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Pauli crystal melting in shaken optical traps
by Jiabing Xiang, Paolo Molignini, Miriam Büttner, Axel U. J. Lode
This is not the latest submitted version.
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Submission summary
| Authors (as registered SciPost users): | Paolo Molignini · Jiabing Xiang |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2204.10335v2 (pdf) |
| Code repository: | http://ultracold.org |
| Data repository: | https://gitlab.com/Jiabing/pauli-crystals |
| Date submitted: | 2022-09-02 12:50 |
| Submitted by: | Xiang, Jiabing |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Pauli crystals are ordered geometric structures that emerge in trapped noninteracting fermionic systems due to their underlying Pauli repulsion. The deformation of Pauli crystals - often called melting - has been recently observed in experiments, but the mechanism that leads to it remains unclear. We address this question by studying the melting dynamics of N=6 fermions as a function of periodic driving and experimental imperfections in the trap (anisotropy and anharmonicity) by employing a combination of numerical simulations and Floquet theory. Surprisingly, we reveal that the melting of Pauli crystals is not simply a direct consequence of an increase in system energy, but is instead related to the trap geometry and the population of the Floquet modes. We show that the melting is absent in traps without imperfections and triggered only by a sufficiently large shaking amplitude in traps with imperfections.
Author comments upon resubmission
Dear editor,
We thank you for sending over the referees report and the referees for their positive comments on our manuscript. In the answer below we address point by point all the questions raised by the referees.
Referee#1
RC#1(Referee Comments #1):
I encourage authors to extend a comment on applicability of the presented formalism to studies of interating systems.
AR(Author Response):
We thank the referee for pointing out the interesting question of exploring the role of interactions in the stability and dynamics of Pauli crystals. We do agree that this is an intriguing direction to extend the present study via a future work on interacting and/or bosonic and/or multicomponent indistinguishable particles. We mention this point in the outlook. For the sake of simplicity, the present study focuses on the noninteracting case. However, the MCTDH method is completely general and can also be used in systems with any kind of two-particle interaction.
RC#2:
More detailed discussion of the recognition function might be of some interest - in particular a remark of how to define the recognition function for systems not possessing axial symmetry.
AR:
We think it is difficult to define the recognition function for systems not possessing axial symmetry. In our simulation case, the particle positions are distributed more or less uniformly in a circle around the center of mass in each single shot. In the melting process, the configuration density transforms into a more uniform pattern in angular distribution. Therefore, the recognition function reduces monotonically. However, when single-shot images are asymmetrical or even drastically anisotropic, the configuration density might not transform into a uniform pattern in angular distribution and the recognition function might not change monotonically when the correlation structure deforms. Hence, the change of the recognition function cannot be used to reveal whether the melting occurs directly. For systems not possessing axial symmetry, it’s probably simpler to observe the pattern of the configuration density directly to judge whether the melting occurs.
Referee#2
RC#1:
What do the authors mean by higher-order spatial correlations in the excited states ? Does the higher-order refer to the excited states or to something else ?
AR:
We thank the referee for raising this point. The term "higher-order spatial correlations" means "N-body correlations". We have replaced "higher-order spatial correlations" with "N-body correlations" to avoid this confusion.
RC#2:
The degree of anharmonicity in Eqs. (5) and (6) seems to be different. In Eq. (5) only quartic anharmonicity is taken into account, whereas in Eq. (6), which is closer to the experimental setup, higher anharmonicities contribute. Employing the same expression in Eq. (5) as well, without the anisotropy, would allow an even more direct comparison between all the potentials.
AR:
The referee correctly points out that the Gaussian trap implicitly contains higher-order anharmonicities. However, we checked that higher anharmonicities play a lesser role compared to the quartic term. In other words, a quartic term is enough to trigger melting, which was the focus of our work. At the same time, we have done simulations for the Gaussian trap without anisotropy (not shown in the manuscript) and the Pauli crystal melting occurs anyway. This means that the anharmonicities indeed play an important role for the melting in the Gaussian trap, and a systematic study of anharmonicities in different potentials could be the subject of future work.
RC#3:
What confuses me is the observation of the Pauli crystal only in the configuration density. Why this structure is not visible in the one-body reduced density as well ? I would expect that some small density humps would reveal the most probable positions of a single fermion, as shown in the reference Phys. Rev. A 99, 013605 (2019).
AR:
The one-body reduced density can indeed reveal the most probable positions of a single fermion. But it can not reveal the correlations between fermions, namely, the relative positions with the maximum probability. See Fig. 2. These correlations revealed by the configuration density are the characteristic of Pauli crystals.
RC#4:
I think the presentation would be more smooth if the discussion before Eq. (7) referred to Fig. (2) instead of Fig. (3). Hence the definitions of the angular distributions and the discussion of Fig. (3) would come more natural.
AR:
We thank the referee for this suggestion. "Fig.(3)" was replace by "Fig.(2)" in III.B.Recognition function.
RC#5:
The analysis for the six fermions is carried out beyond the Hartree-Fock method by employing 7 single-particle orbitals within the MCTDH-X method. Do the results and the overall phenomenology remain the same by accommodating the six particles into 6 single-particle orbitals ?
AR:
Yes. Theoretically, the N single-particle orbitals are enough for N particles (M=N) in our simulation. However, the case M=N might be numerically unstable in some situations (e.g. some matrices might be hard to invert numerically). Therefore, the orbital number M was set to 7 to avoid numerical instability in the simulations.
RC#6:
I think the caption in Table III should state explicitly that this categorization pertains to the case of 6 non-interacting fermions.
AR:
We appreciate the referee for this kind recommendation. The caption was revised.
RC#7:
What is the contribution of the single-particle orbitals in the ground states presented in Fig. 4 ? Is the Pauli crystal structure observed at that level, or one has still to employ the configuration density to see these patterns ?
AR:
For seven single-particle orbitals, only the lowest six single-particle orbitals are occupied in the ground states presented in Fig.4. We still must employ the configuration density to see these patterns. See response to comment 5 above.
RC#8:
It would be maybe more useful to include the time in ms in Figs. 4, 5 and 6.
AR:
We thank the referee for this suggestion. However, we think it is more consistent to keep time in dimensionless units like all the other quantities presented in the manuscript. At the same time, we have clarified how long the total time interval is in ms in the last paragraph of the section Appendix B Parameter Setting.
RC#9:
In Fig. 5 (a), it is shown that the recognition function of all potentials except from the harmonic one, decreases below the threshold identified from the bosonic counterpart of the fermionic system. However, it seems that for the anisotropic potential, there are revivals, and the recognition function increases well above the threshold. Is there some particular reason for that effect ?
AR:
The anisotropic potential shows a lot of "breathing patterns" in the density, in the sense that it expands and contracts along the direction where the confinement is weaker due to the anisotropy. It might well be that it takes longer to melt the crystals in that case because the system can bounce back to the initial configuration more efficiently (hence the revivals).
RC#10:
I think it would be better to group figures 5 (a) and 6 together, and present figure 5(b) separately. This particular ordering seems to fit better with the discussion in the last paragraph of Section IV A.
AR:
We appreciate the referee for this kind recommendation. The order of plots was revised following the referee’s suggestion.
RC#11:
I think the sentence right after Eq. (A5) is incomplete.
AR:
We appreciate the referee for pointing out this problem. The sentence was revised.
RC#12:
There is a missing imaginary unit in Eq. (C1).
AR:
We appreciate the referee for pointing out this typo. The equation was revised.
RC#13:
Do the authors think that this analysis would shed further light to the dynamical formation of star patterns, Faraday waves and fireworks for interacting bosonic particles ?
AR:
We thank the referee for this insightful comment. Indeed we believe these features could be probed with our recognition function. We have added a corresponding remark in our conclusions.
RC#14:
What is the variance of the single shots employed in the manuscript ? Similarly to what has been observed in Phys. Rev. Lett. 118, 013603 (2017), can this quantity be linked to the fragmentation of the system ?
AR:
We think the variance of the single-shot images cannot quantify the melting of a Pauli crystal, since it only quantifies the average distance from the origin point for each particle. We are concerned more with the structure of the configuration density. For example, the extension of the configuration density for the system in the isotropic harmonic trap is largest at the end of modulation in all traps. That means that the variance of the single-shot image is largest in the isotropic harmonic trap, but the fermion system in the isotropic harmonic trap doesn’t melt.
RC#15:
This question goes beyond the scope of this work, but do the authors expect a similar behavior of the dynamical melting process for a different number of fermions, e.g. three as in the experiment. Furthermore, what do the authors think about the few-to- many-body crossover?
AR:
The fermionic system with three non-interacting particles produces results that are qualitatively similar to the ones of the six-fermion system. In fact, the occurrence of a Pauli crystal should not depend on the particle number. The number of fermions only determines the structure of the Pauli crystal (how many shells, geometric configuration etc.). See the examples of N = 10 and N = 15 in EPL (Europhysics Letters), 115(2), 20012 and Symmetry, 12(11), 1886.
As the particle number increases, the shell number of the correlation structure increases as well. Since the outermost shell arranges many more particles, the variation of particle density in outermost shell is flat. Only the innermost shell with few particles has a distinct structure. For N = 10, the structure of inner shell is same as the N = 3. For N = 15, the structure of inner shell is same as the N = 6.
The N=3 case is a little bit limited because, due to the low number of particles, it even shows some structure stemming from random particle positions, and it requires even more precision in the measurements, which is already a challenging experimental task.
RC#16:
This remark also goes beyond the scope of the work. Would the authors expect Pauli crystals to form in a box potential. If so, would the melting mechanism be the same ?
AR:
We think Pauli crystals can form in any attractive potential since the main ingredient for the realization of Pauli crystals is the coexistence of Pauli repulsion and an external confinement. The potential determines the structure of the most probable configuration( Gajda M, Mostowski J, Pylak M, et al. Pauli Crystals–Interplay of Symmetries[J]. Symmetry, 2020, 12(11): 1886). Therefore, we would expect the same melting mechanism in a box potential.
We thank you for sending over the referees report and the referees for their positive comments on our manuscript. In the answer below we address point by point all the questions raised by the referees.
Referee#1
RC#1(Referee Comments #1):
I encourage authors to extend a comment on applicability of the presented formalism to studies of interating systems.
AR(Author Response):
We thank the referee for pointing out the interesting question of exploring the role of interactions in the stability and dynamics of Pauli crystals. We do agree that this is an intriguing direction to extend the present study via a future work on interacting and/or bosonic and/or multicomponent indistinguishable particles. We mention this point in the outlook. For the sake of simplicity, the present study focuses on the noninteracting case. However, the MCTDH method is completely general and can also be used in systems with any kind of two-particle interaction.
RC#2:
More detailed discussion of the recognition function might be of some interest - in particular a remark of how to define the recognition function for systems not possessing axial symmetry.
AR:
We think it is difficult to define the recognition function for systems not possessing axial symmetry. In our simulation case, the particle positions are distributed more or less uniformly in a circle around the center of mass in each single shot. In the melting process, the configuration density transforms into a more uniform pattern in angular distribution. Therefore, the recognition function reduces monotonically. However, when single-shot images are asymmetrical or even drastically anisotropic, the configuration density might not transform into a uniform pattern in angular distribution and the recognition function might not change monotonically when the correlation structure deforms. Hence, the change of the recognition function cannot be used to reveal whether the melting occurs directly. For systems not possessing axial symmetry, it’s probably simpler to observe the pattern of the configuration density directly to judge whether the melting occurs.
Referee#2
RC#1:
What do the authors mean by higher-order spatial correlations in the excited states ? Does the higher-order refer to the excited states or to something else ?
AR:
We thank the referee for raising this point. The term "higher-order spatial correlations" means "N-body correlations". We have replaced "higher-order spatial correlations" with "N-body correlations" to avoid this confusion.
RC#2:
The degree of anharmonicity in Eqs. (5) and (6) seems to be different. In Eq. (5) only quartic anharmonicity is taken into account, whereas in Eq. (6), which is closer to the experimental setup, higher anharmonicities contribute. Employing the same expression in Eq. (5) as well, without the anisotropy, would allow an even more direct comparison between all the potentials.
AR:
The referee correctly points out that the Gaussian trap implicitly contains higher-order anharmonicities. However, we checked that higher anharmonicities play a lesser role compared to the quartic term. In other words, a quartic term is enough to trigger melting, which was the focus of our work. At the same time, we have done simulations for the Gaussian trap without anisotropy (not shown in the manuscript) and the Pauli crystal melting occurs anyway. This means that the anharmonicities indeed play an important role for the melting in the Gaussian trap, and a systematic study of anharmonicities in different potentials could be the subject of future work.
RC#3:
What confuses me is the observation of the Pauli crystal only in the configuration density. Why this structure is not visible in the one-body reduced density as well ? I would expect that some small density humps would reveal the most probable positions of a single fermion, as shown in the reference Phys. Rev. A 99, 013605 (2019).
AR:
The one-body reduced density can indeed reveal the most probable positions of a single fermion. But it can not reveal the correlations between fermions, namely, the relative positions with the maximum probability. See Fig. 2. These correlations revealed by the configuration density are the characteristic of Pauli crystals.
RC#4:
I think the presentation would be more smooth if the discussion before Eq. (7) referred to Fig. (2) instead of Fig. (3). Hence the definitions of the angular distributions and the discussion of Fig. (3) would come more natural.
AR:
We thank the referee for this suggestion. "Fig.(3)" was replace by "Fig.(2)" in III.B.Recognition function.
RC#5:
The analysis for the six fermions is carried out beyond the Hartree-Fock method by employing 7 single-particle orbitals within the MCTDH-X method. Do the results and the overall phenomenology remain the same by accommodating the six particles into 6 single-particle orbitals ?
AR:
Yes. Theoretically, the N single-particle orbitals are enough for N particles (M=N) in our simulation. However, the case M=N might be numerically unstable in some situations (e.g. some matrices might be hard to invert numerically). Therefore, the orbital number M was set to 7 to avoid numerical instability in the simulations.
RC#6:
I think the caption in Table III should state explicitly that this categorization pertains to the case of 6 non-interacting fermions.
AR:
We appreciate the referee for this kind recommendation. The caption was revised.
RC#7:
What is the contribution of the single-particle orbitals in the ground states presented in Fig. 4 ? Is the Pauli crystal structure observed at that level, or one has still to employ the configuration density to see these patterns ?
AR:
For seven single-particle orbitals, only the lowest six single-particle orbitals are occupied in the ground states presented in Fig.4. We still must employ the configuration density to see these patterns. See response to comment 5 above.
RC#8:
It would be maybe more useful to include the time in ms in Figs. 4, 5 and 6.
AR:
We thank the referee for this suggestion. However, we think it is more consistent to keep time in dimensionless units like all the other quantities presented in the manuscript. At the same time, we have clarified how long the total time interval is in ms in the last paragraph of the section Appendix B Parameter Setting.
RC#9:
In Fig. 5 (a), it is shown that the recognition function of all potentials except from the harmonic one, decreases below the threshold identified from the bosonic counterpart of the fermionic system. However, it seems that for the anisotropic potential, there are revivals, and the recognition function increases well above the threshold. Is there some particular reason for that effect ?
AR:
The anisotropic potential shows a lot of "breathing patterns" in the density, in the sense that it expands and contracts along the direction where the confinement is weaker due to the anisotropy. It might well be that it takes longer to melt the crystals in that case because the system can bounce back to the initial configuration more efficiently (hence the revivals).
RC#10:
I think it would be better to group figures 5 (a) and 6 together, and present figure 5(b) separately. This particular ordering seems to fit better with the discussion in the last paragraph of Section IV A.
AR:
We appreciate the referee for this kind recommendation. The order of plots was revised following the referee’s suggestion.
RC#11:
I think the sentence right after Eq. (A5) is incomplete.
AR:
We appreciate the referee for pointing out this problem. The sentence was revised.
RC#12:
There is a missing imaginary unit in Eq. (C1).
AR:
We appreciate the referee for pointing out this typo. The equation was revised.
RC#13:
Do the authors think that this analysis would shed further light to the dynamical formation of star patterns, Faraday waves and fireworks for interacting bosonic particles ?
AR:
We thank the referee for this insightful comment. Indeed we believe these features could be probed with our recognition function. We have added a corresponding remark in our conclusions.
RC#14:
What is the variance of the single shots employed in the manuscript ? Similarly to what has been observed in Phys. Rev. Lett. 118, 013603 (2017), can this quantity be linked to the fragmentation of the system ?
AR:
We think the variance of the single-shot images cannot quantify the melting of a Pauli crystal, since it only quantifies the average distance from the origin point for each particle. We are concerned more with the structure of the configuration density. For example, the extension of the configuration density for the system in the isotropic harmonic trap is largest at the end of modulation in all traps. That means that the variance of the single-shot image is largest in the isotropic harmonic trap, but the fermion system in the isotropic harmonic trap doesn’t melt.
RC#15:
This question goes beyond the scope of this work, but do the authors expect a similar behavior of the dynamical melting process for a different number of fermions, e.g. three as in the experiment. Furthermore, what do the authors think about the few-to- many-body crossover?
AR:
The fermionic system with three non-interacting particles produces results that are qualitatively similar to the ones of the six-fermion system. In fact, the occurrence of a Pauli crystal should not depend on the particle number. The number of fermions only determines the structure of the Pauli crystal (how many shells, geometric configuration etc.). See the examples of N = 10 and N = 15 in EPL (Europhysics Letters), 115(2), 20012 and Symmetry, 12(11), 1886.
As the particle number increases, the shell number of the correlation structure increases as well. Since the outermost shell arranges many more particles, the variation of particle density in outermost shell is flat. Only the innermost shell with few particles has a distinct structure. For N = 10, the structure of inner shell is same as the N = 3. For N = 15, the structure of inner shell is same as the N = 6.
The N=3 case is a little bit limited because, due to the low number of particles, it even shows some structure stemming from random particle positions, and it requires even more precision in the measurements, which is already a challenging experimental task.
RC#16:
This remark also goes beyond the scope of the work. Would the authors expect Pauli crystals to form in a box potential. If so, would the melting mechanism be the same ?
AR:
We think Pauli crystals can form in any attractive potential since the main ingredient for the realization of Pauli crystals is the coexistence of Pauli repulsion and an external confinement. The potential determines the structure of the most probable configuration( Gajda M, Mostowski J, Pylak M, et al. Pauli Crystals–Interplay of Symmetries[J]. Symmetry, 2020, 12(11): 1886). Therefore, we would expect the same melting mechanism in a box potential.
List of changes
1. We replaced the term "higher-order spatial correlations" with "N-body correlations" in the I.Introduction section.
2. We improved the referring way to Fig. (2) and Fig. (3) in the discussion before Eq. (7).
3. The caption of Table III was revised.
4. We grouped figures 5 (a) and 6 together, and presented figure 5(b) separately.
5. We complemented the sentence right after Eq. (A5).
6. The missing imaginary unit in Eq. (C1) was added.
7. We added remarks about the generality of MCTDH method and the dynamical formation of star patterns, Faraday waves and fireworks for interacting bosonic particles in the V.Conclusion section.
Current status:
Has been resubmitted