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## Gross-Pitaevskii equation from the Hartree ansatz

Here, the Gross-Pitaevskii equation [15, 16] is introduced, constituting a closed equation for the macroscopically occupied single particle mode #0(%r, t) below Tc, called the “condensate wave function”. Assuming N particles in the Bose gas to share the same, in general time dependent single particle mode #0(%r, t), the Hartree ansatz [15] can be used to derive the Gross-Pitaevskii equation. The first quantized N-particle Hamiltonian HN for N atoms interacting via two body collisions in the Bose gas is given by HN = %N k=1 ())))* %p2 k 2m +Vext(%rk) +,,,,- + g 2 % k!l 4(%rk −%rl) .

### Theories of condensate growth

There exist different quantum kinetic theories to describe the time evolution of the average con- densate fraction during Bose-Einstein condensation. Summarizing their relevant results in a short fashion, we demonstrate that – to our knowledge – none of the theories could so far address the dynamics of the full condensate number distribution during Bose-Einstein condensation which highlights the condensate formation process of the atoms in the gas below Tc. This is because either the full quantum problem is in most cases impossible to solve numerically, the total particle number is not conserved, or/and condensate formation is studied in terms of the quantum Boltz- mann equation. Moreover, we notice that there exists no quantitative master equation theory for closed and interacting dilute Bose gases below Tc imposing particle number conservation onto the state of the system.

#### Condensate growth from quantum Boltzmann equation

Many works focus on Bose-Einstein condensation in terms of the quantum Boltzmann equation (QBE) which describes the kinetics of a quantum gas in terms of time dependent particle number occupations f%k(t). More explicitly, theQBE is given by 0 f%k 0t = % %l, % m,%n C (%k,%l, %m,%n) 9f %m f%n( f%l + 1)( f%k + 1) − f%k f%l ( f0.

**Pioneering works of Levich and Yakhot**

First investigations on Bose-Einstein condensation have been performed by Levich and Yakhot [60]

using the QBE in order to study the dynamics of a gas in a box coupled to a bath of fermionic particles below the critical temperature. In this first study, the authorsmade the important assertion that there exist two macroscopically distinct stages of condensate formation, the first being a fast equilibration of the gas’ high energetic part within a few two body collision times (thermalization), &col ! 50− 100 ms, and the second stage, the actual formation of the condensate4 – highlighting a clear separation of the time scale &col for thermal equilibration of the non-condensate part of the gas from the time scale &0 for Bose-Einstein condensation.

**Predictions of Kagan, Svistunov and Shlyapnikov**

First considerations on Bose-Einstein condensation by Svistunov [61] were also conducted for the simplified case of a Bose gas in a box, replacing the terms ( f%l + 1) by 1, and thus assuming f%l ≪1 in Eq. (2.17). Within this approximation, Svistunov was led to an analytical solution for the distribution function f0(E, t) being a function of energy E. According to Svistunov, the distribution f0(E, t) propagates from high energies to the ground state energy within a time scale &col after it returns to the initial energy region. The predictions of Svistunov correspond to our observation that particles are transported from the non-condensate modes towards the condensate mode (with net positive current towards the condensate mode below Tc), until a slow (linear) convergence5 into a detailed balance particle flow (compare Chapters 6, 8).

Subsequently [61, 62], Kagan, Svistunov and Shlyapnikov studied the dynamics of condensate growth in more detail again showing that there exist two distinct stages of condensate formation: Initially, the non-equilibrium state of the gas rapidly equilibrates and implies the transport of high- energetic particles to the low-energy region, occuring on the average time scale &col of two body collisions in the gas, which is equivalent to the observations of Levich and Yakhot [61, 62] (see Section 2.4.2) and the results of Holland, Walser and Cooper [59, 63] (see Section 2.4.4). Our theory will show that this separation of time scales – theoretically found by Levich et al., Walser et al., and experimentally confirmed by Miesner et al. [64] – enables and justifies the derivation of a Markov quantum master equation. The second step comprises the experimentally observable condensate formation process, where average macroscopic occupation of the ground state mode occurs within a time scale &0, which ismuch longer than the time scale of two body collisions, &col. Kagan, Svistunov and Shlyapnikov cannot give a number for the time scale &0, howeverproviding a qualitative understanding of Bose-Einstein condensation. In contrast to &0, the time scale &col ! 50 − 100 ms can be theoretically estimated for a thermal gas (see Chapter 3). A direct monitoring of the full quantum distribution during the second stage of Bose-Einstein condensation in real-time is displayed in Chapter 6.

**Kinetic evolution obtained from Holland, Williams and Cooper**

In an early work of Holland, Williams and Cooper [59], the kinetics of condensation formation are studied in a harmonic trap using a simulation procedure of the QBE within the ergodic assumption. Therein, the authors find the characteristic dynamical behavior of exponential condensate growth, f0(t) = f0(,) 9 1 − e−t/&0 : , (2.20).

where f0(,) is the equilibrium condensate occupation (which depends on specific parameters such as temperature, trap frequencies, etc.), and &0 ! 1. . .4 s, the characteristic time scale of condensate formation, being extracted from the condensate growth curves obtained from the exact numerical propagation of the QBE. The times scales are in agreement with our results in Chapters 6, and Eq. (2.20) qualitatively resembles the condensate growth Eq. (6.6) for the average condensate population as predicted by our master equation theory of Bose-Einstein condensation. The exact numerical propagation of Eq. (2.17) is possible to be carried out for small particle numbers of the order of N = 102−103, entailing thedynamics of the expectation value of the average condensate occupation, f0(t). As evident fromEq. (2.17), the full quantumstate of the Bose gas cannot be reproduced from theQBE. The equilibrium occupations of non-condensate single particle modes f%k(,) are found [59] to be in accordance with Bose-Einstein statistics (including the discrete nature of single particle levels (see Chapter 1)), and hence with the results of our quantum master equation theory (see Chapter 8). Again, the important implication of the QBE for our quantum master equation of Bose-Einstein condensation is the separation of time scales between the thermal equilibration in the gas from the condensate formation time. The time scale for equilibration in the non-condensate, i.e. the high-energetic part of the gas, is [59, 63, 65] of the order of the average time scale for two body collisions, &col ! 50−100 ms, whereas condensation formation is predicted to last a few seconds.

**Condensate and non-condensate subsystems**

Motivated by the separation of time scales (see Section 3.1), we split the N-particle Bose gas into a condensate and a non-condensate subsystem. For this purpose, the second quantized field is decomposed into #ˆ (%r) = #0(%r)aˆ0 +#ˆ &(%r) . (3.5).

Here, #0(%r) denotes the condensate wave function, which we quantify by the Gross-Pitaevskii equation (see Eq. (2.16) of Chapter 2.3). The operator ˆ a0 annihilates a particle in the condensate mode. On the other hand, #ˆ &(%r) = ! k!0 #k(%r) ˆ ak denotes the non-condensate field operator, with annihilation operators ˆ ak of the single particle modes #k(%r), which are by definition orthogonal to the condensate mode #0(%r) (see Section 3.2.3 and Chapter 4).

Corresponding to the splitting of the second quantized field in Eq. (3.5), the Hamiltonian in Eq. (3.4), including two body interactions, falls into ˆH = ˆ H0 + ˆ H& + ˆV0& , (3.6).

where ˆ H0 and ˆ H& denote the condensate and the non-condensate Hamiltonian1, respectively, and ˆV0& the various two body interaction processes between condensate and non-condensate. The latter can be classified as single particle events (!N0 = −!N& = ±1, labeled by !), pair events (!N0 = −!N& = ±2, labeled by ») and scattering events (!N0 = !N& = 0, labeled by #), according to the net exchange of condensate particles !N0 per two body interaction process.

**Table of contents :**

Introduction to the thesis

Motivation of this thesis

How to model Bose-Einstein condensation microscopically?

Outline of the thesis

**I CONCEPTS OF ULTRACOLD MATTER THEORY **

**1 Bose-EinsteincondensationinidealBosegases **

1.1 What is a Bose-Einstein condensate?

1.2 What is quantum ergodicity?

1.3 Original prediction of Bose-Einstein condensation

1.4 Experimental state-of-the-art

1.5 Bose-Einstein condensation in harmonic traps

1.5.1 Grand canonical ensemble

1.5.2 The canonical ensemble

1.6 Bose-Einstein condensation in position space

**2 InteractingBose-Einsteincondensates **

2.1 S-wave scattering approximation

2.2 Hamiltonian for two body interactions

2.3 Gross-Pitaevskii equation from the Hartree ansatz

2.4 Theories of condensate growth

2.4.1 Condensate growth from quantum Boltzmann equation

2.4.2 Pioneering works of Levich and Yakhot

2.4.3 Predictions of Kagan, Svistunov and Shlyapnikov

2.4.4 Kinetic evolution obtained from Holland, Williams and Cooper

2.4.5 Stoof’s contribution

2.4.6 Quantum kinetic theory

Survey: Which current aspects can we adopt to monitor the many body dynamics during

Bose-Einstein condensation?

**II QUANTUM MASTER EQUATION OF BOSE-EINSTEIN CONDENSATION **

**3 Concepts,basicassumptionsandvalidityrange **

3.1 Motivation for master equation: Separation of time scales

3.2 Modeling of many particle dynamics

3.2.1 Two body interactions in dilute gases

3.2.2 Condensate and non-condensate subsystems

3.2.3 Thermalization in the non-condensate

3.3 N-body Born-Markov ansatz

3.3.1 General Born-Markov ansatz

3.3.2 Born ansatz for gases of fixed particle number

3.3.3 Markov approximation for a Bose-Einstein condensate

3.4 Limiting cases and validity range

3.4.1 Dilute gas condition

3.4.2 Perturbative limit

3.4.3 Thermodynamic limit

3.4.4 Semiclassical limit

3.4.5 Physical realization of limiting cases

**4 Quantizedfields,twobodyinteractionsandHilbertspace **

4.1 Definition of the condensate

4.2 Interactions between condensate and non-condensate

4.2.1 Separation of the second quantized field

4.2.2 Decomposition of the Hamiltonian

4.2.3 Two body interaction processes

4.3 Hamiltonian of the non-condensate background gas

4.3.1 Diagonalization of the non-condensate Hamiltonian

4.3.2 Perturbative spectrum of non-condensate particles

4.4 Hilbert spaces

4.4.1 Single particle Hilbert space

4.4.2 Fock-Hilbert space

4.4.3 Fock-Hilbert space of states with fixed particle number

**5 LindbladmasterequationforaBose-Einsteincondensate **

5.1 Evolution equation of the total density matrix

5.2 Time evolution of the reduced condensate density matrix

5.2.1 N-body Born ansatz

5.2.2 Evolution equation for the condensate

5.3 Contribution of first order interaction terms

5.3.1 General operator averages in the Bose state

5.3.2 Vanishing of linear interaction terms

5.4 Dynamical separation of two body interaction terms

5.5 Lindblad operators and transition rates

5.5.1 Lindblad evolution term for single particle processes (!)

5.5.2 Lindblad evolution term for pair processes (« )

5.5.3 Evolution term for scattering processes (#)

5.6 Quantum master equation of Lindblad type

**III Environment-induced dynamics in Bose-Einstein condensates **

**6 MonitoringtheBose-Einsteinphasetransition **

6.1 Dynamical equations for Bose-Einstein condensation

6.1.1 Master equation of Bose-Einstein condensation

6.1.2 Growth equations for average condensate occupation

6.1.3 Condensate particle number fluctuations

6.2 Bose-Einstein condensation in harmonic traps

6.2.1 Monitoring of the condensate number distribution

6.2.2 Dynamics of the condensate number variance

6.2.3 Average condensate growth from the thermal cloud

6.3 Comparison of formation times to state-of-the-art

6.4 Modified condensate growth equation

**7 Transiton ratesf or Bose-Einstein condensation **

7.1 Single particle (!), pair (« ) and scattering (#) rates

7.1.1 Single particle feeding and loss rate

7.1.2 Pair feeding and loss rates

7.1.3 Two body scattering rates

7.2 Depletion of the non-condensate

7.3 Detailed particle balance conditions

7.4 Single particle, pair and scattering energy shifts

7.5 Transition rates and energy shifts in the perturbative limit

7.5.1 Leading order of transition rates

7.5.2 Leading order energy shifts

7.6 Generalized Einstein de Broglie condition

**8 Equilibrium properti sofadilute Bose-Einstein condensate **

8.1 Equilibrium steady state after Bose-Einstein condensation

8.2 On the quantum ergodicity conjecture

8.3 Exact condensate statistics versus semiclassical limit

8.3.1 Condensate particle number distribution

8.3.2 Average condensate occupation and number variance

8.3.3 Shift of the critical temperature

8.4 Analytical scaling behaviors in the semiclassical limit

8.4.1 Condensate and non-condensate particle number distribution

8.4.2 Average condensate occupation and number variance

8.4.3 Higher order moments of the steady state distribution

**9 Final conclusions **

9.1 Master equation of Bose-Einstein condensation

9.2 What is Bose-Einstein condensation?

9.3 Outlook

**A Importantproofsandcalculations **

A.1 Correlation functions of the non-condensate field

A.2 Detailed balance conditions

A.3 Occupation numbers of the non-condensate

A.4 Proof of uniqueness of the Bose gas’ steady state

A.5 Non-condensate thermalization

**Bibliography **