Soliton resuscitations: asymmetric revivals of the breathing mode of an atomic bright soliton in a harmonic trap

TL;DR

The paper investigates how a bright soliton in a quasi-one-dimensional Bose-Einstein condensate breathes and loses atoms via dispersion, and how placing the soliton in a shallow harmonic trap makes the emitted atoms return, producing non-Markovian, reviving breathing dynamics. By linearizing the dynamics around the soliton and applying matched asymptotics, the authors derive a discrete BdG spectrum in the trap, with explicit expressions for the mode frequencies Ω_n and the normalization Z_n, and they show how the breathing amplitude B(t) decomposes into a sum over these modes. The key finding is that the resuscitation envelopes are asymmetrical due to a spectral distortion in the above-gap spectrum, captured analytically by a closed-form approximation involving Δ_n and a large phase Nπ/ε^2; this reveals a rich, memory-bearing open-system behavior even in a simple soliton-trap setting. The results provide a robust mean-field background for future exploration of quantum many-body effects and non-Markovian dynamics in trapped solitons, and they connect spectral structure to observable revivals via a tractable analytic framework.

Abstract

An atomic bright soliton realised in a quasi-one-dimensional Bose-Einstein condensate can be considered as an open quantum system. The soliton's breathing mode, for example, is damped by emission of atoms from the soliton to spatial infinity, which thus acts as a Markovian environment for the soliton. If the soliton is held in a shallow harmonic trap, however, the environment becomes non-Markovian: emitted atoms oscillate in the trap and eventually return to the soliton, interfering with it, producing periodic revivals of the breathing mode ("resuscitations"). The amplitude envelopes of these breathing revivals shows a curious asymmetry, with a gradual increase in breathing amplitude followed by sudden drop in amplitude that becomes more and more pronounced in later revivals. We explain this asymmetrical revival pattern in the non-Markovian revivals by deriving a close analytical approximation to the Bogoliubov-de Gennes frequency spectrum for the weakly trapped soliton.

Figures (5)

• Figure 1: Small-amplitude soliton breathing in mean field theory, with no trapping potential. The gas density in the attractively interacting condensate is self-bound in a $\mathrm{sech}^2$ profile; the width and peak density of the spatial profile oscillate periodically in time, while the oscillation amplitude decays due to dispersion.
• Figure 2: Asymmetrical resuscitations of soliton breathing in a harmonic trap of frequency $\omega$, as shown in the peak condensate density $|\psi(0,t)|^2$ as a function of time in trap periods. The evolution is given by the one-dimensional Gross-Pitaevskii equation from the non-stationary initial wave function $\psi(x,0)=A(\tilde{\kappa}/\sqrt{g})\mathrm{sech}(\tilde{\kappa} x)$ for $\tilde{\kappa} = \sqrt{50}a_0^{-1}$, where $a_0=\sqrt{\hbar/(2M\omega)}$ (with atomic mass $M$) is the harmonic trap width. The case $A=1$ would be a perfect bright soliton, with no breathing; in this case $A=1.01$, providing a small-amplitude breathing excitation at the chemical potential frequency $\hbar\kappa^2/(2M)=A^4(\tilde{\kappa} a_0)^2\omega\doteq 52\omega$. The amplitude initially decays in an envelope that closely fits the $t^{-1/4}$ steepest descents approximation of Eqn. (\ref{['B0SD']}) (dashed curves), but then the breathing amplitude revives: it recovers to a maximum amplitude approximately every half-period of the harmonic trap. In fact the first revival begins noticeably sooner than a trap half-period; note as well the "trumpet-shaped" asymmetrical envelopes of the revived oscillations.
• Figure 3: Condensate density in space and time, for an initially excited breathing mode in a harmonic trap with frequency $\omega$.
• Figure 4: The same resuscitation sequence shown in Fig. \ref{['trump']} is here rendered faintly in the background, for the same $\delta = 10^{-2}$ and $\varepsilon = (1+\delta)^{-2}/\sqrt{50}\doteq 0.14$. Superimposed in four time intervals around the revival times $t=N\pi/\omega$ are the Bogoliubov-de Gennes approximations $(1+\delta)^2[1-\delta B_N(t-N\pi/\omega)]$, where the integral approximation (\ref{['BNtau']}) has been used for $B_N(\Delta t)$, with the integrals evaluated numerically.
• Figure 5: The factor $X(\nu)$ that appears in the small-argument limit of the parabolic cylinder functions in Eqn. (\ref{['ABT']}) approaches the limit 1 quickly in $\nu$.