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Identifying diffusive length scales in one-dimensional Bose gases
by Frederik Møller, Federica Cataldini, Jörg Schmiedmayer
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Frederik Skovbo Møller · Jörg Schmiedmayer |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2312.14007v2 (pdf) |
| Date submitted: | 2024-01-19 17:55 |
| Submitted by: | Møller, Frederik Skovbo |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Experimental |
Abstract
In the hydrodynamics of integrable models, diffusion is a subleading correction to ballistic propagation. Here we quantify the diffusive contribution for one-dimensional Bose gases and find it most influential in the crossover between the main thermodynamic regimes of the gas. Analysing the experimentally measured dynamics of a single density mode, we find diffusion to be relevant only for high wavelength excitations. Instead, the observed relaxation is solely caused by a ballistically driven dephasing process, whose time scale is related to the phonon lifetime of the system and is thus useful to evaluate the applicability of the phonon bases typically used in quantum field simulators.
Current status:
Reports on this Submission
Strengths
1. Thorough investigation of relevance/irrelevance of diffusive terms in the GHD equation
2. Comparison between theory and experimental results
3. An important conclusion is reached: diffusive terms are irrelevant for the description of existing experiments on 1D Bose gases
Weaknesses
1. The choice of the quantifier $\Gamma$ introduced in Eq. (19) is arbitrary. A different quantifier would presumably lead to a different diagram in Fig. 2, and possibly to different conclusions (?)
Report
The authors investigate the importance of the diffusive corrections to Euler-scale Generalized Hydrodynamics derived in Refs. [11,12]. Section 2 briefly introduces the main definitions and formulas from Refs. [11,12], and the linearization of Ref. [47]. Section 3 focuses on a quantifier (Eq. (19) and Fig. 2) of the relevance of diffusive effects in states at thermal equilibrium. The diagram in Fig. 2 suggests that diffusive effects are maximal in the regime of intermediate interaction and intermediate temperature, and at the crossover between quasicondensate and ideal Bose gas regime, which is accessible to experiments.
Then Section 4 presents a comparison of Euler-scale GHD simulations (no diffusion) with experimental results, showing excellent agreement, and the authors then argue that the inclusion of diffusive corrections would change nothing.
This is nice work, and I think the main conclusion is highly relevant for the description of existing experiments on the 1D Bose gas. I recommend that the manuscript be published in Scipost Physics, maybe after a few modifications to clarify some points.
Suggested modifications:
1. The presentation is a bit schizophrenic. A lot of emphasis is put on the diffusive corrections to GHD (sections 2 and 3), while the ultimate conclusion is that they do not matter at all in the comparison with experimental data (section 4). What really matters is the non-linear dispersion of the quasi-particles. I think this message could be stressed more, perhaps by highlighting this in the title, or maybe by changing the order of the sections (the experimental results of Section 4, which match Euler-scale GHD perfectly, could be discussed earlier).
2. The quantity $\Gamma$ introduced in Eq. (19) seems very arbitrary. It is an inverse length, therefore by dimensional analysis it must scale as $\Gamma = n f(\gamma, \mathcal{T})$, for some unknown function $f$. But why compare $\Gamma$ to $c$, especially in the TG regime where $c$ diverges and is not particularly meaningful: What if one plots $\Gamma/n$ instead of $\Gamma/c$? This would change figure 2; would it also change the conclusion? What if one exchanges the numerator and denominator in the definition (19), namely if one looks at $\int d \theta \rho_{\rm p}(\theta) \frac{D (\theta)}{|v(\theta)|}$? Also, in view of the fact that what really matters in the end (Section 4) is the curvature of the quasi-particle dispersion, wouldn't it make more sense to compare $D(\theta)$ to $\partial_\theta v(\theta)$, as opposed to $v(\theta)$ itself?
More generally, the question about Section 3 is: how do the conclusions of that Section depend on the choice of the quantifier (19)?
3. It is unclear how the experimental parameters given in (a)-(b)-(c) in page 9 are obtained. How is the initial temperature of the gas estimated? Is the temperature used as a fit parameter in the GHD calculation in Figs. 5(a),5(b),5(c) ? Or is it estimated independently?
4. The formulas in Section 2 are a little bit obscure. For instance, Eq. (2) uses the inverse notation for the dressing, but then the dressing is introduced a few lines below in Eq. (4) and is used in most other formulas: Why not use the same notation everywhere? Section 2.1 does not read well, the authors could perhaps elaborate a bit on the physical meaning of the different formulas. Or if this is just a summary of formulas that have appeared elsewhere and that are not really used in the main text of the paper, then some formulas could perhaps be moved to an appendix, I think it could improve readability.
Requested changes
See suggestions above
Report
In this work authors study the Generalized Hydrodynamics (GHD) description of 1 d ultra-cold atomic gases. Their aim to characterise regimes of the gas in which the diffusion effects of GHD are important and then use this result to describe the dynamics following a single-mode quench.
The question how important is the diffusion in the linearized dynamics was already analysed in ref. [47]. Here, the authors build on that result and furthermore introduce a new measure $\Gamma$ to quantify the magnitude of the diffusive corrections. Following on that, they evaluate $\Gamma$ for thermal states and conclude that the diffusive corrections are the most important in the regime of intermediate $\gamma$ (the dimensionless coupling constant of the Lieb-Liniger model). and intermediate temperatures. The regime of intermediate $\gamma$ is known for being "most correlated" hence this finding is not unexpected.
In the second part of the manuscript, authors look at a recent experiment - a single-mode quench - and demonstrate that the resulting dynamics can be effectively described by a purely ballistic GHD. They also confirm that the experimental system is in the regime, that according to the theoretical understanding developed earlier, should indeed evolve ballistically. Finally, the authors show that even after tuning the experimental parameters to the most diffusive case, the effect of the diffusion on the time evolution of the density is negligible. Still the effect is visible in the full filling function of the state which makes the author conclude that in principle there could be better observables to witness the diffusion however they are difficult to implement experimentally.
The paper is well written and scientifically sound. I think it qualifies for the publication in SciPost Physics Core.
Requested changes
I think the paper could be published as it is.
While reading the manuscript I spotted two typos:
1) double 'and' in the caption of Fig. 2
2) double 'also' in the last paragraph before Sec. 4.2
Author: Frederik Skovbo Møller on 2024-03-13 [id 4364]
(in reply to Report 1 on 2024-02-26)We thank the Referee for their report and comments.
Author: Frederik Skovbo Møller on 2024-03-13 [id 4365]
(in reply to Report 2 on 2024-03-09)We thank the Referee for their report and comments. Following their suggestions, we have made appropriate modifications to the manuscript. In response to the suggested modifications:
1) Firstly, the observed relaxation in the experiment is a consequence of the variation in propagation velocity of the rapidity states making up the perturbation. This variation is not necessarily due to curvature of the quasi-particle dispersion, as it is only required that the effective velocity is a monotonically increasing function of rapidity. Instead, the main cause is the spread in rapidity states occupied, which increases with temperature. This is in contrast to a Bogoliubov description, where all particles occupying a given k-mode propagate with the same velocity. The relaxation of the mode following dephasing of rapidity states was already discussed in the original paper of the experiment (Ref. [24]), albeit not in as much detail as here. However, Ref. [24] did not analyze the contribution of diffusive effects. Therefore, the focus of this manuscript is mainly on the role of diffusion, in which thermodynamic regimes it most influences transport, and how one would/could observe it experimentally. Nevertheless, we acknowledge that the presentation is a little scattered, and we have made some restructuring of the manuscript to better reflect its aim. Section 2 is now focused on the propagation of quasi-particles, and thus the transport properties of a system. Section 3 focuses on the dynamics of single momentum modes of the gas, how these modes are affected by diffusion, and how they relate to the phonon basis of effective field theory descriptions.
2) The choice of $\Gamma$ is far from arbitrary: It describes the average distance traveled by quasi-particles relative to the variance of their trajectory, weighted according to the distribution of particles. Therefore, it characterizes the quasi-particle propagation in a given state; since the quasi-particles are responsible for the spreading of operators in integrable systems, calculating $\Gamma$ yields a quantitative measure of the type of transport in the system. In the manuscript, we have added a paragraph before the introduction of $\Gamma$ to properly explain and motivate the quantity.
We find that for fixed values of $\gamma = c/n$, the measure $\Gamma$ scales linearly with $c$. Hence, the rescaled measure $\Gamma/c$ is determined by just the two parameters $\gamma$ and $\mathcal{T}$ - similarly to thermal states of Lieb-Liniger phase diagram. Note that $\Gamma$ already contains a factor $1/n$ as part of the weighted average over quasi-particles.
An exchange in the numerator and denominator in the definition of $\Gamma$ would produce an equivalent measure, where a higher diffusive broadening of the quasi-particle trajectory would be indicated by a larger number. However, this would then be opposite to the critical mode $j^\star$, where a lower value signifies that diffusion occurs at lower modes/larger length scales, i.e. that diffusion is more prevalent in the system. For consistency, we have chosen $\Gamma$ such that larger values of $\Gamma$ corresponds to a higher critical mode.
3) The temperature and coupling strengths are obtained from separate measurements through density ripples thermometry and measurement of the transverse trapping frequency, respectively. A sentences has been added to detail this. Given these two parameters and the initial atomic density, the initial quasi-particle distribution can be calculated assuming a thermal state.
4) Since several review of GHD have been written already, section 2.1 is meant to just present a simple explanation of the origin of ballistic and diffusive dynamics in integrable systems and list the relevant equations. For more information, we now explicitly refer the reader to in-depth reviews on the subject. All equations currently featuring in section 2.1 are used throughout the manuscript, hence we would prefer to keep them in the main text and not in an appendix. Following the restructuring of the manuscript, the previous sections 2.2 and 2.3 have now been merged into the new section 3. This means that several of the equations in section 2.1 are immediately used in the following section, improving readability. Finally, note that the integral operators in Eq. (2) are not the same as the dressing. In shorthand notation, the dressing reads $g^{\mathrm{dr}} = \left( \mathbf{1} - \frac{\mathbf{\Delta} \vartheta}{2 \pi}\right)^{-1} g$, which indeed looks very similar to Eq. (2), except the order of the scattering kernel and filling has been reversed. This subtle difference changes the action of the operator.