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

### Submission summary

 As Contributors: Isabelle Bouchoule Arxiv Link: https://arxiv.org/abs/1806.08759v2 Date submitted: 2018-07-04 Submitted by: Bouchoule, Isabelle Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics

### Abstract

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.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Submission 1806.08759v2 (4 July 2018)

## Invited Reports on this Submission

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### Strengths

1- Significant extension and generalization of previous theoretical models, which enables the interpretation of recent experiments studying cooling through losses in cold atomic gases.
2- Comprehensive, timely and relevant.
3- Clear explanation of all steps of the derivation.

### Weaknesses

None (see below for minor questions)

### Report

The authors present a general formalism for the cooling of quantum gases through uniform particle loss. While often associated with heating it is shown that under certain conditions few-particle loss processes can also lead to cooling. The study is motivated by a series of recent experiments, e.g. in the groups of Schmiedmayer and Bouchoule. As such, it is timely and relevant.

The present work is particularly interesting because the existing formalism to describe such cooling is significantly extended to cover general N-body losses, as well as arbitrary dimensions and realistic trapping potentials. For example, the cooling of a 1D gas through 3-body losses has recently been studied in an experiment by some of the authors for the first time. The results show good agreement with the theoretical predictions outlined in the present study. It will be very interesting to check experimentally if the predictions also hold in 3D situations, where evaporation (i.e. cooling through selective removal of particles with above average energy) will compete with the mechanism studied here (uniform loss of particles).

The study is well written and clearly explains all assumptions and steps of the derivation, as well as some limits of the theory. In particular, it nicely describes the inherent competition between the cooling of the low energy modes and their heating through shot noise.

### Requested changes

A few minor questions:
1- Footnote 1: It is not quite clear to me what exactly the remark on Ito / Stratonovich formalisms refers to here? Could the authors briefly elaborate to make this clearer for the general reader?
2- Is there a simple way to estimate for which experimental parameters one would expect the assumption of Poissonian, uncorrelated shot noise to hold for the loss process?
3- In realistic experiments there will often be different loss processes present at once (e.g. three-body loss from interatomic collisions due high atomic densities, and one-body loss from background gas collisions), each with their own asymptotic temperature. What would one expect to see in these cases?
4- Is there a deeper meaning / interpretation behind the asymptotic temperature? Particularly Fig. 1b seems to suggest that the cooling exhibits some universal properties?
5- It would be nice to discuss if the gases become more degenerate during the cooling process. Can new regimes be reached, e.g. for 1D Bose gases?

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### Author Isabelle Bouchoule on 2018-09-10

(in reply to Report 1 on 2018-07-26)

We thank the referee for her/his careful reading of the manuscript
and we are glad to hear that she/he appreciates the presented work.

1. In the stochastic equation for $N$ (Eq. 1), the variance of the random
variable $d\xi$ depends on $N$ (see Eq. 2). For a differential equation
for $N$ of the form $dN = -A(N)dt + d\xi$ with an $N$-dependent stochastic
term (namely $\langle d\xi(t)d\xi(t') \rangle = f(N) \delta(t-t')$), one
should specify whether the stochastic equation is written in the Stratonovich
or in the Ito formalism. However, since N(t) presents only small fluctuations
around its mean value $N_0(t)$, one can replace $N$ by $N_0(t)$ when computing the
stochastic term. Then, the resulting equation for $\delta N = N - N_0$ has
a stochastic term which does not depend on $\delta N$ (see Eq. 3). In such
a case the Ito and Stratonovich formalisms are equivalent.

In order not to confuse the reader not familiar with stochastic differential
equations, we prefer to remove the footnote. The reader aware of the subtle
details of stochastic equations will figure out himself that differences
between Ito and Stratonovich equations are irrelevant here.

2. The loss process may be assumed to be of Poissonian nature, as long as we
consider it on a time interval dt small enough so that the number of lost
atoms is very small compared to $N_0$: then the loss rate is constant
during the time interval dt, which amounts to a Poissonian process.

3. The question of the presence of multiple loss processes could a priori
be addressed within our formalism, provided Eq.20 is modified to account for
each loss process. Then Eq.23 would be modified as well. We foresee that
the temperature will still take values close to $mc^2$. We believe that such
a question, while interesting and relevant, deserves a separate study,
and we prefer to skip it in the present paper.

4. Unfortunately, we do not have a simple argument that shows that the
ratio $k_B T/(mc^2)$ goes towards a stationary value as a result of losses.
In the introduction and later in the text (after Eq.3 and after Eqs.23-26),
we emphasize the underlying physics: cooling due to the decrease of density
fluctuations on the one hand, and heating due to the stochastic nature of losses,
on the other. To our knowledge however, only quantitative calculations permit to
conclude that the relevant quantity, which takes a stationary value, is the
ratio $k_B T/(mc^2)$.

5. In the conclusion, we added a brief discussion what
may eventually happen with the ultracold gas as the density and temperature are
lowered. It indeed crucially depends on the system dimensionnality whether quantum
degeneracy increases or decreases.