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Quantum chaos in interacting BoseBose mixtures
by Tran Duong AnhTai, Mathias Mikkelsen, Thomas Busch, Thomás Fogarty
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Authors (as registered SciPost users):  Thomas Fogarty · Duong AnhTai Tran 
Submission information  

Preprint Link:  https://arxiv.org/abs/2301.04818v1 (pdf) 
Date submitted:  20230119 07:57 
Submitted by:  Tran, Duong AnhTai 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
The appearance of chaotic quantum dynamics significantly depends on the symmetry properties of the system, and in cold atomic systems many of these can be experimentally controlled. In this work, we systematically study the emergence of quantum chaos in a minimal system describing onedimensional harmonically trapped BoseBose mixtures by tuning the particleparticle interactions. Using an advanced exact diagonalization scheme, we examine the transition from integrability to chaos when the intercomponent interaction changes from weak to strong. Our study is based on the analysis of the level spacing distribution and the distribution of the matrix elements of observables in terms of the eigenstate thermalization hypothesis and their dynamics. We show that one can obtain strong signatures of chaos by increasing the intercomponent interaction strength and breaking the symmetry of intracomponent interactions.
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Reports on this Submission
Anonymous Report 2 on 2023425 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2301.04818v1, delivered 20230425, doi: 10.21468/SciPost.Report.7100
Strengths
 This is a very thorough paper that addresses a lot of the natural questions involving thermalization of fewboson systems.
 The authors find a number of interesting points where the system does not immediately develop chaotic level statistics.
Weaknesses
 In some sense all the phenomena addressed in this paper are finitesize crossovers. It is clearly true that in a finitesize system that is nearly integrable, there will be regimes where the level statistics is not properly chaotic. The overall motivation for characterizing these crossovers is a bit opaque to me.
 It is not clear to me why fitting to the Brody distribution adds any insight.
Report
On the whole I think this is a solid numerical paper that will be interesting to specialists in the field.
Anonymous Report 1 on 2023321 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2301.04818v1, delivered 20230321, doi: 10.21468/SciPost.Report.6939
Strengths
1 This manuscript constitutes a very thorough investigation of the integrabilitychaos transition in an empirically relevant quantum system: a twocomponent onedimensional Bose gas
2 Authors diligently and properly used all the standard measures of quantum ergodicity. This paper could be used in a reading material for a class on quantum nonequilibrium.
Weaknesses
1 Unless I am mistaken, authors overlooked another integrable point: $g_{A} = g_{B} = g_{AB}$ (see e.g. [Y.Q. Li, S.J. Gu, Z.J. Ying, and U. Eckern, Exact results of the ground state and excitation properties of a twocomponent interacting Bose system, Europhys. Lett., 61, 368 (2003).]). Curiously, no corresponding dip on the diagonal, at either 2ac or 3ac is visible. Yet, I think it would be important to plot the level spacing statistics for, say, 3+3 , $g_{A} = g_{B} = g_{AB} = 10$ and assess if it is "more Poissonian" than $g_{A} = g_{B} = 10, g_{AB} = 1.5$ and $g_{A} = g_{B} = 10, g_{AB} = 20$ .
2 It should be noted that the onedimensional systems particles of the same mass can not fully serve as paradigmes of shortrangeinteracting gases. Unlike in two and three spatial dimensions where thermailzation occurs in a few twobody collision per particle [H. Wu, C.J. Foot, Direct simulation of evaporative cooling, J. Phys. B 29, L321 (1996).], in one dimension thermalization times are purely quantum entities, and thermalization takes longer than a few collisions. Onedimensional systems can be made more generic by introducing mass defects ( see e.g. [Z. Hwang, F. Cao, and M. Olshanii, Traces of Integrability in Relaxation of OneDimensional TwoMass Mixtures, J. Stat. Phys. 161, 467 (2015); Dmitry Yampolsky, N. L. Harshman, Vanja Dunjko, Zaijong Hwang, Maxim Olshanii, Quantum Chirikov criterion: Two particles in a box as a toy model for a quantum gas, SciPost Phys. 12, 035 (2022).] ). I would suggest comparing the relaxation times of Figs. 57 with the time scale associated with the twobody collisions: $\tau_{\text{2B}} \sim 1/(n v)$, where $n$ is the number density and $v \sim \sqrt{\epsilon /m}$, with $\epsilon$ being the total energy per particle.
Report
This is an important, rigorous, and thorough paper that deserves to be published. It will serve as a good reference text on quantum ergodicity.
Both of my comments are suggestions for improvement, and as such, they are optional.
Requested changes
There is no requested changes.
Optional changes:
1 Add a level spacing statistics histogram for $g_{A} = g_{B} = g_{AB} = 10$ and compare it to the $g_{A} = g_{B} = 10, g_{AB} = 1.5$ and $g_{A} = g_{B} = 10, g_{AB} = 20$ histograms;
2 Comment on an integrable point $g_{A} = g_{B} = g_{AB}$, regardless of whether it is visible in the Brody and kurtosis data or not.
3 Compare the relaxation times of Figs. 57 with the time scale associated with the twobody collisions. Cite [H. Wu, C.J. Foot, Direct simulation of evaporative cooling, J. Phys. B 29, L321 (1996); Z. Hwang, F. Cao, and M. Olshanii, Traces of Integrability in Relaxation of OneDimensional TwoMass Mixtures, J. Stat. Phys. 161, 467 (2015); Dmitry Yampolsky, N. L. Harshman, Vanja Dunjko, Zaijong Hwang, Maxim Olshanii, Quantum Chirikov criterion: Two particles in a box as a toy model for a quantum gas, SciPost Phys. 12, 035 (2022).]]
Author: Duong AnhTai Tran on 20230515 [id 3670]
(in reply to Report 1 on 20230321)
We would like to thank the referees for reviewing our manuscript and for finding it a thorough investigation of chaos in fewbody systems. In the following we address the comments of the referees and note the changes made to manuscript.

This is an important, rigorous, and thorough paper that deserves to be published. It will serve as a good reference text on quantum ergodicity. Our reply: We thank the referee for their time and interest into our work, and for their positive report.

Unless I am mistaken, authors overlooked another integrable point: $g_A=g_B=g_{AB}$ (see e.g. [Y.Q. Li, S.J. Gu, Z.J. Ying, and U. Eckern, Exact results of the ground state and excitation properties of a twocomponent interacting Bose system, Europhys. Lett., 61, 368 (2003).]). Curiously, no corresponding dip on the diagonal, at either 2ac or 3ac is visible. Yet, I think it would be important to plot the level spacing statistics for, say 3+3, $g_A=g_B=g_{AB}=10$ and assess if it is "more Poissonian" than $g_A=g_B=10,g_{AB}=1.5$ and $g_A=g_B=10,g_{AB}=20$. Optional change: Comment on an integrable point $g_A=g_B=g_{AB}$, regardless of whether it is visible in the Brody and kurtosis data or not. Our reply: We thank the referee for pointing this out and below we show the level spacing distributions for $g_A=g_B=g_{AB}=10$ and compare it to the cases shown in the manuscript $g_A=g_B=10,g_{AB}=1.5$ and $g_A=g_B=10,g_{AB}=20$. The distributions show that there is no obvious difference between the $g_A=g_B=g_{AB}=10$ case and those with $g_{AB}\neq 10$. This is also confirmed by Fig.~2(ac), Fig.~3(ac) in the manuscript, with the point of equal interactions $g_A=g_B=g_{AB}$ not exhibiting any noticeable distinct features. Furthermore, we have recalculated the distributions of $U_{mn}$ as shown in Fig. 4 for unequal interactions $g_{A}\neq g_{AB}$ and the results are qualitatively similar to those presented in the manuscript (Please see the attached file for the comparison). While the references provided by the referee suggest that the point $g_A=g_B=g_{AB}$ is integrable, this is only true for free space, not for harmonically trapped systems which we are considering. In the introduction of the model we noted that "the harmonic trap breaks integrability, and the system can thermalize, albeit on long timescales owing to the proximity of the system to integrability [K. F. Thomas, M. J. Davis and K. V. Kheruntsyan, Thermalization of a quantum Newton’s cradle in a onedimensional quasi condensate, Phys. Rev. A 103, 023315 (2021)]". Our final conclusion is that the point $g_A=g_B=g_{AB}$ in our model is not integrable although it is special for having SU(2) symmetry. To make this more clear we have added a brief discussion of this to the text.

It should be noted that the onedimensional systems particles of the same mass can not fully serve as paradigmes of shortrangeinteracting gases. Unlike in two and three spatial dimensions where thermailzation occurs in a few twobody collision per particle [H. Wu, C.J. Foot, Direct simulation of evaporative cooling, J. Phys. B 29, L321 (1996).], in one dimension thermalization times are purely quantum entities, and thermalization takes longer than a few collisions. Onedimensional systems can be made more generic by introducing mass defects ( see e.g. [Z. Hwang, F. Cao, and M. Olshanii, Traces of Integrability in Relaxation of OneDimensional TwoMass Mixtures, J. Stat. Phys. 161, 467 (2015); Dmitry Yampolsky, N. L. Harshman, Vanja Dunjko, Zaijong Hwang, Maxim Olshanii, Quantum Chirikov criterion: Two particles in a box as a toy model for a quantum gas, SciPost Phys. 12, 035 (2022).] ). I would suggest comparing the relaxation times of Figs. 57 with the time scale associated with the twobody collisions. Our reply: We thank the referee for pointing out the relation between the thermalization time and the twobody collision time in onedimensional systems. We have checked and found that the thermalization time of the dynamics shown in Figs.~5,6,8 ($t_{rel}\sim 100$) is approximately two orders of magnitude larger than the twobody collision time ($t_{col}\sim 1$). In the revised version of our manuscript, we have mentioned the comparison and cited the suggested references.
Changes to the manuscript:

In second paragraph in Introduction on page 2, we added a new reference about chaos in bosonic mixtures [34].

In section 3.2.1 on page 8 we added: "We also note that in contrast to interacting twocomponent systems in free space [60], the trapped system we consider does not contain an integrable point at $g_A = g_B = g_{AB}$ . While the system does possess SU(2) symmetry at this point, there is nothing evident in the energy spacing statistics or the kurtosis that differentiates it from any other point along the diagonal $g_A = g_B$."

In section 3.2.2 on page 13 we added the sentence: "In addition, we note that the thermalization time of the dynamics shown in Figs. 5 and 6 is approximately two orders of magnitude larger than the twobody collision time [61–63]."

At the end of Appendix A on page 18 we added the sentence: "Although the idea of employing the energytruncated Hilbert space and the effective interaction for obtaining the intercomponent integrals $W^{AB}_{ijk\ell}$ has also been used in [76], we remark that in our improved Exact Diagonalization scheme, we extend this by utilizing the effective interaction for both intra and intercomponents and take the symmetry of the manybody Fock basis into account."
Author: Duong AnhTai Tran on 20230515 [id 3669]
(in reply to Report 2 on 20230425)We would like to thank the referees for reviewing our manuscript and for finding it a thorough investigation of chaos in fewbody systems. In the following we address the comments of the referees and note the changes made to manuscript.
In some sense all the phenomena addressed in this paper are finitesize crossovers. It is clearly true that in a finitesize system that is nearly integrable, there will be regimes where the level statistics is not properly chaotic. The overall motivation for characterizing these crossovers is a bit opaque to me. Our reply: The referee is correct that the results shown in the work can mostly be attributed to finite size effects which are of course inherent in such small systems. However, this does not imply that these are not interesting regimes to study. Motivated by advances in trapping on control of single [Endres \textit{et al}, Science \textbf{354} 1024 (2016)] and fewbody systems [Bayha \textit{et al}, Nature 587, 583–587 (2020)] there are many recent works exploring the emergence of chaos in small systems [Zisling \textit{et al}, SciPost Phys. \textbf{10}, 088 (2021), Wittmann \textit{et al}, Physical Review E \textbf{105}, 034204 (2022), Yampolsky \textit{et al}, SciPost Phys. \textbf{12}, 035 (2022)], generally finding that $3$ or more particles are needed along with a broken symmetry induced by unequal masses or external potentials. In this work we go in a different direction and focus on inducing chaos through interaction effects in experimentally attainable twocomponent systems.
It is not clear to me why fitting to the Brody distribution adds any insight. Our reply: To study the emergence of chaos in these BoseBose mixtures we firstly rely on the energy spacing statistics which is a wellknown indicator of chaos. While examples are shown in the manuscript it is not possible to show these distributions for the entire parameter space. Therefore, we use a fitting to the Brody distribution which allows to concisely differentiate between regions of WignerDyson and Poissonian statistics. We believe this fitting succinctly highlights the chaotic and integrable regions of the parameter space which are then studied more rigorously in the following sections.
Changes to the manuscript:
In second paragraph in Introduction on page 2, we added a new reference about chaos in bosonic mixtures [34].
In section 3.2.1 on page 8 we added: "We also note that in contrast to interacting twocomponent systems in free space [60], the trapped system we consider does not contain an integrable point at $g_A = g_B = g_{AB}$ . While the system does possess SU(2) symmetry at this point, there is nothing evident in the energy spacing statistics or the kurtosis that differentiates it from any other point along the diagonal $g_A = g_B$."
In section 3.2.2 on page 13 we added the sentence: "In addition, we note that the thermalization time of the dynamics shown in Figs. 5 and 6 is approximately two orders of magnitude larger than the twobody collision time [61–63]."
At the end of Appendix A on page 18 we added the sentence: "Although the idea of employing the energytruncated Hilbert space and the effective interaction for obtaining the intercomponent integrals $W^{AB}_{ijk\ell}$ has also been used in [76], we remark that in our improved Exact Diagonalization scheme, we extend this by utilizing the effective interaction for both intra and intercomponents and take the symmetry of the manybody Fock basis into account."