Controlling superfluid flows using dissipative impurities

This is very interesting research. At this point I have some questions about technical aspects of the modelling.

As a general comment, the motivation of an SGPE via the Glauber-P function, along the lines of the Stoof derivation, is, to me at least, quite unnatural. A consistent derivation of the Bose-gas reservoir interaction can be carried out in the truncated Wigner approximation, as developed in

[1] C. W. Gardiner and M. J. Davis, The Stochastic Gross–Pitaevskii Equation: II, Journal of Physics B: Atomic, Molecular and Optical Physics 36, 4731 (2003).

The theory is developed by introducing a formal energy cutoff to define the reservoir, an approach with a number of advantages. First, it is clear what is meant by the low energy "c-field" (everything below the cutoff). Second, the noise terms do not suffer any ultra-violet divergence. Third, and most interesting for the present work, there are two classes of interaction appearing in the resulting stochastic projected Gross-Pitaevskii equation (SPGPE).

1. Number-damping interactions in which particles are exchanged with the reservoir (incoherent region) until equilibrium is reached.

2. Energy-damping interactions in which only energy is exchanged with the reservoir. This interaction generates a term of potential type, and an associated (multiplicative) noise. There is a specific way that the potential depends on the state of the field (roughly speaking, the divergence of the c-field current), but the general form is similar to that presented in this work, but with an additional energy damping term. It is worth pointing out that this interaction doesn't appear in the Stoof SGPE, due to the specific choice of a very low cutoff energy (the condensate energy); this choice is linked to avoiding the issue of how to implement the cutoff in practice, a problem that is solved in [1].

Several later works have further developed the formalism, and carried out numerical simulations of the complete theory, e.g

S. J. Rooney, P. B. Blakie, and A. S. Bradley, Stochastic Projected Gross-Pitaevskii Equation, Physical Review A 86, 053634 (2012). A. S. Bradley, S. J. Rooney, and R. G. McDonald, Low-Dimensional Stochastic Projected Gross-Pitaevskii Equation, Physical Review A 92, 033631 (2015). R. G. McDonald, P. S. Barnett, F. Atayee, and A. Bradley, Dynamics of Hot Bose-Einstein Condensates: Stochastic Ehrenfest Relations for Number and Energy Damping, SciPost Physics 8, 029 (2020).

For a review, see P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, Dynamics and Statistical Mechanics of Ultra-Cold Bose Gases Using c-Field Techniques, Advances in Physics 57, 363 (2008).

So my questions are about the SGPE used in the modelling, and the nature of the point contact:

1. Given that the equation (1) isn't strictly the usual reservoir interaction SGPE, or SPGPE, I wonder if this is really the clearest way to motivate the use of Equation (1)?

2. It is clear that if the interaction is not coming from a reservoir interaction, but rather just a noisy potential, then damping would not occur. On the other hand, if the impurity is dissipative, as in the current work, then the damping and noise terms are essential and the relationship between them is important. Without damping it is not possible for the field to reach equilibrium with the impurity. What is the physical reason why there is no explicit damping term associated with the impurity noise?

3. Without a microscopic proof of the SGPE used to model the point contacts, the theoretical background reads as phenomenology. This is of course fine, but is there a fundamental reason why a derivation can't be carried out or is challenging?

4. Along similar lines, why is Stratonovich used over Ito? The motivation appears to be that the mean field equation has an associated loss term if one starts from Stratonovich. But if (1) is reinterpreted as Ito, then the mean field equation (3) would be lossless for any potential. Ito SGPE would seem a clearer choice on physical grounds for a noisy potential, namely that the potential and the field are uncorrelated in the interval dt. By choosing Stratonovich there is a correlation between the field and noise, and an effective loss term that doesn't come from a microscopic reservoir theory. So: how physical is the proposed noise?

5. Choosing a Stratonovich interpretation leads to the argument below Eq (3) for an effective loss describing "nothing else than the scattering of particles out of the condensate into excited modes of the Bose gas" However, if this is physically accurate it must be possible to capture such a loss using a number-damping theory, rather than the number-conserving interaction used in this work. My main question is whether Stratonovich SGPE is really the right choice for a noisy potential? It would be helpful to give some further discussion around the use of (1), and what the physical implications are. For example, in Ref. [53] the noisy defect is introduced as phenomenology, while in Ref. [54] a one-body loss term is presented as the result of a Born-Markov master equation. In the present manuscript one gets the impression that there are deeper arguments, but is not clear what they are. It would be helpful to the reader to either say it is phenomenology, or summarize the derivation, or at least the physical setup for the derivation.

6. In deriving (2) there is now an effective damping term in the frame of the impurity, and the system could eventually come into equilibrium in that frame. Is this to be expected physically? Or are there obvious grounds to discount such a final state due to the nature of the impurity?