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arXiv:1112.3724 · Physical Review A

Elementary modes of excitation caused by the quadratic Zeeman term and the sensitivity of spin structures of small spin-2 condensates against the magnetic field

Y. Z. He, Bao Cheng-guang

Published Dec 16, 2011• 4 citations
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Abstract

The response of spin-2 small condensates to an external magnetic field $B$ is studied. The parameters of the interaction are considered as variable. The emphasis is placed on clarifying the modes of excitation caused by the quadratic Zeeman term. The theoretical method used is beyond the mean-field theory. A set of eigenstates with the $\text{U}(5)\ensuremath{\supset}\text{SO}(5)\ensuremath{\supset}\text{SO}(3)$ symmetry is introduced to facilitate the analysis. To obtain a quantitative evaluation on the response, the fidelity susceptibility and the $B$-dependent average populations of spin components have been calculated. Mostly the particle number $N=30$ is assumed. The effect with a larger or smaller $N$ is also considered. It was found that the sensitivity of the response depends strongly both on the interaction and on the inherent symmetry.

DOI
10.1103/physreva.84.063620

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The response of spin-2 small condensates to an external magnetic field BBB is studied. The parameters of the interaction are considered as variable. The emphasis is placed on clarifying the modes of excitation caused by the quadratic Zeeman term. The theoretical method used is beyond the mean-field theory. A set of eigenstates with the U(5)\ensuremath⊃SO(5)\ensuremath⊃SO(3)\text{U}(5)\ensuremath{\supset}\text{SO}(5)\ensuremath{\supset}\text{SO}(3)U(5)\ensuremath⊃SO(5)\ensuremath⊃SO(3) symmetry is introduced to facilitate the analysis. To obtain a quantitative evaluation on the response, the fidelity susceptibility and the BBB-dependent average populations of spin components have been calculated. Mostly the particle number N=30N=30N=30 is assumed. The effect with a larger or smaller NNN is also considered. It was found that the sensitivity of the response depends strongly both on the interaction and on the inherent symmetry.
The spin structures of small spin-1 condensates ( N ⩽ 1000) under a magnetic field B have been studied beyond the mean field theory. Instead of the spinors, the many body spin eigenstates have been obtained. We have defined and calculated the spin correlative probabilities to extract information from these eigenstates. The correlation coefficients and the fidelity susceptibility have also been calculated. Thereby, the details of the spin structures responding to the variation of B can be better understood. In particular, from the correlation coefficients, which are the ratio of the two-body probability to the product of two one-body probabilities, strong correlation domains of B are found. The emphasis is placed on the sensitivity of the condensates toward B . No phase transitions in spin structures are found. However, abrupt changes in the derivatives of observables (correlative probabilities) are found in some particular domains of B . In these domains, the condensates are highly sensitive to B . The effect of temperature is considered. The probabilities defined in the paper can work as a bridge to relate theories and experiments. Therefore, they can be used to discriminate various spin structures and refine the interactions.
For studying the spin evolution of a condensate of spin-1 atoms, the total magnetization MMM relative to the ZZZ axis of quantization is prepared at a given value previously. During the evolution, MMM would not be conserved if an inclined magnetic field is applied (i.e., the direction of the field deviates from the ZZZ axis). In this case, the inclined field can cause a strong spin evolution via the linear Zeeman term even the field is very weak (say, close to 0.01 mG). An approach beyond the mean-field theory is proposed to study the evolution of small 87Rb^{87}\text{R}\text{b}87Rb condensates under the weak inclined fields. The time-dependent populations of spin components are given in analytical forms. The evolution is found to be highly sensitive to the magnitude and direction of the field.
We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor F=1 Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicates the absence of metastability.
We show that the classical dynamics underlying the mean-field description of homogeneous mixtures of spinor F=1F=1F=1 Bose-Einstein condensates in an external magnetic field is integrable as a consequence of number conservation and axial symmetry in spin space. The population dynamics depends only on the quadratic term of the Zeeman energy and on the strength of the spin-dependent term of the atom-atom interaction. We determine the equilibrium populations as function of the ratio of these two quantities and the miscibility of the hyperfine components in the ground state spinors are thoroughly discussed. Outside the equilibrium, the populations are always a periodic function of time where the periodic motion can be a libration or a rotation. Our studies also indicate the absence of metastability.
The exact eigenspectra and eigenstates of spin-1 and spin-2 Bose-Einstein condensates (BECs) are found, and their response to a weak magnetic field is studied and compared with their mean-field counterparts. Whereas mean-field theory predicts the vanishing population of the zero magnetic-quantum-number component of a spin-1 antiferromagnetic BEC, the component is found to become populated as the magnetic field decreases. The spin-2 BEC exhibits an even richer magnetic response due to quantum correlations among three bosons.
The field dependence of the magnetic susceptibility and collective mode frequencies of superfluid 3B^{3}\mathit{B}3B are calculated perturbatively in (\ensuremath{\gamma}H/\ensuremath{\Delta}) using the quasiclassical theory. With the aid of this perturbation theory, we interpret the gap distortion, nonlinear susceptibility, and collective excitations of the order parameter in terms of additional correlations between Cooper pairs that are induced by a magnetic field. We also calculate the dispersion splittings of the real squashing modes in a magnetic field. In addition to the nonlinear Zeeman shifts of these modes, which occur in strong magnetic fields (\ensuremath{\gamma}H\ensuremath{\sim}0.10\ensuremath{\Delta}), there are low-field (\ensuremath{\gamma}H\ensuremath{\sim}q2{q}^{2}q2vf2{v}_{f}^{2}vf2​/\ensuremath{\Delta}) nonlinearities of the collective mode frequencies resulting from the field dependence of the quantization axis. Our results for all of these response properties of 3B^{3}\mathit{B}3B depend upon a small number of material parameters; thus measurements of these properties can provide detailed information on the quasiparticle interactions.