motionp

3 breathing= ω2

ω1χK (6.1)

70

6.2 Results

0 5 10 15 20 251

2

3

4

5

6

7

ω1t

√ ⟨ x2⟩ /x

0

FIGURE 6.1: Numerical breathing oscillations of the two condensates which parameters corre-sponds to the second entry of table 6.1. Red line represents the bigger one, black line the smallerone.

whereω2 is the frequency of the trap during the evolution in real time, χ is the parameter

describing the modification of the frequency and K describes the mode. Fitting ⟨x2⟩ and⟨x2

2

⟩with a sinusoidal function we have directly access to the values of ω that we can

compare with the theoretical ones.

In table 6.1 and 6.2 are shown some results for the breathing mode and the CM one

at different values of the ratio of the mass, of the interacting constants and of the fre-

quencies. The accordance between theoretical and numerical values is very good. We

also note that the theoretical previsions are systematically smaller than the numerical

one for the breathing mode and, except in one case, the opposite happens for the CM

mode. We will treat this aspect in the next. In figure 6.1 and 6.2 we show two examples

of oscillation of the two condensates for the breathing and the CM mode. All data shown

until this point are taken at values of interacting constants for which the system is not

phase separated, we treat the phase separation in a second moment.

Having checked the validity of equation (6.1) in the case N1 À N2, for g22 fixed and

for oscillation frequencies not too much small, we now want to better understand the

limit of this formula.

71

Numerical results

0 5 10 15 20 25

−10

−5

0

5

10

ω1t

⟨ x⟩ /

x 0

FIGURE 6.2: Numerical CM oscillations of the two condensates which parameters correspondsto the first entry of table 6.2. Red line represents the bigger one, black line the smaller one.

We start analysing the behaviour of the breathing oscillations of the second conden-

sate as the parameter χ goes to zero. As we can see from figure 6.3, getting close to zero

the frequency becomes smaller and becomes harder and harder to extrapolate it from

the curve. From χ' 0.25 it loses completely the characteristics of an oscillation and one

can no more assign to it a frequency. Observing the time evolution of the densities of

the two component, we see that at small values of χ the condensate 2, after the initial

broadening, exits from the bigger one and can no longer return to the centre of the trap

remaining broadened on a wide region.

In principle nothing forbids that phase separation happens before the frequency

(6.1) goes to zero. We recall that we have phase separation for ∆ = g11g22 − g 212 < 0 so,

keeping g11 and g12 fixed, we can decrease g22 up to and below the critical value g 212/g11.

In figure 6.5 we show the values of frequency resulting from varying g22 from the ratio

1 : 0.648 : 0.260 to the ratio 1 : 0.648 : 3.637 for the breathing mode. We can see that the

numerical values of the frequency are in good agreement with the theoretical prediction

until we are not to much close to the phase separation. At ∆ = 0 the theoretical value

differ from the numerical one of about 15% and for ∆ < 0 this difference explodes (at

72

6.2 Results

0 5 10 15 20 25

2

4

6

8

10

ω1t

√ ⟨ x2⟩ /x

0χ= 0.00χ= 0.22χ= 0.32χ= 0.40χ= 0.45χ= 0.59

FIGURE 6.3: Breathing oscillations of the condensate 2 for different values of χ. Other parame-ters are those of the third entry of table 6.1 where only m2 changes in order to obtain the desiredvalue of χ.

∆=−0.24 we have ω= 2.59, point out of the figure).

In figure 6.6 we show instead the values of frequency resulting from varying g22 from

the ratio 1 : 0.648 : 0.078 to the ratio 1 : 0.648 : 6.500 for the CM mode. Also in this case the

numerical values are in good agreement with the theoretical ones until we reach phase

separation, at this point the frequency starts to decrease very rapidly.

These behaviours are strictly related to the observation that our numerical frequen-

cies are always bigger (breathing mode) or smaller (CM mode) than the theoretical ones.

Equation 6.1 has in fact been obtained under the assumption that the condensate 1 is

unaffected by the the presence of the second one. This clearly is not correct as one can

see from figure 6.4.

In the case of breathing the bigger condensate has a deflection on the central region

where also the smaller one is present, here its density decreases and this hole is seen

by the second one as an additional potential which increases its confinement and con-

sequently its oscillation frequency. Clearly the effect is more dramatic in case of phase

separation (see figure 6.4).

73

Numerical results

−10 0 100

2

4

6

8

10

12

−5 5

x/x0

|Ψi|2

−10 0 10−5 5

x/x0

FIGURE 6.4: Ground state density profiles of the two component at phase separation and mixedphase before the relaxation of the second trap. Values of parameters are those of the third entryof table 6.1 with interacting constants ratio 1 : 0.648 : 0.260 and 1 : 0.648 : 1.455 respectively.

−0.2 0 0.2 0.4 0.6 0.8

1

1.1

1.2

1.3

1.4

1.5

1.6

0

1.03

Phase Separation (∆= 0)

∆

ω

FIGURE 6.5: Variation of the breathing frequency with g22 expressed as ∆ = ∆/(g11g22 + g 212)

with g11 and g12 kept fixed. Other parameters are those of the third entry of table 6.1. Verticalline represents the phase separation point, horizontal one the theoretical value of the frequency.Curve is only a guide to the eye.

74

6.2 Results

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0

0.59

Phase Separation (∆= 0)

∆

ω

FIGURE 6.6: Varying of the CM frequency with g22 expressed as ∆=∆/(g11g22+g 212) with g11 and

g12 kept fixed. Other parameters are those of the third entry of table 6.2. Vertical line representsthe phase separation point, horizontal one the theoretical value of the frequency. Curve is only aguide to the eye.

In the case of CM mode instead, the hollow on the bigger condensate moves together

with the smaller one resulting in an effective reduced mass and consequently in a re-

duced frequency.

Both this effects are of second order compared to the first order modification of the

frequency (6.1). This can be seen also in tables 6.1 and 6.2 where the differences of the

theoretical frequencies from the non interacting ones are of order ∼ 0.1 while the differ-

ences from the numerical values from the theoretical ones are of order ∼ 0.01.

Remains to be discussed the behaviour of our predictions for different values of N2.

Results are plotted in figure 6.8, we see how the frequency initially decrease increasing

the number of particles of the smaller condensate. But from N2/N1 = 0.30 the curves

become difficult to fit because now the motion of one component is much more affected

by the motion of the other and this causes a broadening of the curves. This effect is in

any case less dramatic than the one we obtained approaching the phase separation or

decreasing the parameter χ.

75

Numerical results

0

2

4

6

8

10

12

|Ψi|2

0

2

4

6

8

10

12

|Ψi|2

−15 −10 −5 0 5 10 150

2

4

6

8

10

12

x/x0

|Ψi|2

FIGURE 6.7: Evolution in real time of the two condensates in a phase separated regime. Ratioof coupling constant is 1 : 0.648 : 0.078, values of other parameters are those of the third entry oftable 6.2.

76

6.2 Results

0 2 4 6 8 10 121

2

3

4

5

6

ω1t

√ ⟨ x2⟩ /x

0

N2/N1 = 0.05 N2/N1 = 0.10 N2/N1 = 0.20N2/N1 = 0.30 N2/N1 = 0.50 N2/N1 = 1.00

FIGURE 6.8: Breathing oscillation for different value of N2/N1. Values of other parameters arethose of fourth entry of table 6.1.

Finally we show in figure 6.7 a particular case of CM oscillation in which we have an

highly separated regime (ratio of coupling constant is 1 : 0.648 : 0.078). In this case the

density profiles of the two gases, except for the moving back and forth, change very little

during the oscillation. Such density profiles may suggest that we have created a soliton

but analysing the phase of the two wave functions we do not find the characteristic jump.

Despite this we think this is a type dark-bright solitons described in [7] whit a jump in

phase different from π. Alternatively it can be a kind of self-trapping effect that prevents

the smaller condensate to expand during the oscillation. In any case this is only a starting

point for future insights in order to better understand this behaviour.

77

CONCLUSION

In this work we studied the ground state and the dynamics of a mixture of two Bose gases

passing from relatively high temperatures, where quantum effects are not so important,

to very low temperatures, where the quantum behaviour of matter becomes crucial. We

specialized in treating the more complex case in which the gases are trapped by means of

harmonic potentials and, where relevant, we also described how things change in pass-

ing to one spatial dimension or in considering the impurity limit where one gas is much

less dense than the other.

The motivation for starting this thesis on this subject was an experiment realized at

LENS laboratory of Firenze by Catani et al. [8]. They studied the breathing oscillation of

a potassium impurity (41K) in a one dimensional tube filled by rubidium atoms (87Rb) at

different values of the rubidium-potassium coupling constant.

We initially focused on the case in which the two gases do not interact with each

other and we studied the breathing oscillation of the impurity after the sudden releas-

ing of the trap as a function of the temperature. In doing so we used the formalism of

instantaneous ladder operators.

For the description of the high temperature behaviour we adopted a purely classical

treatment using the kinetic theory of gases and the Boltzmann equation. We developed

a system of Boltzmann equations for the distribution functions of the gases, coupled

by the collision integral term. Using the method of averages we recast this system to a

6-equations one which describes the time evolution of some dynamical quantities in-

cluding the square radii of the clouds. The system thus obtained has been numerically

solved with parameters corresponding to the experiment, for both one and three dimen-

sions.

We then studied the T = 0 case applying the Bose-Einstein condensation theory. First

of all we derived the equations for the ground state density profiles under the Thomas-

Fermi approximation and with the assumption that one component is much less dense

79

Conclusion

than the other. Passing to the dynamics of the mixture and keeping these approxima-

tions, we obtained a relation describing the oscillation modes of the smaller component

within the bigger one. This is the main result of this thesis and predicts a decrease in

frequency whose intensity depends on the first component coupling constant, on the

interspecies coupling constant and on the ratio of mass and trapping frequencies. We

also derived a Bogoliubov-like dispersion law for the homogeneous case that relates the

sound velocities of the two condensates with the new excitation energy. We observed

that the vanishing of the excitation energy is a phenomenon whose causes are very dif-

ferent for the two cases. In the second one it is clearly related to the phase separation

while in the first one it happens independently on the value of the coupling constant of

the second condensate.

Coming back to the experiment and comparing it with the various models we have

developed, we found that no one of these models totally describes the experimental data.

The BEC theory predicts a shift of frequency that has not be observed while the classical

gas kinetic theory fits the data only assuming temperatures much higher than experi-

mentally measured. We concluded then that the system studied in [8] is not completely

classical nor completely condensed. Other groups have tried to describe this experiment

[8, 18] but until this moment no one has obtained a full accordance with the experimen-

tal data.

Finally we used a numerical simulation to confirm the dispersion relation obtained.

Our theoretical predictions are in good agreement with the numerical ones if we are not

in a phase separated state. We found only small deviations from the theory and we think

that these deviations can be described theoretically considering the next order of approx-

imation for condensate’s densities. For the phase separated regime a new description

must instead be developed. We also checked the limit of validity of our approximation

increasing the number of particles of the smaller condensate, we found that our previ-

sions are still good for N1/N2 ' 4.

Directions for future work

As next developments for this work we can extend our code to the three dimensional case

to study deformed traps and compare effective 1D results with real 1D ones. We can also

develop a dispersion relation that takes into account the deformation in density of the

bigger condensate pushing forward the approximation order and then compare this new

results with the numerical ones.

80

Conclusion

From a broader point of view a natural extension of this work can be the study of the

oscillation at phase separation. Our assumption in fact does not permit to correctly de-

scribe this situation and for a future calculation the hypothesis of phase separated regime

must be introduced from the beginning. Also the creation of a dark-bright solitons is an

interesting topic and remains to be verified. In addition the experiment of Catani et al.

[8] remains to be explained.

81

AP

PE

ND

IX

AUSEFUL FORMULAS IN CALCULATING

COLLISION INTEGRALS

A.1 Momentum gained by a particle in a collision

We would find an expression for the momentum gained by a particle in a collision. We

use the transformations (3.12) and the conservation law on v0 to obtain:

m1(v ′1 − v1) = m1

(v ′

0 +µ

m1v ′

r

)−m1

(v0 + µ

m1vr

)=−µ(vr − v ′

r ) (A.1)

A.2 Kinetic energy gained by a particle in a collision

We would find a useful expression for the kinetic energy gained by a particle in a collision.

In doing so we start recalling some definitions of section 3.3.2:

A(v0, vr )− A(v0, v ′r ) = 2µv0(vr − v ′

r )

m1v21 −m2v2

2

2= v2

0

2(m1 −m2)+2µv0vr +

µ2v2r

2

(1

m1− 1

m2

)≡ A(v0, vr )

83

Useful formulas in calculating collision integrals

Using a modified energy conservation equation m1v ′1

2 −m1v21 = m2v2

2 −m2v ′2

2 we get

m1(v ′1

2 − v21) = 1

2

(m1v ′

12 −m1v2

1 +m2v22 −m2v ′

22)

= A(v ′0, v ′

r )− A(v0, vr )

=−[A(v0, vr )− A(v0, v ′

r )]

=−2µv0(vr − v ′r ) (A.2)

A.3 Centre of mass kinetic energy

In this section we will show how to derive the expression for the kinetic energy in the

centre of mass coordinates v0 and vr :

M v20 +µv2

r = m1v21 +m2v2

2 = m1v ′1

2 +m2v ′2

2 (A.3)

= (m1v1 +m2v2)2

m1 +m2+ m1m2(v1 − v2)2

m1 +m2

= 1

M

[m2

1v21 +m2

2v22 +2m1m2v1v2 +m1m2(v2

1 + v22)−2m1m2v1v2

]= 1

M

[m1v2

1(m1 +m2)+m2v22(m1 +m2)

]= m1v2

1 +m2v22 .

84

AP

PE

ND

IX

BCOLLECTIVE EXCITATIONS OF AN

ULTRACOLD ATOMS GAS

In this appendix we investigate the collective oscillations of a cloud of ultracold atoms

whose chemical potential behaves like

µ=µ−Vext = Anγeq.

Inserting an isotropic harmonic potential of frequency ωho and inverting this equation

we obtain the equilibrium density of the gas as a function of r

neq(r ) =(µ

A

)1/γ (1− r 2)1/γ

and the derivative of µ with respect of neq

µ′(r ) = Aγnγ−1eq = A1/γγµ1−1/γ (

1− r 2)1−1/γ.

We now insert these expressions into equation (4.19) obtaining

r(1− r 2)F ′′(r )+2

[A−Br 2]F ′(r )+Cr F (r ) = 0.

where

A = l +1

B = l +3− 1

γ

85

Collective excitations of an ultracold atoms gas

C = 2

γ

(ω

ωho

)2

−2(2l +3)+ 2

γ(l +3)

The only difference from the case of a Bose gas is in the parameters A,B ,C . So, expand-

ing F (r ) in power of r , we obtain the same condition on the coefficients of the expansion

that we have obtained in chapter 4, that is equation (4.21) that we report here for conve-

nience.

cn+2 = cnn(n −1)+2Bn −C

(n +2)(n +1+2A)

If we want odd solutions to our problem we keep only odd coefficients so we substi-

tute n with 2n+1. In order to have a polynomial of order 2n+1 the coefficient c2n+3 must

be zero and this gives us the dispersion relation(ω

ωho

)2

= γ(2n +1)

(l +n +3− 1

γ

)+γ(2l +3)− (l +3)

The interesting values of γ are in this case 1 and 2/3 and we obtain respectively

(ω

ωho

)2

=2n2 +2nl +5n +2l +2 if γ= 1,

l + 43 n(n + l +2) if γ= 2/3.

If we want even solution we substitute n with 2n. Also in this case in order to have a

polynomial of order 2n the coefficient c2n+2 must be zero and we obtain the dispersion

relation (ω

ωho

)2

= γn

(2n +2l +5− 2

γ

)+γ(2l +3)− (l +3)

The interesting values of γ are in this case 1 and 2/3 and we obtain respectively

(ω

ωho

)2

=2n2 +2nl +3n + l if γ= 1,

13 [l +4n(n + l +1)]−1 if γ= 2/3.

86

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ACKNOWLEDGEMENTS

Giunto alla fine di questo mio lavoro voglio ringraziare tutti coloro che mi hanno sup-

portato in questi mesi.

Un grazie particolare va alla mia famiglia che è sempre stata al mio fianco in ogni

situazione, al mio relatore Alessio Recati per essere riuscito a trasmettermi tutto il suo

entusiasmo verso questa materia, a Robin Scott per la pazienza e il tempo che mi ha

dedicato, a tutti i miei amici per i bei momenti passati assieme.

Infine un grande grazie va ad Alice per essere sempre stata dalla mia parte, per non

aver mai smesso di spronarmi, per esserci sempre stata; a lei dedico questa tesi.

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