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arXiv:cond-mat/0604671 · Physical Review A

Theory of ultracold superstrings

Michiel Snoek, Stefan Vandoren, Henk T. C. Stoof

Published Sep 12, 2006• 22 citations
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Abstract

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.

DOI
10.1103/physreva.74.033607

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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.
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 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.
Supersymmetry is assumed to be a basic symmetry of the world in many high-energy theories, but none of the superpartners of any known elementary particle have been observed yet. We argue that supersymmetry can also be realized and studied in ultracold atomic systems with a mixture of bosons and fermions, with properly tuned interactions and single particle dispersion. We further show that in such nonrelativistic systems supersymmetry is either spontaneously broken or explicitly broken by a chemical potential difference between the bosons and fermions. In both cases the system supports a sharp fermionic collective mode similar to the Goldstino mode in high-energy physics, due to supersymmetry. We also discuss possible ways to detect this mode experimentally.
We discuss ultracold Fermi gases in two dimensions, which could be realized in a strongly confining one-dimensional optical lattice. We obtain the temperature versus effective interaction phase diagram for an s-wave superfluid and show that, below a certain critical temperature Tc, spontaneous vortex-antivortex pairs appear for all coupling strengths. In addition, we show that the evolution from weak-to-strong coupling is smooth, and that the system forms a square vortex-antivortex lattice at a lower critical temperature TM.
Fermions localized within vortex cores can form one-dimensional Fermi liquids. The nonzero density of states in these Fermi-liquids can lead to instability of the symmetric structure of the vortex core. We consider a symmetry breaking which is obtained due to spontaneous admixture of the spin-triplet p-wave component of the order parameter in conventional s-wave vortex in the presence of magnetic field. This occurs at low enough temperature and leads to an asymmetric shape of the vortex core. Similar phenomenon of the symmetry breaking induced by the core fermions occurs in electroweak ZZZ-strings (S.G. Naculich, Phys. Rev. Lett., 75 (1995), 998).