Evolution of the average populations of spin components of spin-1 Bose-Einstein condensates beyond mean-field theory

Abstract

An analytical formula is obtained to describe the evolution of the average populations of spin components of spin-1 atomic gases. The formula is derived from the exact time-dependent solution of the Hamiltonian HS = cS 2 without using approximation. Therefore it goes beyond the mean field theory and provides a general, accurate, and complete description for the whole process of nondissipative evolution starting from various initial states. The numerical results directly given by the formula coincide qualitatively well with existing experimental data, and also with other theoretical results from solving dynamic differential equations. For some special cases of initial state, instead of undergoing strong oscillation as found previously, the evolution is found to go on very steadily in a very long duration.PACS numbers: 03.75. Fi, 03.65. Fd The liberation of the freedoms of spin of atoms in optical traps [1,2,3,4,5] opens a new field, namely spin dynamics of condensates, which is promising for super-high precise measurement, quantum computation, and quantum information processing. [6,7,8] Recently, the evolution of spinor condensates has been extensively studied experimentally and theoretically. [9,10,12,13,14] Initially, the condensate was prepared in a Fock-state or a coherent state confined in an optical trap. Then, due to the spin-dependent interaction, the system begin to evolve where a pair of atoms with spin components 1 and -1 can jump to 0 an 0, and vice versa, via scattering. Finally the system will arrive in equilibrium, however the process is not smooth. In 1998, the average population of each of the spin components µ =1, 0, and -1 was found to depend sensitively on initial states and may oscillate strongly with time. [12]. This finding was further confirmed by a number of research groups. In 2006, in the study of the probability of finding a given number of bosons in a given µ state, the "quantum carpet" spin-time structure was found. [14] These findings show the amazing peculiarity of the spin dynamics. Related theoretic calculations are mostly based on the mean field theory. Although, in a number of particular cases, theoretical results compares qualitatively well with experimental data, the underlying physics remains to be further clarified. This paper is a study of the evolution of the average populations. We shall go beyond the mean field theory but use strict quantum mechanic many-body theory with a full consideration of symmetry. Instead of solving dynamic differential equations under specified initial condition, we succeed to derive a general analytical formula to describe rigorously the whole process of evolution (non-dissipative) and is valid for all possible initial status. This is reported as follows.It is first assumed that the initial state of N spin-1 * Corresponding author: stsbcg@mail.sysu.edu.cn atoms is a Fock-state with populations N 1 , N 0 and N −1 , the magnetization M = N 1 − N −1 . When N and M are given, the Fock-state can be simply denoted as |N 0 . Let the part of the Ham...

A 2-Level Condensate with Tunable and Sharp Susceptibility Against the Magnetic Field

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Modification of spin mixing of spinor BEC by cavity QED coupling

“…where M = M + 1. The matrix element between Fock state and total spin eigenstate S, M − 1|N 0 , M − 1 is given in reference [15]. After extracting the spin state of the Nth atom from the total spin eigenstate |S ′ , M − µ; {µ} with equation ( 6), we can calculate the evolution of population of each spin component as…”

“…One of the most interesting phenomenon of spinor BEC is spin mixing, where the average atomic number of each spin component evolute versus time [13,14]. The pattern of time evolution is found to be decided by symmetric property and initial condition of the system [15]. If the spinor BEC is placed inside a far off resonant cavity, the atoms will couple to the cavity mode, which change the symmetric of the system as well as the effective coupling strength between atoms.…”