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arXiv:2011.05135 · New Journal of Physics

Zero-energy modes of two-component Bose–Bose droplets

Paweł Zin, Maciej Pylak, Mariusz Gajda

Published Feb 10, 2021• 12 citations
arXivHTMLPDF

Abstract

Abstract Bose–Bose droplets are self-bound objects emerging from a mixture of two interacting Bose–Einstein condensates when their interactions are appropriately tuned. During droplet formation three continuous symmetries of the system’s Hamiltonian are broken: translational symmetry and two U(1) symmetries, allowing for arbitrary choice of phases of the mean-field wavefunctions describing the two components. Breaking of these symmetries must be accompanied by appearance of zero-energy excitations in the energy spectrum of the system recovering the broken symmetries. Normal modes corresponding to these excitations are the zero-energy modes. Here we find analytic expressions for these modes and introduce Hamiltonians generating their time evolution—dynamics of the droplet’s centers of mass as well as dynamics of the phases of the two droplet’s wavefunctions. When internal types of excitations (quasiparticles) are neglected then the very complex system of a quantum droplet is described using only a few ‘global’ degrees of freedom—the position of the center of mass of the droplet and two phases of two wave-functions, all these being quantum operators. We believe that our work might be useful in describing in a relatively easy way the low energy collisions of quantum droplets in situations where coherent flow of atoms between the droplets takes place.

DOI
10.1088/1367-2630/abe482

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Abstract Bose–Bose droplets are self-bound objects emerging from a mixture of two interacting Bose–Einstein condensates when their interactions are appropriately tuned. During droplet formation three continuous symmetries of the system’s Hamiltonian are broken: translational symmetry and two U(1) symmetries, allowing for arbitrary choice of phases of the mean-field wavefunctions describing the two components. Breaking of these symmetries must be accompanied by appearance of zero-energy excitations in the energy spectrum of the system recovering the broken symmetries. Normal modes corresponding to these excitations are the zero-energy modes. Here we find analytic expressions for these modes and introduce Hamiltonians generating their time evolution—dynamics of the droplet’s centers of mass as well as dynamics of the phases of the two droplet’s wavefunctions. When internal types of excitations (quasiparticles) are neglected then the very complex system of a quantum droplet is described using only a few ‘global’ degrees of freedom—the position of the center of mass of the droplet and two phases of two wave-functions, all these being quantum operators. We believe that our work might be useful in describing in a relatively easy way the low energy collisions of quantum droplets in situations where coherent flow of atoms between the droplets takes place.
We study coherent dynamics of two interacting Bose-Bose droplets by means of the extended Gross-Pitaevskii equation. The relative motion of the droplets couples to the phases of their components. The dynamics can be understood in terms of the evolution of zero-energy modes recovering symmetries spontaneously broken by the mean-field solution. These are translational symmetry and two U(1) symmetries, associated with the phases of the droplets' two components. A phase-dependent interaction potential and double Josephson-junction equations are introduced to explain the observed variety of different scenarios of collision. We show that the evolution of the droplets is a macroscopic manifestation of the hidden dynamics of their phases. The occurrence of nondissipative drag between the two supercurrents (Andreev-Bashkin effect) is mentioned.
Abstract Laser induced transitions between internal states of atoms have been playing a fundamental role to manipulate atomic clouds for many decades. In absence of interactions each atom behaves independently and their coherent quantum dynamics is described by the Rabi model. Since the experimental observation of Bose condensation in dilute gases, static and dynamical properties of multicomponent quantum gases have been extensively investigated. Moreover, at very low temperatures quantum fluctuations crucially affect the equation of state of many-body systems. Here we study the effects of quantum fluctuations on a Rabi-coupled two-component Bose gas of interacting alkali atoms. The divergent zero-point energy of gapless and gapped elementary excitations of the uniform system is properly regularized obtaining a meaningful analytical expression for the beyond-mean-field equation of state. In the case of attractive inter-particle interaction we show that the quantum pressure arising from Gaussian fluctuations can prevent the collapse of the mixture with the creation of a self-bound droplet. We characterize the droplet phase and discover an energetic instability above a critical Rabi frequency provoking the evaporation of the droplet. Finally, we suggest an experiment to observe such quantum droplets using Rabi-coupled internal states of K39 atoms.
These lecture notes contain an introduction to quantum simulation of bosonic systems in the continuum, focusing on weakly interacting Bose-Bose mixtures with competing mean-field interactions. When the values of such interactions are fine-tuned to almost completely cancel the mean-field energy, quantum fluctuations become apparent and dominate the behavior of the system, stabilizing an ultradilute quantum liquid phase. An analogous situation appears in single-component dipolar quantum gases. We review the mechanism that gives rise to this exotic quantum liquid, which can form droplets that are self-bound in the absence of any external confinement, and discuss their properties and dynamics in both the mixture and the dipolar cases. In dipolar gases, arrays of dipolar droplets stabilized by quantum fluctuations can establish global phase coherence and form supersolids. In bosonic mixtures, supersolidity can emerge already at the mean-field level through spin-orbit coupling. We discuss the properties of such spin-orbit-coupled supersolids, comparing them to their dipolar counterparts. Specifically, we focus on their periodic density modulation, phase coherence, and peculiar excitation spectrum, which hosts both superfluid and crystal excitations. Finally, we conclude by discussing open research directions in the areas of quantum liquid droplets and spin-orbit-coupled supersolids, in particular at the interface of the two research topics.
We study liquid quantum droplets in a mixture of two-component Bose-Einstein condensates under variable confinement introduced along one or two spatial dimensions. Although the atom-atom scattering has a three-dimensional character, discreteness of the available modes in the reduced dimension(s) strongly influences the zero-point energy, which for a weakly interacting three-dimensional system is given by the Lee-Huang-Yang term. Here, we generalize the result of Lee, Huang, and Yang and calculate the higher order corrections to the mean-field energy for a single component Bose-Einstein condensate under condition of a tight confinement in one or two spatial directions. We apply our results to the two-component mixture of bosonic species under conditions favorable for the creation of quantum liquid droplets, which constitute a perfect playground for studies of the beyond mean-field effects. Our findings allow one to analyze the properties of quantum droplets at the dimensional crossover and provide demanding conditions for accessing quasilow dimensions. They also pave a way for experimental exploration of this state of matter in reduced dimensions.
Quantum droplets are ultradilute liquid states which emerge from the competitive interplay of two Hamiltonian terms, the mean-field energy and beyond-mean-field correction, in a weakly interacting binary Bose gas. We relate the formation of droplets in symmetric and asymmetric two-component one-dimensional boson systems to the modulational instability of a spatially uniform state driven by the beyond-mean-field term. Asymmetry between the components may be caused by their unequal populations or unequal intra-component interaction strengths. Stability of both symmetric and asymmetric droplets is investigated. Robustness of the symmetric solutions against symmetry-breaking perturbations is confirmed.
Abstract We consider a two-dimensional self-bound quantum droplet, which consists of a mixture of two Bose–Einstein condensates. We start with the ground state and then turn to the rotational response of this system, in the presence of an external (harmonic) potential. We identify various phases, depending on the atom number, the strength of the external confinement and the angular momentum. These include center of mass excitation, ghost vortices, as well as vortices of single and multiple quantization. According to our results, this is an excellent system for the study of superfluid states.