Density Resolved Wave Packet Spreading in Disordered Gross-Pitaevskii Lattices

*Reply to Referee 1

We thank the referee for the careful reading of the manuscript and the positive statement about our work.

1. "Unfortunately, the authors only present evidence for the existence of the strong chaos regime, but not a systematic study of the conditions where this regime occurs. It was argued in earlier publications that strong chaos should occur when the norm density exceeds the mean frequency spacing in the localisation volume. This could be checked numerically (a) by looking at different initial conditions and (b) by looking at the time when the strong chaos crosses over to the weak chaos which should inevitably occur during the wave packet expansion, but no systematic quantitative study is presented."

   Earlier publications ignored the impact of the energy density, hence focusing on a WC/SC condition involving norm density only. We show that the energy density ignoring is wrong, as are the conclusions based on that. As we explain in the manuscript, a clean WC/SC scenario can be expected only on the h=0 line. On a qualitative level, we find that the mean frequency spacing argument is valid. Deviating from the h=0 line inevitably leads the dynamics into self-trapping or Lifshits regimes at later times, which can be much larger than the times accessible in our computations. The wave packet dynamics in these regimes are discussed separately and show substantial deviations from the simplified SC/WC picture. We think that our results present a significant new level of understanding of the spreading problem.

2. "As far as the energy is concerned, no condition is given at all, even as a conjecture. For this reason, the paper seems to me more suitable for publication in Scipost Physics Core rather than in Scipost Physics."

   We do not agree with that statement. We investigate the self-trapping and Lifshits regimes with conditions h >0 and h<0 respectively.

   Keeping the same norm density and same disorder strength, we showed how the change in h(t=0) affects the spreading quantitatively in Fig. 5, and for the same disorder realization in Fig 2. We showed weak and strong chaos results for h=0 to make sure the wave packet does not enter self-trapping and Lifshits regimes during the spreading. However, the conditions to separate different regimes are complicated as mentioned above. We do not present an explicit distinction between different regimes, because 1) one has to know all conditions to separate them including a theoretically established relation between two densities, and 2) all the crossovers between different regimes do not show any clear-cut behavior to determine the separation precisely, instead they exhibit a quite slow transition process.

3. "Could the authors explain in more detail their reasons for choosing the method of minimising the Hamiltonian in Sec. 3? I always thought that the most standard way would be to use some gradient-based method. Also, I would naively expect the authors' method to be efficient in minimising local configurations, but not in equilibrating populations of two distant Lifshitz minima. I am not questioning the authors' results, which are correctly verified by the local chemical potential test, but I guess many readers (including myself) would appreciate some idea of why this method was chosen."

   To find the ground state, we chose MATLAB for convenience which offers to minimize functions of many variables with "fminsearch", based on the Nelder-Mead simplex algorithm. We obtained the ground state results (checked with mu) in a reasonable time, e.g. < 2 minutes for N=1000 ( 2 days for N=50000), by using minimization windows. For small system sizes, we compared our results to standard gradient method results and found very good agreement. We did not check how gradient methods will perform for larger system sizes. We added a comment on the efficiency of our method to Sec.3 in the manuscript.

4. "In Sec. 7, I am a bit surprised by the authors' statement that in the non-Gibbs regime the norm density in the spreading fraction cannot be determined and the regime cannot be characterised. Why is it impossible to determine the norm in the self-trapped fraction and subtract it?"

   The main reason is that we do not know how to identify the condensed fraction. This is already a problem for equilibrium dynamics. In our case, we are dealing with a non-equilibrium spreading process, which worsens the situation. Seemingly self-trapped fractions appear to rejoin the spreading fraction at later times, with new self-trapped fluctuations popping up. We added a clarification in the text.

5. "Also, the wave packet second moment growth is supposed to characterise the spreading fraction, since the self-trapped fraction is frozen, isn't it?"

   Asymptotically, yes. But note that the self-trapped part might not be entirely frozen, see above.

6. The exponent in Fig. 5(d) does not seem to saturate. Is it possible to go to longer times and see if it goes beyond 1/3 (maybe by choosing different initial conditions)?

   The self-trapping data in Fig. 5 are replaced with a slightly longer run of 100 realizations. The exponents are clearly below 1/3.

*Reply to Referee 2

We thank the reviewer for the thorough reading of the manuscript and encouraging comments.

1. "The paper is clearly written, and I have only one suggestion: place the "trajectories" on the plane (a,h) for the scenaria presented in figs 3-5 also on Fig 2. Or, if this is difficult, place at least the initial conditions for the runs at W=4 as markers on Fig. 2, this will help readers to arrange the simulations to the theoretical considerations."

   Thanks for the useful suggestion. We now added the spreading trajectories of weak and strong chaos in Fig.1 for the data shown in figures 3,4 with W=4. We also pinned the initial states of ST and LP from figures 2 and 5 to Fig. 1. Since the wave packet is sparse for ST and LP, and P is not increasing in time, their trajectory is unknown.

- Fig. 1 is updated with an inset, the caption is modified.
- In Sec.3, we added a comment on the efficiency of our method.
- In Sec.7, we extended our reasoning of not extracting the self-trapped norm.
- Fig.5 is updated with a longer run of the self-trapped curves, as requested by referee 1.