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Vortex matter and ultracold superstrings in optical lattices

Michiel Snoek

Published Jun 23, 2006• 0 citations
OpenAlex

Abstract

The combination of a rotating cigar-shaped Bose-Einstein condensate with a one-dimensional optical lattice gives rise to very interesting physics. The one-dimensional optical lattice splits the Bose-Einstein condensate into two-dimensional pancake-condensates, each containing a small number of particles and coupled by tunneling processes. The rotation gives rise to the appearance of quantized vortices. The quantum fluctuations of these vortex lines are strongly enhanced because of the small number of particles in each pancake-condensate. For slow rotation the system contains a single vortex-line. The transverse fluctuations of this vortex-line are bosonic modes and obey the same equations as a non-relativistic bosonic string in four dimensions. When fermionic atoms are bound to the vortex-core, it is possible to tune the system parameters such that the dispersions of the fermionic and bosonic modes are identical and the system is supersymmetric. This is called the ultracold superstring and it corresponds to the non-relativistic limit of a four-dimensional superstring. The supersymmetry is experimentally observable by performing density measurements. Moreover, it stabilizes the vortex-line from dissipation. Dissipation means that the vortex-line moves out of the center of the condensate. Since this is directly visible experimentally, this system allows for the investigation of the dynamics of sypersymmetry breaking. For fast rotation a vortex lattice appears. Because of the optical lattice, the modes of the optical lattice get a dispersion in the axial direction of the condensate. When the ratio of the number of particles to the number of vortices is small enough, the vortex lattice melts. Because of the small number of particles in each pancake-condensate, the particle density is inhomogeneous. Therefore also the vortex-lattice melting occurs homogeneous and phase coexistence between a vortex crystal and a vortex liquid occurs. By looking into the correlation functions between the vortex positions, it is possible to distinguish different phases within the vortex crystal and within the vortex liquid.

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The combination of a rotating cigar-shaped Bose-Einstein condensate with a one-dimensional optical lattice gives rise to very interesting physics. The one-dimensional optical lattice splits the Bose-Einstein condensate into two-dimensional pancake-condensates, each containing a small number of particles and coupled by tunneling processes. The rotation gives rise to the appearance of quantized vortices. The quantum fluctuations of these vortex lines are strongly enhanced because of the small number of particles in each pancake-condensate. For slow rotation the system contains a single vortex-line. The transverse fluctuations of this vortex-line are bosonic modes and obey the same equations as a non-relativistic bosonic string in four dimensions. When fermionic atoms are bound to the vortex-core, it is possible to tune the system parameters such that the dispersions of the fermionic and bosonic modes are identical and the system is supersymmetric. This is called the ultracold superstring and it corresponds to the non-relativistic limit of a four-dimensional superstring. The supersymmetry is experimentally observable by performing density measurements. Moreover, it stabilizes the vortex-line from dissipation. Dissipation means that the vortex-line moves out of the center of the condensate. Since this is directly visible experimentally, this system allows for the investigation of the dynamics of sypersymmetry breaking. For fast rotation a vortex lattice appears. Because of the optical lattice, the modes of the optical lattice get a dispersion in the axial direction of the condensate. When the ratio of the number of particles to the number of vortices is small enough, the vortex lattice melts. Because of the small number of particles in each pancake-condensate, the particle density is inhomogeneous. Therefore also the vortex-lattice melting occurs homogeneous and phase coexistence between a vortex crystal and a vortex liquid occurs. By looking into the correlation functions between the vortex positions, it is possible to distinguish different phases within the vortex crystal and within the vortex liquid.
We investigate quantum fluctuations of a vortex lattice in a one-dimensional optical lattice for realistic numbers of particles and vortices. Our method gives full access to all the modes of the vortex lattice and we discuss in particular the Bloch bands of the Tkachenko modes. Because of the small number of particles in the pancake Bose-Einstein condensates at every site of the optical lattice, finite-size effects become very important. Therefore, the fluctuations in the vortex positions are inhomogeneous and the melting of the lattice occurs from the outside inwards. By looking into correlations between neighboring vortices, we identify new solid and liquid phases. Tunneling between neighboring pancakes substantially reduces the inhomogeneity as well as the size of the fluctuations.
We investigate quantum and temperature fluctuations of a vortex lattice in a one-dimensional optical lattice. We discuss in particular the Bloch bands of the Tkachenko modes and calculate the correlation function of the vortex positions along the direction of the optical lattice. Because of the small number of particles in the pancake Bose-Einstein condensates at every site of the optical lattice, finite-size effects become very important. Moreover, the fluctuations in the vortex positions are inhomogeneous due to the inhomogeneous density. As a result, the melting of the lattice occurs from the outside inwards. However, tunneling between neighboring pancakes substantially reduces the inhomogeneity as well as the size of the fluctuations. On the other hand, nonzero temperatures increase the size of the fluctuations dramatically. We calculate the crossover temperature from quantum melting to classical melting. We also investigate melting in the presence of a quartic radial potential, where a liquid can form in the center instead of at the outer edge of the pancake Bose-Einstein condensates.
The combination of a vortex line in a one-dimensional optical lattice with fermions bound to the vortex core makes up an ultracold superstring. We give a detailed derivation of the way to make this supersymmetric string in the laboratory. In particular, we discuss the presence of a fermionic bound state in the vortex core and the tuning of the laser beams needed to achieve supersymmetry. Moreover, we discuss experimental consequences of supersymmetry and identify the precise supersymmetry in the problem. Finally, we make the mathematical connection with string theory.
We propose a setup with ultracold atomic gases that can be used to make a nonrelativistic superstring in four spacetime dimensions. In particular, we consider for the creation of the superstring a fermionic atomic gas that is trapped in the core of a vortex in a Bose-Einstein condensate. We explain the required tuning of experimental parameters to achieve supersymmetry between the fermionic atoms and the bosonic modes describing the oscillations in the vortex position. Furthermore, we discuss the experimental consequences of supersymmetry.
This body of work examines the non-equilibrium dynamics of vortex lattice carrying Bose–Einstein condensates. We solve the mean-feld Gross–Pitaevskii equation for a two-dimensional pancake geometry, in the co-rotating frame within the limit of high rotation frequencies. The condensate responds to this by creating a large periodic lattice of vortices with 6-fold triangular symmetry. By applying two distinct perturbations to this lattice, we examine the resulting effects on the vortices during time evolution. The frst perturbation involves applying an optical potential with matching geometry to the vortex lattice. We observe the appearance of interference fringes, and we show that these can be described by moiré interference theory. This is backed up by a decomposition of the kinetic energy spectra of the condensate. The applied perturbation only modifes the condensate density, with the vortex positions largely unaffected. From this we conclude that the vortex lattice is very stable and robust against phononic disturbances. Next, by removing vortices at predefned positions in the lattice using phase imprinting techniques, we examine the resulting order of the lattice. By performing this we generate stable topological defects in the crystal structure. The resulting lattice remains highly ordered in the presence of low numbers of these defects, where crystal structure and order of the lattice shows to be highly robust. By varying the type of imprinted phases we can create controllable degrees of disorder in the lattice. This disorder is analysed using orientational correlations, Delaunay triangulation, and Voronoi diagrams of the vortex lattice, and demonstrates a method for examining order and generating disorder in vortex lattices in Bose–Einstein condensates. All work described makes extensive use of GPU computing techniques, and allows for the simulation of these systems to be realised in short times. The implementation of the calculations using GPU computing are also discussed, where the software is shown to be the fastest of its kind out of the independently tested software suites.
In this review, we give an overview of the experimental and theoretical advances in the physics of quantized vortices in dilute atomic-gas Bose--Einstein condensates in a trapping potential, especially focusing on experimental research activities and their theoretical interpretations. Making good use of the atom optical technique, the experiments have revealed many novel structural and dynamic properties of quantized vortices by directly visualizing vortex cores from an image of the density profiles. These results lead to a deep understanding of superfluid hydrodynamics of such systems. Typically, vortices are stabilized by a rotating potential created by a laser beam, magnetic field, and thermal gas. Finite size effects and inhomogeneity of the system, originating from the confinement by the trapping potential, yield unique vortex dynamics coupled with the collective excitations of the condensate. Measuring the frequencies of the collective modes is an accurate tool for clarifying the character of the vortex state. The topics included in this review are the mechanism of vortex formation, equilibrium properties, and dynamics of a single vortex and those of a vortex lattice in a rapidly rotating condensate.