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Conservation of angular momentum in BoseEinstein condensates requires manybody theory
by Kaspar Sakmann, Jörg Schmiedmayer
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Submission summary
Authors (as registered SciPost users):  Kaspar Sakmann · Jörg Schmiedmayer 
Submission information  

Preprint Link:  http://arxiv.org/abs/1802.03746v1 (pdf) 
Date submitted:  20180213 01:00 
Submitted by:  Sakmann, Kaspar 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Conservation of angular momentum is a fundamental symmetry in rotationally invariant quantum manybody systems. We show analytically that GrossPitaevskii meanfield dynamics violates this symmetry, provide its parametric dependence and quantify the degree of the violation numerically. The results are explained based on the timedependent 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 manybody level by providing according manybody simulations.
Current status:
Reports on this Submission
Strengths
1 the idea of comparing explicitly the GP approach with its multiorbital extension for a relevant quantity as the angular momentum L_z and its fluctuations L_z^2
Weaknesses
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.
Report
From the title and the abstract the reader infer that the aim of the present manuscript is to show that the mean field GrossPitaevskii (GP) description of a BoseEinstein 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 multiorbital approach — it is possible to
obtain the proper dynamics, which in the case study correspond to noevolution, 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 meanfield 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 multiorbital 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.
Author: Kaspar Sakmann on 20181213 [id 385]
(in reply to Report 2 on 20180426)added pdf of response to referee 2
Attachment:
Author: Kaspar Sakmann on 20181212 [id 374]
(in reply to Report 2 on 20180426)
Referee 2 writes: ''1  the aim of the manuscript  as highlighted very strongly also in the title and abstract  is misleading to say the least \dots From the title and the abstract the reader infer that the aim of the present manuscript is to show that the mean field GrossPitaevskii (GP) description of a BoseEinstein 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 $\langle L_z \rangle$  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.''
Answer: There appears to be a misunderstanding: for the timeindependent case meanfield violations of conservation laws have been known for a very long time. We show the same is true for the timedependent case, explain the origin rigorously and provide quantitative predicitions. Referee 2 states that angular momentum is conserved by the GP meanfield dynamics because the expectation value $\langle L_z \rangle$ is timeindependent. This is not correct. In quantum mechanics $\frac{d}{dt}\langle L_z \rangle=0$ is only a necessary, but not a sufficient condition for the conservation of angular momentum. Let us explain.

''... the author do not mean the conservation of the angular momentum $\langle L_z \rangle$ ....'' Observables in quantum mechanics are represented by hermitian operators. Angular momentum is represented by the operator $L_z$, not by its expectation value $\langle L_z\rangle$.

The condition for an observable $A$ to be conserved is $[H,A]=0$.

$\frac{d}{dt}\langle L_z\rangle=0$ follows from $[H,L_z]=0$. It is a necessary, but not a sufficient condition for the conservation of angular momentum. See points 4, 5, 6.

It is incorrect to conclude that an observable $A$ is conserved solely on the basis that $\frac{d}{dt}\langle A\rangle=0$. To see this consider a single particle at rest in free space. The Hamiltonian is $H=p^2/2$. The observable $x$ is {\it not} conserved: $[H,x]\neq0$. Nevertheless $\frac{d}{dt}\langle x\rangle=0$. Please see section 2.5 in the revised version for more details.

$[H,L_z]=0$ implies $\frac{d}{dt}\langle L_z^n\rangle=0$ for all $n$, as shown in Eq. (9). Another proof, taken from a textbook, can be found in the Appendix of the revised manuscript. The probability distribution of measurements of $L_z$ is only stationary if $\frac{d}{dt}\langle L_z^n\rangle=0$ for all $n$. This is proven in Eq. (18). Also this proof is taken from a textbook. Any $\frac{d}{dt}\langle L_z^n\rangle\neq 0$ is a violation of $[H,L_z]=0$ and is experimentally detectable by measuring $L_z$, see section 2.5 and the new Fig. 1.

Ground state meanfield solutions have been known to violate conservation laws since 1963 for HartreeFock and at least since 1975 for GP meanfield. Methods for restoring these symmetries have been textbook material since at least 1980. There is nothing ''misleading'' about stating that ''the mean field GrossPitaevskii (GP) description of a BoseEinstein condensate (BEC) is unable to properly take into account angular momentum conservation''. It is a wellestablished fact. We only provide new results for the timedependent case. See the newly rewritten introduction for details.

We provide an explanation why satisfying conservation laws in the dynamics is only possible on the manybody level, see section 4.3 and 4.4, as well as parametric dependencies of the violations and how to fix the problem. None of this has been published before.
Referee 2 writes: ''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 meanfield approach. It would be rather surprising the opposite.''
Answer: Referee 2 is right that it is very well known that fluctuations are not accurately described by the GP meanfield or any other meanfield. However, fluctuations as such are not the topic of this work. The question we answer is what kind of approximations are capable of satisfying conservation laws such as angular and linear momentum conservation. It turns out that conservation laws can only be satisfied on the manybody level. This is not a priori clear. But we go far beyond this. Specifically, our findings are that angular momentum conservation is not only violated in the stationary case as has long been known, but that angular momentum conservation is violated by the GP timedynamics. We provide quantitative predictions, including parametric dependencies for specific examples in a parameter regime, where one would expect the GP meanfield to provide very accurate predictions: the depletion of the condensate is less than $5\times 10^{4}$. But most importantly we provide an explanation of this violation based on the variational principle. We even go further by gradually restoring the symmetry through extensive manybody simulations. None of the above has been published anywhere and it is very surprising that restoring this fundamental symmetry in the timedynamics takes a tremendous computational effort. Please also see the new example we provide to demonstrate a violation of linear momentum conservation in section 8.
Referee 2 writes: ''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.''
Answer: 1. A single counter example is enough to prove a hypothesis wrong. We have provided such a counter example: GP theory violates angular momentum conservation. There is no need for further cases. Nevertheless, we have included another case, see the new section 8.

We selected an analytically and numerically tractable case, to provide the parametric dependencies. This is why we chose the dipole mode.

Referee 2 asks for another example. As is clear from the theory we provide in section 4, angular momentum is nothing special. We therefore decided to provide an additional example that shows that the GP meanfield also violates momentum conservation, $[H,P]=0$. Please see the new section 8.

Referee 2 asks for higher moments. While no higher orders are needed to prove the violations we went the extra mile for referee 2 and evaluated $\langle L_z^3\rangle_{GP}$ and $\langle P^3\rangle_{GP}$ in Appendix B. As expected they are timedependent as well.
Referee 2 writes: ''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.''
Answer: The referee is right that we did not use Bogolyubov theory. We have explained our findings based on the variational principle, i.e. the most fundamental level of explanation possible, see section 4. Unlike Bogolyubov theory or the linearized GP approach our approach does not rely on the assumption of a small depletion and linearization around a meanfield state. We use the full equations of motion. Obviously rigorous results are always preferable over approximate treatments when it is possible to obtain them. Thereby, we have excluded the possibility for loopholes. Had we only relied on linearized versions of the equations of motion, it could not be excluded that a violation of angular momentum conservation is merely a consequence of the linearization approximation.
Referee 2 writes: ''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).''
Answer: 1. $\langle L_z\rangle=0$ is exactly constant in the dynamics. Even in the numerics at a level of the numerical precision $<10^{8}$. In contrast, $\langle L_z^2\rangle$ varies over $\pm 13\%$, a difference of at least seven(!) orders of magnitude.

As stated in the abstract, we chose very weak interaction strength and practically no depletion ($<5\times 10^{4}$) to make the conditions as favorable as possible for the GP meanfield. Nevertheless, angular momentum conservation is heavily violated.

Arbitrarily strong violations: $\frac{d}{dt}\langle L_z^2\rangle$ can take on any value between $\infty$ and $\infty$ depending on the values chosen for $x_0, \sigma_0, p_0$, see Eq. (50). The violation grows linearly with the initial displacement from the center of the trap $x_0$, linearly with the initial momentum $p_0$, linearly with the GP nonlinearity parameter $\lambda$ and so on. It is no problem at all to find larger violations. To illustrate this fact we have included a new Fig. 2 to show these parametric dependencies.

In section 8 we now provide an example for the violation of linear momentum conservation where $\langle P^2\rangle$ grows quickly by about 600% instead of staying constant.

Note: many experiments work in the opposite limit where the kinetic energy is much smaller than the interaction energy. The violation grows with the interaction strength, see Eq. (50).
Conclusion: Referee 2 claims (incorrectly) that angular momentum is conserved by the GP meanfield dynamics, because $\frac{d}{dt}\langle L_z\rangle=0$. However, as we have shown, this is not enough in quantum mechanics: all moments $\langle L_z^n\rangle$ need to be timeindependent. Meanfield violations of conservation laws have been known for many decades for stationary states. Here, we treat the timedependent case. We explain from first principles why conservation laws can only be satisfied on the manybody level in the dynamics. In order to avoid further confusion we have included all necessary background material that illustrates these points. Following referee 2's requests we included another example, showing the violation of momentum conservation by the GP meanfield and provide results for higher order momenta.
Strengths
1 Clear and pedagogical
2 Quantify in an interesting case the deviations from the exact result for the second momentum of L_z
Weaknesses
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.
Report
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 twofold: from one side I think the paper is remarkably clear, even pedagogical, with a true effort to be selfcontained. From the other side I do not agree with the emphasis of the paper. Let me explain better this point: when one does meanfield 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 meanfield, like correlations. So, there are quantities which are simply not exactly or badly captured by meanfield. 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 BoseEinstein condensates requires manybody 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 meanfield, 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 threedimensional 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
Author: Kaspar Sakmann on 20181213 [id 383]
(in reply to Report 1 on 20180425)added pdf of response to referee 1.
Attachment:
Author: Kaspar Sakmann on 20181213 [id 382]
(in reply to Report 1 on 20180425)
Referee 1 writes: ''1  Emphasis not clear, or misleading according me. ... The Authors shows that $\frac{d}{dt}\langle L_z^n\rangle$ 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.''
Answer: There appears to be a misunderstanding: for the timeindepdendent case meanfield violations of conservation laws have been known for a long time. We show the same is true for the timedependent case, explain the origin rigorously and provide quantitative predicitions. Referee 1's conclusion that angular momentum is conserved if $\frac{d}{dt}\langle L_z\rangle=0$, but $\frac{d}{dt}\langle L_z^n\rangle\neq0$ for $n>1$, is not correct in quantum mechanics. Let us explain.

Observables in quantum mechanics are represented by hermitian operators. Angular momentum is the operator $L_z$, it is not the expectation value $\langle L_z\rangle$.

The condition for an observable $A$ to be conserved is $[H,A]=0$.

$\frac{d}{dt}\langle L_z\rangle=0$ follows from $[H,L_z]=0$. It is a necessary, but not a sufficient condition for the conservation of angular momentum. See points 4, 5, 6.

It is incorrect to conclude that an observable $A$ is conserved solely on the basis that $\frac{d}{dt}\langle A\rangle=0$, as done by referee 1. To see this consider a single particle at rest in free space. The Hamiltonian is $H=p^2/2$. The observable $x$ is not conserved: $[H,x]\neq0$. Nevertheless $\frac{d}{dt}\langle x\rangle=0$. Please see the new section 2.5 of the revised manuscript for more details.

$[H,L_z]=0$ implies $\frac{d}{dt}\langle L_z^n\rangle=0$ for all $n$, as shown in Eq. (9). Another proof, taken from a textbook, can be found in Appendix A. The probability distribution of measurements of $L_z$ is only stationary if $\frac{d}{dt}\langle L_z^n\rangle=0$ for all $n$. This is proven in Eq. (18). Any $\frac{d}{dt}\langle L_z^n\rangle\neq 0$ is a violation of $[H,L_z]=0$ and is experimentally detectable by measuring $L_z$, see section the new section 2.5 and the new Fig. 1.

Ground state meanfield solutions have been known to violate conservation laws since 1963 (HartreeFock) and at least since 1975 (GP meanfield). Even the methods to restore these broken symmetries have been textbook material since at least 1980. It is a wellestablished fact that the GP meanfield violates conservation laws. We only provide new results for the timedependent case. See the introduction for the historical context.

We provide an explanation why satisfying conservation laws in the dynamics is only possible on the manybody level, see section 4.3 and 4.4, as well as parametric dependencies of the violations and how to fix the problem. None of this has been published before.
The above points are discussed in detail in the new version of the manuscript, including proofs, examples and references. Satisfying conservation laws in BEC dynamics requires manybody theory. We have changed the abstract to emphasize that we explain why this is the case.
Referee 1 writes: ''Actually I am even surprised that the error for n=2 is relatively small like the one shown in Fig.4. [now Fig.6]}''
Answer: 1. $\langle L_z\rangle=0$ is exactly constant in the dynamics. Even in the numerics at a level of the numerical precision $<10^{8}$. In contrast, $\langle L_z^2\rangle$ varies over $\pm 13\%$, a difference of at least seven(!) orders of magnitude.

As stated in the abstract, we chose very weak interaction strength and practically no depletion (<$5\times 10^{4}$) to make the conditions as favorable as possible for the GP meanfield. Nevertheless, angular momentum conservation is substantially violated.

Arbitrarily strong violations: $\frac{d}{dt}\langle L_z^2\rangle$ can take on any value between $\infty$ and $\infty$ depending on the values chosen for $x_0, \sigma_0, p_0$, see Eq. (50). The violation grows linearly with the initial displacement from the center of the trap $x_0$, linearly with the initial momentum $p_0$, linearly with the GP nonlinearity parameter $\lambda$ and so on. It is no problem at all to find larger violations. To illustrate this fact we have included a new Fig. 2 to show these parametric dependencies.

Many experiments work in the opposite limit where the kinetic energy is much smaller than the interaction energy. The violation grows with the interaction strength, see Eq. (50).
Referee 1 writes: ''Let me come to the main point: ... 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.''
Answer: 1. We thank the referee for suggesting to look at the conservation of linear momentum as an additional example. Our calculations are general. There is no conceptual difference between angular and linear momentum conservation. We picked violations of angular momentum conservation out of personal preference. The GP dynamics also violates momentum conservation. We have included a specific example for a dynamic violation of momentum conservation by meanfield in a new section 8.
 Following referee 1's suggestion we now first discuss the (non)conservation of general onebody operators $A$. Later we specialize to $A=L_z$ and $A=P$ and provide specific examples. However, we have kept the main focus of the paper on angular momentum.
Referee 1 writes: ''... I am also concerned about the fact that the title would lead the reader to think that angular momentum is not conserved in meanfield, 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 threedimensional GP theory'', I would say that "constitute explicit violations of the conservation of higher momenta ($n \geq 2$) of angular momentum''.''
Answer: As discussed above, Referee 1's conclusion that angular momentum is conserved when $\frac{d}{dt}\langle L_z\rangle=0$, but $\frac{d}{dt}\langle L_z^n\rangle=0$ for $n>1$ is not correct.

The condition for angular momentum conservation is $[H,L_z]=0$, which implies $\frac{d}{dt}\langle L_z^n\rangle=0$ for all $n$. Therefore angular momentum is not conserved in the GP meanfield dynamics.

The full probability distribution of the measured values of $L_z$ involves all moments $\langle L_z^n\rangle$. If any of these is timedependent, so is this probability distribution. See Eq. (18) as well as the new sections 2.4 and 2.5.

In quantum mechanics it is not correct to conclude that an observable is conserved only because its expectation value $\langle A\rangle$ is timeindependent. It is important to realize that even for observables $A$ that are not conserved, $[H,A]\neq0$, one can have $\frac{d}{dt}\langle A\rangle=0$. We have now included a single particle counter example, for details see the new section 2.5.
Conclusion: Referee 1 concludes (incorrectly) from $\frac{d}{dt}\langle L_z\rangle=0$ that angular momentum is conserved by the GP meanfield dynamics, whereas for conservation of $L_z$ in quantum mechanics all moments $\langle L_z^n\rangle$ have to be timeindependent, which is not the case. Stationary meanfield violations of conservation laws have been long known. We treat the timedependent case. We explain from first principles why conservation laws can only be satisfied on the manybody level in the dynamics. In order to avoid further confusion we have included the relevant background material that illustrates these points. Furthermore, we included an example for the violation of linear momentum conservation by the GP meanfield, as was asked for by Referee 1 as well as expressions for higher order momenta.
Anonymous on 20180515 [id 253]
1) That the meanfield equations most often possess solutions which break the symmetries of the manybody Hamiltonian is well known in the literature of chemistry, nuclear physics, and condensed matter; examples of such symmetries are the spin and angularmomentum symmetries.
2) The term "conservation of angularmomentum symmetry" usually refers to the fact that the angularmomentum 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 realnumber value.
3) What is not as widely known is the manybody theory that further restores the broken symmetries. In the context of condensedmatter and trapped ultracold atoms, this twostep symmetry breakingsymmetry 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, https://doi.org/10.1088/00344885/70/12/R02
4) The fact that the GrossPitaevskii solutions describing vortices break the angularmomentum symmetry has been explicitly discussed in a rapid communication, "Symmetryconserving vortex clusters in small rotating clouds of ultracold bosons," Phys. Rev. A 78, 011606(R) – Published 28 July 2008, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.78.011606
The beyondmeanfield step of restoring the broken angularmomentum symmetry was also explicitly presented.