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Equilibration of quantum cat states

by Tony Jin

Submission summary

As Contributors: Tony Jin
Arxiv Link: https://arxiv.org/abs/2003.04702v3 (pdf)
Date accepted: 2020-07-02
Date submitted: 2020-06-18
Submitted by: Jin, Tony
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Quantum Physics
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 back-up our analysis with numerical results obtained on the XXX spin chain with a random field along the z-axis 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.

Ontology / Topics

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Quantum spin chains Quantum states

Published as SciPost Phys. 9, 004 (2020)



Author comments upon resubmission

All comments are made in the list of changes.

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 blue-shaded 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 blue-shaded region in the top-right 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 operator-absolute 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 off-diagonal 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 off-diagonal 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 non-local fluctuations of the off-diagonal components in the steady-state that are fixed by the initial quantum coherences." The "off-diagonal components" of what?
The sentence
«we will see that such states present non trivial, possibly non-local fluctuations of the off-diagonal
components in the steady-state 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 non-local fluctuations of the off-diagonal components in the steady-state 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 non-degenerate. Then, $\mathbb{E}[\rho_{0}]$ is just the diagonal ensemble, i.e the density matrix in which one has set all the off-diagonal 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. Fully-degenerate spectrum corresponds in general to chaotic or non-integrable 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 steady-state 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 non-local, 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 non-locality, 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 off-diagonal 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 off-diagonal 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, few-body 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, few-body 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, few-body operators have thermal distributions in the thermodynamic”
limit.

8) Also on page 9 the author writes: "Typically, for finite-size integrable systems for example, one expects the existence of long-lived oscillations that prevents the system from equilibrating [30, 31]." I don't think this is what generally happens in finite-size 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 finite-size integrable systems.

I changed

“Typically, for finite-size integrable systems for example, one expects the existence of long-lived oscillations that prevents the system from equilibrating \cite{XYmodel,BlochBECrevival}.”

into

“For instance, for finite-size integrable systems, there might be long-lived oscillations that prevents the system from equilibrating \cite{XYmodel,BlochBECrevival}.”


Reports on this Submission

Anonymous Report 2 on 2020-6-30 Invited Report

Report

The author has addressed appropriately all my comments. I recommend publication of the manuscript in the present form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2020-6-21 Invited Report

  • Cite as: Anonymous, Report on arXiv:2003.04702v3, delivered 2020-06-21, 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:
"Fully-degenerate spectrum corresponds in general to chaotic or non-integrable systems..."
the author means:
"In general chaotic or non-integrable systems have a fully non-degenerate spectrum..."

I should add that this entire comment seems a bit off-track to me.

  • validity: high
  • significance: high
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

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