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arXiv:1302.7214 · Journal of Statistical Mechanics Theory and Experiment

Sine-Gordon model coupled with a free scalar field emergent in the low-energy phase dynamics of a mixture of pseudospin-$\frac{1}{2}$ Bose gases with interspecies spin exchange

Li Ge, Yu Shi

Published Oct 23, 2012• 0 citations

Abstract

Using the approach of low-energy effective field theory, the phase diagram is studied for a mixture of two species of pseudospin- Bose atoms with interspecies spin exchange. There are four mean-field regimes on the parameter plane of g e and g z , where g e is the interspecies spin-exchange interaction strength, while g z is the difference between the interaction strength of interspecies scattering without spin exchange of equal spins and that of unequal spins. Two regimes, with | g z | > | g e |, correspond to ground states with the total spins of the two species parallel or antiparallel along the z direction, and the low-energy excitations are equivalent to those of two-component spinless bosons. The other two regimes, with | g e | > | g z |, correspond to ground states with the total spins of the two species parallel or antiparallel on the x y plane, and the low-energy excitations are described by a sine-Gordon model coupled with a free scalar field, where the effective fields are combinations of the phases of the original four boson fields. In (1 + 1)-dimension, they are described by Kosterlitz–Thouless renormalization group (RG) equations, and there are three sectors in the phase plane of a scaling dimension and a dimensionless parameter proportional to the strength of the cosine interaction, both depending on the densities. The gaps of these elementary excitations are experimental probes of the underlying many-body ground states.

DOI: 10.1088/1742-5468/2012/10/p10020

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Sine-Gordon model coupled with a free scalar field emergent in the low-energy phase dynamics of a mixture of pseudospin-

\frac{1}{2}

Bose gases with interspecies spin exchange

Using the approach of low-energy effective field theory, the phase diagram is studied for a mixture of two species of pseudospin- Bose atoms with interspecies spin exchange. There are four mean-field regimes on the parameter plane of g e and g z , where g e is the interspecies spin-exchange interaction strength, while g z is the difference between the interaction strength of interspecies scattering without spin exchange of equal spins and that of unequal spins. Two regimes, with | g z | > | g e |, correspond to ground states with the total spins of the two species parallel or antiparallel along the z direction, and the low-energy excitations are equivalent to those of two-component spinless bosons. The other two regimes, with | g e | > | g z |, correspond to ground states with the total spins of the two species parallel or antiparallel on the x y plane, and the low-energy excitations are described by a sine-Gordon model coupled with a free scalar field, where the effective fields are combinations of the phases of the original four boson fields. In (1 + 1)-dimension, they are described by Kosterlitz–Thouless renormalization group (RG) equations, and there are three sectors in the phase plane of a scaling dimension and a dimensionless parameter proportional to the strength of the cosine interaction, both depending on the densities. The gaps of these elementary excitations are experimental probes of the underlying many-body ground states.

We develop a low energy effective field theory of a mixture of two species of pseudospin- atoms with interspecies spin exchange, in addition to density–density interaction. This approach is beyond the single-orbital-mode approximation. In a wide parameter regime, it indicates the existence of the four elementary excitations, especially the gapped mode due to interspecies spin exchange. The spectrum of the effective spin Hamiltonian yielded by the single-mode approximation can be obtained by quantizing the homogeneous excitation, which is spin excitation and is the long-wavelength limit of the gapped mode among the elementary excitations. These low energy excitations may be experimentally detected by using Bragg spectroscopy.

Number and spin densities in the ground state of a trapped mixture of two pseudospin-

\frac{1}{2}

Bose gases with interspecies spin-exchange interaction

We consider the ground state of a mixture of two pseudospin- Bose gases with interspecies spin exchange in a trapping potential. In the mean field approach, the ground state can be described in terms of four wavefunctions governed by a set of coupled Gross–Pitaevskii-like (GP-like) equations, which differ from the usual GP equations in the existence of an interference term due to spin-exchange coupling between the two species. Using these GP-like equations, we calculate such ground state properties as chemical potentials, density profiles and spin density profiles, which are directly observable in experiments. We compare the cases with and without spin exchange. It is demonstrated that the spin exchange between the two species lowers the chemical potentials, tends to equalize the wavefunctions of the two pseudospin components of each species, and, thus, homogenizes the spin density. The novel features of the density and spin density profiles can serve as experimental probes of this novel Bose system.

We consider a mixture of two species of spin-1 atoms with interspecies spin exchange, which may cooperate or compete with intraspecies spin exchanges and thus dramatically affect the ground state. It represents a new class of bosonic gases differing from single-species spinor gases. We determine the exact ground states in several parameter regimes and study the composite structures using the generating function method generalized here to be applicable to a mixture of two species of spinor gases. The most interesting phase is the so-called entangled Bose-Einstein condensation (BEC), which is fragmented BEC with quantum entanglement between the two species and with both interspecies and intraspecies singlet pairs. For comparison, we also apply the generating function method to a mixture of two species of pseudospin-

\frac{1}{2}

atoms, for which the total spin quantum number of each species is fixed as half of the atom number, in contrast with the case of spin

1

, for which it is a variable determined by energetics. Consequently, singlet pairs in entangled BEC of a pseudospin-

\frac{1}{2}

mixture are all interspecies. Interspecies spin exchange leads to novel features beyond those of spinor BEC of a single species of atoms as well as mixtures without interspecies spin exchange.

We consider a mixture of two species of spin-1 atoms with both interspecies and intraspecies spin exchanges in a weak magnetic field. Under the usual single mode approximation, it can be reduced to a model of coupled giant spins. We find most of its ground states. This is a complicated problem of energy minimization, with three quantum variables under constraints, i.e., the total spin of each species and the total spin of the whole mixture, as well as four parameters, including intraspecies and interspecies spin coupling strengths and the magnetic field. The quantum phase diagram is very rich. Compared with the case without a magnetic field, the ground states are modified by a magnetic field, which also modifies the ground state boundaries or introduces new crossover regimes on the phase diagram. Without interspecies spin coupling, the quantum phase transitions existing in absence of a magnetic field disappear when a magnetic field is applied, which leads to crossover regimes in the phase diagram. Under ferromagnetic interspecies spin coupling, the ground states remain disentangled no matter whether there is a magnetic field. For antiferromagnetic interspecies spin coupling, a magnetic field entangles the ground states in some parameter regimes. When the intraspecies spin couplings are both ferromagnetic, the quantum phase transition between antiferromagnetic and zero interspecies spin couplings survives the magnetic field. When the intraspecies spin couplings are both antiferromagnetic, a magnetic field induces new quantum phase transitions between antiferromagnetic and zero interspecies spin couplings.

We consider the effective spin Hamiltonian describing a mixture of two species of pseudo-spin-

\frac{1}{2}

Bose gases with interspecies spin exchange. First we analyze the stability of the fixed points of the corresponding classical dynamics, of which the signature is found in quantum dynamics with a disentangled initial state. Focusing on the case without an external potential, we find all the ground states by taking into account quantum fluctuations around the classical ground state in each parameter regime. The nature of entanglement and its relation with classical bifurcation is investigated. When the total spins of the two species are unequal, the maximal entanglement at the parameter point of classical bifurcation is possessed by the excited state corresponding to the classical fixed point which bifurcates, rather than by the ground state.