Theoretical Research On Superfluid Properties Of Ring-Shaped Bose-Einstein Condensates

With the development of cold atoms and their trapping technology,ring-shaped Bose-Einstein condensates have made a series of achievements in exploring superfluid properties,especially the experimental observation of hysteresis and the current-phase relation,which is considered to be an important progress in the development of atom superconducting quantum interferometer devices.Based on the analytical solutions of Jacobian elliptic function of Gross-Pitaevskii equation,the superfluid properties of ring-shaped Bose-Einstein condensates are systematically studied in this paper,including the critical velocity of superfluid decay,hysteresis,current-phase relation and so on.We generalize the analytical solution of Gross-Pitaevskii equation describing the basic properties of Bose-Einstein condensates to the case of periodic boundary conditions.We find that the stationary solutions of rotational states with different topological winding number will exhibit the characteristics of symmetry and periodicity.At the same time,the relations between physical quantities in different reference systems are discussed.According to the "swallow tail" structure of energy spectrum in the rotating reference system,we can get the critical velocity of superfluid decay and the area of hysteresis loop.The study shows that the hysteresis phenomenon is the manifestation of superfluid characteristics,while the superfluid characteristics of Bose-Einstein condensates originate from the non-linear interaction.With the increase of the non-linear interaction or the decrease of the barrier height,the area of hysteresis loop will increase.Therefore,the parameter range of the hysteresis phenomenon can be obtained.With the analytical solution,we have a deeper understanding of the measurements of hysteresis and current-phase relation in ring-shaped Bose-Einstein condensates by NIST experimental group in the United States.Without any fitting parameters,the results of the analytical solution agree qualitatively with the experimental results in the area of hysteresis loop varying with the height of barrier.After the strength of non-linear interaction is modified by the loss of the number of particles,in terms of the slope and area of the hysteresis loop and the current-phase relation,the agreement between the analytical solution and the experimental results is greatly improved.At the same time,we try to use Leggett's toy model to understand the physical nature of hysteresis.In the aspect of studying the current-phase relation of weak links,our analytical results show that there exists single-valued and multi-valued current-phase relation,which is basically the same as that with open boundary conditions;however,due to the influence of periodic boundary conditions and finite size,only plane wave solutions are allowed when the barrier in ring-shaped Bose-Einstein condensate is high,while the current-phase relation is not a cosine function when the barrier is zero.For ring-shaped Bose-Einstein condensates,the parameter ranges of different current-phase relation are given.Some comparisons with the experimental results are made.Inspired by the experiment of the hysteresis of cold atoms in double wells,we find that at some special rotational speeds,the order parameters of ring-shaped Bose-Einstein condensates exhibit a pitchfork bifurcation with changing the barrier height,accompanied by a continuous quantum phase transition with parity-symmetry breaking.The preliminary results show that the phase transition is a second-order quantum phase transition.At the critical barrier height and rotating speed,the chemical potential of the ground state develops a cusp,reflecting the characteristics of topological quantum phase transition.Finally,when the barrier height is zero,the second-order quantum phase transition can occur between the plane wave solution and the soliton solution;but when the barrier height is not zero,the phase transition will not occur,and the bifurcation type of the chemical potential will also change.In conclusion,by using the analytical solution of Gross-Pitaevskii equation,we can explain the superfluid characteristics of ring-shaped Bose-Einstein condensates.It can help us better to understand the rich physical images contained in this simple system,and lay a foundation for further understanding the superfluid phenomenon and related physics.