## SciPost Submission Page

# On the low-energy description for tunnel-coupled one-dimensional Bose gases

### by Yuri D. van Nieuwkerk, Fabian H. L. Essler

#### This is not the current version.

### Submission summary

As Contributors: | Fabian Essler · Yuri Daniel van Nieuwkerk |

Arxiv Link: | https://arxiv.org/abs/2003.07873v1 (pdf) |

Date submitted: | 2020-03-19 01:00 |

Submitted by: | van Nieuwkerk, Yuri Daniel |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

We consider a model of two tunnel-coupled one-dimensional Bose gases with hard-wall boundary conditions. Bosonizing the model and retaining only the most relevant interactions leads to a decoupled theory consisting of a quantum sine-Gordon model and a free boson, describing respectively the antisymmetric and symmetric combinations of the phase fields. We go beyond this description by retaining the perturbation with the next smallest scaling dimension. This perturbation carries conformal spin and couples the two sectors. We carry out a detailed investigation of the effects of this coupling on the non-equilibrium dynamics of the model. We focus in particular on the role played by spatial inhomogeneities in the initial state in a quantum quench setup.

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### Submission & Refereeing History

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## Reports on this Submission

### Anonymous Report 1 on 2020-3-22 Invited Report

- Cite as: Anonymous, Report on arXiv:2003.07873v1, delivered 2020-03-22, doi: 10.21468/SciPost.Report.1589

### Report

The paper addresses an interesting problem of the coupling between symmetric and antisymmetric modes in a system consisting of two elongated, tunnel-coupled degenerate Bose gases and their dynamics out of equilibrium. To tackle this problem, the authors opt for the self-consistent time-dependent harmonic approximation (SCTDHA). The anharmonic Hamiltonian is approximated by a harmonic one with time-dependent coefficients determined by the field correlations. This approximate quantum Hamiltonian describes scattering of elementary excitations on mean-field-like fluctuations as well as driving and squeezing of the elementary modes in a self-consistent way. If the initial state is Gaussian then it remains Gaussian for all times under the evolution determined by the SCTDHA equations. This makes possible to evaluate all the time-dependent coefficients of the approximate harmonic model.

The antysimmetric modes are described by the sine-Gordon Hamiltonial, the symmetric modes are free bosons with linear dispersion. The product of the density fluctuation in the symmetric mode and the cosine of the phase in the antisymmetric mode is considered as the main coupling term.

In my opinion, the paper doesn't represent a real breakthrough in understanding the out-of-equilibrium dynamics in an extended bosonic Josephson junction. The qualitative picture has become clear a long ago, the key remaining open problem is to find the best quantitative method for the cases where the well-established mean-field approach isn't sufficient and the quantum effects become important.

My first point of criticism concerns the choice of the relevant perturbation. Besides the coupling given by Eq. (11), there is also coupling of the density fluctuation to the square of the phase gradient in each of the two elongated Bose gases. This term has been known not only to dominate the dynamics in a single 1D system at finite temperature [A. F. Andreev, Sov. Phys. JETP 51, 1038 (1980)], but also to destroy the coherence between totally decoupled ($t_\perp =0$) quasicondensates [see A. A. Burkov, M. D. Lukin, and E. Demler, PRL 98, 200404 (2007); however, their estimated decoherence time seems to be too short in comparison to what we can expect from an extension of Andreev's analysis]. For non-zero coupling, this term becomes comparable to that considered by the authors at wavelength of order of $\sqrt{\hbar ^2/(mt_\perp )}$ or shorter. In other words, the assumption made by the authors with respect to the relevant perturbation holds at unrealistically low (below $t_\perp $) temperatures only.

Secondly, the SCTDHA doesn't seem to be the best method to solve the given problem. On the one hand, there are methods of the quantum field theory to derive the quantum Boltzmann equation for a 1D system of bosons out of equilibrium [M. Buchhold and S. Diehl, EPJD 69, 224 (2015)]. On the other hand, the usual mean-field approximation suffices to describe results of many experiments in 1D. The authors should have compared their results with the results of solving numerically two coupled Gross-Pitaevskii equations in 1D with stochastic initial conditions and possibly to the results of the truncated Wigner approach to understand the influence of the quantum noise [perhaps, the latter approach works well in 1D on longer times then it does in higher dimensions]. I would be not surprised if the results of the SCTDHA and the numerical solution of the mean-field equations [following from the semiclasscal version of Eq. (3)] averaged over the ensemble of initial conditions sampling the displaced thermal state turn out to be very close. Indeed, the initial state discussed in Section 4.1 corresponds to the state after a fast splitting that undergoes a rapid prethermalization, see Ref. [17]. But the prethermalized state contains not only zero-point oscillations of the postquench Hamiltonian, but also large semiclassical excitations corresponding to the temperature of order of the chemical potential. This means that the quantum noise plays an important role only at large momenta, where Eq. (6) is no more valid anyway.

The same applies to the choice of the boundary conditions. One can set inhomogeneous initial conditions in any finite-size system. In that sense there is no qualitative difference between the hard-wall boundary conditions Eq. (4) and periodic boundary conditions over a finite period $L$ (a system on a ring). The authors should carefully distinguish between the effects of the longitudinal trapping that causes inhomogeneity of the background density profile and of the finite size only (with $\rho_0 =$const).

The third weak point of the paper is the absence of analytic estimations (even roughest ones) of the lifetime of an elementary excitations and of the typical time of equilibration between the symmetric and antisymmetric subsystems. Sometimes it is possible to use scaling arguments for the quantum Boltzmann equations, sometimes an empirical analysis of numerical results yields a formula. And such an estimation is usually very helpful for understanding the physics of a system. The paper will gain much novelty and significance if the authors summarize their gained knowledge in few simple expressions.

Finally, there are quite a few typos in the paper. For example,

(i) the word "harmonic" is omitted in p. 3 in the line where the abbreviation SCTDHA is introduced;

(ii) the word "profiles" is repeated twice in the caption to Fig. 1;

(iii) proper names such as Josephson or Gordon are spelled with the lower-case first letter in some reference titles (pp. 22-27);

(iv) the notation for the tunnel coupling strength is inconsistent, sometimes $T_\perp $ is used, sometimes $t_\perp $.

To summarize, a major revision of the paper is needed before its resubmission.

### Requested changes

1- The authors should consider the coupling of the density fluctuation to the square of the phase gradient in the coupled Bose gases and either include it into the perturbation or clearly define the parameter range where this type of perturbations can be ignored.

2- The authors should compare the SCTDHA results to the numerical simulations of the two coupled 1D Gross-Pitaevskii equations (without further simplifications) with stochastic boundary conditions sampling the Gaussian initial state.

3- If an unexplainable discrepancy between the SCTDHA and the mean-field theory will be found, it will be highly desirable to compare the SCTDHA to the truncated Wigner method (the simplest quantum method).

4- The distinction between the effects of the longitudinal trapping potential, on the one hand, and of the finite system size (and the related discreteness of the spectrum), on the other hand, should be clarified. If possible, the results for the hard-wall boundary conditions and for the periodic boundary conditions should be compared.

5- If possible, analytic estimations of the relaxation (or energy-flow) time scale(s) should be provided.

6- The authors should carefully proof-read the paper before resubmission to correct the typos and other small inconsistencies.

(in reply to Report 1 on 2020-03-22)

We thank the referee for their report and for their time in reading our work. However, we strongly disagree with the criticisms the referee raised and below respond to them point by point. We also wish to note that the referee's suggestions are mostly far beyond the scope of our work and while they address issues related to our work, they essentially constitute separate research projects.

The main point we would like to make is that the referee's report focusses on what they think should be done in order to develop a quantitative description of the experiments carried out by the Vienna group, rather than engaging with our work, which addresses a problem that is motivated by said experiments but deals with a much narrower set of theoretical issues.

As we have made very clear in our manuscript much of the interest in the Josephson oscillation experiments conducted by the Vienna group arises from their potential to simulate non-equilibrium dynamics in a (weakly perturbed) sine-Gordon QFT. In particular, it is believed that an appropriate choice of experimental parameters will allow accessing precisely this regime. An important question is how far from this parameter regime the existing experiments are. The aim of our work is to take a first step towards answering this question,

starting from the low energy scaling regime where the sine-Gordon QFT applies. We identify the leading perturbing operator in this regime and study its effects, and we analyze the effects of spatial inhomogeneity (which are present in the actual experiments). Our work is the first to study both these effects. Rather than engaging with our work in the context it applies to, namely the low energy scaling regime, the referee views our work strictly in the context of the existing experiments and possible approximate approaches for describing them. We feel that the referee thus misunderstands in an essential way the purpose of our work, and its utility for understanding the role of corrections to the sine-Gordon model description in the low energy regime (which we expect to be realized in future experiments).We now address the referee's specific comments in turn.

"In my opinion, the paper doesn't represent a real breakthrough in understanding the out-of-equilibrium dynamics in an extended bosonic Josephson junction. The qualitative picture has become clear a long ago, the key remaining open problem is to find the best quantitative method for the cases where the well-established mean-field approach isn't sufficient and the quantum effects become important."We disagree. There is no known theory that gives a satisfactory account of the damping observed in the experiments. There simply isn't any "well-established mean-field approach" that applies to the problem at hand (we comment more on this below). The main interest in this problem arises from the fact that in the simplest approximation it can be mapped onto a sine-Gordon model, which is a paradigmatic relativistic QFT that exhibits many interesting phenomena in equilibrium. The experiments by the Vienna group have opened the exciting possibility of having a quantum simulator for non-equilibrium physics in this QFT. As far as we know our work is the first to address the issue of how important effects beyond the simple sine-Gordon description will be.

"My first point of criticism concerns the choice of the relevant perturbation. Besides the coupling given by Eq. (11), there is also coupling of the density fluctuation to the square of the phase gradient in each of the two elongated Bose gases. This term has been known not only to dominate the dynamics in a single 1D system at finite temperature [A. F. Andreev, Sov. Phys. JETP 51, 1038 (1980)], but also to destroy the coherence between totally decoupled ($t_{\perp}=0$) quasicondensates [see A. A. Burkov, M. D. Lukin, and E. Demler, PRL 98, 200404 (2007); however, their estimated decoherence time seems to be too short in comparison to what we can expect from an extension of Andreev's analysis]. For non-zero coupling, this term becomes comparable to that considered by the authors at wavelength of order of $\sqrt{\hbar^{2}/ (m z_{\perp})}$ or shorter. In other words, the assumption made by the authors with respect to the relevant perturbation holds at unrealistically low (below $t_{\perp}$) temperatures only."We do not think that this is a fair criticism. Firstly, we are of course fully aware that there are other perturbations, including the one mentioned by the referee. This is very clearly stated in our manuscript. We have in fact begun to study such terms, but their analysis turns out to be technically difficult as they require a non-trivial renormalization. The reason for retaining the particular term we consider is clearly explained in our manuscript (end of Sec. 2.1, paragraph of Eq. (11)): it is the perturbation with the smallest scaling dimension apart from the cosine term. The referee seems to completely ignore this discussion.

As we have repeated above, our key motivation for studying the Josephson oscillation problem is its relation to a sine-Gordon QFT. As always in bosonization this relation a priori applies in an appropriate scaling limit around a quantum critical point (a 2-component Luttinger liquid in the case at hand). The perturbed sine-Gordon model applies in the vicinity of this critical point. The renormalized coupling constants of various perturbations in this regime should be calculated by renormalization group methods. This is why strongly irrelevant operators, like the one mentioned by the referee, are expected to be less important in this regime (as their renormalized coupling constants flow to zero) than perturbations with very low scaling dimensions such as the one we keep. The works cited by the referee deal with very different models/regimes, which do not have a strongly relevant cosine perturbation that dominates the physics. These works therefore do not apply to the regime we are interested in, namely energy scales comparable to the mass gap of the sine-Gordon model in equilibrium. The energy density we use in our work corresponds to a temperature of $5 \, \mathrm{nK}$, which is indeed of the order of the (zero temperature) mass gap in our model.

"Secondly, the SCTDHA doesn't seem to be the best method to solve the given problem. On the one hand, there are methods of the quantum field theory to derive the quantum Boltzmann equation for a 1D system of bosons out of equilibrium [M. Buchhold and S. Diehl, EPJD 69, 224 (2015)]. On the other hand, the usual mean-field approximation suffices to describe results of many experiments in 1D. The authors should have compared their results with the results of solving numerically two coupled Gross-Pitaevskii equations in 1D with stochastic initial conditions and possibly to the results of the truncated Wigner approach to understand the influence of the quantum noise [perhaps, the latter approach works well in 1D on longer times then it does in higher dimensions]. I would be not surprised if the results of the SCTDHA and the numerical solution of the mean-field equations [following from the semiclasscal version of Eq. (3)] averaged over the ensemble of initial conditions sampling the displaced thermal state turn out to be very close. Indeed, the initial state discussed in Section 4.1 corresponds to the state after a fast splitting that undergoes a rapid prethermalization, see Ref. [17]. But the prethermalized state contains not only zero-point oscillations of the postquench Hamiltonian, but also large semiclassical excitations corresponding to the temperature of order of the chemical potential. This means that the quantum noise plays an important role only at large momenta, where Eq. (6) is no more valid anyway."We think that this criticism is unjustified for the following reasons. The SCTDHA is a sensible and well-motivated approximation

in the regime we are studying. This is because the regime of the sine-Gordon model relevant to the experiments is close to the semi-classical limit, in which the solution of the (nonlinear) classical equations of motion would already provide a good approximation. The SCTDHA considers self-consistent Gaussian fluctuations around the solutions of the classical equations of motion. Given that we are starting in a Gaussian state, this is expected to provide a good approximationat short times, which is the regime of interest to us here. We fail to understand the referee's objections to this approximation.The other approaches the referee mentions are very different in nature. We agree that it would be interesting to analyze the (perturbed) sine-Gordon model by a quantum Boltzmann equation. Given that we are interested in effects of spatial inhomogeneities this is numerically very expensive, and would constitute an interesting and worthwhile, but entirely separate, research project.

The referee also asks us to compare our results to "solving numerically two coupled Gross-Pitaevskii equations in 1D with stochastic initial conditions and possibly to the results of the truncated Wigner approach to understand the influence of the quantum noise." We frankly do not see how doing this could help answering the questions we are addressing in our paper. As we have stressed repeatedly, our work is concerned with the experiments only in so far as they motivate the study of the perturbed sine-Gordon model, describing a particular scaling limit of the original problem. The approaches mentioned by the referee are uncontrolled approximations to the original problem that has been studied experimentally. We see no reason why the results of the approaches mentioned by the referee should agree quantitatively with ours, or how (dis)agreements can help us answer our research question, which is whether sector mixing and spatial inhomogeneity lead to important effects in the regime of the sine-Gordon model relevant to experiments.

"The same applies to the choice of the boundary conditions. One can set inhomogeneous initial conditions in any finite-size system. In that sense there is no qualitative difference between the hard-wall boundary conditions Eq. (4) and periodic boundary conditions over a finite period L (a system on a ring)."We strongly object to this comment. The point is that there are observable

quantitativedifferences between hard wall and periodic boundary conditions. These are clearly seen at times late enough for correlations to have spread through the system. This is in fact shown in our manuscript, where we compare the results for periodic boundary conditions (PBC) to those for hard-wall boundary conditions, in Sec. (4.3.1) and Fig. 2. One of the reasons for working with hard wall boundary conditions is precisely to ascertain the form and size of these effects, and find out whether these might play a role in the effects of the mixing term or in the damping of Josephson oscillations."The authors should carefully distinguish between the effects of the longitudinal trapping that causes inhomogeneity of the background density profile and of the finite size only (with $\rho_{0}=$ const)."We do indeed carefully distinguish between these effects in our manuscript. The referee appears to have missed this. We compare the results for periodic boundary conditions (PBC) to those for hard-wall boundary conditions, in Sec. (4.3.1) and Fig. 2. Deviations from the PBC result are clearly visible after the traversal time $L/(2v)$. We feel that the discussion is quite clear, but would be happy to emphasize this point in our paper more strongly if required.

"The third weak point of the paper is the absence of analytic estimations (even roughest ones) of the lifetime of elementary excitations and of the typical time of equilibration between the symmetric and antisymmetric subsystems. Sometimes it is possible to use scaling arguments for the quantum Boltzmann equations, sometimes an empirical analysis of numerical results yields a formula. And such an estimation is usually very helpful for understanding the physics of a system. The paper will gain much novelty and significance if the authors summarize their gained knowledge in few simple expressions."This comment suggests that the referee has failed to appreciate a key aspect of our work. The SCTDHA is designed to describe the

short timebehaviour of the system and is then a well motivated approximation as our initial state is Gaussian. As we have discussed in our previous work at late times the SCTDHA cannot be expected to work quantitatively because higher cumulants become appreciable and need to be taken into account. The Josephson oscillations occur on a short time scale and are therefore in principle accessible to the SCTDHA. If equilibration between the symmetric and antisymmetric subsystems occurs (for which we supply no evidence, and which could only happen in a time-averaged fashion as the system is finite), this would happen at late time scales, for which the SCTDHA is simply not expected to hold. We therefore strongly object to the referee's criticism, as it asks us to make statements about timescales which lie beyond the scope of our work (and outside of the regime of applicability of our method, and in fact any other method that has been brought to bear on this problem).We are not entirely sure about what the referee has in mind with regard to their request to provide analytic results for the lifetime of elementary excitations. We interpret it as follows: one might envisage developing a quantum Boltzmann approach that starts with the Klein-Gordon plus free boson theory, and then treats the higher powers of $\phi$ and sector coupling terms as perturbations. While we agree that it would be interesting to carry out such an analysis, it is completely different in nature and spirit from what we do in our work and constitutes a separate research project.

"To summarize, a major revision of the paper is needed before its resubmission."We strongly disagree for the detailed reasons we have given above. We think the points raised by the referee are aimed at either studying a different regime of the model (higher temperatures, late times) or posing interesting but very different research questions (e.g., studying Gross-Pitaevskii theory to model split Bose gases). Such questions are aimed at directly modelling the experiments, rather than answering the research question we have posed, which is to understand the effects of perturbations to the sine-Gordon model at short times and low temperatures, in view of the experiments. We feel that the report therefore does not engage in any substantial way with the work we have actually done, making it seem like a list of suggestions for the research projects we should have performed instead.