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Equilibration of quantum cat states
by Tony Jin
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Tony Jin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2003.04702v3 (pdf) 
Date accepted:  20200702 
Date submitted:  20200618 02:00 
Submitted by:  Jin, Tony 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study the equilibration properties of isolated ergodic quantum systems initially prepared in a cat state, i.e a macroscopic quantum superposition of states. Our main result consists in showing that, even though decoherence is at work in the mean, there exists a remnant of the initial quantum coherences visible in the strength of the fluctuations of the steady state. We backup our analysis with numerical results obtained on the XXX spin chain with a random field along the zaxis in the ergodic regime and find good qualitative and quantitative agreement with the theory. We also present and discuss a framework where equilibrium quantities can be computed from general statistical ensembles without relying on microscopic details about the initial state, akin to the eigenstate thermalization hypothesis.
Author comments upon resubmission
List of changes
I thank both referees again for their very useful comments and criticisms.
Below I make a list of the changes made in the manuscript that address the requests formulated by the referees point by point.
Report 2
1  Discussion of numerical example is a bit imprecise (or sketchy). For example the author refers to "mean eigenvalue spacing" as the `mean ratio of consecutive level spacings'. Please be more precise, as the mean level spacing is an irrelevant quantity.
Thank you for this precision, this has been corrected.
2  It is not clear in Figure 2 what is meant by error bars (mentioned in the main text)? I guess there is no "disorder averaging" in numerics, but what is the meaning of the blueshaded region in the top right panel?
I changed «Details on how these values are obtained and the meaning of the error
bars are given in app.C»
to «Details on how these values and the confidence intervals are obtained are given in app C»
I added a sentence in the caption
«The blueshaded region in the topright panel is the consequence of oscillations occurring on a much shorter time scale.»
I do not comment this shorter time scale further because the theory only says something about the amplitude of these fluctuations and not on their dynamical behavior (though it would be something interesting to invest).
3  The meaning of operatorabsolute value (Q) in the last display equation at the end of page 6 is not explained.
Q is not the operator, it’s its expectation value (operators are denoted with an hat). Q is just the absolute value of Q.
4  In discussion of ETH is section 4 it is not clear why ETH  which is a statement about the distribution of matrix elements of local observables  should depend on the initial state (of course it does through \delta E, in particular when one would want to apply it to cat states where no longer \delta E << E, but I guess this should be explicitly stated).
Yes, this is exactly the point.
In the discussion, I modified the paragraph
«The important remark is that for the situations we looked at in the
paper, this has no reason to be true anymore, essentially because
we don't expect the offdiagonal correlations to be exponentially
suppressed as there is no notion of ''narrow window'' around a given
energy for the cat states considered.»
into
«In the ETH, the role of the initial state is restricted to fixing the energy scales $E$ and $\Delta E$. The important remark is that ,for the cat states, there is no notion of ''narrow window'' around a given energy anymore, hence we don't expect the ETH to apply. One illustration of that is the fact that offdiagonal correlations are not exponentially suppressed for cat states.»
Report 1
1) On page 2, I do not understand this sentence: "We will see that such states present non trivial, possibly nonlocal fluctuations of the offdiagonal components in the steadystate that are fixed by the initial quantum coherences." The "offdiagonal components" of what?
The sentence
«we will see that such states present non trivial, possibly nonlocal fluctuations of the offdiagonal
components in the steadystate that are fixed by the initial quantum coherences.»
has been modified to
«Working in the Hamiltonian basis, we will see that such states present non trivial, possibly nonlocal fluctuations of the offdiagonal components in the steadystate that are fixed by the initial quantum coherences.»
2) On page 4, what is the "diagonal ensemble"? I also do not understand why the case in which the Hamiltonian is the identity operator ("one energy sector that is the whole Hilbert space") is called the "usual microcanonical ensemble." If the Hamiltonian is the identity operator there is no dynamics so I do not see the point of highlighting that "extreme case".
I modified the whole paragraph to make the motivations more explicit :
«Before going on, let's notice two extreme cases of interest : the first case is when all the energy levels are nondegenerate. Then, $\mathbb{E}[\rho_{0}]$ is just the diagonal ensemble, i.e the density matrix in which one has set all the offdiagonal components to zero. The second case is when there is only one energy sector that is the whole Hilbert space itself. We then have $\mathbb{E}[\rho_{0}]=\frac{1}{d}\mathbb{I}$. In this case, the density matrix tells us that all states with the same energy $E$ have the same probability weight, i.e it is the microcanonical ensemble. Fullydegenerate spectrum corresponds in general to chaotic or nonintegrable systems, so one should expect that the diagonal ensemble describes accurately the steady state of such systems \cite{reviewthermalization}. However, in practice, we know that equilibrium states of isolated system are accurately described by the microcanonical ensemble which corresponds to the steadystate of a fully degenerate spectrum. To go from the first ensemble to the second is not a trivial task which requires additional assumptions. We will discuss this point in more details in the section \ref{Discussion}.»
3) Throughout the manuscript instead of the word "moment" the author incorrectly uses the word "momenta". The first place where I identified this was in the first line on page 5.
This has been corrected
4) I suggest the author to number all the equations. Below I have a comment about an equation that is not numbered.
This has been done.
5) I would not call the state for protocol II (unnumbered equation) a "classically mixed state", and I would not call ^Q an observable. ^Q is a highly many body operator. Has the author found similar signatures to the ones in the right panels in figure 2 in a local operator?
I added a few sentences to discuss this :
After introducing \hat{Q}
“Note that $Q$ is nonlocal, in the sense that it has non zero support on the whole physical space.”
At the end of section 3
“Let us add a remark here. In general because of its high degree of nonlocality, it is not expected that $Q$ might be a suitable observable for experimental measurements. But similar qualitative statements about fluctuations should apply for any observables that couple the different energy sectors. For instance, as suggested at the end of \cite{GullansHuse}, one could imagine doing an interference experiment between two parts of the system far part and look at the fluctuations of the pattern.”
6) On page 8, what the author describes as the ETH is not what I understand as the ETH. I understand the ETH as a statement about matrix elements of observables that has nothing to do with the initial states. In that sense the discussion in the last paragraph of page 8 does not appear to be an equivalent statement to the ETH but just the ETH applied to the states the author is studying. I should add that ^Q does not look to me like an operator that would fulfill the ETH.
This meets points 3) of referee 2
In the discussion, I modified the paragraph
«The important remark is that for the situations we looked at in the
paper, this has no reason to be true anymore, essentially because
we don't expect the offdiagonal correlations to be exponentially
suppressed as there is no notion of ''narrow window'' around a given
energy for the cat states considered.»
into
«In the ETH, the role of the initial state is restricted to fixing the energy scales $E$ and $\Delta E$. The important remark is that for the cat states, there is no notion of ''narrow window'' around a given energy anymore, hence we don't expect the ETH to apply. One illustration of that is the fact that offdiagonal correlations are not exponentially suppressed for cat states.»
7) On page 9, the author writes "Typicality states that for all eigenstates of the energy window, fewbody operators have thermal distributions in the thermodynamic limit." This is not what I understand as typicality, which I don't think says anything about eigenstates but rather about pure states that are random superpositions of eigenstates in the energy window.
I changed
“Typicality states that for all eigenstates of the energy window, fewbody operators have thermal distributions in the thermodynamic limit.”
into
“Typicality states that for all pure states that are random superpositions of eigenstates of the energy window, fewbody operators have thermal distributions in the thermodynamic”
limit.
8) Also on page 9 the author writes: "Typically, for finitesize integrable systems for example, one expects the existence of longlived oscillations that prevents the system from equilibrating [30, 31]." I don't think this is what generally happens in finitesize integrable systems, which I believe equilibrate so long as they are not too small or the initial state is not too special. There is a long literature on this involving the generalized Gibbs ensemble in finitesize integrable systems.
I changed
“Typically, for finitesize integrable systems for example, one expects the existence of longlived oscillations that prevents the system from equilibrating \cite{XYmodel,BlochBECrevival}.”
into
“For instance, for finitesize integrable systems, there might be longlived oscillations that prevents the system from equilibrating \cite{XYmodel,BlochBECrevival}.”
Published as SciPost Phys. 9, 004 (2020)
Reports on this Submission
Anonymous Report 1 on 2020621 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2003.04702v3, delivered 20200621, doi: 10.21468/SciPost.Report.1776
Report
The author addressed most of my points. The ones not addressed may be considered a matter of interpretation so I recommend publication.
I think by:
"Fullydegenerate spectrum corresponds in general to chaotic or nonintegrable systems..."
the author means:
"In general chaotic or nonintegrable systems have a fully nondegenerate spectrum..."
I should add that this entire comment seems a bit offtrack to me.