Non-equilibrium evolution of Bose-Einstein condensate deformation in temporally controlled weak disorder

We are grateful to the Referee for recognizing the value of our work and suggesting its publication in the SciPost journal. We are also thankful for the given remarks, which refined our paper and made it more accessible to a wider audience. Below we address the specific changes, in a point-by-point manner:

1) In section 3, immediately after the equation (24), we mention potential technical and physical constraints concerning the disorder switch on time in the experimentally feasible cold-atom setups.

2) Higher than the second order terms for q(t) would, of course, assume the knowledge of \Psi_2(x,t), \Psi_3(x,t), etc. More concretely, the third order correction for q(t) involves \Psi_2(x,t) and \Psi_2^*(x,t). The perturbative procedure yields the equations for \Psi_2(x,t) and \Psi_2^*(x,t) of the same form as (11), but with the source terms of the second order which involve u(x) and \Psi_1(x,t). The corresponding solutions would contain nested temporal convolutions and, thus, would not lead to any practical analytic final result. Therefore, we did not pursue their definitive evaluation. However, one can state with certainty that higher than second order correlation functions of the disorder would become important. Based on the finite second order corrections to the equilibrium wave function available in the literature, we expect that at least the third order correction for q(t) would be finite.

3) This important point is discussed at the end of section 3.3, where it is shown that for the disorder to be weak it is necessary that its Larkin length is much larger than the condensate healing length.

We thank the Referee for a positive report, appreciation of our work and useful suggestions that considerably improved our manuscript and extended its range of applicability. Below we make a point-by-point reply to the specific Referee’s comments.

1) The correspondence of the equilibrium part to adiabatic switching on is now explicitly stated in the abstract.

2) We have now used the more appropriate wording: “after explicitly separating out the initial phase evolution exp(-i\mu_0 t/\hbar).”

3) The trapped case is now discussed in the new section 4. Therein, we demonstrate that the homogeneous time-dependent results can readily be extended to the case of inhomogeneous condensates in disordered traps, provided that the disorder correlation length is much smaller than the spatial extent of the atomic cloud. The backbone of such an extension is the local density approximation discussed in our previous publication New J. Phys. 22, 033021 (2020). A new figure 5 is added to stress that the homogeneous analysis can be quite valuable in reliably estimating the overall effect of weak disorder in the inhomogeneous trapped settings, provided that the appropriate quantities are properly matched.

4) We definitely agree with the Referee that a comparison with the Gross-Pitaevskii (GP) simulations would be very useful. Since there are nowadays freely available numerical codes for that purpose, it could be relatively easy to do so, at least from the coding side. However, here we would need full-fledged 3D time-dependent simulations, with the addition of the statistical sampling of the disorder distribution. According to Figure 1, the time propagation until stationarity in a single disorder realization would have to be many orders of magnitude larger than the characteristic mean-field time scale. Thus, the complete comparison would require considerable time and extensive computational resources. Moreover, the full GP simulations would open the door into the strong disorder regime, which is beyond the scope of the present paper. Therefore, we choose to leave this for future work.

In line with the aforementioned changes, we have adapted the text of the manuscript for consistency and readability. Also, we have added 4 new references [47], [48], [51], [52].