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Immersing carbon nano-tubes in cold atomic gases

We investigate the sympathetic relaxation of a free-standing, vibrating carbon nano-tube that is mounted on an atom chip and is immersed in a cloud of ultra-cold atoms. Gas atoms colliding with the nano-tube excite phonons via a Casimir-Polder potential. We use Fermi's Golden Rule to estimate the relaxation rates for relevant experimental parameters and develop a fully dynamic theory of relaxation for the multi-mode phononic field embedded in a thermal atomic reservoir. Based on currently available experimental data, we identify the relaxation rates as a function of atom density and temperature that are required for sympathetic ground state cooling of carbon nano-tubes.


I. INTRODUCTION
In 1959, Richard Feynman gave the visionary talk "There's plenty of room at the bottom" [1] and drew the road map for the coming quantum technologies. Half a century later, we are still amidst the silicium era and enjoy many gadgets based on lithographically manufactured nano-electronics. In terms of miniaturization, many sectors of nano-technology [2] have reached the proverbial quantum bottom, but today, this mile stone is recognized rather as a resource for novel devices [3,4] than the ultimate limit.
In this context, carbon nano-tubes [20,21] are a particularly promising form of rolled-up, mono-layered carbon sheets. These families of fullerenes [22] combine fascinating aspects of electrical conductivity, exceptional mechanical tension moduli with the ability to be grown on demand, or to be deposited mechanically on atomic chips [23][24][25]. Such features render carbon nano-tubes on atom chips ideal candidates to explore quantum mechanical limits [11,13,26,27], to construct single atom detectors [28] with them, or to hybridize them with matter waves [29].
Polarizable particles in front of surfaces are affected by Casimir-Polder potentials [30][31][32]. The measurement of such minute forces with ultra-cold atoms is of high interest [33][34][35][36]. In particular, the interaction and trapping of atoms in front of nano-tubes has been explored [37,38].
However, so far only a few experiments have been performed with carbon-nano tubes immersed in ultra-cold alkali gases [29,39].
In this article, we will combine aspects of the quantum motion of carbon nano-tubes as well as the quantum degeneracy of the matter waves to examine the prospects for the sympathetic cooling due to the Casimir-Polder interaction of a free-standing carbon nano-tube mounted on an atom-chip. In Sec. II, we discuss a micromechanical model of the oscillating carbon nano-tube and quantize the phononic excitations. We introduce the energies of the atomic gas, phonons, and atom-phonon interaction. Within the Heisenberg picture, we model the temporal evolution and obtain equilibrium relaxation rates versus the temperature for different densities of the atomic cloud in Sec. III. This approach is generalized in Sec. IV, where we formulate a fully dynamical theory for the density operator of the phonons. We conclude the theoretical analysis with a numerical study of the cooling efficiency of a carbon nano-tube due to the interaction with the cold atoms. Two short appendices summarize properties of the carbon nano-tubes and thermodynamic correlation functions for degenerate bosonic gases.

II. ATOMS HITTING CARBON NANO-TUBES
In the present section, we will introduce the basic mechanical features of single-walled carbon nano-tubes. In particular, we will consider the experimental setup of P. Schneeweiss et al. [29], where free-standing carbon nano-tubes are grown on an atomic chip [23,25] and interact with ultra cold-atoms in ultra-high vacuum systems. From a continuum model of a free-standing carbon nano-tube, we derive a quantized description of the phononic excitations. This carbon nano-tube is embed-ded in an ultra-cold atomic bosonic gas at temperatures above the critical temperature for Bose-Einstein condensation (BEC). In particular, we model the interaction between a carbon nano-tube and a 87 Rb alkali atom by a Casimir-Polder interaction potential [29]. This approach leads to a total energy for the carbon nano-tube immersed in an atomic bath.

A. Vibrating carbon nano-tube
The vibrations of a carbon nano-tube can be well described by the Euler-Bernoulli model of an oscillating beam [13,20,40,41]. Here, we use a real, transversal displacement field u(z, t) ≡ u x (z, t)e x + u y (z, t)e y to represent the two-dimensional bending of an elastic beam of the length L, which is aligned along the surface-normal (z-axis) of an atomic chip. The transverse polarization directions are denoted by e x and e y . To be precise, we have to specify the boundary conditions u(0, t) = ∂ z u(0, t) = 0, for the fixed end on the chip and ∂ 2 z u(L, t) = ∂ 3 z u(L, t) = 0 on the loose end. In the case of small displacements, the vibration of the tube follows from the linear Euler-Bernoulli equation The physical parameters are [41][42][43] if we can determine the eigenfrequencies ω l and the phononic eigenmodes φ l (z) of the problem. From the axial symmetry of Eq. (1), it follows that the transverse oscillation frequencies must be degenerate for both polarization directions. The complex amplitudes β lσ are determined from the initial conditions. This model of an oscillating beam, together with the boundary conditions leads to a self-adjoint eigenvalue problem for the modes with respect to the real scalar product Explicitly, the eigenmodes read φ l (z) =ã l [(cos κ l L + cosh κ l L)(cos κ l z − cosh κ l z) + (sin κ l L + sinh κ l L)(sin κ l z − sinh κ l z)], (4) where the normalization constantsã l , defined by Eq. (A1), are proportional to the harmonic oscillator length a l ≡ /ω l ρ c L.
However, a mechanical oscillation is only possible, if the wave numbers κ l satisfy the condition cos (κ l L) cosh (κ l L) = −1.
At first glance, the solution of this equation looks like a formidable task, but in fact the phononic wave numbers can be approximated quite well [27] by an equidistant array κ l ∼ = π(l + 1/2)/L, for l ≥ 0. From the solution of Eq. (1), we obtain finally the particle-like dispersion relation for the phononic frequency ω l of mode l. Such a quadratic phonon dispersion relation can be derived also microscopically from the zone-folding method [44], considering the lattice symmetries of graphene [27].
Having determined the eigenmodes of the tube, we represent the energy of the phonon field as a separable sum of independent oscillator modes, which is necessary for the quantization of the phononic fieldû in terms of bosonic excitations [b lσ ,b † l ′ σ ′ ] = δ ll ′ δ σσ ′ , acting on the multi-mode phononic Fock-states | . . . , n lσ , . . . . Thus, we obtain the Hamiltonian operator of the vibrating carbon nano-tube.

B. Cold atomic gas
Atoms colliding with the carbon nano-tube are modeled as a homogeneous gas of scalar bosons represented in a discrete plane-wave basis as [â k ,â † k ′ ] = δ kk ′ . This representation implies periodic boundary conditions for all spatial fields, e. g., the atomic fieldΨ(r + e i L i ) =Ψ(r), where L x , L y and L z are the lengths of the quantization box.
Normalized plane-waves r|k ≡ exp (ikr)/ √ V, are then a complete set of basis functions within the quantization volume V ≡ L x L y L z . For the length of the quantization volume in the z−direction, we choose the length L z = L of the carbon nano-tube.
This spatial periodicity leads to a discrete set of atomic wave vectors k i = 2πn i /L i , n i ∈ Z. Hence, we obtain for the spatial amplitude of the atomic field the expression and the density of the atomic gas follows aŝ In this article, we consider atomic temperatures above the Bose-Einstein condensation temperature, where corrections to the ideal gas energy are minute. Therefore, we only keep in the energy the kinetic energŷ of the gas.

C. Casimir-Polder potential
The interaction of a neutral, polarizable atom in front of a surface is described by the Casimir-Polder theory [31,32,37,[45][46][47]. In principle, a full three-dimensional geometric modeling of the Casimir-Polder potential surface would be in place to describe all details of the interaction. However, we assume for simplicity that the dominant contribution to the dynamics is obtained by the axis-symmetric, translationally invariant part of the Casimir-Polder potential V (r) = V (ρ), where ρ ≡ x 2 + y 2 is the radius in cylindrical coordinates with the z-axis oriented along the tube.
Recent experiments [29,36], have provided evidence that the dominant contribution is given by an inverse power-law term ∼ C 5 /ρ 5 . However, in order to keep the model flexible, we approximate the Casimir-Polder potential as an inverse power series for distances well above the tube's physical radius ρ > R.
For example, in case of a single-walled carbon nano-tube R = 1 nm. The different contributions to the short-range part of the potential are parameterized by real coefficients C n , starting at least with terms beyond n m > 2. The surface physics (adhesion/adsorption) at ρ = R is unknown but should be of no concern for the calculation of the collisional relaxation. Thus, we assume that the potential is real and vanishes within the tube V (ρ ≤ R) = 0.
The later analysis requires the Fourier-transform of the Casimir-Polder potential, where V (q) is the twodimensional, axis-symmetric Fourier-transform of the Casimir-Polder-potential. If the size of the periodic box is much larger than the finite range of the potential, one can simply extend the integration limits to the full plane With defining a dimensionless wave numberq = qR, the expression for the partial potential amplitudes is formed with the generalized hyper-geometric function 1 F 2 (a 1 ; {b 1 , b 2 }; z), as well as the gamma function Γ(z) [48]. For each positive integer n this function can be represented explicitly in terms of Bessel functions. In Fig. 1, we depict the behavior of the potential amplitude V n (q) in its dependence on the dimensionless momentumq for the powers n = 3 − 7. We notice the typical oscillatory Bessel-like behavior with a weak dependence on n.
In order to obtain the interaction energy between atoms and the tube, we have to sum the Casimir-Polder potential weighted by the atomic density over all space and over the whole length of the tube leading to the ex-pressionĤ We note thatĤ i is given by a convolution integral. For computational purposes, it is more convenient to transform it into the Fourier domain where it becomes a separable sum of products of atomic and tube variables. Indeed, by introducing the atomic density fluctuationsN q as the Fourier transform of the atomic density and a translation operator of the tube, we can express the interaction energy aŝ Here, we have defined the frequency w q ≡ V q / √ V to measure the strength of the Casimir-Polder potential.

D. Total Energy
Therefore, the total Hamiltonian of the system is the sum of the phonon energyĤ c , the atomic kinetic energyĤ a , and the atom-phonon interaction energyĤ i . Recalling the interaction energy Eq. (20), we notice that the homogeneous background contribution at q = 0, depends only on the atomic particle numberN ≡N 0 . Hence, it is convenient to shift the individual energies by this amount, that iŝ This energy renormalization amounts to dropping the term q = 0 in the interaction and to off-set the atomic dispersion relation by ε k → ε k + w 0 .
Experimentally relevant temperature scales are given by the cryogenically cooled atom chip, where the carbon nano-tube is mounted on, corresponding to T c = 4 K, and by the temperature T a = 100 nK of the atomic gas.
In Appendix A and Ref. [27], the maximal thermal displacement u of the tube as a function of temperature is estimated. Clearly, u decreases with decreasing the temperature of the chip. On the other hand, the thermal de Broglie wavelength λ dB of the atomic gas increases at lower temperatures. Hence, if we are in a regime where u ≪ λ dB , we can approximate the tube's displacement operator by the first two terms of the Taylor series of Eq. (19).

III. STATIC RELAXATION RATE
We are now ready to proceed with the calculation of the static relaxation rate using first-order timedependent perturbation theory. From the solution of the Heisenberg equations for the phononic fields, we can derive the time-dependent occupation number of the lowest phononic mode. Then, we use Fermi's Golden Rule [49], to calculate the relaxation rate as a function of the temperature of the atomic gas. Our analysis demonstrates the possibility of carbon nano-tube cooling in the experiment.

A. Heisenberg equations of motion
By evaluating the Heisenberg equations of motion for the atomic and phononic fields, i. e., i where the constants I l ≡ L 0 dzφ l (z)/ √ 2L and J l ≡ a 2 l /4 reflect the spatial extent of the l-th phononic mode.
From the formal solution of Eq. (25) for the atomic variables, we obtain that is a free evolution of the atomic field plus corrections. First, we are only considering the two lowest degenerate phononic modes with l = 0. Excited-state modes with l > 0 will contribute little to the collisionally induced excitation rate considering the principle of energy conservation.
The two ground-state modes can be combined into a vectorb Within the rotating-wave approximation, we drop the counter-rotating term in Eq. (26) and obtain a Langevin equationḃ for the phononic Kubo-oscillators with a hermitian frequency tensor which results in pressure-broadened oscillation frequencies, and a stochastic forcê The linear inhomogeneous Kubo equation (29), can be solved formally aŝ by introducing the retarded propagator for times t ≥ t 1 with the initial conditionÛ Ω (t, t) = 1. Thus, we have obtained a linear-response model for the carbon nano-tube immersed in an atomic gas.

B. Occupation number
In the present subsection, we assume that the statê ρ tot ≡ρ a ⊗ρ c of atoms and phonons is initially uncorrelated. Furthermore, we consider atoms that are prepared in a grand canonical ensembleρ a at a temperature T a ≡ 1/k B β a without macroscopic motion. Then, the number of the atoms with wave number k is given by where we have introduced the chemical potential µ, or alternatively the fugacity η ≡ e βaµ , to constrain the particle number to N as shown in App. B. Moreover, the tube is prepared in its phononic vacuum ρ c ≡ |0, 0 0, 0|, representing the ground-state for the x− and y−polarization of the l = 0 modes.
As an observable, we consider the unpolarized occupation of the degenerate ground-state manifold. From Eq. (32), it follows that the occupation (36) is determined solely by the force-correlation function. In the evaluation of Eq. (36), we have used the Born approximation keeping only processes up to second order in the Casimir-Polder potential amplitude w q . In turn, the force-correlation function depends fundamentally on the density-correlation function N q (t 1 )N k (t 2 ) , which by means of space-time Fourier-transformation is linked to the dynamic structure factor [50]. For the free fields of Eq. (27), we have evaluated this finite temperature correlation function in App. B 2. After performing the time integration, we obtain for the occupation the second-order perturbative result Here, we have introduced the wave vector q ⊥ perpendicular to the tube axis by the relation q 2 ⊥ = q 2 x + q 2 y , and the frequency difference ∆ qk ≡ ω 0 + ω k−q − ω k due to inelastic scattering between the initial and the final state.
Assuming that the temperature T a of the gas is above the temperature T BEC for Bose-Einstein condensation, that is T a > T BEC , the thermal occupation satisfies the inequality n k−q < 1. Consequently, we can disregard this bosonic enhancement of scattering in Eq. (37) and retain only Now, we evaluate the occupation number in the continuum limit k → d 3 k V/(2π) 3 and transform the kintegral to cylindrical coordinates (k, φ, k z ) with the k z -axis aligned to q, which yields with ∆ qkz ≡ ω 0 + ω q − qk z /m and q ≡ q 2 ⊥ + q 2 z . For an atomic gas at a temperature T a > T BEC , the fugacity obeys the inequality η < 1. Hence, we can Taylorexpand the logarithm in Eq. (39) and arrive at where we have introduced the convolution integral Integrals of this type are the essence of Fermi's Golden Rule [51]. If either the width q/ √ jβ a m of the thermal Gaussian, or its central frequency ω 0 + ω q are far bigger than the width 2π/t of the sinc-function localized at ∆ = 0, i. e., t ≫ min ( √ jβ a m/q, 2π/ω 0 ), then we can approximate the integral by evaluating the Gaussian function at the maximum of the sinc-function as This expression displays the familiar linear increase of the excitation as a function of time.

C. Relaxation rate
The evaluation of relaxation rates is always based on time-dependent perturbation theory, either in the Schrödinger-or in the Heisenberg picture, and a subsequent evaluation of the amount of excitation observed after turning on the interaction for some time. The linear temporal increase of the excitation then translates into a rate coefficient.

General expression
Substituting Eq. (42) into Eq. (40) and taking the time-derivative at the initial instant, Γ v 0 ≡ṗ v 0 (t = 0), we obtain the excitation rate We note the reappearance of the two-dimensional Fourier-transform of the Casimir-Polder potential Eq. (15), depicted in Fig. 1.
In terms of the dimensionless wave numberq = qR of Eq. (15), we can rewrite Eq. (43) as Here, we gathered all the constants into the prefactor A l ≡ 8πmI 2 l L/ 3 λ 5 dB and introduced the sharply peaked function that depends on the characteristic ratio κ ≡ 4 √ πR/λ dB of the tube's radius R to the thermal de Broglie wavelength λ dB ≡ 2πβ a /m. Indeed, δ (l) j (q) decreases very rapidly asq → 0, as well as forq → ∞ and we will treat it like a deltadistribution compared to the smooth potential in the integral of Eq. (44). We can determine its extremum from ∂qδ (l) j (q) = 0, and find its location at Therefore, we approximate the integral of Eq. (44) with the mean value theorem of integral calculus which with the help of the known integral formula [48] ∞ 0 dq δ (0) finally yields the excitation rate Clearly, the first j = 1 term dominates the sum.

Fermi's Golden Rule
A simple view of the previous result comes from considering the unpolarized total transition rate for the inelastic process induced by the Casimir-Polder potential of Eq. (22). This expression represents the signal current of a fictitious detector that records all final states of the atoms and averages it with the thermal distribution of initial states. Here, the initial state |i = |1 k i , 0 σi describes a single atom with wave vector k i and the carbon-nano tube in its ground state. This state has an energy ε i = 2 2m k 2 i . As we are interested in inelastic collisions, we consider the final state |f = |1 k f , 1 σ f with one phonon in the σ-polarized mode ω 0 . Now, this state has an energy ε f = ω 0 + 2 2m k 2 f . With a few algebraic transformations, we obtain the excitation rate from Fermi's Golden Rule and it coincides with the first term j = 1 of Eq. (49) In order to bring out the essential physics, one can approximate the fugacity by e βaµ ≈ nλ 3 dB for temperatures T a > T BEC (comp. Eq. (B3)). This simplifies the rate to Thus, decreasing the thermal energy of the atoms below the excitation energy, i. e., k B T a < ω 0 , suppresses the inelastic relaxation rate exponentially..

Application to current experiment
In order to evaluate the relaxation rate Eq. (49) explicitly, we need to know the exact form of the Casimir-Polder potential of Eq. (15). Recent measurements of the interaction potential between the carbon nano-tube and the atomic gas [29] have shown that the effect is very well described by the contribution C 5 /ρ 5 with a numerical coefficient C 5 ≡ 6×10 −65±1 Jm 5 . Thus, we approximate the Casimir-Polder potential Eq. (15) only by that term, i. e., and V 5 (q) =q As the R ≪ λ dB , we need to retain only lowest term in this expression and find the following approximation for the rate All other relevant parameters of the carbon nano-tube and the atomic gas are listed in Apps. A and B. Now, figure 2 depicts the dependence of the full excitation rate Γ v 0 of Eq. (49) on the scaled temperature T a /T BEC of the atomic cloud for different atomic densities. We note that the excitation rate depends strongly on the temperature, as well as on the density of the gas.
For the given parameters, we find the rates of Eq. (49) and Eq. (54) are indistinguishable. When the excitation rates are of the order of the oscillation frequency or below Γ v 0 < ω 0 , the Fermi-Golden-Rule approach is suitable. Therefore, we conclude that cooling of the carbon nano-tube to the ground mode is feasible in current experimental situations.

IV. DYNAMICS OF RELAXATION
In this section, we will generalize the previous considerations to determine the relevant time scales and study the dynamics of the relaxation process with a master equation approach. Here, the phonons are considered as the system and the atoms are the bath. This analysis yields (i) a dynamic picture of the carbon nano-tube cooling, (ii) its approach towards equilibrium, and (iii) its dependence on temperature.

A. Master equation
In the interaction picture, the time evolution of the total density operatorρ tot of phonons and atoms is described by the von-Neumann equatioṅ As usual, the interaction picture is obtained from the Schrödinger picture using the free atom-phonon propa- and To study the phonon dynamicsρ(t) ≡ Tr a [ρ tot (t)], we have to average the state of total system over the atomic bath. Considering the deviation ∆ρ(t) ≡ρ(t + ∆t) −ρ(t) for a short time interval ∆t, we find from an iterated formal solution of Eq. (55) the expression Within the Born-Markov approximation [51], we derive a master equation for the coarse-grained rate of change ∆ρ(t) ≈ ∆t dρ dt . For the particular atom-phonon interaction Hamiltonian given by Eq. (57), we obtain the master equation . In progressing from Eqs. (61) to (62), we have not included linear contributions in the interaction as there is no deterministic motion. Moreover, in evaluating thermal contributions for T a > T BEC , we have retained only the dominant contributions and encountered the same thermal density-fluctuation correlation function N q (t 1 )N k (t 2 ) , as in Eq. (36) and discussed in App. B 2. Since the thermal correlation functions also decay quickly, it is very well justified to extend the upper limit of the τ -integration to infinity.

B. Ground state excitation rate
As an application of this master equation, we consider a carbon nano-tube that is initially in the multi-mode phononic vacuumρ(t = 0) ≡ |0, . . . 0, . . .|. Then, we immerse it into the bath of atoms at temperature T a , and observe the decrease of the ground-state occupation ρ v (t) ≡ 0|ρ(t)|0 . In this case, the excitation rate into any available phononic mode l ≥ 0 is given by Clearly, the excitation rate is just the negative depletion rate of the ground-state occupation, i. e., Γ v ≡ −ρ v (t). As in Sec. II D, we assume that the thermal de Broglie wavelength of the atomic gas is much longer than the tube's oscillation amplitude, that is λ dB ≫ u. Hence, we can approximate the translation operator of Eq. (59), as in Eq. (24), by a second-order Taylor series, and find After performing the time integrations and transforming the discrete wave number sums in the continuum limit, we arrive for the vacuum excitation rate at Theq-integration is approximated in a way completely analogous to Eq. (47) and we obtain the expression for the phononic excitation rate out of the vacuum. Clearly, the sum in Eq. (67) is dominated by the contribution of the lowest phononic mode l = 0, that is This expression for the rate agrees with the first term of the extended thermal series Eq. (49).

C. Finite temperature thermalization rate
In this subsection, we generalize the previous calculation by assuming that the carbon nano-tube is initially close to a thermal state at temperature T c ≡ 1/k B β c , which is different from the temperature T a ≡ 1/k B β a of the atomic bath. The rate of change of the occupation p l,σ ≡ p l,σ = b † l,σbl,σ of the mode (l, σ) is given bẏ p l,σ = Tr p l,σρ (t) .
Now, we use the master equation Eq. (62) and assume in the evaluation of the averages a canonical density operator parametrized by a time-dependent coefficient β c (t), as outlined in App. B 3. In the continuum-limit, we obtain from Eq. (69) for the unpolarized occupation numbers p l ≡ p l,x + p l,y of the l-th level the rate equatioṅ with temperature dependent rate coefficients Integrating overq similarly to Eqs. (44) and (47), we find withq (l) 1 given by Eq. (46). As expected, this expression generalizes Eqs. (49) and (67), as it allows for a finite temperature of the tube that is different from the atomic gas, i. e., β c = β a , as well as for excitations into all phononic modes l ≥ 0.
If the carbon nano-tube is hotter than the atomic bath, that is β c < β a , then the relaxation rate of Eq. (72) is positive and leads, according to Eq. (70), to a cooling of the tube's phonons. If the carbon nano-tube is colder than the atoms, i. e., β c > β a , the rate is negative and leads to a heating of the tube. Finally, all rates vanish, when β c = β a , as required for a thermodynamic equilibrium.

V. CONCLUSION
Using time-dependent perturbation theory with finite temperature ensembles, we have calculated the excitation rate of a free-standing single-walled carbon nanotube immersed in a bath of neutral bosonic atoms. The interaction between the carbon nano-tube and the atoms was modeled by a generic Casimir-Polder potential series. We have assumed that the temperature of the atoms was above the Bose-Einstein condensation transition temperature as the collisional relaxation at the MHz level is insensitive to the phase coherence of the bath.
For the numerical evaluation of the excitation rates, we have used experimentally determined values of the Casimir-Polder potential between a thermal-, as well as a Bose-Einstein condensed 87 Rb gas [29]. In this situation, an inverse power-law ∼ C 5 /r 5 is very well-suited to approximate this potential. With this analysis and the current data, we find that cooling of the free-standing carbon nano-tube to the phononic ground state due to the Casimir-Polder interaction with a cold atomic gas is feasible.
We emphasize that the excitation rate depends strongly on the temperature and the density of the atomic cloud. The form of the interaction potential, for example interferences between different potential contributions, can influence the rate as well. Hence, more extensive experimental data is needed.
Typical mechanical parameters for single-walled carbon nano-tube are summarized in Tab In the low temperature approximation of Eq. (24), we have assumed that the spatial excursions of the nanotube are much less than the thermal de Broglie wavelength of the atomic gas. Thus, we summarize [27] the maximal thermal excursions u = ∆ 2û Tc of the tip of the tube z = L for different temperatures of the tube T c in Tab In this appendix, we summarize the central results of equilibrium thermodynamics of bosonic fields that were employed in the main sections of our article.

Bose-Einstein condensation
The state of the free atomic gas [52] is described by the density operator of the grand canonical ensemblê held at a temperature T a ≡ 1/k B β a and maintaining an average particle number N . The Hamiltonian is denoted byĤ, Ω is the grand canonical potential and µ is a chemical potential, which is determined self-consistently through the particle number constraint Here n 0 ≡ n k=0 is the occupation of the ground-state andÑ (β a ) denotes the number of thermal particles. Introducing the fugacity as η ≡ e βaµ , one finds for the occupation number of the ground state, n 0 ≡ η/(1 − η) andÑ (β a ) = Vg 3/2 (η)/λ 3 dB for the number of particles in the excited states. The poly-logarithmic function is defined by In the thermodynamical limit, when particle number and volume increase at constant particle density, i. e., n = lim N,V→∞ N/V = const., one can identify two cases. The first corresponds to particle densities below the critical density, that is, n < g 3/2 (1)/λ 3 dB . There, the occupation number of the ground state n 0 is negligibly small and the total number of atoms is equal to the number of particles in the excited states N ≈Ñ (β a ). In this limit of a dilute quantum gas, the de Broglie wavelength is much smaller than the average distance between the particles. Therefore, the gas behaves classically.
The second regime occurs for densities larger than the critical density n > g 3/2 (1)/λ 3 dB . In this case the fugacity attains its maximum value η → 1. The occupation of the ground state is not negligible anymore and increases with the decrease of temperature where we have introduced the critical BEC temperature T BEC ≡ 2π 2 mk B n g 3/2 (1) 2/3 .
(B5) Figure 3 presents the behavior of the gas in these two cases. For temperatures below the critical temperature, the ground-state occupation (dashed-dotted line) becomes macroscopic and the fugacity (solid line) is practically one. Above the critical temperature, the fugacity decreases with the increase of temperature and the number of particles in the excited states (dashed line) is equal to the total number of particles.   From this analysis, we can obtain the values of the thermal de Broglie wavelength, summarized in Tab. III, for a range of atomic densities at the critical temperature.

Thermal correlation functions
In this part of the appendix we focus on the quantum averages used in the master equation Eq. (62). In particular, we derive single-time averages and two-time correlation functions.

a. Single-time average
With the choice of the interaction Hamiltonian Eq. (57), and assuming that the atomic bath is in a grand-canonical state without macroscopic motion, we find a vanishing quantum average Tr a ρ aĤi = 0, because â † k−qâk = 0, for q = 0. Therefore, the trace over the single commutator in the first term of Eq. (61), will vanish -which means that we have incorporated this energy shift in the unperturbed Hamiltonian of the atoms. This result is well-known in first-order timeindependent perturbation theory.

b. Two-time correlations
The two-time density-fluctuation correlation function N q (t 1 )N k (t 2 ) evaluated at temperature T a , emerges ubiquitously in field theory and is proportional to the dynamic structure factor [50]. In the case of stationary, translationally invariant systems, it simplifies to where the Kronecker delta enforces momentum conservation and we find the thermal correlation function g q (τ ) ≡ k e i(ω k−q −ω k )τ n k−q (1 + n k ) (B7) with thermal occupations n k defined in Eq. (34). For further discussions, we separate the ground-state contribution from the sum and approximate the remainder within the continuum-limit. For T a > T BEC , we find n k < 1 and we will take only the linear term in Eq. (B7) into account. In this regime, we can also approximate the Bose-Einstein distribution by the Maxwell-Boltzmann distribution. Moreover, the occupation number in the lowest level is negligible and the correlation function can be written as