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arXiv:2305.09960 · Physical Review A

Scattering of one-dimensional quantum droplets by a reflectionless potential well

Xiaoxiao Hu, Zhiqiang Li, Yu Guo, Yajiang Chen, Xiaobing Luo

Published Nov 3, 2023• 9 citations
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Abstract

We investigate both analytically and numerically the scattering of one-dimensional quantum droplets by a P\"oschl-Teller reflectionless potential well, confirming that there is a sharp transition between full reflection and full transmission at a certain critical incident speed for both small droplets and large flat-top droplets. We observe sharp differences between small quantum droplet scattering and large quantum droplet scattering. The scattering of small quantum droplets is similar to that of solitons, where a spatially symmetric trapped mode is formed at the critical speed, whereas for large quantum droplets a spatially asymmetric trapped mode is formed. Additionally, a nonmonotonic dependence of the critical speed on the atom number is identified: On the small-droplet side, the critical speed increases with the atom number, while in the flat-top regime, the critical speed decreases with increasing atom number. Strikingly, the scattering excites internal modes below the particle-emission threshold, preventing the quantum droplets from emitting radiation upon interaction with the potential. Analysis of the small-amplitude excitation spectrum shows that as the number of particles increases, it becomes increasingly difficult to emit particles out of the droplet during scattering, while radiation from solitons cannot be completely avoided. Finally, we study the collision of two quantum droplets at the reflectionless potential, revealing the role of the $\ensuremath{\pi}$-phase difference generator played by the reflectionless potential.

DOI
10.1103/physreva.108.053306

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We investigate both analytically and numerically the scattering of one-dimensional quantum droplets by a P"oschl-Teller reflectionless potential well, confirming that there is a sharp transition between full reflection and full transmission at a certain critical incident speed for both small droplets and large flat-top droplets. We observe sharp differences between small quantum droplet scattering and large quantum droplet scattering. The scattering of small quantum droplets is similar to that of solitons, where a spatially symmetric trapped mode is formed at the critical speed, whereas for large quantum droplets a spatially asymmetric trapped mode is formed. Additionally, a nonmonotonic dependence of the critical speed on the atom number is identified: On the small-droplet side, the critical speed increases with the atom number, while in the flat-top regime, the critical speed decreases with increasing atom number. Strikingly, the scattering excites internal modes below the particle-emission threshold, preventing the quantum droplets from emitting radiation upon interaction with the potential. Analysis of the small-amplitude excitation spectrum shows that as the number of particles increases, it becomes increasingly difficult to emit particles out of the droplet during scattering, while radiation from solitons cannot be completely avoided. Finally, we study the collision of two quantum droplets at the reflectionless potential, revealing the role of the \ensuremathπ\ensuremath{\pi}\ensuremathπ-phase difference generator played by the reflectionless potential.
We investigate theoretically and numerically quantum reflection of dark solitons propagating through an external reflectionless potential barrier or in the presence of a position-dependent dispersion. We confirm that quantum reflection occurs in both cases with sharp transition between complete reflection and complete transmission at a critical initial soliton speed. The critical speed is calculated numerically and analytically in terms of the soliton and potential parameters. Analytical expressions for the critical speed were derived using the exact trapped mode, a time-independent, and a time-dependent variational calculations. It is then shown that resonant scattering occurs at the critical speed, where the energy of the incoming soliton is resonant with that of a trapped mode. Reasonable agreement between analytical and numerical values for the critical speed is obtained as long as a periodic multi-soliton ejection regime is avoided.
We analyze quantum droplets formed in a two-dimensional symmetric mixture of Bose-Einstein condensed atoms. For sufficiently large atom numbers, these droplets exhibit a flattop density profile with sharp boundaries governed by surface tension. Within the bulk of the droplet, traveling matter waves-localized density dips-can propagate at constant velocity while maintaining their shape. Using numerical simulations and qualitative analysis, we investigate the rich phenomenology that arises when such excitations reach the boundary of a finite droplet. We show that they can emit a small outgoing droplet, excite internal modes of the host soliton, or, in the case of vortex-antivortex pairs, split into individual vortices propagating backward near the edge. Furthermore, we demonstrate that traveling waves can be dynamically generated near the boundary through the collision of distinct droplets, and we discuss their trajectories and interactions.
The structure and dynamics of one-dimensional binary Bose gases forming quantum droplets is studied by solving the corresponding amended Gross-Pitaevskii equation. Two physically different regimes are identified, corresponding to small droplets of an approximately Gaussian shape and large ``puddles'' with a broad flat-top plateau. Small droplets collide quasielastically, featuring the solitonlike behavior. On the other hand, large colliding droplets may merge or suffer fragmentation, depending on their relative velocity. The frequency of a breathing excited state of droplets, as predicted by the dynamical variational approximation based on the Gaussian ansatz, is found to be in good agreement with numerical results. Finally, the stability diagram for a single droplet with respect to shape excitations with a given wave number is drawn, being consistent with preservation of the Weber number for large droplets.
The collisional dynamics of two symmetric droplets with equal intraspecies scattering lengths and particle number density for each component is studied by solving the corresponding extended Gross—Pitaevskii equation in two dimensions by including a logarithmic correction term in the usual contact interaction. We find the merging droplet after collision experiences a quadrupole oscillation in its shape and the oscillation period is found to be independent of the incidental momentum for small droplets. With increasing collision momentum the colliding droplets may separate into two, or even more, and finally into small pieces of droplets. For these dynamical phases we manage to present boundaries determined by the remnant particle number in the central area and the damped oscillation of the quadrupole mode. A stability peak for the existence of droplets emerges at the critical particle number N c ≃ 48 for the quasi-Gaussian and flat-top shapes of the droplets.
We examine a one-dimensional binary mixture of ultradilute quantum droplets in the presence of a central potential defect. The properties of the potential strength and the number of atoms exhibit distinct polarization transitions and low-lying ollective excitation spectra. The balance of (attractive) Lee-Huang-Yang quantum fluctuations and repulsive mean-field interactions results in a self-bound quantum droplet in a potential well. However, the potential barrier causes the polarization transition with the number of atoms, which is reflected in the excitation spectrum as a discontinuity and softening of the quasiparticle modes. We reveal that the critical number of atoms for the transition decreases as the attractive intercomponent interaction increases. Finally, the quench time dynamics of the interaction in potential barrier and well show the localization and diffusive fragmented droplets of two-component systems.
Abstract Solitonic excitations of the one-dimensional quantum droplets are obtained, which smoothly connect vacuum with the flat-top droplet, akin to compactons in classical liquids. These solitons are of the kink type, necessarily residing on a constant pedestal, determined by the mean-field (MF) repulsion and beyond mean field quantum correction and having exactly one-third of the uniform condensate amplitude. Akin to the kinks, the propagating modes occur in pairs and are phase-locked with the background. The lowest chemical potential and condensate amplitude at the flat-top boundary matches with the self-trapped quantum droplet. More general excitations of analogous kind are obtained through the Möbius transform, which connect the required solutions to elliptic functions in general.