Exact results on the dynamics of a multicomponent Bose-Einstein condensate

Abstract

We study the time-evolution of the two dimensional multicomponent Bose-Einstein condensate in an external harmonic trap with arbitrary time-dependent frequency. We show analytically that the time-evolution of the total mean-square radius of the wave-packet is determined in terms of the same solvable equation as in the case of a single-component condensate. The dynamics of the total mean-square radius is also the same for the rotating as well as the non-rotating multi-component condensate. We determine the criteria for the collapse of the condensate at a finite time. Generalizing our previous work on a single-component condensate, we show explosion-implosion duality in the multi-component condensate.PACS numbers: 03.75. Fi, 05.45.Yv, 03.65.Ge The successful creation and observation of BoseEinstein condensation(BEC) in dilute alkali atoms have opened up a plethora of new possibilities to test, otherwise intractable, many-body quantum phenomenon in the laboratory [1]. The Gross-Pitaevskii equation(GPE), the mean-field description of the BEC, is successful enough in explaining most of the observed results as well as predicting new phenomenon. The methods involved in studying the GPE are mainly numerical and/or approximate: perturbative and variational. The exact and analytical results of a nonlinear equation, if known, not only act as a guide to determine the validity of different approximate and numerical methods; they also give rise to new, counter-intuitive results in some cases. Unfortunately, no exact solution of GPE is known except for in one dimension.The two dimensional GPE, like its counterparts in higher dimensions, is not exactly solvable. However, due to an underlying dynamical O(2, 1) symmetry [2], the time-evolution of certain moments related to the two dimensional GPE can be described exactly [3]. This result is valid even if the condensate is considered in a timedependent harmonic trap. This leads to the prediction of explosion-implosion duality [4] and extended parametric resonance [4,5] in the two dimensional BEC. Both of these phenomenon are universal for any non-relativistic theory having dynamical O(2, 1) symmetry [3,4,6]. Interestingly enough, apart from the two dimensional BEC, the same explosion-implosion duality can also be observed in supernova explosion and in laser induced implosion in plasma [7,8]. This shows the importance of exact methods, based on an underlying symmetry, in relating diverse areas of physics such as the BEC and the supernova explosion.The results described above are for a single-component condensate, where the spin degree's of freedom have been frozen though the use of a magnetic trap. Recently, the spinor condensate with independent spin degree's of freedom has also been created and observed in the laboratory [9]. Similarly,the two-component condensate, where two different hyperfine states of the same atomic species are condensed simultaneously, has also been experimentally realized [10]. The spinor condensate [11][12][13][14] and the two-component condensate [15] ...

Ground States and Dynamics of Multicomponent Bose--Einstein Condensates

“…For numerical solutions of time-dependent GPE for finding dynamics of BEC, Bao et al [6] presented a time-splitting spectral (TSSP) method, Ruprecht et al [42] used the Crank-Nicolson finite difference method, Cerimele et al [16,17] proposed a particle-inspired scheme. Up to now, only a few numerical simulations on multi-component BEC [30,18,25].…”

Self-similar solutions and collective coordinate methods for nonlinear Schrödinger equations

“…Finally, the so-called moment method allows in some specific cases to obtain rigorous results for the evolution of integral quantities related to the solution of Eq. (1) [8,9,10].…”

Ground-State Properties and Collective Excitations in a 2D Bose-Einstein Condensate with Gravity-Like Interatomic Attraction

“…These modes depend on scattering length a in the TF-G regime unlike the ordinary TF regime. We show that the monopole mode exists for negative values of S unlike the situation in 3D [4].…”

“…In this system, kinetic energy and gravity-like attractive interaction forms a stable configuration without a trap potential [1,3]. Ghosh [4] has studied the collective excitation frequencies of this system in 3D within the time-dependent variational method. He has shown that variational analysis agrees very well with the results of Giovanazzi et al [3] in which the sum-rule approach was used.…”