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Equilibration of quantum cat states
by Tony Jin
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|As Contributors:||Tony Jin|
|Arxiv Link:||https://arxiv.org/abs/2003.04702v2 (pdf)|
|Date submitted:||2020-04-10 02:00|
|Submitted by:||Jin, Tony|
|Submitted to:||SciPost Physics|
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.
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Anonymous Report 2 on 2020-6-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2003.04702v2, delivered 2020-06-01, doi: 10.21468/SciPost.Report.1727
1 - Rather fresh and original approach to a very timely problem
2 - The main statements and definitions concisely formulated
3 - Presentation has a very good balance between heuristic presentation and formal derivations and proofs (mainly in appendices)
1 - Presentation of numerics is a bit "too sketchy" and imprecise
2 - Text needs a bit more of (language) polishing
This is a very nice paper and I strongly recommend its publication. It presents an approach to quantum ergodicity from the viewpoint of fluctuations of physical observables. After a careful thought, the result may not be very surprising, but it is very nicely and clearly formulated. First of all, even the standard definition of quantum ergodicity is given in very general and flexible formulation, allowing for any number of conserved quantities.
The author then generalizes the ergodic-time-aveaging projector to fluctuating observables which is formulated in a tensor product Hilbert space
of n-copies of the original system. Not only the result for the second moment (n=2) is compactly written down, also the general result for any moment order n is derived in appendix. What is remarkable is, that this time-averaging projectors in higher tensor-product spaces are never really ergodic (even for a nicely ergodic system) but they generically keep memory of the initial state. This is demonstrated in a numerical example of XXX chain in a random field, starting from initial cat states.
I particularly find interesting the discussion at the end in relation to integrable systems. Specifically, the presentation as it is formulated refers to systems with finite Hilbert spaces, where one can formulate ergodicity through spectral decomposition. It would be interesting to think of generalizing this to truly extended (infinite) quantum systems, where locality or non-locality of conserved operators would become important.
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.
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?
3 - The meaning of operator-absolute value (|Q|) in the last display equation at the end of page 6 is not explained.
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).
Anonymous Report 1 on 2020-5-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2003.04702v2, delivered 2020-05-16, doi: 10.21468/SciPost.Report.1690
Jin's work addresses an interesting question, how can one identify cat states after a system has equilibrated? He shows that the time fluctuations in the steady state distinguish cat states from other more traditional states.
The results reported in the paper look technically correct but I think the presentation can be improved. I found some statements that may not be correct. Below I point out some of my concerns and questions in the order they appear in the manuscript. They are not ranked by importance.
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?
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".
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.
4) I suggest the author to number all the equations. Below I have a comment about an equation that is not numbered.
5) I would not call the state for protocol II (unnumbered equation) a "classically mixed state", and I would not call $\hat Q$ an observable. $\hat Q$ 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?
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 $\hat Q$ does not look to me like an operator that would fulfill the ETH.
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.
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.