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.
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.
Quantum droplets represent a novel state of matter achieved by tailoring a binary Bose–Einstein condensate, where the collapse driven by self-attraction is arrested by quantum effects. This opens the possibility of producing interesting novel physical phenomena, which require accurate mathematical models for its correct description. In this work, we apply variational procedures to the mean field equation governing two-dimensional symmetric quantum droplets, to find simple yet very precise formulae for the droplet profile and the relationships between its main parameters, such as internal and external radii of the bright rings, the chemical potential or the number of atoms of the droplets. The comparison with direct numerical calculations demonstrates that our model provides a highly accurate description of the self-trapped eigenstates with flat-top profiles. These results should aid in the exploration of additional effects in liquid-like solitons and vortex states endowed with angular momentum.
Recent theoretical and experimental results show that one-dimensional (1D) weakly interacting atomic Bose-Bose mixtures with repulsive interspecies mean field (MF) interaction are stabilized by attractive quadratic beyond-mean-field (BMF) effects into self-bound quantum droplet (QD) in free space. Here, we construct an exact analytical model to investigate the structure and dynamics of QDs in presence of external harmonic confinement by solving the 1D extended Gross-Pitäevskii equation (eGPE) with temporal variation of MF and BMF interactions. The model provides the analytical form of wavefunction, phase, MF and BMF nonlinearities. The generation of QDs and interesting droplet to soliton transition in presence of regular/expulsive parabolic traps by taking the comparable MF and BMF interactions are illustrated. We derive the phase diagram of the droplet-soliton phase transition between amplitude of MF, BMF interactions and harmonic oscillator frequency. The strength and form of oscillator frequency are identified as key parameter for tuning the compression, fragmentation and transport of droplets. Finally, the stability of the obtained solutions are confirmed from Vakhitov-Kolokolov (VK) criterion and are found stable.
Abstract Supersolid is an exotic state of matter, showing crystalline order with a superfluid background, observed recently in dipolar Bose–Einstein condensate in a trap. Here, we present exact Bloch wave function of the self-trapped supersolid phase, in the presence of mean-field and beyond mean-field interaction. Our general solutions of the amended nonlinear Schrödinger equation are obtained through Möbius transform, connecting a wide class of supersolid solutions to the ubiquitous cnoidal waves. The solutions yield the supersolid phase in the self-trapped quantum matter, where an array of quantum droplets exist, accompanied by a constant condensate. For the supersolid phase, the chemical potential for one class of solutions is the same as that of self-trapped quantum droplets, and is lower for the general non-perturbative solution. Due to the destabilizing effects of fluctuations on long range order in one dimension, the realization of the supersolid phase may be possible in a finite system.
Motivated by recent experiments on light-matter interacting systems, we investigate a dilute Bose gas and self-bound quantum droplets in a one-dimensional optical lattice coupled to a lossy cavity mode. Using a classical-field approach, we determine the stationary states and collective excitations of this non-equilibrium system. Apart from the usual Bogoliubov modes, we identify a polariton-like gapped excitation, the frequency of which softens as a precursor of the density ordering transition. Moreover, its relaxation time diverges as the critical point is approached, signaling the non-equilibrium nature of this transition. Dynamically, this polariton-like mode can be probed by inducing cavity field fluctuations, which in turn generates spatio-temporal oscillations of both the condensate and droplet states. In the droplet regime, we also analyze the bound modes which bear the characteristics of such non-equilibrium self-bound state. In addition, we uncover solitonic non-equilibrium states, including condensate with kink-like configuration and double-droplet state, and investigate their robustness following a sudden quench. Remarkably, although these states become unstable beyond a critical coupling, they continue to manifest in the dynamics, akin to the scarring phenomena. Our results demonstrate that dissipative cavity coupling provides a versatile route for exploring rich non-equilibrium dynamics of condensates and quantum droplets within experimentally accessible settings.
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.
Abstract We study the dynamics of rogue wave excitations and quasi-transitions in one-dimensional quantum droplets, whose homogeneous-density flat top provides an applicable background wave for nonlinear excitation. Using a quenching technique and a localized space modulation technique for the nonlinear effect coefficient, the rogue wave is excited on droplets. The excitation modes of rogue waves can be transitioned to dispersive shock waves or Kuznetsov–Ma breathers by regulating their modulation amplitude and width, and their maximal density and width can be regulated by modulation their amplitude. Meanwhile, two Gaussian modulations for droplets also bring the second-order rogue wave’s dynamics, whose maximal amplitude can approach 3.6 times that of the amplitude of droplets. Our results provide guidance for rogue wave excitation and transition on self-bound background waves and deepen our understanding of quantum droplets.