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arXiv:cond-mat/0203052 · Physical Review A

Theory of spin-2 Bose-Einstein condensates: Spin correlations, magnetic response, and excitation spectra

Masahito Ueda, Masato Koashi

Published May 29, 2002• 163 citations
arXivHTMLPDF

Abstract

The ground states of Bose-Einstein condensates (BECs) of spin-2 bosons are classified into three distinct phases (ferromagnetic, ``antiferromagnetic,'' and cyclic) depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while low-lying excitation spectra are investigated in the thermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied magnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet ``pairs'' undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson ``trios'' condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one single-particle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.

DOI
10.1103/physreva.65.063602

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The ground states of Bose-Einstein condensates (BECs) of spin-2 bosons are classified into three distinct phases (ferromagnetic, antiferromagnetic,'' and cyclic) depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while low-lying excitation spectra are investigated in the thermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied magnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet pairs'' undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson ``trios'' condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one single-particle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.
We develop a mean-field theory for Bose-Einstein condensation of spin-1 atoms with internal degrees of freedom. It is applicable to nonuniform systems at finite temperatures with a plausible feature of satisfying the Hugenholtz-Pines theorem and various conservation laws simultaneously. Using it, we clarify thermodynamic properties and the excitation spectra of a uniform gas. The condensate is confirmed to remain in the same internal state from T=0 up to TcT_{c}Tc​ for both antiferromagnetic and ferromagnetic interactions. The excitation spectra of the antiferromagnetic (ferromagnetic) interaction are found to have only a single gapless mode, contrary to the prediction of the Bogoliubov theory where three (two) of them are gapless. We present a detailed discussion on those single-particle excitations in connection with the collective excitations.
Motivated by recent experiments [Chen et al., Phys. Rev. Lett. 121, 113204 (2018); Chen et al., Phys. Rev. Lett. 121, 250401 (2018)], we investigate the low-lying excitation spectrum of the ground-state phases of spin-orbital-angular-momentum-coupled (SOAM-coupled) spin-1 condensates. At vanishing detuning, a ferromagnetic SOAM-coupled spin-1 Bose-Einstein condensate (BEC) can have two ground-state phases, namely, coreless and polar-core vortex states, whereas an antiferromagnetic BEC supports only polar-core vortex solution. The angular momentum per particle, longitudinal magnetization, and excitation frequencies display discontinuities across the phase boundary between the coreless vortex and polar-core vortex phases. The low-lying excitation spectrum evaluated by solving the Bogoliubov--de Gennes equations is marked by avoided crossings and hence the hybridization of the spin and density channels. The spectrum is further confirmed by the dynamical evolution of the ground state subjected to a perturbation suitable to excite a density or a spin mode and a variational analysis for the density-breathing mode.
Abstract We obtain and justify a many-body Hamiltonian of pairwise interacting spin-1 atoms, which includes eight generators of the SU(3) group associated with spin and quadrupole degrees of freedom. It is shown that this Hamiltonian is valid for non-local interaction potential, whereas for local interaction specified by s -wave scattering length, the Hamiltonian should be bilinear in spin operators only (of the Heisenberg type). We apply the obtained Hamiltonian to study the ground-state properties and single-particle excitations of a weakly interacting gas of spin-1 atoms with Bose–Einstein condensate (BEC) taking into account the quadrupole degrees of freedom. It is shown that the system under consideration can be in ferromagnetic, quadrupole, and paramagnetic phases. The corresponding phase diagram is constructed and discussed. The main characteristics such as the density of the grand thermodynamic potential, condensate density, and single-particle excitation spectra modified by quadrupole degrees of freedom are determined in different phases.
We have numerically solved the low-energy excitation spectra of ferromagnetic Bose-Einstein condensates subject to dipolar interparticle interactions. The system is assumed to be harmonically confined by purely optical means, thereby maintaining the spin degree of freedom of the condensate order parameter. Using a zero-temperature spin-1 model, we solve the Bogoliubov excitations for different spin textures, including a spin-vortex state in the absence of external magnetic fields and a rapidly rotating polarized spin texture in a finite homogeneous field. In particular, we consider the effect of dipolar interactions on excitations characteristic of ferromagnetic condensates. The energies of spin waves and magnetic quadrupole modes are found to increase rapidly with the dipolar coupling strength, whereas the energies of density oscillations change only slightly.
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.
Selected topics on spinor-dipolar aspects of Bose-Einstein condensation are overviewed. Spinor aspects include spin correlations, fragmentation, dynamical symmetries, and quantum mass acquisition. Dipolar aspects include magnetostriction, d-wave collapse, roton-maxon spectrum, supersolidity, and ferrofluidity. Finally, spinor-dipolar aspects concern the Einstein–de Haas effect and spontaneous mass and spin currents in the ground state. Introduction Bose-Einstein condensation is a phenomenon in which a macroscopic number of particles occupy the same single-particle state as a consequence of symmetrization of a many-body wave function and thereby quantum effects are amplified to the macroscopic level. The amplified single-particle state serves as a complex order parameter whose phase behaves as an emergent thermodynamic quantity, with its temporal and spatial variations giving the chemical potential and the superfluid velocity, respectively. If the interparticle interaction is repulsive, a Bose-Einstein condensate (BEC) acquires stability against excitations out of the condensate because they would cost the Fock exchange energy. This rigidity of the order parameter or the condensate wave function endows a BEC with several remarkable transport properties which are collectively known as superfluidity. In particular, the superfluid fraction can be 100% at zero temperature even though the condensate fraction is depleted due to interparticle interactions. Interparticle interactions also make excited particles phase-locked to the condensate, leading to Bogoliubov quasiparticles, which are the manifestation of phase-coherent particle– hole excitations. When constituent particles have spin degrees of freedom, a BEC features magnetism and spin nematicity which conspire with superfluidity to produce a rich phase diagram. In the absence of an external magnetic field, the mean-field groundstate phase diagram includes two [1, 2], five [3, 4, 5], and eleven phases [6, 7] for spin-1, 2, and 3 BECs, respectively. BECs with spin degrees of freedom are called spinor condensates. Moreover, the magnetic dipole–dipole interaction (MDDI) between atoms lends distinct twists to spinor condensates because it is long-ranged and anisotropic in sharp contrast with other interactions, which are short-ranged and isotropic. In this chapter, we will present a brief overview on the spinordipolar aspects of BEC. An early experimental work of spinor BECs is reviewed in Ref. [8]. Later developments with an emphasis on theoretical aspects are described in Ref. [9]. Many-body physics of polarized dipolar gases is reviewed in Ref. [10].