Spinor Bose–Einstein condensates
Introduction
Gaseous Bose–Einstein condensate (BEC) was first created using atoms in single spin states of rubidium 87 (87Rb) [1], sodium 23 (23Na) [2], and lithium 7 (7Li) [3]. In these systems, only those atoms in a week-field seeking state were magnetically trapped, and therefore, their spin degrees of freedom were frozen. A spinor Bose–Einstein condensate (BEC), namely a BEC with spin internal degrees of freedom, was first realized in a gas of spin-1 23Na atoms confined in an optical dipole trap in 1998 [4], opening up a new research arena of ultracold atomic systems. Confined in an optical trap, the direction of atomic spins can change due to the interparticle interaction. Consequently, the order parameter of a spin- BEC has components that can vary over space and time, producing a very rich variety of spin textures. In contradistinction to a spinor BEC, a BEC of atoms in a single spin state is referred to as a scalar BEC. This article provides an overview of the physics of spinor BECs.
Bose–Einstein condensation is a genuinely quantum-mechanical phase transition in that it occurs without the help of interaction. However, in the case of spinor BECs, several phases are possible below the transition temperature , and which phase is realized at what temperature does depend on the nature of the interaction. The general form of the Hamiltonian of spinor BECs is discussed in Section 2. With the spin and gauge degrees of freedom, the full symmetry of the system above is . Below the transition temperature, the symmetry is spontaneously broken in many different ways: the number of ground-state phases in the absence of an external magnetic field is two for the spin-1 case [5], [6], three for the spin-2 case [7], [8], and eleven for the spin-3 case [9], [10]. The number of phases further increases under an external magnetic field. Such a wealth of possible phases make a spinor BEC a fascinating area of research for quantum gases. Another striking consequence of the spin degrees of freedom is the spin dynamics. The populations of magnetic sublevels can change via spin-exchange collisions. For example, in a spin-1 spinor BEC, two atoms in the magnetic sublevel can coherently and reversibly scatter into a pair of atoms in the and states, and vice versa. So far, spin-exchange collisions have been observed in systems of spin-1 87Rb [11], spin-1 23Na [12], [13], spin-2 87Rb [11], [14], [15], and spin-3 chromium 52 (52Cr) [16]. The basic properties of the mean-field spinor condensates, such as ground-state phase diagrams and spin dynamics, are reviewed in Section 3, and the experimental achievements are summarized in Section 4.
The BEC responds to an external perturbation in a very unique manner. Even if the perturbation is weak, the response may be nonperturbative; for example, the phonon velocity depends on the scattering length in a nonanalytic manner. The low-lying excitations of spinor BECs are described by the Bogoliubov theory, as discussed in Section 5. A spinor BEC can be experimentally prepared in an unstable stationary state. The dynamics starting from such an unstable state can be understood in terms of a growth of unstable Bogoliubov modes or the dynamical instability which may be triggered by quantum fluctuations. In Section 5 we discuss how topological defects are nucleated after a quantum phase transition in the context of the dynamical instability.
The magnetic moment of an atom causes the magnetic dipole–dipole interaction (DDI). The long-range and anisotropic nature of DDI leads to new phenomena even in a spin-polarized BEC [17], [18]. Although the strength of the magnetic DDI is usually the smallest among all the energy scales involved, it plays a pivotal role in a spinor BEC in producing the spin texture—the spatial variation of the spin direction. In particular, the magnetic DDI couples spin and orbital angular momenta under an ultralow magnetic field (below ), which spontaneously generates superfluid flow of atoms. Properties of spin-polarized and spinor–dipolar BECs are discussed in Section 6.
The dynamics of spinor BECs may be understood more intuitively if the equations of motion are expressed in terms of physical quantities such as superfluid velocity, spin superfluid velocity, magnetization, and nematic directors, as discussed in Section 7. We will see that the space and time dependences of spin configurations naturally generate a geometric gauge field.
One of the hallmarks of superfluidity manifests itself in its response to an external rotation. In a scalar BEC, the system hosts vortices that are characterized by the quantum of circulation, , where is the Planck constant and is the mass of the atom. The origin of this quantization is the single-valuedness of the order parameter. As mentioned above, however, in spinor BECs, the gauge degree of freedom is coupled to the spin degrees of freedom. This spin–gauge coupling gives rise to some of the unique features of the spinor BEC. For example, the fundamental unit of circulation can be a rational fraction of , and when two vortices collide, they may not reconnect unlike the case of the scalar BEC but form a rung vortex that bridges the two vortices. Vortices of spinor BECs are discussed in Section 8.
In Section 9, we discuss symmetry properties of mean-field ground states. The ground-state phases in spinor BECs differ from each other in gauge, spin-rotation, or their combined symmetries. For example, the ferromagnetic phase has the SO(2) spin–gauge coupled rotational symmetry, whereas the spin-2 cyclic phase has the symmetry of tetrahedron. We will show that ground-state order parameters can be found from the symmetry consideration without minimizing the mean-field energy.
Once we know the symmetry property of the condensed phase under consideration, the homotopy theory tells us possible types of topological excitations. In general, the direction of the spin can vary rather flexibly over space and time. Nonetheless, the global configuration of the spin texture must satisfy certain topological constraints. This is because the order parameter in each phase belongs to a particular order-parameter manifold whose symmetry leads to a conserved quantity called a topological charge. Such a topological constraint determines the nature of topological excitations such as line defects, point defects, Skyrmions, and knots. Elements of homotopy theory with applications to spinor BECs are reviewed in Section 10.
In the last part of this review, we discuss the ground states and spin dynamics beyond the mean-field theory by exactly diagonalizing the many-body Hamiltonian. In some parameter regions, the many-body spin correlations dramatically alter the ground states of spinor BECs and fragmented ground states arise so as to recover the SO(3) spin rotational symmetry as discussed in Section 11.
In view of these ongoing developments, it now seems appropriate to consolidate the knowledge that has been accumulated over the past decade. In this paper, we provide an overview of the basics and recent developments on the physics of spinor and dipolar BECs. Some topics and related problems which we do not cover in the main text, such as finite-temperature effects, low-dimensional systems, optical lattice, and spin–orbit coupling are overviewed in Section 12.
One of the major topics that is not treated in this review is fictitious spin systems such as a binary mixture of hyperfine spin states [19], [20]. Since the intra- and interspecies interactions of this mixture are almost identical, this system has an approximate SU(2) symmetry and is regarded as a pseudo-spin-1/2 system. It has some intrinsic interest because it offers such unique phenomena as interlaced vortex lattices and vortex molecules caused by the Josephson coupling. A comprehensive review of this subject is given in Ref. [21].
This paper is organized as follows. Section 2 describes the fundamental Hamiltonian of the spinor BEC. Section 3 develops the mean-field theory of spinor condensates and discusses the ground-state properties and spin dynamics of the spin-1, 2, and 3 BECs. Section 4 summarizes the experimental achievements so far. Section 5 develops the Bogoliubov theory of the spinor BEC. Section 6 provides an overview of the dipolar BEC and the spinor–dipolar BEC. Section 7 derives the hydrodynamic equations of motion of superfluid velocity and magnetization. Section 8 discusses various types of vortices that can be created in spinor BECs. Section 9 classifies the ground-state order parameters based on group theory and discusses the symmetry property of each phase. Section 10 examines the topological aspects of spinor BECs. Possible topological excitations such as non-Abelian vortices are investigated using homotopy theory. Section 11 reviews the many-body aspects of spinor BECs. Section 12 summarizes the main results of this paper and discusses possible future developments.
Section snippets
Single-particle Hamiltonian
The fundamental Hamiltonian of a spinor BEC can be constructed quite generally based on the symmetry argument. We consider a system of identical bosons with mass and spin that are described by the field operators , where denotes the magnetic quantum number. The field operators are assumed to satisfy the canonical commutation relations where is the Kronecker delta which takes on the value 1 if
Number-conserving theory
The mean-field theory is usually obtained by replacing the field operator with its expectation value . This recipe, though widely used and technically convenient, has one conceptual difficulty; that is, it breaks the global U(1) gauge invariance, which implies that the number of atoms is not conserved. However, in reality, the number of atoms is strictly conserved, as are the baryon (proton and neutron) and lepton (electron) numbers. In fact, it is possible to construct the mean-field
Experiments on spinor Bose–Einstein condensates
A spinor condensate was first realized by Stamper-Kurn et al. [4] by using a spin-1 23Na condensate confined in an optical dipole trap, and the spin-dependent interaction coefficient was estimated from the domain size [12]. The ground-state phase diagram in the space of linear () and quadratic () Zeeman energies (see Fig. 3) was theoretically predicted and experimentally verified for from an analysis of spin-domain structures subject to a magnetic field gradient [12]. By changing the
Bogoliubov Hamiltonian and its diagonalization
Quantum and thermal fluctuations as well as external perturbations induce excitations from the mean-field ground state. When the excitations are weak, they can be described by the Bogoliubov theory. We express the field operator as a sum of its mean-field value and the deviation from it, : Here, the mean-field part can be calculated from the Gross–Pitaevskii theory described in Section 3. The basic idea of the Bogoliubov theory is to substitute Eq. (188) in
Dipolar Bose–Einstein condensates
In this section, we consider the dipole–dipole interaction (DDI) that, unlike the -wave contact interaction, is long-range and anisotropic. The DDI between alkali atoms is thousand times smaller than the short-range interaction for the background scattering length. However, recent experimental developments, such as the realization of BECs of 52Cr [64], [134], 164Dy [135], and 168Er [136], creation of ultracold molecules [137], [138], [139], [140], and precision measurements [78], [141] and
Hydrodynamic equations
In this section, we discuss basic properties of mass current and spin current of spinor condensates by deriving the hydrodynamic equations of motion for supercurrent and magnetization.
Vortices and hydrodynamic properties
A scalar BEC can host only one type of vortex, that is, a U(1) vortex or a gauge vortex. However, a spinor BEC can host many different types of vortices. The properties of a vortex can be characterized by looking at how the order parameter changes along a loop that encircles the vortex. For the case of a scalar BEC, the order parameter is a single complex function which can be written as where is the particle number density. The corresponding superfluid velocity is
Symmetry classification
When a system undergoes Bose–Einstein condensation, certain symmetries of the original Hamiltonian are spontaneously broken. The classification of symmetry breaking can be carried out systematically using a group-theoretic method, and the resulting symmetry of the order parameter determines the types of possible topological excitations, as discussed in the next section. In this section, we introduce the concept of the order-parameter manifold and present a brief overview of some basic notions
Topological excitations
Bose–Einstein condensates can accommodate topological excitations such as vortices, monopoles, and Skyrmions. These topological excitations are diverse in their physical properties but have one thing in common; they can move freely in space and time without changing their characteristics that are distinguished by topological charges. The topological charges take on discrete values and have very distinct characteristics independently of the material properties. It is these material-independent
Many-body theory
In this section, we examine the many-body spin states of spin-1 and 2 BECs by assuming that a single spatial mode is shared by all spin states, namely, with the single-mode approximation (SMA, see Section 3.6.1). According to the discussion in Section 3.6.1, the field operator of a spin- BEC in the SMA can be expressed as We describe the Hamiltonian in terms of and investigate the many-body states.
Summary and future prospects
In the present paper, we have reviewed the basic knowledge concerning spinor Bose–Einstein condensates (BECs) that has been accumulated thus far. The fundamental characteristics of spinor BECs are the rotational invariance, the coupling between the spin and gauge degrees of freedom, and magnetism arising from the magnetic moment of the spin.
The rotational invariance and gauge invariance determine the microscopic Hamiltonian of the spinor BEC, as discussed in Section 2. The mean-field theory
Acknowledgments
We acknowledge the fruitful collaborations with P. Blair Blakie, Michikazu Kobayashi, Shingo Kobayashi, Kazue Kudo, Muneto Nitta, Nguyen Thanh Phuc, Hiroki Saito, Satoshi Tojo, Shun Uchino, and Zhifang Xu. MU acknowledges the Aspen Center for Physics, where part of this work was carried out. This work was supported by Grants-in-Aid for Scientific Research (Nos. 22340114, and 22740265), a Grant-in-Aid for scientific Research on Innovative Areas “Topological Quantum Phenomena” (No. 22103005), a
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