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arXiv:cond-mat/0601695 · Physical Review Letters

Vortex-Lattice Melting in a One-Dimensional Optical Lattice

Michiel Snoek, Henk T. C. Stoof

Published Jun 15, 2006• 17 citations
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Abstract

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.

DOI
10.1103/physrevlett.96.230402

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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 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.
Using path integral formalism, we show that the Abrikosov-Tkachenko vortex lattice may equivalently be understood as an array of Josephson junctions. The Tkachenko modes are found to be basically equivalent to the low energy excitations (Goldstone modes) of an ordered state. The calculated frequencies are in very good agreement with recent experimental data. Calculations of the fluctuations of the relative displacements of the vortices show that vortex melting is a result of quantum fluctuations around the ordered state due to the low energy excitations (Tkachenko modes)and occurs when the ratio of the kinectic energy to the potential energy of the vortex lattice is 0.001.
We study the vortex lattice dynamics in the presence of a single impurity as well as random impurities or disorder. We show that in the presence of a single impurity, the vortex lattice gets distorted and the distortion depends on the position of the single impurity with respect to the positions of the vortices in the impurity-free Abrikosov vortex lattice and also on the strength of the impurity potential. We then show that a type of giant hole with hidden vortices inside it can be created in the vortex lattice by a cluster of impurities. In the presence of random impurity potential or disorder, the vortex lattice melts. We show that the vortex lattice also melts in the presence of the pseudorandom potential generated by the superposition of two optical lattices. The absence of long-range order in the melted vortex lattice is demonstrated from the structure factor profile and the histogram of the distances between each pair of vortices.
In the present work we introduce and explore a technique for the efficient removal of vortices from an atomic Bose-Einstein condensate, through the application and subsequent removal of a one-dimensional optical lattice. We showcase a prototypical experimental realization of the technique that motivates a detailed theoretical study of vortex removal mechanisms. Through simulations of the condensate dynamics during application of the optical lattice, we also discover a vortex removal mechanism that arises in narrow, optical-lattice-induced atomic density channels for which the channel width is on the order of the nominal vortex core size and healing length. This mechanism involves the density profile typically associated with a vortex core spatially separating from the phase singularity associated with the vortex. By analyzing numerical experiments covering a wide range of variations of the optical lattice amplitude and fringe periodicity, we identify the existence of an optimal set of parameters that enables the efficient removal of all vortices from the condensate. This analysis paves the way for further studies aimed at understanding vortex dynamics in narrow channels, and adds to an experimental toolkit for working with vortices and controlling the dynamical states of condensates.
We report the observation of vortex nucleation in a rotating optical lattice. A 87Rb Bose-Einstein condensate was loaded into a static two-dimensional lattice and the rotation frequency of the lattice was then increased from zero. We studied how vortex nucleation depended on optical lattice depth and rotation frequency. For deep lattices above the chemical potential of the condensate we observed a linear dependence of the number of vortices created with the rotation frequency, even below the thermodynamic critical frequency required for vortex nucleation. At these lattice depths the system formed an array of Josephson-coupled condensates. The effective magnetic field produced by rotation introduced characteristic relative phases between neighboring condensates, such that vortices were observed upon ramping down the lattice depth and recombining the condensates.