QUANTUM MASTER EQUATION OF BOSE-EINSTEIN CONDENSATION 

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Bose-Einstein condensation in ideal Bose gases

We recall the technical terms “Bose-Einstein condensation”, and “quantum ergodicity”, before the reader is introduced into the experimental state-of-the-art. Einstein’s original prediction of Bose-Einstein condensation is summarized in a short fashion to demonstrate the link of the Einstein de Broglie condition1 to the first experimental observations of condensate formation [23, 24, 25]. Thereupon, the canonical and the grand canonical statistical ensembles are implemented as state-of-the-art theoretical techniques to access the condensate particle number statistics of non-interacting bosonic gases below the critical temperature Tc for Bose-Einstein condensation.

What is a Bose-Einstein condensate?

Encyclopic definition: “When a gas of bosonic particles is cooled below a critical temperature Tc, it condenses into a Bose-Einstein condensate. The condensate consists of a macroscopic number of particles, which are all in the ground state of the system. Bose-Einstein condensation (BEC) is a phase transition, which does not depend on the specific interactions between particles. It is based on the indistinguishability and wave nature of particles, both of which are at the heart of quantum mechanics [26].”
We shall recall here that the purpose of the present thesis is to directly model the microscopic condensate number distribution during the Bose-Einstein phase transition under inclusion of both the wave nature and the indistinguishability of the quantum particles. Within our theory, we will theoretically proof that the equilibrium steady state indeed depends on the specific (nonlinear) form of the interactions, nevertheless recovering the statistics of a thermal state for an ideal gas!

What is quantum ergodicity?

The expression “ergodicity” refers to a concept of classical statistical mechanics. Introduced by Ludwig Boltzmann in the nineteenth century [13, 14], a system which behaves ergodically is ment to sample each point in phase space equally over time, so that each state with the same energy has equal probability to be populated. Boltzmann showed that his conjecture applies for a gas of non-interacting, classical particles, subject to the condition of fixed energy and fixed particle number, evolving to a maximum entropy thermal state under the assumption of molecular chaos. However, some examples from classical statistical mechanics are known to be non-ergodic (e.g. strictly integrable systems) and do not relax into a thermal state, even after infinitely long times, such as a chain of coupled, one-dimensional harmonic oscillators [11]. Even less is known about the accuracy of the thermal state ansatz for quantum systems with finite particle number (where the density matrix does not necessarily factorize into different partitions such as condensate and non-condensate), especially for weakly interacting, quantum degenerate bosonic gases. So far, the ergodicity conjecture has been proven [27] only for ideal quantum gases coupled to an external heat reservoir. For an ideal gas, it is intuitive that the steady state of the non-interacting particles being in contact with a heat reservoir is a thermal state – independent of the condensate non-condensate interaction strength – since entirely the coupling to the external heat reservoir (which itself is in a thermal state) thermalizes the system. In contrast, the equilibrium steady state of a weakly interacting Bose gas which undergoes condensation because of atomic collisions as predicted by our master equation theory still depends on the specific nonlinearity of the atomic interactions: A question to be answered in the present thesis is hence whether a weakly interacting gas of finite particle number below Tc really relaxes towards a thermal Boltzmann state of an ideal quantum gas [27], in the limit of very weak interactions, as presumed by the theory of thermodynamics?

Original prediction of Bose-Einstein condensation

In the 20’s of the twentieth century, Einstein predicted [28, 29] what we call today “Bose-Einstein condensation”: a macroscopic number expectation value of a single particle quantum state, in a gas of N indistinguishable, non-interacting bosonic particles.
The heart of Bose’s contribution [30, 31] to Bose-Einstein condensation was to treat a photon gas as an ensemble of indistinguishable bosonic particles, inspiring Einstein to apply [28, 29] Bose’s statistics [30, 31] equivalently to ideal monoatomic gases enclosed in a volume V. This led him to the Bose-Einstein distribution function Nl = exp[ +βη%] 1 Equation (1.1) refers to the average occupation number N%l of a single particle state with energy η%l = 2|%l|2/2m, where %l = (lx,ly,lz ) is a particle’s wave vector in each spatial direction x, y, and z, β = (kB T) 1 the inverse thermal energy of the gas, and α a Lagrangian multiplier. For a gas at thermal equilibrium, α can be interpreted [10] as the product of the inverse thermal energy β and the chemical potential µ of the gas, defined by µ= β1 ∂lnZ (N,T) . (1.2)
In Eq. (1.2), Z (N,T) denotes the partition function of N indistinguishable, non-interacting bosonic atoms at temperature T, i.e. the number of different available microstates to the system, see Eqs. (1.7, 1.9). In thermodynamic terms, µ is the change of the Helmholtz free energy F = β 1lnZ (N,T) with the particle number, being proportional to the change in Boltzmann’s entropy S = kBlnZ (N,T). Einstein speculated that the equilibrium state of a Bose gas – which is the state of maximum entropy and minimum free energy according to the postulates of thermodynamics [10] – reveals that all particles in the gas “condense” into the same quantum state, if the number of particles in the
gas tends to infinity. Indeed, in the limit N at fixed temperature, we notice that the number of available microstates Z (N,T) in the gas does (intuitively) no longer change significantly with the particle number, so that the chemical potential in Eq. (1.2) approaches the single particle ground state energy of the gas, being zero for a non-interacting gas in a box.
According to Eq. (1.1), Einstein recognized that macroscopic average ground state occupation should especially occur for high particle densities2 at fixed temperature. This can be retraced by imposing that the number of particles in the gas be constant, and by summing Eq. (1.1) over all possible values of l except the condensate single particle mode, l = (0,0,0) 0. Replacing the summation by an integration over the density of states g( ) = Vm3/2/21/2 2 3 1/2 (see Section 7.3)
and taking the limit µ 0 (reflecting the behavior of µ in Eq. (1.2) in the limit N ), the ground state occupation number in Eq. (1.1) diverges, if we match the Einstein de Broglie .
Equation (1.3) arises from the requirement that the integral over all non-condensate single particle occupations in Eq. (1.1) equals the total number of particles, N, at the critical point of the phase transition. Here, ( ) = k=1 k is the Riemann Zeta function, see Table 7.1, = N/V the (homogeneous) atomic density of the gas, and T is the de Broglie wavelength of the particles: (T)= » 2 2 # 1/2 mkBT
Equation (1.3) indicates in particular that Bose-Einstein condensation occurs, if the wavelength (T) of the quantum particles in the gas becomes larger than their mean inter particle distance.
By default, this condition is interpreted as the wave length of the atoms in the gas getting infinitely large such that all particles are supposed to overlap and to form a giant matter wave, the condensate. The first monitoring of the microscopic quantum dynamics in this thesis (see part III) reflects that the reaching of the Einstein de Broglie condition leads to fulminating non-condensate number fluctuations and an average macroscopic ground state occupation. Our microscopic, many particle picture thus partially reproduces the idealized, intuitive picture of the condensate to consist of one giant matter wave, however, reflecting the actual balancing process of particle flow towards and out of the condensate mode, garnished by large quantum fluctuations characteristic for the Bose-Einstein phase transition.
Note that the Bose-Einstein phase transition is in particular defined in the thermodynamic limit, N ,V , with = const., meaning that the particle number and the quantization volume simultaneously tend to infinity, such as to keep the atomic density and the critical temperature Tc fixed. In this limit, the result obtained in Eq. (1.3) becomes exact (recompensating the approximation for the density of states g( ) to be continuous, see Chapter 8), defining analytically the transition temperature Tc for Bose-Einstein condensation in a uniform,3 non-interacting Bose gas: 3uniform non-interating gas in a box of volume V Tc = kB (3/2)2/3m
How was Einstein led to Eq. (1.1)? Having a look to the original predictions of Bose-Einstein condensation [28, 29], we recognize that the major underlying assumption is the indistinguishability
of particles: The number of quantum cells (in phase space) with energies between l and l +Δ is z = 2 V (2m)3/2 1/2Δ (1.6)
According to Bose’s previous analysis, Einstein infered [28] that the number of possibilities to distribute Nl indistinguishable particles over zl cells within the infinitesimal energy interval Δ is given by Z = (Nl +zl 1)! . (1.7)
This can be understood as follows [27]: Consider Nl particles (drawn as a one-dimensional sequence of dots), and zl lines which represent the different cells (as vertical lines creating a certain partition of the one-dimensional row). The number of positions carrying a label in this one-dimensional row is Nl + zl 1, so that the number of different configurations having Nl dots in Nl + zl 1 labels equals the number of different microstates, which is exactly the binomial coefficient in Eq. (1.7).
Taking into account all different energies l , the total number of microstates is Z (N,T) = l Zl , assuming that the state of the gas factorizes. Then, Einstein adopts the definition [10] of Boltzmann’s entropy, S = kB lnZ (N,T), where kB is the Boltzmann constant, which (with the above partition function) leads to the entropy [10] S = kB %l &Nl ln « 1 + zl #+zl ln  » N #’ .
Equation (1.1) is subsequently derived from maximizing S (by setting the first order variation of S to zero), under the constraint that l Nl = N and l Nl l = E. Hence, Einstein derived Eq. (1.1) by assuming a unique maximum entropy equilibrium state which can be factorized, treating the particles in the gas as indistinguishable, and neglecting number and energy fluctuations.
What does hence happen, Einstein asked, if the particles are considered as distinguishable? In that case, the number of possibilities to distribute Nl on zl cells is simply Z l = (z )Nl , (1.9) that means, each of the Nl particles has the same probability of occupying any cell zl , irrespectively of a single particle state’s occupation with energy l , and l ! k. Again, taking into account all energies as in Eq. (1.8), care has to be taken that a microstate with {Nl1 ,Nl2 ,…} particles occupying the cells {zl1 ,zl2 ,…} can be realized in N!/ l Nl ! different ways, considering for a moment the particles as distinguishable. Hence, the total number of states is given by Z ( N T ) = Z = N ! l ( z l ) Nl N , which yields the Boltzmann entropy $ S = kB N ln N + Nl ln  » zl +Nl (1.10) by taking the natural logarithm. Equation (1.10) indicates that the resulting entropy cannot be correct, i.e., the number of possible microstates is overcounted. This is because the first term in Eq. (1.10) is proportional to N ln N – contradicting the extensivity property [10] of the thermodynamic entropy, S ( N1 +µN2) = S (N1) +µS (N2). Moreover, modeling the limit of zero temperature by setting N0N, and N l 0+, for all l ! (0,0,0), the expression in Eq. (1.8) for indistinguishable particles gives the correct limit S0+ (as imposed by the 3rd law of thermodynamics [10]), whereas Eq. (1.10) for distinguishable particles leads to kBN ln N.
The main assertion of Bose and Einstein in a nutshell was thus that radiation can be treated as a photon gas, with the same specific combinatoric results induced by indistinguishability.

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Experimental state-of-the-art

As reported in Section 1.3, Einstein’s original prediction refered to a gas of non-interacting particles
in the thermodynamic limit N ,V , with = const. Thus, his prediction could not be taken for granted to work also for finite, interacting Bose gases in harmonic, typically anisotropic traps.4 The solidification of almost all materials at typical densities required at usual (e.g. room) temperatures for the reaching of Einstein’s condition in Eq. (1.3) is the major problem of realizing Bose-Einstein condensation experimentally [33].
To achieve Bose-Einstein condensation in the laboratory, the atomic ensemble is therefore brought to extremly low atomic densities by laser cooling [34] and is rapidly cooled hereupon to very low temperatures by evaporative cooling techniques [35, 36]. By this means, the gas has no time to solidify, whereas Einstein’s condition in Eq. (1.3) can still be matched. Typical densities and temperature ranges required to achieve Bose-Einstein condensation are [15, 23, 24, 32]: 1012 1015cm 3 and T 20 nK 1 µK . (1.11)
First observations of Bose-Einstein condensation in the laboratory were reported in 1995, for the alkali species 87Rb [23] in the group of Eric Cornell and Carl Wieman, at the Joint Institute for Laboratory Astrophysics [23], for 23Na [24] in the group of Wolfgang Ketterle, at the Massachusetts Institute of Technologies [32], and for 7Li [37] at RICE university. Up to date, Bose-Einstein condensation has been experimentally proven to exist in 1H, 7Li, 23Na, 39K, 52Cr, 85Rb, 133Cs, 170Yb and 4He [15].
Except for the species 4He [38, 39], which obeys – contrarily to all other summarized candidates – very strong interactions between its atomic constituents in the Bose condensed phase, the typical atomic density of a Bose-Einstein condensate is surprisingly dilute: At the center of the trap, where
the highest atomic density (the condensate) is located, it is of the order of 1012 1015 cm 3. In comparison, the density of air molecules at room temperature and atmospheric pressure is about four to seven orders of magnitudes larger [15]. A direct quantitative measure for the diluteness of a Bose gas is the gas parameter ξ = a!1/3 (where a is the s-wave scattering length, see Section 2.1), typically of the order ξ = a!1/3 10 2 ≪ 1 , (1.12) for a dilute Bose-Einstein condensate. Thus, the experimental path of producing Bose-Einstein condensates becomes theoretically noticeable as a small parameter in our master equation ansatz in Part II of the thesis: The dilute gas parameter ξ = a!1/3 will be identified in the derived transition rates for particle exchange between the non-condensate and the condensate, and is employed to quantify condensate formation times in a perturbative approach for the condensate wave function. The same applies for the condensate and non-condensate steady state number distributions.
In the remainder of the thesis, state-of-the-art experimental parameters such as those of the early experiments on Bose-Einstein condensation [23, 24, 32] are used for quantitative calculations of condensate formation times and particle number distributions during and after condensate formation. A recollection of relevant experimental parameters is shown in Table 1.1.

Bose-Einstein condensation in harmonic traps

In order to describe the statistics of a bosonic gas in an external confinement, the original analysis of Bose-Einstein statistics for uniform gases needs to be extended to harmonic traps. This is realized within the quantum version of the canonical and the grand canonical ensemble, which are conventually used to describe the statistics of non-interacting bosonic gases [10].
In classical thermodynamics, the two ensembles are equivalent in the thermodynamic limit of large particle numbers. Note, however, that an unsolved problem in the theory of quantum degen-erate gases below the critical temperature is that the canonical and the grand canonical ensemble lead to different predictions for the condensate statistics, even in the thermodynamic limit [27]. Therefore, the results on condensate statistics below Tc obtained from the grand canonical and the canonical ensemble shall be contrasted: Although both ensembles predict the same expectation value of the condensate particle number in the thermodynamic limit (and similar occupation for finite particle numbers), the grand canonical ensemble features the so called “fluctuation catastro-phe” (divergence of the condensate particle number variance in the thermodynamic limit) below Tc . Hence, it is the canonical ensemble which is in accordance with experimentally observed scenarios for condensate particle number expectation values and condensate number variances below the critical temperature.5

Grand canonical ensemble

We consider a gas of non-interacting atoms in a harmonic trapping potential, described by the first quantized Hamiltonian h1(r) = 1 .px2 +p2y +pz2/+ 1 m( x2x2 + 2y y2 + z2z2) 1 ( x + y + z ) , (1.13) with trapping frequencies = ( x , y, z ), momenta p = (px,py,pz) of the atoms in the three different spatial directions r = (x,y,z) of Euklidian space R3. In Eq. (1.13), the zero point energy is substracted for convenience. The eigenvectors vectors | l of h1(r) are labeled by the three component vector l = (lx ,ly,lz), with li N0. For non-interacting systems, the single particle eigenstates r| l in position representation are given by r| l = = lξ 2 1/4 2 2 Hlξ(Lξξ) , (1.14) where Lξ = mωξ/ is the width of the harmonic oscillator ground state, and the Hlξ(Lξξ) denote Hermite polynomials [40]. The corresponding single particle eigenenergies η%l read η% = lx ωx +ly ωy +lz ωz . (1.15)
Since the particles do not interact by assumption, particle exchange between atoms occupying the different single particle eigenmodes |χ%l is a consequence of coupling the gas to an external heat reservoir. In addition to the energy exchange, the grand canonical ensemble assumes particle exchange with the external reservoir to account for fluctuations of the total number of particles as sketched in Fig. 1.1.

Table of contents :

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 durin
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 TransitonratesforBose-Einsteincondensation
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 EquilibriumpropertiesofadiluteBose-Einsteincondensate
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 Finalconclusions
9.1 Master equation of Bose-Einstein condensation
9.2 What is Bose-Einstein condensation?
9.3 Outlook
Appendix
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 

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