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arXiv:2007.13946 · Physical Review A

Excitations and number fluctuations in an elongated dipolar Bose-Einstein condensate

Sukla Pal, D. Baillie, P. B. Blakie

Published Oct 7, 2020• 22 citations
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Abstract

We study the properties of a magnetic dipolar Bose-Einstein condensate in an elongated (cigar shaped) confining potential in the beyond quasi-one-dimensional regime. In this system the dipole-dipole interactions develop a momentum dependence related to the transverse confinement and the polarization direction of the dipoles. This leads to density fluctuations being enhanced or suppressed at a length scale related to the transverse confinement length, with local atom number measurements being a practical method to observe these effects in experiments. We use mean-field theory to describe the ground state, excitations, and the local number fluctuations. Quantitative predictions are presented based on full numerical solutions and a simplified variational approach that we develop. In addition to the well-known roton excitation, occurring when the dipoles are polarized along a tightly confined direction, we find an ``antiroton'' effect for the case of dipoles polarized along the long axis: A nearly noninteracting ground state that experiences strongly repulsive interactions with excitations of sufficiently short wavelength.

DOI
10.1103/physreva.102.043306

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We study the properties of a magnetic dipolar Bose-Einstein condensate in an elongated (cigar shaped) confining potential in the beyond quasi-one-dimensional regime. In this system the dipole-dipole interactions develop a momentum dependence related to the transverse confinement and the polarization direction of the dipoles. This leads to density fluctuations being enhanced or suppressed at a length scale related to the transverse confinement length, with local atom number measurements being a practical method to observe these effects in experiments. We use mean-field theory to describe the ground state, excitations, and the local number fluctuations. Quantitative predictions are presented based on full numerical solutions and a simplified variational approach that we develop. In addition to the well-known roton excitation, occurring when the dipoles are polarized along a tightly confined direction, we find an ``antiroton'' effect for the case of dipoles polarized along the long axis: A nearly noninteracting ground state that experiences strongly repulsive interactions with excitations of sufficiently short wavelength.
Abstract We develop a variational theory for a dipolar condensate in an elongated (cigar shaped) confinement potential. Our formulation provides an effective one-dimensional extended meanfield theory for the ground state and its collective excitations. We apply our theory to investigate the properties of rotons in the system comparing the variational treatment to a full numerical solution. We consider the effect of quantum fluctuations on the scattering length at which the roton excitation softens to zero energy.
The effect of dipolar orientation with respect to the condensate plane on the mean-field dynamics of dipolar Bose-Einstein condensates in a pancake-shaped confinement is discussed. The stability of a quasi-two-dimensional condensate, with respect to the tilting angle, is found to be different from a two-dimensional layer of dipoles, indicating the relevance of the transverse extension while characterizing two-dimensional dipolar systems. An anisotropic excitation spectrum exhibiting a highly tunable, rotonlike minimum can arise entirely from the dipole-dipole interactions, by tilting the dipoles. At the magic angle and in the absence of contact interactions, the long-wavelength excitations are not phononlike and always unstable. The post-roton-instability dynamics, in contrast to phonon instability, in a uniform condensate, is featured by a transient, defect-free, stripe pattern, which eventually undergoes local collapses, and driving the condensate back into the stable regime can make them sustained for longer. Hopping between stripes has been observed before it melts into a uniform state in the presence of dissipation. Finally, we discuss a class of solutions, in which a quasi-two-dimensional condensate is self-trapped in one direction, as well as a regime of interaction parameters, including attractive short-range interactions, at which a two-dimensional anisotropic soliton can be stabilized, and we show that a chromium condensate with a relatively small number of atoms is well suited for this.
We study a Bose-Einstein condensate of 52Cr^{52}\mathrm{Cr}52Cr atoms confined in a toroidal trap with a variable strength of sss-wave contact interactions. We analyze the effects of the anisotropic nature of the dipolar interaction by considering the magnetization axis to be perpendicular to the trap symmetry axis. In the absence of a central repulsive barrier, when the trap is purely harmonic, the effect of reducing the scattering length is a tuning of the geometry of the system from a pancake-shaped condensate when it is large to a cigar-shaped condensate for small scattering lengths. For a condensate in a toroidal trap, the interaction in combination with the central repulsive Gaussian barrier produces an azimuthal dependence of the particle density for a fixed radial distance. We find that along the magnetization direction the density decreases as the scattering length is reduced but presents two symmetric density peaks in the perpendicular axis. For even lower values of the scattering length we observe that the system undergoes a dipolar-induced symmetry breaking phenomenon. The whole density becomes concentrated in one of the peaks, resembling an origin-displaced cigar-shaped condensate. In this context we also analyze stationary vortex states and their associated velocity fields, finding that these also show a strong azimuthal dependence for small scattering lengths. The expectation value of the angular momentum along the zzz direction provides a qualitative measure of the difference between the velocity in the different density peaks.
We derive rigorous one- and two-dimensional mean-field equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. We show how the dipolar interaction modifies the contact interaction of the strongly confined atoms. In addition, our equations introduce a nonlocal potential, which is anisotropic for pancake-shaped condensates. We propose to observe this anisotropy via measurement of the condensate aspect ratio. We also derive analytically approximate density profiles from our equations. Both the numerical solutions of our reduced mean-field equations and the analytical density profiles agree well with numerical solutions of the full Gross-Pitaevskii equation while being more efficient to compute.
In a recent experiment, it has been found that by tuning the interaction strength of a dipolar Bose-Einstein Condensate confined in a quasi-1D geometry orthogonal to the polarization direction, the roton gap vanishes, while density modulations are induced in the trapped gas. This suggests the possibility of realizing a supersolid in dipolar quantum gases. Inspired by this experiment, we simulate a dipolar BEC in a ring geometry by confining the dipoles in a ”tube” orthogonal to the polarization direction and enforcing periodic boundary conditions. Our calculations are based on a Density Functional approach where the total energy density functional includes the beyond-mean-field correction of Lee-Huang-Yang, which partially accounts for quantum fluctuations. We study the ground state properties of the system by evolving in imaginary time the EulerLagrange equation derived from the variational principle. We also study the excitation spectrum of the system by solving the corresponding Bogoliubov-de Gennes equations. We find that the calculated excitation spectrum shows indeed a roton minimum, and the roton gap can become vanishingly small by reducing the effective scattering length. As the roton gap disappears, we find that the system spontaneously develops in its ground-state a periodic structure formed by denser clusters of atomic dipoles immersed in a dilute superfluid background. We find that this structure shows the hallmarks of a supersolid system, i.e. (i) a finite non-classical translational inertia and (ii) the appearance, besides the phonon mode, of the Nambu-Goldstone gapless mode corresponding to fluctuations in the phase, and related to the spontaneous breaking of the gauge symmetry.
We develop theory for a flattened dipolar Bose–Einstein condensate produced by harmonic confinement along one direction. The role of both short-ranged contact interactions and long-ranged dipole–dipole interactions is considered, and the dipoles are allowed to be polarized along an arbitrary direction. We discuss the symmetry properties of the condensate and the part of the excitation spectrum determining stability, and introduce two effective interaction parameters that allow us to provide a general description of the condensate properties, rotons, and stability. We diagonalize the full theory to obtain benchmark results for the condensate and quasiparticle excitations, and characterize the exact mean field stability of the system. We provide a unified formulation for a number of approximate schemes to describe the condensate and quasiparticles, including the standard quasi-two-dimensional approximation, two kinds of variational ansatz, and a Thomas–Fermi approximation. Some of these schemes have been widely used in the literature despite not being substantiated against the exact theory. We provide this validation and establish the regimes where the various theories perform well.