Asymptotic temperature of a lossy condensate

We thank a lot the referee for her/his work on our manuscript and for her/his comments and remarks that enable us to improve the paper. We answer below to all comments of the referee.

1) We agree that introducing the notation $n_{qBEC}$ for the calculated theoretical profile of a quasi-BEC was not necessary and even maybe confusing. We no longer introduce this notation.

2) The factor 0.7 we choose to extract $y_{\infty}$ from our data is somehow arbitrary. For times $t>0.7\Gamma$, each data set reached, within our measurement precision, a stationary value. For smaller times, and for some data set (data set 1 and 5), we exclude, with this criteria, data points that might show temporal evolution. We added a sentence in the text: “For all data sets $y$ is about stationary for times $t>0.7/\Gamma$ and we note $y_\infty$ the mean value of $y$ for times $t>0.7/\Gamma$.”

3) Indeed the energy independent loss process is counter-intuitive. The loss rate per atom does not depend on the position of the atom, its potential energy nor the density it sees. However, in presence of repulsive interactions between atoms, this does not imply that the energy per atom – for the remaining atoms-- stays constant during the loss process. First, as the mean density decreases, the interaction energy of the remaining atoms decreases. This however is not sufficient to explain the cooling of the phonon modes. To understand the origin of the cooling, let us consider a density fluctuation $\delta n(z_i)$ at position $z_i$ which is larger than the average density $n_0$: the loss rate per unit length at $z_i$ – equal to $n(z_i) \Gamma$ – is larger than the loss rate per unit length in a zone where the density is $n_0$. In other words, if one considers a lost atom, it has more chance to come from a region of high density than from a region of low density. Thus losses tend to decrease density fluctuations. Since the interaction energy is higher in zone of high densities that in zone of low density, the energy associated to density fluctuations, has decreases: this is at the origin of the cooling of the phononic modes.

In the introduction, we added the following sentences to give a hint of the origin of cooling : “The dissipative term is responsible for a cooling: although the loss process is homogeneous, losses per unit length occur at a higher rate in regions of higher densities-- just because there are more atoms-- which leads to a decrease of density fluctuations and thus of their associated interaction energy. On the other hand, the stochastic nature of losses tends to increase density fluctuations and thus the interaction energy; this corresponds to a heating term. As a result of the competition between both effects, it has been predicted that phononic collective modes at large times acquire a temperature $k_B T$ that decreases in proportionally to the energy scale $mc^2$ where $m$ is the atomic mass and $c$ the speed of sound.”

Requested changes: we made all requested changes.

List of changes :
-Second paragraph of the introduction : We added a sentence to give a hint on the --counter-intuitive-- cooling mechanism
-Page 3, in the description of the analysis of the mean density profile : we remove the notation $n_{qBEC}$
-En of page 3 : we added references for the formula \mu=\hbar\omega_\perp(\sqrt{...)
-Page 5, introduction of $y_{\infty}$ : We added a sentence to justify the introduction of a minimum time $t\Gamma$
-We corrected the typos and orthographic errors pointed out by the referee