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arXiv:cond-mat/0402052 · Physical Review A

Quantum theory of a vortex line in an optical lattice

J.-P. Martikainen, Henk T. C. Stoof

Published May 18, 2004• 24 citations
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Abstract

We investigate the quantum theory of a vortex line in a stack of weakly coupled two-dimensional Bose-Einstein condensates, that is created by a one-dimensional optical lattice. We derive the dispersion relation of the Kelvin modes of the vortex line and also study the coupling between the Kelvin and quadrupole modes. We solve the coupled dynamics of the vortex line and the quadrupole modes, both classically as well as quantum mechanically. The quantum mechanical solution reveals the possibility of generating nonequilibrium squeezed vortex states by strongly driving the quadrupole modes.

DOI
10.1103/physreva.69.053617

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We investigate the quantum theory of a vortex line in a stack of weakly coupled two-dimensional Bose-Einstein condensates, that is created by a one-dimensional optical lattice. We derive the dispersion relation of the Kelvin modes of the vortex line and also study the coupling between the Kelvin and quadrupole modes. We solve the coupled dynamics of the vortex line and the quadrupole modes, both classically as well as quantum mechanically. The quantum mechanical solution reveals the possibility of generating nonequilibrium squeezed vortex states by strongly driving the quadrupole modes.
Using a variational ansatz for the wave function of the Bose-Einstein condensate, we develop a quantum theory of vortices and quadrupole modes in a one-dimensional optical lattice. We study the coupling between the quadrupole modes and Kelvin modes, which turns out to be formally analogous to the theory of parametric processes in quantum optics. This leads to the possibility of squeezing vortices. We solve the quantum multimode problem for the Kelvin modes and quadrupole modes numerically and find properties that cannot be explained with a simple linear-response theory.
We investigate the low-lying excitations of a stack of weakly coupled two-dimensional Bose-Einstein condensates, which is formed by a one-dimensional optical lattice. In particular, we calculate the dispersion relations of the monopole and quadrupole modes, both for the ground state as well as for the case in which the system contains a vortex along the direction of the lasers creating the optical lattice. Our variational approach enables us to determine analytically the dispersion relations for an arbitrary number of atoms in every two-dimensional condensate and for an arbitrary momentum. We also discuss the feasibility of observing our results experimentally.
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.