Cooling phonon modes of a Bose condensate with uniform few body losses

We present a general analysis of the cooling produced by losses on condensates or quasi-condensates. We study how the occupations of the collective phonon modes evolve in time, assuming that the loss process is slow enough so that each mode adiabatically follows the decrease of the mean density. The theory is valid for any loss process whose rate is proportional to the $j$th power of the density, but otherwise spatially uniform. We cover both homogeneous gases and systems confined in a smooth potential. For a low-dimensional gas, we can take into account the modified equation of state due to the broadening of the cloud width along the tightly confined directions, which occurs for large interactions. We find that at large times, the temperature decreases proportionally to the energy scale $mc^2$, where $m$ is the mass of the particles and $c$ the sound velocity. We compute the asymptotic ratio of these two quantities for different limiting cases: a homogeneous gas in any dimension and a one-dimensional gas in a harmonic trap.


Introduction
Despite their extensive use as quantum simulators or for quantum sensing, the temperatures reached in ultracold gases are not fully understood. Careful analyses of the cooling mechanisms have a long tradition in the cold atoms community, and the corresponding temperature limits constitute important benchmarks. The role of atom losses, however, is not yet elucidated, although such processes often play a role in quantum gas experiments. Different loss processes may occur. One-body processes are always present, their origin could be for instance a collision with a hot atom from the residual vapour. The familiar method of evaporative cooling involves losses that depend on the particle energy, a case we exclude in this paper. For clouds trapped in an internal state which is not the lowest energy state, such as low-field seekers in a magnetic trap, two-body (spin flip) collisions may provide significant loss. Finally, three-body processes where atoms recombine into strongly bound dimers are always present and are often the dominant loss mechanism. The effect of one-body losses for an ideal Bose gas was investigated in [1]. Loss processes involving more than one body are a source of heating for trapped thermal clouds, since they remove preferentially atoms in dense regions where the potential energy is low [2]. Here we are interested in the effect of losses in Bose condensates or quasi-condensates, and we focus on low energy collective modes, whose physics is governed by interactions between atoms.
One-body losses have recently been investigated for one-dimensional (1D) quasi-condensates [3][4][5][6]. Quasi-condensates characterise weakly interacting 1D Bose gases at low enough temperature: repulsive interactions prevent large density fluctuations such that the gas resembles locally a Bose Einstein condensate (BEC), although it does not sustain true long-range order [7,8]. The above studies have focussed on low-energy excitations in the gas, the phonon modes. These correspond to hydrodynamic waves propagating in the condensate, where longwavelength phase (or velocity) modulations are coupled to density modulations. On the one hand, losses reduce density fluctuations and thus remove interaction energy from each phonon mode. This decrease in energy, and thus of quasiparticle occupation, amounts to a cooling of the modes. On the other hand, the shot noise due to the discrete nature of losses feeds additional density fluctuations into the gas. This increases the energy per mode and amounts to heating. Theoretical studies [4][5][6], valid for one-body losses in 1D homogeneous gases, predict that as a net result of these competing processes, the system is cooling down in such a way that the ratio between temperature k B T and the chemical potential µ becomes asymptotically a constant (equal to 1). Many questions remain open. For instance, the role of longitudinal confinement has not been elucidated. Moreover, theoretical predictions for higher-body loss processes are lacking, although cooling by three-body losses was recently demonstrated experimentally [9].
In this paper, we generalise the theoretical results for one-body losses in homogeneous 1D gases and extend the analysis to a BEC or a quasicondensate in any dimension, for any j-body loss process, and for homogeneous gases as well as clouds confined in a smoothly varying trapping potential. We concentrate on phonon modes and the loss rate is assumed small enough to ensure adiabatic following of each mode. Low-dimensional systems are realised experimentally by freezing the transverse degrees of freedom with a strong transverse confinement. However, in many experiments the interaction energy is not negligible compared to the transverse excitation frequencies such that the freezing is not perfect. The interactions then broaden the wave function in the transverse directions, and longitudinal phonon modes are associated with transverse breathing [10][11][12]. Our theory can take this into account with a modified equation of state: the quantities µ and mc 2 , where m is the atomic mass and c the sound velocity, equal for a strong transverse confinement, no longer coincide. We find that the evolution produced by losses is better described by a constant ratio k B T /(mc 2 ) instead of k B T /µ, as observed experimentally in our group [9]. The asymptotic ratio k B T /(mc 2 ) is computed for a few examples.

Model
We consider a condensate, or quasi-condensate, in dimension d = 1, 2 or 3. The gas is either homogeneous or trapped in a smoothly varying potential V (r). We assume it is subject to a j-body loss process of rate constant κ j : the number of atoms lost per unit time and unit volume is κ j n j where n is the density. This density includes fluctuations of quantum and thermal nature, and its average profile is denoted n 0 (r, t). Instead of using involved powerful theoretical techniques such as the truncated Wigner approach [13,14], we compute the effect of losses in this paper with a spatially coarse-grained approach that does not rely on involved theory and in which the approximations are made transparent. For the same pedagogical reason, we explicitly construct the phase-density representation of the collective excitations of the gas, in a similar way as is done for instance in [15].

Stochastic dynamics of the particle density
Let us first consider the sole effect of losses and fix a cell of the gas of volume ∆, small enough so that the density of the (quasi)condensate is about homogeneous in this volume, but large enough to accommodate many atoms. The atom number in the cell is N = N 0 + δN where N 0 = n 0 ∆ and δN ≪ N 0 since the gas lies in the (quasi)condensate regime. (We drop the position dependence n 0 = n 0 (r) for the moment.) Since typical values of δN are much smaller than N 0 , one can assume without consequence that δN is a variable that takes discrete values between −∞ and ∞. Hence, one can define a phase operator θ, whose eigenvalues span the interval [0, 2π[ and that is canonically conjugate to δN . Losses will affect both the density fluctuations and the phase fluctuations.
We first concentrate on the effect of losses on density fluctuations. Consider a time step dt, small enough that the change dN in atom number is much smaller than N , but large enough such that dN is much larger than 1. After the time step, we have where Here, dξ is a random number with vanishing mean value that translates the shot noise associated with the statistical nature of losses 1 . The number of loss events during the small step dt is Poisson distributed so that the variance of dξ relates to the mean number of lost atoms by the factor j coming from the fact that at each event, j atoms are lost. The evolution of fluctuations in the atom number is obtained from dδN = dN − dN 0 , where dN 0 is the change of the mean number, equal to dN 0 = −K j N j 0 dt in the lowest order in δN . Expanding N j in Eq.(1) to first order in δN , we obtain the following evolution for the density fluctuation δn = δN/∆: dδn = −jκ j n j−1 0 δn dt + dη where dη = dξ/∆ is a random variable of variance dη 2 = jκ j n j 0 dt/∆. The first term in the r.h.s, the drift term, decreases the density fluctuations. It will thus reduce the interaction energy associated to fluctuations in the gas and produce cooling. The second term on the other hand increases the density fluctuations in the gas which leads to heating.

Shot noise and phase broadening
We now compute the effect of losses on the phase fluctuations, following an approach similar to Ref. [16]. For this purpose, one imagines that one records the number of lost atoms during dt. This measurement increases the knowledge about N , and thus δN . To quantify this increase of knowledge, we use the Bayes formula where P(δN ) is the initial probability of having an atom number N = N 0 +δN , and P(N l |δN ) is the probability that a number N l of atoms will be lost, given that the initial atom number was N 0 + δN . Finally, P(δN |N l ) is the probability that the final number is N 0 − N l + δN , knowing the fact that N l atom have been lost. As argued above, the Poissonian nature of the loss process and the assumption that the number of lost atoms is large compare to one, imply the Gaussian distribution where N = N 0 + δN and σ 2 l = jK j N j 0 dt. Expanding N j around N j 0 and introducing δN = where Thus, according to Eq.(4), the width of the distribution in δN is multiplied by a function of rms width σ δN after recording the number of lost atoms. This narrows the number distribution and must be associated with a broadening in the conjugate variable, θ, lest the uncertainty relations are violated. The phase broadening must be equal to This spreading of the phase results from the shot noise in the loss process. In the following, keeping in mind that only length scales larger than the interparticle distance have to be considered, we go to the continuous limit. The factors 1/∆ in the variance for dη in Eq.(3) and in the phase diffusion of Eq.(8) then turn into Both diffusion terms are due to the quantised nature of the bosonic field, namely the discreteness of atoms. Their effects become negligible compared to the drift term in Eq. (3) in the classical field limit, i.e. n 0 → ∞ at fixed typical density fluctuations δn/n 0 . Note finally that these results could also have been obtained using a truncated Wigner approach [13,14], using approximations based on the relation δn ≪ n 0 . Before going on, let us make a remark concerning gases in reduced dimension. An effective 1D (resp. 2D) gas is obtained using a strong transverse confinement in order to freeze the transverse degree of freedom: the atoms are in the transverse ground state of the confining potential, of wave function ψ(x ⊥ ). In the case of j-body losses with j > 1, the loss process a priori modifies the transverse shape of the cloud since it occurs preferentially at the center, where the density is the highest. In other words, it introduces couplings towards transverse excitations. We assume here the loss rate to be much smaller than the frequency gap ω ⊥ between the transverse ground and first excited states. Then the coupling to transverse excitations has negligible effects, and the above analysis of the effect of losses also holds for the effective 1D (resp. 2D) gas, provided j is the rate constant coefficient for the 3D gas.

Collective excitations
Let us now take into account the dynamics of the gas. Under the effect of losses the profile n 0 (r, t) evolves in time and, except for a homogeneous system, a mean velocity field appears, generated by a spatially dependent phase θ 0 (r, t). Here we assume the loss rate is small enough so that, at any time, n 0 (r) is close to the equilibrium profile. We moreover assume the potential varies sufficiently smoothly such that the equilibrium profile is obtained with the local density approximation. Then, at any time, n 0 (r) fulfills µ(n 0 (r)) = µ p − V (r) (11) where µ(n) is the chemical potential of a homogeneous gas of density n and µ p is the peak chemical potential, which fixes the total atom number 2 . In most cases µ = gn where g is the coupling constant. In 3D condensates, g = 4π 2 a/m where a is the scattering length describing low-energy collisions. In situations where two (resp. one) degrees of freedom are strongly confined by a transverse potential of frequency ω ⊥ , µ depends on a, on the linear (resp. surface) density n, and on ω ⊥ . As long as ω ⊥ ≫ µ, the transverse cloud shape is close to that of the transverse ground state 3 , and one recovers the expression µ = gn where the effective 1D (resp. 2D) coupling constant g depends only on a and on ω ⊥ [17,18]. At large densities, ω ⊥ ∼ µ, the transverse degrees of freedom are no longer completely frozen: interactions broaden the transverse wave function, and µ is no longer linear in n [11,12]. We discuss one example in Sec.3.2.
To treat the dynamics around the average density n 0 (r, t), a Bogoliubov approximation is valid since the gas is in the (quasi)condensate regime: one can linearise the equations of motion in the density and phase fluctuations δn(r) and ϕ(r) = θ − θ 0 [15,19]. These equations involve the mean velocity field ∇θ 0 /m. Here we assume the loss rate is small enough so that such terms are negligible. We moreover consider only length scales much larger than the healing length. Then, as detailed in Appendix A, the dynamics of δn(r) and ϕ(r) is governed by the hydrodynamic Hamiltonian Here the speed of sound c = c(r) is related to the local compressibility, mc 2 = n 0 ∂ n µ, evaluated at n 0 (r). At a given time, H hdyn can be recast as a collection of independent collective modes. The collective modes are described by the eigenfrequencies ω ν and the real functions g ν [details in Appendix B]. They obey and are normalised according to Then The dimensionless canonically conjugate quadratures x ν and p ν are related to δn and ϕ respectively. More precisely, At thermal equilibrium, the energy in the mode ν is equally shared between both quadratures and, for temperatures T ≫ ω ν , one has H ν = T .

Evolution of the excitations
Let us consider the effect of losses on the collective modes. The loss process modifies in time the mean density profile and thus the two functions of r, n 0 and c, that enter into the Hamiltonian Eq. (12). We however assume the loss rate is very low compared to the mode frequency and their differences ω ν − ω ν ′ , so that the system follows adiabatically the effect of these modifications. As a consequence, equipartition of the energy holds at all times for any collective mode ν, and the adiabatic invariant A ν = H ν /( ω ν ) is unaffected by the slow evolution of n 0 . The dynamics of A ν is then only due to the modifications of δn(r) and ϕ(r) induced by the loss process (subscript l), namely Injecting Eq. (3) into Eq. (17), we obtain for the 'density quadrature' Using the mode expansion (16) for δn(r) in the first term, we observe the appearance of couplings between modes. In the adiabatic limit (loss rate small compared to mode spacing), the effect of these couplings is however negligible. Then, Eq. (19) leads to Let us now turn to the phase diffusion associated with losses. It modifies the width of the conjugate quadrature p ν , according to The hydrodynamic modes are characterised by low energies, ω ν ≪ mc 2 , when the speed of sound is evaluated in the bulk of the (quasi)condensate. Then d p 2 ν l /dt gives a contribution that scales with the small factor ( ω ν /mc 2 ) 2 compared to the second term of Eq. (20). In other words one expects that the phase diffusion associated to the loss process gives a negligible contribution to the evolution of A ν [Eq. (18)] for phonon modes 4 .
We see from Eq.(20) that the adiabatic invariant A ν is actually changed by j-body losses. We now show that the decrease in the energy per mode H ν is better captured by the energy scale associated with the speed of sound, as their ratio will converge towards a constant during the loss process. More precisely, we introduce where c p is the speed of sound evaluated at the peak density n p . The second expression is valid as long as the phonon modes stay in the classical regime, H ν ≫ ω ν . From Eq. (18) and (20), neglecting the contribution of Eq.(21), we immediately obtain where the dimensionless parameters A, B and C are In general, all of them depend on ν but we omit the index ν for compactness. The term A is the rate of decrease of y ν induced by the reduction of the density fluctuations under the loss process, normalised to κ j n j−1 p . The term B originates from the additional density fluctuations induced by the stochastic nature of the losses. The term C arises from the time dependence of the ratio mc 2 p / ω ν . It is computed using the dependence of mc 2 p / ω ν on the total atom number, the latter evolving according to Eqs. (23-26) constitute the main results of this paper.
We would like at this stage to make a few comments about these equations. First, the factor , although it appears explicitly in the equations, is not relevant since it is canceled by the contained in the normalisation (14) of the mode functions g ν . Second, we note that A, B and C are intensive parameters: they are invariant by a scaling transformation V (r) → V (λr) and depend only on the peak density n p and on the shape of the potential. Finally, Eqs. (23-26) depend on ν and it is possible that the the lossy (quasi-)condensate evolves into a non-thermal state where different modes acquire different temperatures. Such a non-thermal state of the gas is permitted within the linearised approach where modes are decoupled. In the examples studied below, however, it turns out that all hydrodynamic modes share about the same temperature 5 . In the following, we investigate the consequences of Eq. (23-26), considering different situations.

Example: homogeneous gas
In this case, density n 0 and speed of sound c are spatially constant. The collective modes are sinusoidal functions, labelled by ν and of wave vector k ν 6 . The frequencies are given by the acoustic dispersion relation ω ν = c|k ν | and the mode functions g νc,s (r) are normalised to d d r g 2 ν (r) = ω ν mc 2 n 0 (28) 5 In the case of one-body losses, theories that go beyond the hydrodynamic approximation predict nonthermal states to appear, where the high-frequency modes reach higher temperatures than the phonon modes [4,5]. 6 For 1D gases, ν = (p, σ) where p is a positive integer and σ = c or s depending wether we consider cosine or sine modes. The wave-vector is kν = 2pπ/L where L is the length of the box, assuming periodic boundary conditions. This generalises to higher dimensions with ν = (p1, σ1, p2, σ2, p3, σ3) in 3D for instance.

Then Eqs.(23-26) reduce to
which is the same for all modes ν. Let us consider the limit µ = gn 0 , valid in 3D gases, or in low-dimensional gases with strong transverse confinement (negligible broadening of the transverse wave function). Then c ∝ n 1/2 0 and Eq. (29) shows that y tends at long times towards the asymptotic value independent of the mode energy 5 . For one-body losses, one recovers the result y ∞ = 1 [4,5].
In the case of 3-body losses, one finds y ∞ = 3/5. Let us now consider a quasi-low-dimensional gas, where transverse broadening of the wave function cannot be neglected. The logarithmic derivative in Eq.(29) is then no longer constant. We will focus on the case of a quasi-1D gas, as realised experimentally for instance in [9]. The effect of the transverse broadening is well captured by the heuristic equation of state [11,12] where ω ⊥ is the frequency of the transverse confinement and a the 3D scattering length. Inserting into Eq. (29), one can compute the evolution of y. The transverse broadening also modifies the rate coefficient κ j , making it density-dependent. However, re-scaling the time according to u = t 0 κ j (τ )n j−1 p dτ = ln(n 0 (0)/n 0 (t)), Eq. (29) transforms into dy du = −y j − 1/2 + n 0 (0)a e −u 1 + 4n 0 (0)a e −u + j/2 and no longer depends on κ j . Fig.1 shows the solution of this differential equation in the case of 3-body losses, and for a few initial situations, namely different values of y and n 0 a (right plot). The asymptotic value y = y ∞ is always reached at long times since the transverse broadening then becomes negligible. Note that in distinction to pure 1D gases, the effect of transverse broadening allows the system to reach transiently lower scaled temperatures y < y ∞ , even when starting at values of y larger than y ∞ . More precisely, let us denote y min (n 0 ) = j/2/(j − 1/2 + an 0 /(1 + 4an 0 )). When starting with y > y min , the lowest value of y is reached for some (non-vanishing) density, and it falls on the curve y min . For j = 3, one find that y min varies between y ∞ = 0.6 and 6/11 ≃ 0.55. Thus, the coldest temperatures in the course of the loss process never deviate by more than 10% from the asymptotic value 0.6: the impact of transverse swelling is relatively small. Note that, if one considered the scaled temperature T /µ rather than y, much larger deviations would appear.

Example: 1D harmonic trap
We consider a 1D gas confined in a harmonic potential of trapping frequency ω. We assume for simplicity a pure 1D situation with µ = gn = mc 2 . In the Thomas-Fermi approximation, the mean density profile is where n p is the peak density and R = 2gn p /(mω 2 ) is the axial radius of the quasicondensate. From Eq.(13), we recover the known result that the hydrodynamic modes are described by the Legendre polynomials P ν , and the eigenfrequencies are ω ν = ω ν(ν + 1)/2 [7,20]. A trivial calculation using N tot = 4 3 n p R ∝ c 3 p and the substitution z = R cos α gives C = π/2 0 dα sin 2j+1 α = 2/3, 8/15, 16/35 for j = 1, 2, 3. To compute A and B, one needs the exact expression of g ν , which according to the normalisation (14) can be written Inserting this expression, together with Eq. (33), into the integrals (24) and (25), we find that A, B, and C are time-independent. Thus y tends at long times towards the asymptotic value y ∞ = jB/(jA − C). For large ν, one can use the asymptotic expansion [21] P ν (cos α) ≃ 2 π(ν + 1 2 ) sin α with φ ν = (ν + 1 2 )α − 1 4 π. Moreover the fast oscillations of P ν (cos α) can be averaged out in the calculation of the coefficients A and B. Then A and B no longer depend on ν, so that y ∞ is identical for all modes, and we find For one-and three-body losses, this gives y ∞ = 3/4 = 0.75 and y ∞ = 525/748 ≃ 0.701, respectively. This asymptotic result is compared to calculations using the expression Eq. (34) in Fig. 2. We find very good agreement as soon as the mode index is larger than 5.  To conclude this example, we come back to the diffusive dynamics of the 'phase quadratures' p ν we neglected so far. In the case of one-body losses, however, it happens that the integral (21) does not converge: while the mode function g ν (z) [Eq.(34)] remains finite at the condensate border z → ±R, the integrand n j−2 0 (z)g 2 ν (z) is not integrable for j = 1. This is actually an artefact of the hydrodynamic approximation, which breaks down at the border of the condensate.
We have performed numerical calculations of the collective excitations by solving the Bogoliubov equations 7 . The mode functions g ν (z) are defined according to Eq.(61): they extend smoothly beyond the Thomas-Fermi radius and match well with the Legendre polynomials (34) within the bulk of the gas. The resulting values for the parameter B [Eq.(25)] are shown in Fig.3: they depend very weakly on the mode index ν and are well described by the approximate calculation based on the Legendre modes mentioned after Eq.(35) (solid lines). In the lower part of the figure, the corresponding values for the diffusion coefficient originating from phase noise are shown, namely the parameter They remain at least one order of magnitude below. For losses involving more than one particle, the approximation, under which the functions g ν are given by the Legendre polynomials, gives a convergent integral in Eq.(37). The result is shown as solid lines for two-and threebody losses, where we made the additional approximation Eq. (35) on the Legendre functions and we averaged out the oscillating part. We find that the Legendre approximation performs better for three-body losses than for 2-body losses, which is expected since a stronger weight is given to the bulk rather than the edge of the condensate. In conclusion of this numerical study, we verified the validity of the assumption that, for phonon modes, the phase diffusion term gives negligible contribution to the evolution of y. This term becomes noticeable when one leaves the phonon regime ω ν ≪ mc 2 . Then, one should go beyond the hydrodynamic Hamiltonian Eq. (12) to properly compute the mode dynamics.

Conclusion
In this paper, we construct a stochastic model to describe the effect of losses on the hydrodynamic collective modes of condensates or quasicondensates. Explicit formulas for cooling and diffusion of the density and phase quadratures are derived. They provide the behaviour of the mode temperature T with time. We show that T becomes proportional to the energy scale mc 2 where c is the hydrodynamic speed of sound. The asymptotic ratio k B T /(mc 2 ) is computed explicitly in different situations and for different j-body processes. These results are in good agreement with recent experiments in our group [9] where three-body losses provided the dominant loss channel. This work raises many different questions and remarks. First, the effect of losses on high-frequency modes, not described by our hydrodynamic model, leads a priori to higher temperatures, as was investigated for 1D gases subject to one-body losses [5]. The gas then is described by a generalised Gibbs ensemble where different collective modes experience different temperatures. This non-thermal state is even long lived in 1D quasicondensates [5,22]. While the calculations presented here are formally valid for higher dimensions, efficient coupling between modes may reduce their relevance, since such coupling favours a common temperature. It is an open question whether our methods could be extended to the case of evaporative cooling where the one-body loss rate is energy-or position-dependent. This mechanism may play a role in experiments where temperatures as low as k B T ≈ 0.3 mc 2 have been observed, lower than the predicted temperatures for uniform losses [3]. Second, it would be interesting to extend this work to different regimes of the gas. For instance, one may ask how the effect of losses transforms as one goes from a quasi-condensate to the ideal gas regime. The approximation of weak density fluctuations then clearly becomes invalid. One could also investigate losses at even lower densities, where the 1D gas enters the fermionised (or Tonks-Girardeau) regime. Here, ones expects that the losses act in a similar way as in a non-interacting Fermi gas. One-body losses, for example, should then produce heating, since the temperature increases as the degeneracy of an ideal Fermi gas decreases. Finally, it would be interesting to investigate whether the results presented here may also cover interacting Fermi gases in the superfluid regime.

A Reduction to low-dimensional hydrodynamics
As mentioned in the main text, we assume the loss process is slow enough so that, first, the mean profile at each time is very close to the equilibrium profile with the same atom number, and second, we can safely neglect any mean velocity field when computing the time evolution of the fluctuating fields δn, ϕ. The evolution equations ∂δn/∂t and ∂ϕ/∂t are thus, at a given time, equal to those for a time-independent quasi-condensate. In the purely 3D, 2D and 1D cases, for contact interactions, we can use the well known results based on Bogoliubov theory. We then find that the equation of state takes the form µ = gn and ∂δn/∂t and ∂ϕ/∂t derive from Eq. (12) for the long-wavelength modes.
Let us now consider the case where the gas is confined strongly enough in 1 or 2 dimensions, such that the relevant low-lying excitations are of planar or axial nature. We allow, however, for a transverse broadening of the wave function under the effect of interactions. We show below that the equations of motion for the slow phononic modes, for which the transverse shape adiabatically follows the density oscillations, also derive from Eq. (12). The proof given here is complementary to Refs. [10,11] because it does not need an explicit model about the shape of the transverse wave function. In order to simplify the notations, we restrict ourselves to the quasi-1D situation. The derivation can be easily translated to quasi-2D situations.
We thus consider a gas confined in a separable potential consisting of a strong transverse confinement and a smooth longitudinal confinement. The equilibrium density distribution of the quasi-condensate is |φ 0 (x, y, z)| 2 where the real function φ 0 (x, y, z) obeys the stationary Gross-Pitaevskii equation Here g = 4π 2 a/m is the 3D coupling constant with a the zero-energy scattering length. Within the Bogoliubov theory, the evolution of excitations is governed by the equations [19] The field operators are half sum and difference of the fluctuating field operators δψ and δψ † . f + is linked to density fluctuations andf − to phase fluctuations. Since we assume that the axial variation is slow compared to the transverse one, the solution φ 0 can be approximated by a function ψ that depends on the axial coordinate z only via a local chemical potential φ 0 (x, y, z) ≃ ψ(x, y; µ) , µ = µ p − V (z) Here, ψ solves the Gross-Pitaevskii equation for an axially homogeneous system: We again project out the transverse coordinates and use the identities (47). Combining the axial derivatives, we then have These calculations illustrate that the Ansatz of Eq.(43) captures well the axial and transverse dependence of the collective excitations in the low-dimensional gas. Note in particular how the density fluctuations (f + ) are accompanied by density-dependent changes in the transverse wave function.
To make contact with the hydrodynamic Hamiltonian (12), we need to relate F + and F − to the low-dimensional density and phase fields, δn and ϕ. Bogoliubov theory tells us that three-dimensional density fluctuations are linked tof + via δρ = 2φ 0f + . Integrating δρ over the transverse plane, replacingf + by its Ansatz (43) and using Eq. (47), we obtain

B Hydrodynamic Bogoliubov modes
Here we consider low-energy modes of either a three-dimensional gas or low-dimensional gas, whose dynamics is described by the hydrodynamic approximation. More precisely, we diagonalize the Hamiltonian (12), for a given, time-independent, equilibrium profile n 0 (r). From Eq.(12) we derive the evolution equations ∂ ∂t δn/ √ n 0 √ n 0 ϕ = L δn/ √ n 0 √ n 0 ϕ where The factors √ n 0 are convenient to give the two components the same dimension and to symmetrize the differential operator that appears in L. The two equations derived from Eq.(54) correspond to the hydrodynamic equations provided we identify ∇ϕ/m with the velocity: the first one is the continuity equation, the second one gives the Euler equation.
We build the mode expansion on pairs of real functions that form right eigenvectors of L: