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| As Contributors: | Kaspar Sakmann · Jörg Schmiedmayer |
| Arxiv Link: | https://arxiv.org/abs/1802.03746v2 |
| Date submitted: | 2018-12-17 |
| Submitted by: | Sakmann, Kaspar |
| Submitted to: | SciPost Physics |
| Domain(s): | Theoretical |
| Subject area: | Quantum Physics |
We explain from first principles why satisfying conservation laws in Bose Einstein condensate dynamics requires many-body theory. For the Gross-Pitaevskii mean-field we show analytically and numerically that conservation laws are violated. We provide examples for angular momentum and linear momentum conservation. Arbitrarily large violations occur despite negligible depletion and interaction energy. For the case of angular momentum we show through extensive many-body simulations how the conservation law can be gradually restored on the many-body level. Implications are discussed.
We have answered all questions of the referees. There appears to have been a misunderstanding in regards to what a conservation law constitutes. We show that inconsistencies would arise from making d/dt<A>=0 the condition for the conservation of observable A instead of the well-known condition [H,A]=0. We provide the relevant context, put our work in the historical context and show that not only angular momentum, but also linear momentum conservation is violated by the GP mean-field dynamics.
* introduction: discussion of the history of mean-field violations of conservation laws since 1963
* section 2: instead of specializing to just angular momentum we have generalized the theoretical treatment to arbitrary conservation laws
* a new section 2.4: discussion of the full distribution function and its relation to the moment problem
* a new section 2.5: discussion of a Gedankenexperiment using a single particle to show explicitly that d/dt<A> is not a sufficient condition for the conservation of an observable A. Discussion of the inconsistencies that arise if [H,A]=0 is replaced by the weaker condition d/dt<A>=0.
* a new Fig. 1 to illustrate the Gedankenexperiment in section 2.5 for a specific example.
* a new Figure 2 that demonstrates that arbitrarily large violations of angular momentum conservation are possible in GP mean-field dynamics at fixed interaction strength depending on the parameters of the initial state.
* a new section 8 discussing the violation of linear momentum conservation by the GP mean-field dynamics using stronger interaction, including analytical calculations, analogous to the case of angular momentum.
* a new Figure 7 demonstrating the violation of linear momentum conservation by the GP mean-field in a numerical example (a violation of ~600%).
* a new Appendix A containing an alternative proof for the fact that conserved observables have stationary moments, d/dt<A^n>=0 for all n. This proof is taken from the textbook "Quantum Mechanics" by Cohen-Tannoudji, Diu and Laloe.
* a new Appendix B containing analytical results for the third moments d/dt<L_z^3> and d/dt<P^3> in GP mean-field. These are nonzero, just like the second moments.
* a new Appendix C demonstrating that kinietic energy is not conserved for an interacting system even though d/dt<P^2>=0 in the exact dynamics.
1- The results on how much the GP fails in describing the dynamics some higher momenta of L_z and now also for P, and the possibility, with not that many orbitals, to heal at least partially the problem (but only for L_z).
2 - The new version - with the added material -better explains the point of the authors.
3 - In the new version the readability has been improved and it is written in a pedagogical way.
As for the original version the weakness of the manuscript resides in the claim, in the emphasis the authors put on their results, namely that it is surprising that the GP mean-field equation does not conserve the symmetries of the many-body system.
As already pointed out in previous referee's reports and comments the fact that mean-field approaches fail in describing correlations - and therefore the particular instance considered by the authors of higher momenta of observable - is well known.
The weakness is even reinforced in the new version. The authors instead of amending their claim, they put even more emphasis on it.
In the first round the referees have pointed out that it is well known that mean field approaches are not expected to properly describe correlations (also dynamically) and in particular higher momenta of observables. Therefore in general the symmetries of the microscopic Hamiltonian are not satisfied.
Although the authors have not appreciated it, both referees mentioned that the failure of the GPE of properly describing the evolution of <L_z> would have been very surprisingly. Not, however, the failure in describing higher momenta of L_z or any other operator.
In the new version the authors instead of amending the original claim, put all the emphasis - also in the reply to the referee's reports - on explaining that many body theory is needed to preserve/recover the symmetries of the microscopic Hamiltonian.
In this respect let me also notice a small logical loophole in the reply of the authors.
As the author mention it is well known (appendix A is rather useless indeed) that conservation laws related to a generator A implies the d/dt<A^n>=0 for any n. If the latter is not satisfied the global symmetry is not preserved by the approximation used. The author in particular use the fact that for n=2 GP shows some evolution and they claim this effect being surprising and therefore the need to go beyond it.
On the other hand they also write in the reply that:
" [..] it is very well known that fluctuations are not accurately described by the GP mean-field or any other mean-field. However, fluctuations as such are not the topic of this work."
Now fluctuations correspond to the case n=2 and therefore it is also not surprising that GP is not symmetry conserving.
As for the previous round I cannot recommend the publication of the manuscript in its present form,
since the authors' aim is to convey the massage that they have discover that GP mean-field equations do not properly describe higher momenta of observables (and therefore for the observables generators of a symmetry of the Hamiltonian, GP does not preserve the symmetry of it). What it is interesting is to quantify the failure of GP for some quantities and how "hard" is to cure it.
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Concerning the (added) results:
1. The results on <P^2> and the emphasis (in the reply) put by the authors on the deviation of the GP result even when the depletion is negligible, is almost not new.
It has being discuss in the context of uncertainty in ref [38], where the authors offer also an explanation of why the large deviation in the time evolution occurs within a GP approach.
2. Sec 2.5 should be an Appendix if they want to keep it.
3. Appendix A is textbook.
As a final comment let me stress that I do recognize clear merits of the manuscript: it is well written (forgetting the parts concerning their major claim) and with a pedagogical aim; they use some quite general examples to quantify, at least to some extent, the failure of the GP in describing the evolution of n=2 (and n=3) momenta and show how including more orbitals the situation is improved.
I find their results together with the ones of Ref [38] and of the not-cited paper by Klaiman and Cederbaum [PHYSICAL REVIEW A
94, 063648 (2016)] (which in my opinion has to be cited) useful to get a better understanding on the GP mean-field equations.
The paper improved in the resubmitted version, and in particular:
1 - an historic discussion and perspective has been added, clarifying the point raised by the authors;
2 - a discussion of the momentum conservation has been added, again clarifying the point raised by the authors.
In my opinion, the main weakness of the previous version remain unaltered, actually it further increased: the authors put all the emphasis on the violation of the conservation laws by performing mean-field approximations, while it is well known that such approximations do not reproduce correlation functions and higher momenta.
I read with interest both the revised version, and the detailed reply of the authors. To me, the main point of interest of the paper is the quantification of the errors committed by performing the mean-field, both at equilibrium and during the dynamics. However, in the present version the emphasis is almost entirely given to the need of inserting the many-body theory to restore the symmetry. I think the point is far from being unexpected: consider the Ising model in a transverse field (in one or more dimensions, so independently from the fact that is integrable/solvable or not), and then perform mean-field. Everybody would agree that the values for the ground-state energy and the other observables are affected by some error, that errors are committed during the dynamics, and that the higher momenta or the correlations are not well reproduced. The interesting point is to quantify such deviations, and to demonstrate that by a series of controllable approximation one can reduce them - but not that it is needed the full theory to correctly conserve all higher-order expectation values. So the paper is convincing in points like figure 6, where the convergence is studied - not convincing when it is showing in several ways something expected such that there is a failure in the determination of expectation values of observables like A^n.
Finally, despite I agree that mean-field approaches does not reproduce higher correlation functions, a simple example can show that in mean-field expectation values of higher power of observables are conserved. The example is the following: suppose that one solve the time-independent GPE as
H_GPE \psi_n(r)=\mu_n \psi_n(r). It is obvious that choosing as initial condition \psi(r,t=0)=\psi_n(r), then \psi(r,t)=e^{-i \mu_n t/\hbar} \psi_n(r). Compute now the mean-field expectation value of p^{2n}, defined <p^{2n}>_MF=\int dr \psi(r,t)*\ast p^{2m} p^{2n} \psi(r,t). It is clear that <p^{2n}>_MF does not depend on time. No contradiction with the previous statements is present, since <...>_MF is different from <...>_TRUE. This simple example confirm that what is truly interesting, and partly done in this paper, is to quantify the deviation of expectation values in mean-field from the exact ones.
So, from one side it is completely on the author'side to choose the emphasis they want to give. However, from my referee's side, I conclude that with a misdirected emphasis, the readability and the quality of the paper is affected, so that I cannot suggest publication of the paper in the present version.