Time evolution during and after finite-time quantum quenches in the transverse-field Ising chain

Dear Editors,

we thank you for forwarding us the referee reports. We are very happy about the positive reports and thank the referees for their constructive suggestions. We have addressed all suggestions and revised the manuscript accordingly.

Please do not hesitate to contact us if you have further suggestions or questions.

Sincerely yours Tatjana Puskarov Dirk Schuricht —————————————— Reply to referee 1

We thank the referee for his/her careful reading and thoughtful report. We have made some changes according to the suggestions. Below we reply to them in detail.

1) We agree with the impression of the referee. We have revised the 1st paragraph of the introduction to improve the motivation. 2) Unfortunately we do not have a good understanding of the differences between protocols with and without kinks. We have revised the text in Sec. 5.1 to clarify our understanding. 3) It should be noted that the quasiparticles will, at least in the quasi-classical picture underlying the discussion, propagate with the instantaneous velocity v(k,t). Thus particles with the same momentum k will possess identical velocities and hence cannot overtake each other. Of course, it may be possible to devise a protocol to create quasiparticles at a specific momentum and time such that these quasiparticles may overtake other quasiparticles created earlier. However, this is not expected for the generic protocols not considered in our manuscript. In any case, our estimate for t_F should only be understood as a rough one. 4.i) We have indicated the times when the QCP is crossed in the revised figure.
4.ii) In Fig. 10(b) we show the Loschmidt echo for different quench protocols while after Eq. (71) we discuss the findings in the text.

—————————————— Reply to referee 2

We thank the referee for his/her careful reading and thoughtful report. We have made some changes according to the suggestions. Below we reply to them in detail.

Concerning the 2. weakness we would like to point out that the different protocols lead to different behaviour in some quantities like the dependence of the total energy on the quench duration, but that one reason to consider the different protocols is also to show that certain features are independent of the protocol and thus show a certain degree of universality. We have revised the 3rd paragraph of the introduction to stress this aspect.

1) Unfortunately we do not have a good understanding of the behaviour of the total energy as function of the quench duration. In fact, our text in Sec. 5.1 is misleading insofar that also for quenches across the QCP a simple stationary phase approximation cannot be applied, since the dependence on the quench time is rather implicit. We have revised the text in Sec. 5.1 to clarify our understanding. 2) We thank the referee for his/her remark. However, we do not fully understand the argument, since also the k-dependence of the coefficients in the transformation between the post- and pre-quench modes will enter the late-time behaviour. 3) We thank the referee for spotting these typos. We have corrected them. 4) Unfortunately we do not have a good physical picture for the meaning of the critical time t^*. 5) We thank the referee for pointing this out. In order to avoid the introduction of the symmetrised derivative we decided to keep the presentation in its current form. 6) We have revised the conclusion to stress again the motivation for considering different quench protocols.

Summary of changes

-Revised 1st and 3rd paragraph of the introduction.
-Added Ref. 49.
-We have revised the text in Sec. 5.1 to clarify our understanding of the behaviour of the total energy as a function of the quench time.
-Clarified at the end of Sec. 5.2 that all qualitative features of the magnetisation are independent of the finite quench duration.
-Clarified that our estimate for the time t_F is only a rough one.
-Added Ref. 58.
-Added a remark on the experimental observation of dynamical phase transitions as well as Ref. 62.
-We corrected the subindices n->m on page 20.
-Replaced Fig. 11(a) to include the times when the QCP is crossed. Revised the caption accordingly.
-Slightly revised the conclusion as suggested by referee 2.