## SciPost Submission Page

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

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

### Submission summary

As Contributors: | Yuri Daniel van Nieuwkerk |

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

Date submitted: | 2020-03-19 |

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.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

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

### 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.