Dynamics of Hot Bose-Einstein Condensates: stochastic Ehrenfest relations for number and energy damping

We first address objection

_3) the Kohn theorem is violated and this shows the theory is not appropriate for real experiments_

There are a number of systems under current and future study for which the theory is applicable. For our purpose it forms a very useful test for the formalism. We have revised the manuscript to clarify these points.

At the start of section 4 we have made substantial revisions of the text to point out several systems of interest where Kohn’s theorem is not satisfied by the system, and hence the SPGPE reservoir theory provides a useful approximation. We have also emphasized that the SPGPE has been used to give a first-principles treatment of high temperature non-equilibrium experiments. In particular, spontaneous vortex formation was described using one fitted parameter (ref [3] of revised manuscript), and the SPGPE gave a quantitative description of the experiment in ref [17] of the revised manuscript, with no fitted parameters:

"While the time-independent reservoir approximation (TIRA) is not strictly applicable for scalar BEC in the purely harmonic trap, there are a number of physical systems where it is applicable: Kohn’s theorem does not apply to a scalar Bose gas held in a harmonic trap if the trap becomes non harmonic at high energy. The theorem is also inapplicable for a harmonically trapped system if the reservoir consists of second atomic species confined by a different trapping potential, as may occur during sympathetic cooling. Furthermore, any system that is not harmonically trapped will not obey Kohn’s theorem, and is thus potentially amenable to the TIRA. Example systems in non-harmonic traps for which the theory is applicable include vortex decay in hard-wall confinement [46], soliton decay in a 1D toroidal trap (where the present approach was first used) [19], and persistent current formation in a 3D toroidal trap, where SPGPE simulations [17] compare well with experiment [44].

In this work our approach is simply to test the formalism on a simple model system within the TIRA by integrating out the spatial degrees of freedom to find effective stochastic equations of motion for the centre of mass. We stress that the TIRA approached used here is physically valid for (at least) two scenarios if immediate interest: non-harmonic trapping at high energies for a scalar BEC, and sympathetic cooling involving two BEC components in different harmonic traps [14]."

These points of context are also mentioned briefly in footnote 3 (page 14) of the revised manuscript.

We have also substantially revised our conclusions to further clarify the physical applicability of the theory of the harmonically trapped system to real physical systems. In particular:

“We tested our stochastic Ehrenfest equations in two ways. Considering the centre of mass motion of a finite-temperature quasi-1D condensate near equilibrium, we tracked the size of the largest projector corrections and saw they are indeed small. We also compared the steady-state correlations of position and momentum to analytic solutions derived by neglecting the projector corrections, finding excellent agreement. Our chosen test system has the weakness that the centre of mass motion in a purely harmonic trap is not strictly amenable to the reservoir theory due to a violation of Kohn's theorem. However, our treatment is physically relevant for non-harmonic trapping, multicomponent systems, and other systems of interest that physically violate Kohn's theorem, provided a low-energy fraction is harmonically trapped. Indeed, since the thermal equilibrium properties involve small excursions from equilibrium, the confining potential is only required to be _locally_ harmonic near the trap minimum and any number of non-harmonic effects may intrude at larger distances. The centre of mass motion thus provides an excellent formal and numerical test of the SERs, being one of the simplest states of motion to handle analytically. \par We have shown that SERs can be used to obtain analytic equations that agree with numerical solutions of the full SPGPE and offer some physical insight into the open system dynamics. Future work will explore systems involving analytically tractable excitations such as vortex decay in hard-wall confinement [46], soliton [39] and phase-slip dynamics [40] in toroidal confinement, sympathetic cooling [41,42], spinor BECs [23], and quantum turbulence in non-harmonic confinement [44-47].”

We hope that these changes are appropriate to address the remaining concerns regarding physical applicability.

_2) the equilibrium is not properly discusses and if it is the classical (field) equilibrium that I think is the case here, then it is actually not appropriate for the normal state of the gas, which the authors are discussing_

We find this comment unclear. We do not discuss the normal state of the gas directly, as our theory is a stochastic field theory of the low-energy partially degenerate fraction of the gas. There is a classical field equilibrium (within the truncated Wigner classical field approximation), that contains the condensate and a low energy normal fraction, but the latter must be extracted numerically (typically via Penrose-Onsager).

A high-energy normal fraction of the gas appears explicitly via the reservoir, due to the choice of the energy cutoff - it must be chosen to separate the coherent region from the incoherent region of phase space.

The normal fraction of the gas is thus included in the properties of the reservoir (gaussian statistics, chemical potential, temperature, and reservoir interaction rates), formulated using the single-particle Wigner function for the high energy part of the field.

In short, we are not sure we understand the question, but we would be happy to provide an answer to a clarified question.

With Best Regards,

Ashton Bradley (on behalf of the authors).

Revisions:

- Introduced acronym Stochastic Ehrenfest relation (SER)

- Minor change of wording in second to last paragraph of introduction:

“As a test we apply the Ehrenfest relations to the centre of mass fluctuations of a harmonically trapped system tightly confined along two spatial dimensions. We find that the analtyic solution of the SER for the centre of mass is in close agreement with SPGPE simulations. ”

- Replaced center —> centre throughout

- Reordered the wording in the “v) Thermal equilibrium” paragraph at the end of section 3 on page 13, to improve clarity

- Introduced acronym TIRA (Time Independent Reservoir Approximation)

- Start of section 4, revised text:

"While the time-independent reservoir approximation (TIRA) is not strictly applicable for scalar BEC in the purely harmonic trap, there are a number of physical systems where it is applicable: Kohn’s theorem does not apply to a scalar Bose gas held in a harmonic trap if the trap becomes non harmonic at high energy. The theorem is also inapplicable for a harmonically trapped system if the reservoir consists of second atomic species confined by a different trapping potential, as may occur during sympathetic cooling. Furthermore, any system that is not harmonically trapped will not obey Kohn’s theorem, and is thus potentially amenable to the TIRA. Example systems in non-harmonic traps for which the theory is applicable include vortex decay in hard-wall confinement [46], soliton decay in a 1D toroidal trap (where the present approach was first used) [19], and persistent current formation in a 3D toroidal trap, where SPGPE simulations [17] compare well with experiment [44].

In this work our approach is simply to test the formalism on a simple model system within the TIRA by integrating out the spatial degrees of freedom to find effective stochastic equations of motion for the centre of mass. We stress that the TIRA approached used here is physically valid for (at least) two scenarios if immediate interest: non-harmonic trapping at high energies for a scalar BEC, and sympathetic cooling involving two BEC components in different harmonic traps [14]."

These points of context are also mentioned briefly in footnote 3 (page 14) of the revised manuscript.

- Conclusions revised:

“We tested our stochastic Ehrenfest equations in two ways. Considering the centre of mass motion of a finite-temperature quasi-1D condensate near equilibrium, we tracked the size of the largest projector corrections and saw they are indeed small. We also compared the steady-state correlations of position and momentum to analytic solutions derived by neglecting the projector corrections, finding excellent agreement. Our chosen test system has the weakness that the centre of mass motion in a purely harmonic trap is not strictly amenable to the reservoir theory due to a violation of Kohn's theorem. However, our treatment is physically relevant for non-harmonic trapping, multicomponent systems, and other systems of interest that physically violate Kohn's theorem, provided a low-energy fraction is harmonically trapped. Indeed, since the thermal equilibrium properties involve small excursions from equilibrium, the confining potential is only required to be _locally_ harmonic near the trap minimum and any number of non-harmonic effects may intrude at larger distances. The centre of mass motion thus provides an excellent formal and numerical test of the SERs, being one of the simplest states of motion to handle analytically. \par We have shown that SERs can be used to obtain analytic equations that agree with numerical solutions of the full SPGPE and offer some physical insight into the open system dynamics. Future work will explore systems involving analytically tractable excitations such as vortex decay in hard-wall confinement [46], soliton [39] and phase-slip dynamics [40] in toroidal confinement, sympathetic cooling [41,42], spinor BECs [23], and quantum turbulence in non-harmonic confinement [44-47].”