Dynamics of formation of topological defects in continuous phase transitions using Bose gas systemsBackgroundTopological defects are structures formed in quantum fields that undergo `imperfect' symmetry-breaking phase transitions. This type of transition, where a system not only undergoes a change of state but a change of symmetry as well, is very common in physics. The topological defects that are formed give us insight into the dynamics of the phase transition. One example of a topological defect is a vortex in a superfluid, where the phase of the superfluid component varies continuously through two pi about a point where the fluid density is zero.The dynamics of continuous phase transitions has long been an outstanding problem in the field of critical phenomena [1]. Part of the difficulty is that perturbation theory diverges in the vicinity of a critical point, and so calculations become rather difficult. Such phenomena are important in cosmology – e.g. it is suggested that the formation of topological defects in such a transition in the early universe seeded the density fluctuations that have since evolved into galaxies. The initial density of the defects would then correspond to the density of galaxies we see today. It was originally suggested by Tom Kibble in 1976 that at a continuous phase transition there would be sufficient thermal energy about such that domain structures not be fixed, and topological defects would not be frozen in at the critical point [2]. The domains would not become stable until the temperature had decreased to a point where it would be extremely unlikely for such a fluctuation to occur - this is known as the Ginzburg temperature. This argument is based on equilibrium theory, and the density of defects formed is dependent only on the value of the Ginzburg temperature. However, in 1985 Wojciech Zurek developed an alternative scenario based on the theory of critical exponents, which measure how quantities such as the correlation time and coherence length of a system diverge at a critical point. For a phase transition occurring on a finite time scale, at some stage these quantities will be frozen out, and these should determine the size of the domain structures. Thus the speed at which a system passes through a phase transition will determine the density of defects. In the 1990s experiments were performed in superfluid helium that seemed to indicate that the Zurek scenario was the appropriate model; however the results are not entirely conclusive [4,5]. Recent experimental developments in ultra-cold Bose gases offer an alternative physical system where this scenario can be tested. Combined with the development of accompanying microscopic theory from a well-characterized Hamiltonian, this project will should lead to further understanding of such phenomena. ProjectThe aim of this project is to investigate the formation of vortices in quenches of dilute Bose gas systems, and interpret the results in terms of the theory of critical phenomena. In particular, does the Zurek scenario hold true for such systems, and what can it teach us about phase transitions in other systems? One particular goal is propose experiments that could be performed in current Bose gas systems to test this theory.The project will proceed via dynamic simulations of quenches in a degenerate Bose gas. While the exact quantum field equations are impossible to solve numerically, a Wigner function representation can be valid in the regime of the critical point as long as the low energy modes of the system are sufficiently highly occupied. This has been pointed out by a number of authors, and essentially results in a form of Gross-Pitaevskii equation (GPE). In particular the projected Gross-Pitaevskii equation has been extensively studied by Davis et al. [6,7,8], who have shown that it can accurately reproduce equilibrium critical phenomena. Quenches could potentially be performed using this equation of motion. However, very recently a more sophisticated treatment has been proposed by Gardiner and Davis, who have introduced the stochastic Gross-Pitaevskii equation [9]. This is based on a marriage of the projected GPE with quantum kinetic theory, and provides a coupling of the low energy modes of the projected GPE to the high energy modes of the system. These act as a reservoir with a given temperature and chemical potential, and introduce noise and damping into the equation of motion. The external control of the reservoir variables could be used to force the system through the phase transition at a given speed. Initial calculations will be performed for the homogeneous system. This has the advantage that all previous theory has been carried out for such a situation, and also presents relatively few technical challenges. Eventually the project will apply the knowledge gained to begin to apply the method to the trapped Bose gas, and to consider what effects could be studied experimentally in the lab. [1] A. J. Gill, Contemporary Physics 39, 13 (1998). [2] T. W. B. Kibble, J. Phys. A 9, 1387 (1976). [3] W. H. Zurek, Nature 317, 505 (1985). [4] P. C. Hendry, N.S. Lawson, R. A. M. Lee, P. V. E. McClintock and C. D. H. Williams, Nature 368, 315 (1994). [5] V. M. Ruutu, V. B. Elstov, M. Krusius, Yu. G. Makhlin, B. Plaçais, and G. E. Volovik, Phys. Rev. Lett. 80, 1465 (1998). [6] M. J. Davis, S. A. Morgan and K. Burnett, Phys. Rev. Lett. 87, 160402 (2001). [7] M. J. Davis, S. A. Morgan and K. Burnett, Phys. Rev. A 66, 053618 (2002). [8] M. J. Davis and S. A. Morgan, cond-mat/0307155 (2003). [9] C. W. Gardiner and M. J. Davis, cond-mat/0308044 (2003). |
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