Degenerate Bose gases with uniform loss

We theoretically investigate a weakly-interacting degenerate Bose gas coupled to an empty Markovian bath. We show that in the universal phononic limit the system evolves towards an asymptotic state where an emergent temperature is set by the quantum noise of the outcoupling process. For situations typically encountered in experiments, this mechanism leads to significant cooling. Such dissipative cooling supplements conventional evaporative cooling and dominates in settings where thermalization is highly suppressed, such as in a one-dimensional quasicondensate.

Introduction. Engineering dissipation and driving protocols in interacting quantum many-body systems is an important emerging area of out-of-equilibrium physics. On the theoretical side, it has unveiled a series of novel quantum phenomena, from topological states of fermions [1] and the establishment of long-range order of a Bose-Einstein condensate (BEC) in an optical lattice [2], to the dissipative preparation of entangled states [3] and dissipative quantum computations [4]. On the experimental side, dissipation, for example, has been used to create strongly correlated states of matter [5] and to study the dynamics of open quantum systems [6].
In the present letter we develop a model for dissipationdriven cooling, which is based on spatially uniform and coherent atomic loss from a BEC into a continuum of free single-particle modes.
Although reminiscent of standard evaporative cooling [7][8][9], this process is distinctly different in that it neither relies on energy-selective outcoupling nor conventional rethermalization.
The paper is organized as follows: we start by introducing the model and derive an effective stochastic Gross-Pitaevskii equation in the Markovian approximation. Then we linearize it and focus on the experimentally relevant quasi-stationary dissipation process in the low-energy phononic limit. We show that in this regime the elementary excitations are in a thermal state with a time-dependent temperature. Finally, we obtain scaling laws for the temperature and an asymptotic dissipative state.
The model. We consider a gas of degenerate bosonic particles with contact interaction in a flat potential over the length L with periodic boundary conditions. While the proposed cooling mechanism itself is very general, we illustrate it in the case of a one-dimensional (1D) Bose gas in the quasi-condensate regime. This is of direct experimental relevance, as the thermalizing collisions in this system are strongly suppressed, rendering conventional evaporative cooling highly inefficient. This fact enables a direct observation of our mechanism in 1D experiments.
The Hamiltonian of the model in second quantized form is given bŷ The Hamiltonians of the free systemĤ 0 , the contact interactionĤ int , the bath of harmonic modesĤ bath and the coupling to the bathĤ coupling in the momentum space read where k = {0, ± 2π L , ± 4π L , . . .} is the discrete momentum, m is the mass of bosonic atoms, g is the two-body interaction strength, s k are the quantum numbers characterizing the continuum of bath modes with momentum k, g s k are the system-bath coupling coefficients, ω s k are the energies of the bath modes, andâ k ,b s k are the annihilation operators for the momentum mode k of the system and the bath (with an additional quantum number s), respectively. To make the notation more compact, we set = k B = 1, which implies measuring energies and temperatures in frequency units (s −1 ), and masses in s/m 2 . Note the similarity of our Hamiltonian to the one of the Caldeira-Leggett model [10], save the interaction term H int .
Semiclassical treatment. We use the Keldysh technique [11] to develop a consistent semi-classical treatment of the problem. In this framework the expectation values of observables are calculated from closed time contour path integrals over coherent atomic fields φ(x, t) φ =â(x, t) φ , whereâ(x, t) = 1 √ L kâ k (t)e ikx . The basic building blocks are the classical and quantum components of the atomic field, introduced as linear combinations of the field defined on the forward φ + arXiv:1411.4946v1 [cond-mat.quant-gas] 18 Nov 2014 and backward φ − branches of the time contour, φ cl,q = (φ + ± φ − )/ √ 2 . We neglect the back-action of the atomic system on the bath, assuming the latter to be empty, which allows the use of the fluctuation-dissipation theorem for the bath modes. Next, noticing that the bath is quadratic, we integrate it out exactly and focus on the Markovian case, characterized by a constant effective spectral density J(ω, k) = π s g 2 s k δ(ω − ω s k ). The physical implication of this is the applicability of Fermi's golden rule: only the process of the escape of atoms into the continuum of empty modesb s k is considered, and the return probability is neglected.
The Markovian bath approximation is particularly well suited for 1D and 2D systems, whose experimental realizations are characterized by tight confinement in transversal dimensions. The non-Markovianity time (roughly the time it takes for an atom to escape the cloud) is of the order of the inverse trapping frequency in these dimensions and is much shorter than other relevant dynamical time scales.
After the stated approximations the action becomes where φ † = φ * cl φ * q , and γ is the field decay rate. We derive the semiclassical dynamics by expanding the action S in powers of the quantum component φ q . By neglecting all terms beyond linear, we recover the classical dissipative Gross-Pitaevskii equation, which has been previously phenomenologically introduced in experimental studies of localized dissipation [6]. Our method not only gives microscopic justification to this phenomenological model, but is also capable to lead beyond the mean-field approximation.
The dynamics beyond mean-field is influenced by the quantum noise associated with dissipation, which is revealed by expanding (6) in φ q and keeping the terms up to the second order. We achieve this by integrating out the φ * q -φ q component using the Hubbard-Stratonovich transformation [12], introducing an additional complex force field ξ with Gaussian statistics. This leads to a stochastic Gross-Pitaevskii equation (SGPE) where Φ = Φ(x, t) is the solution of the saddle-point equation for the classical component φ cl , and ξ(t) is the Gaussian noise term, which in the Markovian approxima-tion has the correlator of the white noise ξ(t) * ξ(t ) = 2γ δ(t − t ) fixed by the fluctuation-dissipation theorem. The SGPE (7) is similar in form to the conventional Gross-Pitaevskii equation [13], however it captures beyond-mean-field dynamics: the field Φ incorporates quantum fluctuations due to the symmetric Keldysh nor- Φ(x, t)e −ikx dx, and n k (t) is the population of the momentum mode k [11]. This normalization is also responsible for the factor 1 /2 in front of the interaction term in (7).
Bogoliubov theory. The Mermin-Wagner theorem states that there can be no true condensate in 1D in thermodynamic limit due to the increased role of phase fluctuations [14]. Nevertheless, a weakly-interacting Bose gas can enter the regime of a quasicondensate at low temperatures, characterized by small density fluctuations but large phase gradients [15,16]. The field Φ(x, t) in the phase-density representation reads where n 0 (t) = n 0 (0)e −2γt is the decaying mean density, δn(x, t) is the local density fluctuation, K(t) is a homogeneous gauge field, whose derivative plays the role of the chemical potential ∂ t K(t) = µ(t) = gn 0 (t), θ(x, t) is the local phase, and the factor two stems from the Keldysh normalization. The issues concerning the illdefined phase field at short spatial distances are cured by coarse-graining as explained in [15].
Substituting (8) into (7), linearizing the equation with respect to small parameters δn and ∇θ, and performing the linear transformation to an emergent bosonic basis ϕ = ϕ(x, t) = δn(x,t) √ 2n0(t) In the case of a true condensate, such as the one forming in a 3D degenerate bosonic gas and characterized by the macroscopic occupation of the Penrose-Onsager mode, we end up with the same equation (9) as shown in the supplementary material.
To solve (9) we perform an instantaneous unitary transformation to the Bogoliubov basis where ϕ k (t) = 1 √ L ϕ(x, t)e −ikx dx, and the Bogoliubov coefficients v k (t), u k (t) were taken to be real for convenience. The evolution of the quasiparticle field is given by where k (t) = k 2 2m k 2 2m + 2µ(t) is the quasiparticle energy. Eq. (11) is also general and is not limited to the 1D case as long as the dynamics is Markovian.
The first term of Eq. (11) represents the non-adiabatic dynamics due to the time-changing Bogoliubov basis, which mixes the counter-propagating fields χ k and χ * −k . Here we focus on quasi-stationary dissipation characterized by γ 2 k for all momenta k, when the nonadiabaticity can be neglected (see the supplementary material). This kind of dissipation models the continuous outcoupling of atoms from the trap, as it is typically realized during the cooling process in experiments.
Due to the gapless character of the spectrum, there will always be long-wavelength modes not satisfying this adiabaticity criterion in the thermodynamic limit. However, in all experimentally relevant finite-size systems the smallest momentum k min ∼ 1/L is given by the system size, and γ can be tuned to be small enough for adiabaticity to hold.
The last term of Eq. (11) represents the quantum noise transformed into the Bogoliubov basis. The Bogoliubov coefficients u k , v k diverge as |k| −1/2 for small k, which means that by removing atoms we excite long-wavelength phonons. However, as shown in the following, this effect is counterbalanced by the decreasing energy of the modes k (t), allowing to define a common temperature in the universal phononic limit.
Intuitively, this noise is the result of the probabilistic nature of the outcoupling process on the single atom level. It is thus of the same origin as the noise in a fast-split quasicondensate that manifests itself in the dephasing to a prethermalized state [17].
The explicit time-dependence of the Keldysh correlation function of the Bogoliubov modes is given by which in turn gives the quasiparticle modes' occupation numbers n k (t) according to the normalization condition χ k (t)χ k (t) = 2n k (t) + 1. The first term on the righthand side of Eq. (12) represents the decay of quasiparticles given by the classical theory (the dissipative GPE), while the second term represents the correction due to the quantum noise. Universal phononic limit. In the following we focus on the process of dissipation of an initially thermal degenerate cloud in the experimentally relevant low-energy phononic limit, considering the modes with momenta k mc(t), where c(t) = µ(t)/m is the speed of sound. The occupation numbers of the low-energy phonic modes with momenta k T (t)/c(t) are much larger than one, which permits approximating the Bose-Einstein distribution with the classical Rayleigh-Jeans equipartition at a temperature T (t) = kc(t)n k (t). The effect of the dissipation (i.e. the outcoupling of atoms) can thus be under- stood as a removal of phonons from the system, which in the Rayleigh-Jeans approximation is directly equivalent to cooling. We now calculate the ratio of temperature to chemical potential T /µ ( Figure 1) taking into account that in the phononic limit |u k (t)| 2 + |v k (t)| 2 = mc(t)/k, and the speed of sound decays as c(t) ∝ n 0 (t) ∝ e −γt , where again the first on the right-hand side represents the classical part and the second term represents the contribution of the quantum noise. Eq. (13) can be used to derive other important degeneracy criteria such as the scaling of the thermal coherence length λ = 2µ mgT and the Penrose-Onsager mode occupation number N P O = λN/L, with N being the total number of particles.
From Eq. (13) we see that the system evolves towards an asymptotic dissipative state T = µ/2, setting a limit on how far the system can be cooled through uniform dissipation.
The quantum noise influence can be neglected either at short times, slow dissipation rates or at high enough initial temperatures, so the classical approximation and the noiseless dissipative Gross-Pitaevskii equation are applicable as long as To test the predictions of the theory in more detail, we propose to measure the temperature dependence on the mean density, which in the uniform case reads For instance, this could be measured in a 1D quasicondensate prepared on an atom chip [18] which is subject to continuous dissipation through rf-induced outcoupling to untrapped states. To take into account the longitudinal trapping confinement we propose to use the local density approximation [19]. The tight transversal confinement leads to a fast atom escape from the cloud, rendering the dynamics Markovian for the relevant experimental timescales. In fact, temperatures far below the energy splitting of the transversal confinement (where thermalizing two-body collisions freeze out) have previously been reached in this experimental setting [20][21][22]. The mechanism presented in this work could provide a explanation for these observations. Conclusions. We developed a general theoretical description of dissipative cooling of degenerate Bose gases by outcoupling atoms from the condensate. Our model is applicable as long as there is either a true condensate or a quasicondensate, and at sufficiently low temperatures where the Bogoliubov theory remains valid.
We found that the low-momentum phononic modes remain close to the thermal equilibrium during dissipation, and the white quantum noise, stemming from the coupling to a continuum of empty modes, limits the cooling process and leads to an asymptotic dissipative state. We presented scaling laws for temperature dynamics with respect to the mean density, which can be tested in experiment.
We conjecture that the dissipative cooling effect should be readily observable in experiments with onedimensional Bose-Einstein quasicondensates. In higher dimensions the direct application of the model may be limited mainly due to two reasons: the first one is that dissipative cooling may be overshadowed by conventional evaporative cooling due to effective thermalization; the other reason is a possible deviation from Markovianity, caused, for instance, by the finite particle escape time.

Supplementary material Dissipative harmonic oscillator
To gain insight into dissipative cooling of 1D BEC, let's consider first a toy model, namely a single quantum harmonic oscillator with the frequency ω and creation/annihilation operators a † and a, coupled to a bath of harmonic oscillators with frequencies ω s and field operators b † and b. The Hamiltonian of the model in the rotating wave approximation (RWA) is given by This particular Hamiltonian is a close kin to the one of the Caldeira-Leggett model and has been recently considered in detail in the context of a lossy optical cavity in Ref. [23], however here we present some explicit derivations for clarity.
The model is quadratic, so it allows an exact solution. Sure, the Hamiltonian of a BEC is not quadratic, but we will see how it can be reduced to the similar system in the next section.
In a standard Leggett-Caldeira model the system and the bath are coupled through their x coordinates, which implies the standard harmonic oscillator field quadratures and setting m = m s = = 1. In our MLC model the coupling is given by So our modification amounts to an additional coupling through the momentum quadrature.
If the bath modes ω s are in equilibrium, they can be integrated out exactly using the Matsubara technique. Nevertheless, let's solve the system in Keldysh formalism for a possibility to potentially introduce explicit timedependence. In the following we generalize the solution of the Caldeira-Leggett model presented in [11] to the case of complex fields.
The Keldysh action for our system is given by S = S 0 + S bath + S int . The first term is the action for a free bosonic field given by the two-component vector ( ) denotes matrix transpose, and φ are fields given by the coherent states ofâ φ = φ φ . The Green's function for a free field is given by its causal structure with the retarded and advanced components The Keldysh contrubution G −1 K for a free field is only a regularization and plays no role in continuum notation.
The bath contribution is given by the sum of actions of harmonic oscillators, each of them being completely equivalent to S 0 save the different frequency ω → ω s : The interaction Hamiltonian H int = s g s (a † b s + b † s a) translates into the action coupling φ and ϕ fields The interaction is local in time so we have omitted the dt δ(t − t ) terms for brevity. Gaussian integration for ϕ s fields can be performed according to where we notice that J 1 = φ q , J 2 = φ cl and A ij = (−G −1 s ) ij . Integrating out the bath degrees of freedom leaves us with the dissipative action for φ fields The inverse dissipative Green's function has the same causal structure as (21): Retarded and advanced components of the inverse dissipative Green's function (D −1 ) R/A are given by the respective components of G s : where J(ω) = π s g 2 s δ(ω − ω s ) is the spectral density of the bath.

Markovian bath approximation
Let's assume a constant coupling with the bath modes g 2 s = γ, and the continuum of the latter, allowing to write the spectral density as J(ω) = γ, then where C is a real constant used to renormalize the potential ('Lamb's shift'). Other choices of the spectral density will lead to a different type of (possibly non-Markovian) dissipation and noise kernels, but won't change the picture qualitatively. Due to the fact that the loss process is quasi-stationary and the bath is infinite, the latter can be always assumed to be at thermal equilibrium, which leads to the fluctuation-dissipation theorem for the Keldysh component of the bath Green's function This translates into the same expression for the inverse dissipative Green's function We are interested in the case where the bath is initially empty, and in this case the Green's function can be simplified further by taking into account the scale separation between the pump frequency (which was previously eliminated with RWA) and the relevant energy scales of the system and as explained in [23]. This leads to the Markovian fluctuation kernel An intuitive explanation of this Markovian approximation is that the hyperbolic cotangent factor gives the occupation number of those bath modes which are coupled to the system coth 2T = 2n B + 1, where n B is the Bose-Einstein distribution. If all the relevant modes are empty, then n B = 0 and coth 2T = 1.
In time domain all the components of D −1 are local: Finally substituting the found D into the full action S = S 0 + S diss in time representation and noting that the action is completely time-local, we get If only linear terms in φ q ,φ q are kept in the action, the standard saddle-point approximation leads to leading to the exponential decay of the fields This amounts to the classical approximation, where there is no noise in the fields (the variance is zero); for example, a classical x coordinate of a harmonic oscillator would evolve as x(t) ∝φ cl (t) + φ cl (t) ∝ e −γt cos ωt.

Dissipative BEC
A free bosonic field is given by the Hamiltonian where ω k = k 2 /2m, and k is a wavenumber vector. For simplicity we will concentrate on the one-dimensional case, but the developed formalism is applicable to any number of dimensions (extension of this theory to quasicondensates is presented in the last section of the supplementary material).

Constant coupling
We model dissipation by the same construction from the previous section, assuming that each mode of the field is coupled to an independent empty bath of harmonic oscillators, the coupling being independent on k.
Applying the formalism of the previous section we arrive at the dissipative action for the field

Periodic boundary conditions
Making use of the Fourier transformation, dispersion relation and homogeneity, we reformulate the action in the real space where φ cl,q = φ cl,q (r, t).
Short-range interactions can be added the conventional way Semiclassical Gross-Pitaevskii treatment

Saddle-point approximation
The saddle-point solutions are given by requiring that variation of the action S over each of the four fields is zero. The classical solution can be found noticing that the minimization of the action S can be achieved by which leads to the noiseless dissipative GPE where Φ(r, t) is the solution of the saddle-point equation for the "classical" field φ cl . Resulting GPE can be solved by conventional numerical algorithms. For instance, the initial state Ξ cl (t = 0) can be sampled from a thermal distribution (like in the truncated Wigner approach, but neglecting the quantum noise).

Neglecting cubic terms in φ q
Fluctuations around the classical solution can be found by expanding the action by a small parameter φ q . The first approximation would be leaving the quadratic component 2iγφ q φ q in S 0 and neglecting the cubic one from S int , which amounts to keeping the quantum noise due to the dissipation process and neglecting all the processes of Landau, Beliaev damping, particle exchange between the condensate and the quasiparticles and quasiparticle interactions.
We can get rid of the quadratic term by introducing additional fields ξ(r, t),ξ(r, t) and utilizing the Hubbard-Stratonovich transformation The partition function becomes where ∀φ = {φ cl , φ cl ,φ q , φ q }.
The saddle-point equations δS δφ q = 0, δS δφ q = 0 lead to where the complex terms ξ,ξ must be integrated over all possible paths in time with the Gaussian weight e − 1 2γ dt|ξ(t)| 2 . Gaussian statistics implies that ξ,ξ can be represented by correlated complex white noise sources with so the evolution of the field is given by a stochastic Gross-Pitaevskii equation (SGPE). Note the normalization convention stemming from the Keldysh rotation where N is the total number of particles.

Linearized Bogoliubov dynamics
Experimentally relevant predictions, such as temperature decay in time, can be extracted from numerical solutions of the SGPE above, nevertheless, insight into the physics can be gained by studying its linearized dynamics.
In this section we concentrate on a true condensate regime (macroscopic occupation of the Penrose-Onsager mode), which can be achieved in one-and two-dimensional geometries only for very small systems. The extension of the given theory to a 1D quasicondensate is presented in the last section of the supplementary material.
To start, let's split the field into time-dependent homogeneous part (the condensate) and fluctuations around it where K(r, t) is a real field used to fix a particular gauge. It is convenient to utilize a gauge which makes the condensate field Ξ(t) as slow as possible. For instance, if there exists an equilibrium solution Ξ = const, then K(t) = µt with µ = const. The SGPE becomes where in the last step we linearized the equation in terms of small fluctuations ϕ. We also neglected the counterrotation of the noise given by ξ(r, t)e −iK(r,t) due to the fact that in the final results only terms ξ (r, t)ξ(r, t) have physical significance (in the Markovian approximation).

Linearized fluctuations
In our dissipative case we can achieve the slowest evolution of the condensate by gauging out fast phase rotation and leaving only the noiseless exponential decay in time, namely where n 0 is the initial density of the condensate, and the factor two is due to the Keldysh rotation. In the following we assume that the fluctuations are small, so n 0 can approximately stand for the total particle density.

Condensate density ≈ total density
Substituting this ansatz into (52) we recover the dissipative homogeneous gaugė which can be thought of as a time-dependent chemical potential. The linearized SGPE becomes or in the vector form where After the Fourier transformation ϕ(x) = 1 √ L k ϕ k e ikx the equation reads The Hamiltonian can be diagonalized using the standard Bogoliubov rotation (we set u, v = u k , v k ∈ Re for convenience), given by This leads to Note that the original Gaussian noise in the particle basis gets squeezed in Bogoliubov basis, and the non-adiabatic termP −1 ∂ tP mixes χ k andχ −k .

Instantaneous Bogoliubov basis
To simplify following derivations we concentrate on adiabatic dynamics, assuming that there is a scale separation between fast evolution of the Bogoliubov modes χ and slow change of the Bogoliubov coefficients u, v, soP −1 ∂ tP ≈ 0. This adiabatic approximation will be justified in the next section.
Then the linearized SGPE becomes and the evolution of the field is then given by The Keldysh equal-time field correlator for Bogoliubov bosons reads as Substituting the correlator of the Markovian noise we get Quasiparticle occupation numbers n k (t) are respectively given by Note that anomalous densities such as 2m k + 1 = χ k χ k can only decay without noise due to the noise correlator ξ k (t)ξ −k (t) = 0. In the initial thermal state m k = 0, meaning that anomalous correlators do not grow up and do not influence the quasicondensate thermodynamics.

Phononic regime
In the phononic regime, given by n k (t) 1, k (t) ≈ c(t)k, and the non-adiabatic term becomesP so it indeed mixes counter-propagating Bogoliubov modes. We can put this term into the HamiltonianĤ k , introducing a new "dissipation-dressed" Bogoliubov Hamiltonian which has the eigenvalues ± 2 k − γ 2 /4. So we see that non-adiabatic dynamics slows down the rotation of the Boboliubov modes. However, this effect can be neglected as long as k γ/2, which is the adiabatic regime we are exploring in this paper.
Taking into account that we see that the quasiparticle occupation numbers evolve as To evaluate experimetally relevant temperature dependence on the particle number, we note that (1) the initial quasiparticle occupation number in a thermal state at the temperature T 0 is given by the classical Rayleigh-Jeans equipartition T 0 = k c(0) n k (0), (2) the instantaneous speed of sound is proportional to the square root of the condensate density according to mc(t) 2 = µ(t) = gn 0 (t), and (3) the total particle number in case of periodic boundary conditions N (t) ∝ e −2γt ∝ n 0 (t).
This leads to the following scaling law in the case of periodic boundary conditions: The first term is given by the classical noiseless dissipation, and the second is due to the quantum noise from an empty bath according to the fluctuation-dissipation theorem.

Quasicondensate
In this section we generalize the linearization of SGPE (7) to the case of a quasicondensate of a weakly-interacting one-dimensional bosonic gas, namely when there is no single macroscopically occupied mode due to enhanced phase fluctuations. We can take into account the fact that in the weakly-interacting case the density fluctuations are suppressed, and represent the quantum field in phase-density variables: where n 0 (t) is the mean density, δn(x, t) is the local density fluctuation, K(t) is a homogeneous gauge field, which derivative plays the role of the chemical potentialK(t) = µ(t), and θ(x, t) is the local phase, and the factor two stems from the Keldysh normalization.
Substituting (8) into (7) and linearizing it with respect to the small parameters δn(x, t), ∇θ(x, t) and ∇ 2 θ(x, t), then taking into account that the mean density decays exponentiallyṅ 0 (t) = −2γ n 0 (t), and equating the real and imaginary parts to zero separately, we get where we represented the noise term with its real and imaginary parts ξ = η+iζ. Next we notice that for the white noise phase terms e iK(t)−iθ(x,t) are irrelevant, so we omit them for brevity, and pick the gauge fieldK(t) = µ(t) = gn 0 (t), which leads to Finally, we perform the unitary transformation into emergent bosonic basis by leading to which is equivalent to (55) with the chemical potential µ(t) = gn 0 (t) dependent on the mean density instead of the density of the condensate mode. Note that due to Keldysh normalization ϕ = √ 2 b. This proves that the analysis for a true condensate, presented in the previous sections, is fully applicable to the quasicondensate case as long as density fluctutations and phase gradients are small enough.

Markovian approximation
In the current section we derive the applicability criterion of the Markovian approximation for an outcoupling process of a quasicondensate. We formulate this section directly in the second quantized formalism for pedagogical reasons.
The radial wave function of the trapped state is assumed to be Gaussian: ψ ⊥ (x, y) = exp[−(y 2 + z 2 )/(2σ 2 )]/( √ πσ). The linear density of atoms is assumed to be small enough not to increase the width of this Gaussian too strongly compared to the ground state size of the radial trapping harmonic potential: σ = 1/ √ mω ⊥ , where ω ⊥ is the radial trapping potential fundamental frequency (recall that we use units where ≡ 1). We neglect contact interactions of atoms in the untrapped state with trapped atoms, therefore assuming formers' radial wave functions to be plane waves. 1D field operators for trapped atoms and untrapped atoms having the momentum (k y , k z ) in the transverese directions are, respectively,Ψ andΦ ky,kz . The coupled set of equations for them reads as Here the coupling between the trapped and untrapped fields is given by κ ky,kz = Ω √ A dy dz e −ikyy−ikzz ψ ⊥ (y, z), where A is the quantization area in the (y, z)-plane, Ω is the effective Rabi frequency of the multiphoton, microwavedriven transition between the trapped and untrapped states. The difference of the sum of the frequencies of the driving photons and the transition frequency (the multiphoton detuning) is denoted by −∆ω. Since Eq. (85) is linear, it is easy to eliminateΦ ky,kz using Green's function formalism. The resulting equation for Ψ reads as where the quantum noise isς We explicitly indicate the time argument of the fieldsΨ andΦ ky,kz in Eqs. (87) and (88), respectively, when it differs from t. Substituting Gaussian transverse profile into Eq. (89), we obtain The characteristic time scale set by Eq. (90) is 1/ω ⊥ . We assume that all the relevant energy scales of the systems, given by the temperature, the chemical potential and the Rabi frequency Ω are well below ω ⊥ . This allows us to pull Ψ(t ) out of the integral in the r.h.s. of Eq. (87), replacing t by t, and neglecting the small kinetic energy in the longitudinal direction which is the analogue of the Thomas-Fermi approximation. Long-time condition t 1/ω ⊥ allows us to substitute the upper limit of the time integral in the r.h.s. of Eq. (87) by ∞. The imaginary part of ∞ 0 dτ F (τ ) exp(i∆ωτ ) renormalizes the energy of the trapped state and can be incorporated into ∆ω. The real part determines the loss rate where Θ(∆ω) is Heaviside's step function. Finally, we obtain the dissipative equation for the trapped-atom field operator Initially the bath is empty, and we have only vacuum fluctiuations of the untrapped-atom field: Φ ky,kz (x, 0)Φ † k y ,k z (x , 0) = δ ky k y δ kz k z δ(x − x ).