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arXiv:cond-mat/0604505 · Physical Review A

Cyclic phase in F=2 spinor condensate: Long-range order, kinks, and roughening transition

W. V. Pogosov, K. Machida

Published Aug 18, 2006• 5 citations
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Abstract

We study the effect of thermal fluctuations on homogeneous infinite Bose-Einstein condensate with spin $F=2$ in the cyclic state, when atoms occupy three hyperfine states with ${m}_{F}=0,\ifmmode\pm\else\textpm\fi{}2$. We use both the approach of small-amplitude oscillations and mapping of our model on the sine-Gordon model. We show that thermal fluctuations lead to the existence of the rough phase in one- and two-dimensional systems, when the presence of kinks is favorable. The structure and energy of a single kink are found. We also discuss the effect of thermal fluctuations on spin degrees of freedom in $F=1$ condensate.

DOI
10.1103/physreva.74.023611

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We study the effect of thermal fluctuations on homogeneous infinite Bose-Einstein condensate with spin F=2F=2F=2 in the cyclic state, when atoms occupy three hyperfine states with mF=0,\ifmmode±\else\textpm\fi2{m}_{F}=0,\ifmmode\pm\else\textpm\fi{}2mF​=0,\ifmmode±\else\textpm\fi2. We use both the approach of small-amplitude oscillations and mapping of our model on the sine-Gordon model. We show that thermal fluctuations lead to the existence of the rough phase in one- and two-dimensional systems, when the presence of kinks is favorable. The structure and energy of a single kink are found. We also discuss the effect of thermal fluctuations on spin degrees of freedom in F=1F=1F=1 condensate.
We consider the effect of thermal fluctuations on rotating spinor F=1F=1F=1 condensates in axially symmetric vortex phases, when all the three hyperfine states are populated. We show that the relative phase among different components of the order parameter can fluctuate strongly due to the weakness of the interaction in the spin channel. These fluctuations can be significant even at low temperatures. Fluctuations of relative phase lead to significant fluctuations of the local transverse magnetization of the condensate. We demonstrate that these fluctuations are much more pronounced for the antiferromagnetic state than for the ferromagnetic one.
A homogeneous gas of atoms with hyperfine spin F=1F=1F=1 is considered below the critical temperature of the Bose-Einstein condensation. The general structure of the correlation functions (both for density and spin fluctuations) and their connection to the thermal Green's functions are worked out. The damping of the modes is calculated in the random-phase approximation.
Extended Gross-Pitaevskii equations for the rotating F=2F=2F=2 condensate in a harmonic trap are solved both numerically and variationally using trial functions for each component of the wave function. Axially symmetric vortex solutions are analyzed and energies of polar and cyclic states are calculated. The equilibrium transitions between different phases with changing of the magnetization are studied. We show that at high magnetization the ground state of the system is determined by interaction in ``density'' channel, and at low magnetization spin interactions play a dominant role. Although there are five hyperfine states, all the particles are always condensed in one, two, or three states. Two interesting types of vortex structures are also discussed.
We analyze the structure of spin-one Bose-Einstein condensates in the presence of a homogeneous magnetic field. We classify the homogeneous stationary states and study their existence, bifurcations, and energy spectra. We reveal that phase separation can occur in the ground state of antiferromagnetic (polar) condensates while the spin components of the ferromagnetic condensates are always miscible and no phase separation occurs. Our theoretical model, confirmed by numerical simulations, explains that this phenomenon takes place when the energy of the lowest homogeneous state is a concave function of the magnetization. In particular, we predict that phase separation can be observed in a 23Na^{23}\text{N}\text{a}23Na condensate confined in a highly elongated harmonic trap. Finally, we discuss the phenomena of dynamical instability and spin domain formation.
Cold-atom developments suggest the prospect of measuring scaling properties and long-range fluctuations of continuous phase transitions at zero temperature. We discuss the conditions for characterizing the phase separation of Bose-Einstein condensates of boson atoms in two distinct hyperfine spin states. The mean-field description breaks down as the system approaches the transition from the miscible side. An effective spin description clarifies the ferromagnetic nature of the transition. We show that a difference in the scattering lengths for the bosons in the same spin state leads to an effective internal magnetic field. The point at which the internal magnetic field vanishes ( i.e. , equal values of the like-boson scattering lengths) is a special point. We show that the long-range density fluctuations are suppressed near that point, while the effective spin exhibits the long-range fluctuations that characterize critical points. The zero-temperature system exhibits critical opalescence with respect to long-wavelength waves of impurity atoms that interact with the bosons in a spin-dependent manner.
It is investigated theoretically that magnetized Bose-Einstein condensation (BEC) with the internal (spin) degrees of freedom exhibits a rich variety of phase transitions, depending on the sign of the interaction in the spin channel. In the antiferromagnetic interaction case there exist always double BEC transitions from single component BEC to multiple component BEC. In the ferromagnetic case BEC becomes always unstable at a lower temperature, leading to a phase separation. The detailed phase diagram for the temperature vs the polarization, the spatial spin structure, the distribution of non-condensates and the excitation spectrum are examined for the harmonically trapped systems.