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arXiv:2004.09973 · Journal of Physics B Atomic Molecular and Optical Physics

Supersolid behavior in one-dimensional self-trapped Bose–Einstein condensate

Mithilesh K. Parit, Gargi Tyagi, Dheerendra Singh, Prasanta K Panigrahi

Published Apr 12, 2021• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1361-6455/abf6e0

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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.
We consider interacting Bose particles in an external potential. It is shown that a Bose-Einsteincondensate is possible at finite temperatures that describes a supersolid in three dimensions (3D) for a wide range of potentials in the absence of an external potential. However, for 2D, a self-organized supersolid exists for finite temperatures provided the interaction between bosons is nonlocal and of infinitely long-range. It is interesting that in the absence of the latter type of potential and in the presence of a lattice potential, there is no Bose-Einstein condensate and so in such a case, a 2D supersolid is not possible at finite temperatures. We also propose the correct Bloch form of the condensate wave function valid for finite temperatures, which may be used as the correct trial wave function.
Abstract The properties of a supersolid state (SS) in quasi-one-dimensional dipolar Bose–Einstein condensate is studied, considering two possible mechanisms of realization—due to repulsive three-body atomic interactions and quantum fluctuations in the framework of the Lee–Huang–Yang theory. The role of both mechanisms in the formation of SS properties has been emphasized. The proposed theoretical model, based on minimization of the energy functional, allows evaluating the amplitude of the SS for an arbitrary set of parameters in the extended Gross–Pitaevskii equation (eGPE). To explore the dynamics of the SS first we numerically construct its ground state in different settings, including periodic boundary conditions, box-like trap and parabolic potential, then impose a perturbation. In oscillations of the perturbed supersolid we observe the key manifestation of SS, namely the free flow of the superfluid fraction through the crystalline component of the system. Two distinct oscillation frequencies of the supersolid associated with the superfluid fraction and crystalline components of the wave function are identified from numerical simulations of the eGPE.
Motivated by a recent experiment [L. Chomaz et al., Nat. Phys. 14, 442 (2018)], we perform numerical simulations of a dipolar Bose-Einstein condensate (BEC) in a tubular, periodic confinement at T=0T=0T=0 within density functional theory, where the beyond-mean-field correction to the ground-state energy is included in the local density approximation. We study the excitation spectrum of the system by solving the corresponding Bogoliubov--de Gennes equations. The calculated spectrum shows a roton minimum, and the roton gap decreases by reducing the effective scattering length. As the roton gap disappears, the system spontaneously develops a periodic linear structure formed by denser clusters of atomic dipoles immersed in a dilute superfluid background. This structure shows the hallmarks of a supersolid system, i.e., (i) a finite nonclassical translational inertia along the tube axis and (ii) the appearance of two gapless modes, i.e., a phonon mode associated with density fluctuations and resulting from the translational discrete symmetry of the system, and a Nambu-Goldstone gapless mode corresponding to phase fluctuations, resulting from the spontaneous breaking of the gauge symmetry. A further decrease in the scattering length eventually leads to the formation of a periodic linear array of self-bound droplets.
The study of Bose–Einstein condensation (BEC) in lower dimensions plays an important role in understanding the fundamentals of many-body physics as they can be treated theoretically with relative ease and can be verified experimentally. Recently, observation of a liquid-like state in a BEC mixture has been reported along with a theoretical prescription for its observation in the lower dimension. This observation is unique and has serious ramifications in our prevailing conception of the liquid state, which has a deep influence on the van der Waals theory. In explaining the self-bound nature of this state, quantum fluctuation and its fine balance with mean-field (MF) interaction turn out to be playing a key role. Though the experiments are performed predominantly in three dimensions, theoretical studies extend to the lower dimensions. In this brief review, we plan to summarize the recent theoretical advances in droplet research in the lower dimension and elaborate on the description of our contributions. We will mainly focus on analytical results related to this self-bound state in a one-dimension and quasi one-dimension environment. We aim to cover a few results from the family of cnoidal solutions to droplet solutions with smooth transitions between each other, finishing it by carrying a modest discussion on the supersolid phase.
Supersolidity---a coexistence of superfluidity and crystalline or amorphous density variations---has been vividly debated ever since its conjecture. While the initial focus was on helium-4, recent experiments uncovered supersolidity in ultracold dipolar quantum gases. Here we propose a self-bound supersolid phase in a binary mixture of Bose gases with short-range interactions, making use of the nontrivial properties of spin-orbit coupling. We find that a first-order phase transition from a self-bound supersolid stripe phase to a zero-minimum droplet state of the Bose gas occurs as a function of the Rabi coupling strength. These phases are characterized using the momentum distribution, the transverse spin polarization, and the superfluid fraction. The critical point of the transition is estimated in an analytical framework. The predicted density-modulated supersolid stripe and zero-minimum droplet phase should be experimentally observable in a binary mixture of 39K^{39}\mathrm{K}39K with spin-orbit coupling.
Quantum fluctuations can stabilize Bose-Einstein condensates (BEC) against the mean-field collapse. Stabilization of the condensate has been observed in quantum degenerate Bose-Bose mixtures and dipolar BECs. The fine-tuning of the interatomic interactions can lead to the emergence of two new states of matter: liquid-like self-bound quantum droplets and supersolid crystals formed from these droplets. We review the properties of these exotic states of matter and summarize the experimental progress made using dipolar quantum gases and Bose-Bose mixtures. We conclude with an outline of important open questions that could be addressed in the future.