Supersolid phase of a spin-orbit-coupled Bose-Einstein condensate: a perturbation approach

We thank both referees for their careful review of our manuscript, which has been revised to address their remarks. In addition we have made several minor improvements, including a short discussion of the hybridisation of the crystal and superfluid Goldstone modes in the stripe phase (accounted for by second-order corrections in $\hbar \Omega_R / 4 E_R$ to the mode wave functions). We hope that the improved version of our work meets the criteria of interest and importance required for publication in SciPost Physics.

Changes in response to remarks by Prof. Spielman:
- Below Eq. (8) we have briefly commented on the phase diagram for $g_{ss} < 0$;
- Below Eq. (9) we now state that $k_1$ is a parameter to be adjusted to find the ground state;
- In Sec. 3.1 we have enlarged the discussion on the characterisation of the $\Omega_R = 0$ states; to this purpose we have introduced the arbitrary phase $\chi_0$ in the formulas of this section (taken equal to $0$ in the calculations of the Appendices);
- In the paragraph below Eqs. (19)-(21) we have commented on supersolid effects in current ${}^{87}$Rb experiments;
- Below Eqs. (44) and (45) we have commented on the involved dependence of the sound velocity on the interaction parameters, and we have discussed the occurrence of a dynamic instability at the spinodal point;
- On page 17 we have improved the discussion on the spin Goldstone mode at $\Omega_R = 0$ (also recalled in the conclusion);
- We have rewritten Eq. (55) of the previous version eliminating the $q \to - q$ notation.

Changes in response to remarks by Referee 2:
- In the introduction we have added a comment on the stripe phase of two-dimensional dipolar gases;
- At the beginning of Sec. 2.2 we have added a comment on the quantum depletion in our system;
- In the paragraph below Eqs. (19)-(21) we have improved the discussion on the experimental relevance of our results;
- On page 21 we now mention the availability of Monte Carlo calculations in our system (Ref. [54]);
- Again on page 21 we have added a paragraph comparing our results for the superfluid density with those of the Leggett criterion, and added the corresponding curves in Fig. 4 (a short comment has also been added in the conclusion).

Additional changes:
- In the introduction we have quoted further experimental [17,18,27] and theoretical [32,33] results;
- Below Eq. (6) we now explicitly state that the stability conditions $g > 0$ and $g_{dd} > 0$ are assumed;
- In Eq. (26) we now take $\lambda$ complex to eliminate the arbitrary phase $\phi$ introduced in Eqs. (40) of the previous submission;
- In Sec. 4.5 we have restored some missing mass factors;
- On page 20 we now state that the superfluid density in our system is an anisotropic tensor and that Eq. (60) holds in the symmetric intraspecies coupling case;
- In App. B.4 we have added a discussion and a new figure on the hybridisation of the crystal and superfluid modes in the stripe phase; correspondingly, the paragraph below Eqs. (46) in the main text has been amended;
- A few typos in the previous submission have been fixed;
- References [19,37] of the previous submission have been updated.