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Conservation of angular momentum in Bose-Einstein condensates requires many-body theory

by Kaspar Sakmann, Jörg Schmiedmayer

This is not the current version.

Submission summary

As Contributors: Kaspar Sakmann · Jörg Schmiedmayer
Arxiv Link: (pdf)
Date submitted: 2018-02-13 01:00
Submitted by: Sakmann, Kaspar
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Quantum Physics
Approach: Theoretical


Conservation of angular momentum is a fundamental symmetry in rotationally invariant quantum many-body systems. We show analytically that Gross-Pitaevskii mean-field dynamics violates this symmetry, provide its parametric dependence and quantify the degree of the violation numerically. The results are explained based on the time-dependent variational principle. Violations occur at any nonzero interaction strength and are substantial even when the depletion of the condensate and the interaction energy appear to be negligible. Furthermore, we show that angular momentum is only conserved on the full many-body level by providing according many-body simulations.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2018-4-26 Invited Report


1- the idea of comparing explicitly the GP approach with its multi-orbital extension for a relevant quantity as the angular momentum L_z and its fluctuations L_z^2


1 - the aim of the manuscript - as highlighted very strongly also in the title and abstract - is misleading to say the least.
Indeed the authors seem to infer that it is surprising that
the dynamics of the fluctuations of the angular momentum is not taken into account properly by GP approach to a BEC.

However the fact that GP or better mean field approaches cannot, in general, properly describe fluctuations (their static as well) is well known.

2 - (given 1-) only the dipole mode with L_z=0 is considered. And only the second order L_z^2. Further cases are needed.

3 - (given 1-) Comments on the use of Bogolyubov theory or linearised GP approach above the ground state in the linear regime to determine fluctuations - as done in literature - are completely absent.


From the title and the abstract the reader infer that the aim of the present manuscript is to show that the mean field Gross-Pitaevskii (GP) description of a Bose-Einstein condensate (BEC) is unable to properly take into account angular momentum conservation.

Reading the manuscript however it turns out that the author do not mean the conservation of the angular momentum <L_z> — which indeed the author show to be well account for within GP — but of the absence of conservation of the higher order momenta of the the angular momentum distribution. In particular they focus on the dynamics of <L_z^2>, i.e., the dynamics of the fluctuation of the angular momentum. They show how only going beyond GP — they use for their purpose a multi-orbital approach — it is possible to
obtain the proper dynamics, which in the case study correspond to no-evolution, i.e., conservation.

However the fact the GP approach is unable in general to properly describe fluctuations
is well known. It has nothing to do with BEC, but it is true
for any mean-field approach. It would be rather surprising the opposite.

Therefore (in the context of cold gases) no one would even use GP to describe not only the dynamics, but also the static value (as also the author find although without mentioning it, see Fig.3 right panel) of fluctuations.

For the above reason, in its present form, the manuscript does not deserve publication -- and I would even dare to say that it should no be published -- anywhere.

However I think the results on the comparison between GP (or single orbital) and a multi-orbital approach in describing static and dynamics of fluctuations are rather interested once put in the proper context.
Here below just a few points:
1. the fact that the error is not very large even in the dynamics is per se an interesting result (although pointing in the opposite direction of the author`s aim).
2. the fact that the time average of the fluctuations calculated within GP reproduces
the “exact” value of the fluctuations is interested. It is related to the use of the dynamics
of fluctuations above GP equations, for e.g. determine number fluctuations, angular momentum fluctuations, condensate depletion, …
3. The needed number of orbital to recover conservation of L_z^2 is also an interesting information, at least once a few other momenta (as L_z^3 and L_z^4) are analysed. How is the number of needed multi orbital growing? Does there exist any scaling or saturation?
4. How the multi orbital approach compare, especially in the linear regime, with other
more standard approaches to determine fluctuations, as, e.g., Bogolyubov approach?

Obviously my suggestion require a rather radical revision of the manuscript (including title and abstract) and a completely new point of view.
However I believe that it is worth the effort.

Requested changes

1 - completely change the emphasis of the presentation of the results
2 - higher order momenta analysis (scaling, saturation of the orbital...)
3 - comment on or at least mention other works (mainly in linear response) in which fluctuations have been calculated.

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

Author Kaspar Sakmann on 2018-12-13
(in reply to Report 2 on 2018-04-26)

added pdf of response to referee 2



Author Kaspar Sakmann on 2018-12-12
(in reply to Report 2 on 2018-04-26)
reply to objection

Anonymous Report 1 on 2018-4-25 Invited Report


1 Clear and pedagogical
2 Quantify in an interesting case the deviations from the exact result for the second momentum of L_z


1 Emphasis not clear, or misleading according me.
2 No parallel discussion of the behaviour of the expectation values of higher momenta of momentum p is provided.


The Authors study the conservation of angular momentum for weakly interacting Bose gases. After setting the formalism, they write a relation for the time evolution of the expectation values of L_z^n, showing that for n=2 one has that such time derivative is not zero. They then show that adding orbitals these deviations decrease.

My reaction to the paper is two-fold: from one side I think the paper is remarkably clear, even pedagogical, with a true effort to be self-contained. From the other side I do not agree with the emphasis of the paper. Let me explain better this point: when one does mean-field and a quantity is conserved (like the energy), one knows it may be different from the exact one. Nevertheless it may be a good approximation – however there are quantities that cannot be captured in mean-field, like correlations. So, there are quantities which are simply not exactly or badly captured by mean-field. The Authors shows that (d/dt) <L_z^n> is zero for n=1, and not zero for n=2. I conclude that the angular momentum is conserved, but that higher momenta of it are not. Actually I am even surprised that the error for n=2 is relatively small like the one shown in Fig.4. Let me come to the main point: suppose that one computes (d/dt) <p_z^n> and shows that it is zero for n=1, and not zero for n=2. One would entitle the paper “Conservation of momentum in Bose-Einstein condensates requires many-body theory”? Actually, my question is: do it is true that (d/dt) <p_z^n> is zero for n=1, and zero also for n larger that 2, differently from the case considered in the paper? Do p and L_z are different for the purposes of the paper? After all, translational invariance would require exact conservation of all momenta of p (when there is no external potential), and I do not see how the argument would be different. In other words the Authors could/should make similar computations and considerations for the momentum p and the reader would like to understand why the two cases may be different (if they are). I think that such a discussion would be useful for the clarity and the substance of the paper.

As a Referee, I think is completely up to the Authors to choose the title and the emphasis of their paper, but I am also concerned about the fact that the title would lead the reader to think that angular momentum is not conserved in mean-field, while it is. Are the (expectation values of the) higher momenta of L_z that are not conserved. So, when the Authors say “However, equations (37) and (39) are generally not zero and thus constitute explicit violations of the conservation of angular momentum in two- and three-dimensional GP theory”, I would say that “constitute explicit violations of the conservation of higher momenta (n \geq 2) of angular momentum”. If no discussion of the corresponding results for momentum p and no change of emphasis is done (including the title and the abstract), I do not think the paper should be published. For this reason I suggest to the Authors major revisions according the lines discussed in this report.

Requested changes

1 Provide a discussion of the corresponding results for the momentum p in the translational invariant case
2 Reconsider the emphasis of the presentation

  • validity: high
  • significance: ok
  • originality: good
  • clarity: top
  • formatting: excellent
  • grammar: good

Author Kaspar Sakmann on 2018-12-13
(in reply to Report 1 on 2018-04-25)

added pdf of response to referee 1.



Author Kaspar Sakmann on 2018-12-13
(in reply to Report 1 on 2018-04-25)
reply to objection

Login to report


Anonymous on 2018-05-15

1) That the mean-field equations most often possess solutions which break the symmetries of the many-body Hamiltonian is well known in the literature of chemistry, nuclear physics, and condensed matter; examples of such symmetries are the spin and angular-momentum symmetries.

2) The term "conservation of angular-momentum symmetry" usually refers to the fact that the angular-momentum operator ($L_z$ in two dimensions) commutes with the Hamiltonian. As a result, the values of $L_z$ are good quantum numbers (integers); they are not expectation values that can take continuously any real-number value.

3) What is not as widely known is the many-body theory that further restores the broken symmetries. In the context of condensed-matter and trapped ultracold atoms, this two-step symmetry breaking-symmetry restoration theory has been reviewed in "Symmetry breaking and quantum correlations in finite systems: studies of quantum dots and ultracold Bose gases and related nuclear and chemical methods," (2007) Rep. Prog. Phys. 70, 2067,

4) The fact that the Gross-Pitaevskii solutions describing vortices break the angular-momentum symmetry has been explicitly discussed in a rapid communication, "Symmetry-conserving vortex clusters in small rotating clouds of ultracold bosons," Phys. Rev. A 78, 011606(R) – Published 28 July 2008,

The beyond-mean-field step of restoring the broken angular-momentum symmetry was also explicitly presented.