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arXiv:0807.3129

Bose-Einstein condensates of polar molecules: anisotropic interactions = anisotropic mass

Andrei Derevianko

Published Jul 20, 2008• 0 citations
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Abstract

So far the theory of Bose-Einstein condensates (BEC) of polar molecules was based on an ad hoc generalization of equations for spherical atoms. Here I adopt a rigorous pseudo-potential approach to low-energy dipolar interactions and derive a non-linear mean-field Schrodinger equation for a harmonically-trapped condensate. I show that, effectively, the dipolar interactions alter molecular mass. The resulting effective mass is anisotropic: to the leading order the mass is altered only for the motion along the polarizing field. For a typical BEC of spin-polarized magnetically-interacting alkali-metal atoms the effective atomic mass is reduced by 10% from its bare value. For a BEC of polar molecules the mass may be reduced by a factor of a 1,000.

DOI
10.48550/arxiv.0807.3129

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So far the theory of Bose-Einstein condensates (BEC) of polar molecules was based on an ad hoc generalization of equations for spherical atoms. Here I adopt a rigorous pseudo-potential approach to low-energy dipolar interactions and derive a non-linear mean-field Schrodinger equation for a harmonically-trapped condensate. I show that, effectively, the dipolar interactions alter molecular mass. The resulting effective mass is anisotropic: to the leading order the mass is altered only for the motion along the polarizing field. For a typical BEC of spin-polarized magnetically-interacting alkali-metal atoms the effective atomic mass is reduced by 10% from its bare value. For a BEC of polar molecules the mass may be reduced by a factor of a 1,000.
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.
This thesis contains a study of ultracold paramagnetic atoms or polar molecules characterized by a long-range anisotropic dipolar interaction. We particularly focus on two aspects of ultracold dipolar gases. In the first problem the ground state properties of dipolar Bose-Einstein condensates (BEC) are investigated. This problem has gained importance due to recent experimental advances in achieving a condensate of Chromium atoms and ongoing research to produce quantum degenerate polar molecules. In the second problem, we consider possible applications of ultracold polar molecules to rotation sensing and interferometry. First, we concentrate on the interplay between the trapping geometry and dipole-dipole interaction for a polarized dipolar bosonic condensate. As the dipole-dipole interaction is attractive along the polarized direction, the lowest energy state of the BEC is always a collapsed state. However by applying a trapping potential along the polarization direction it is possible to achieve a metastable dipolar BEC. By numerically solving the Gross-Pitaevskii equation, we show that below a critical interaction strength, a metastable state exists depending on the trapping geometry. We also show that a novel feature of dipolar BEC is the appearance of different structural metastable ground states for certain combinations of trapping geometry and particle number. Next, by mixing in single component fermions we show that dipolar BEC can be stabilized against collapse in pancake shaped or cylindrical traps. We also show that the excitation spectrum of the BEC may have a minimum for non-zero momentum, termed a “roton minimum”. This minimum leads to a transition to stable or metastable density-wave states depending on the density of the bosons and boson-fermion interaction strength. In the second problem, we study a proposal for a large-angle coherent beam splitter for polar molecules. By taking into account the effect of a quasi-static external electric field on the rotational levels of the polarized molecules we show that it is possible to coherently split a stationary cloud of molecules into two counter-propagating components. We then investigate the effect of longitudinal acceleration on the transverse motion of the particles, assuming that the longitudinal motion of the molecules can be approximated classically by a wave packet with some mean velocity while the transverse motion is governed by quantum mechanics. We propose a particular time-dependent shape of acceleration to minimize the excitations in the transverse motion. Our theory is also applicable to the general case of particles moving along a circular guide with time-dependent longitudinal velocity. In addition, we include the effects of velocity fluctuations due to noise in the accelerating field.
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This study investigates the dynamical evolution of spin-1 and spin-2 Bose-Einstein Condensates (BECs) when the dipolar interaction is taken into consideration. Using the Single-Mode Approximation (SMA) equations of motion are derived for each condensate component which are then numerically solved. The results convincingly show that the dipolar interaction has a significant effect despite being much weaker than the collision or spin-exchange interaction and that if included magnetic effects are easy to treat. Further, an analytical Mean-Field Theory formulation is used to derive and simplify equations of motion for a spin-1 BEC in 2D (pancake) configuration. Moving to a reference frame that rotates at the Larmor frequency greatly simplifies the equations and removes first-order Zeeman terms, rendering the effects of the dipolar interaction much clearer to observe in the condensate evolution.
The recent creation of a Bose-Einstein condensate (BEC) of dipolar molecules has opened a new frontier for many-body quantum systems in which dipolar interactions can drive novel self-organization phenomena. Here, we observe electrostriction in a molecular BEC, an elliptical deformation driven by anisotropic dipolar interactions. We use double microwave dressing, involving σσσ- and πππ-polarized fields, to control non-axially symmetric dipolar interactions. We compare the experimental observations of electrostriction to a model based on an extended Gross-Pitaevskii equation and find excellent agreement in the regime of weak to moderate interactions. Using electrostriction, we demonstrate that the molecular BEC can be torqued by dynamically changing the orientation of the elliptical σσσ microwave field. This provides a route to setting molecular quantum gases into rotation, opening opportunities to probe vorticity, superfluidity, and supersolidity in strongly dipolar matter.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.