Abstract We study binary Bose–Einstein condensates subject to synthetic magnetic fields in mutually parallel or antiparallel directions. Within the mean-field theory, the two types of fields have been shown to give the same vortex-lattice phase diagram. We develop an improved effective field theory to study properties of collective modes and ground-state intercomponent entanglement. Here, we point out the need to introduce renormalized coupling constants for coarse-grained densities. We show that the low-energy excitation spectra for the two types of fields are related to each other by suitable rescaling with the renormalized coupling constants. By calculating the entanglement entropy, we find that for an intercomponent repulsion (attraction), the two components are more strongly entangled in the case of parallel (antiparallel) fields, in qualitative agreement with recent studies for a quantum (spin) Hall regime. We also find that the entanglement spectrum exhibits an anomalous square-root dispersion relation, which leads to a subleading logarithmic term in the entanglement entropy. All of these are confirmed by numerical calculations based on the Bogoliubov theory with the lowest-Landau-level approximation. Finally, we investigate the effects of quantum fluctuations on the phase diagrams by calculating the correction to the ground-state energy due to zero-point fluctuations in the Bogoliubov theory. We find that the boundaries between rhombic-, square-, and rectangular-lattice phases shift appreciably with a decrease in the filling factor.
Abstract We study binary Bose–Einstein condensates subject to synthetic magnetic fields in mutually parallel or antiparallel directions. Within the mean-field theory, the two types of fields have been shown to give the same vortex-lattice phase diagram. We develop an improved effective field theory to study properties of collective modes and ground-state intercomponent entanglement. Here, we point out the need to introduce renormalized coupling constants for coarse-grained densities. We show that the low-energy excitation spectra for the two types of fields are related to each other by suitable rescaling with the renormalized coupling constants. By calculating the entanglement entropy, we find that for an intercomponent repulsion (attraction), the two components are more strongly entangled in the case of parallel (antiparallel) fields, in qualitative agreement with recent studies for a quantum (spin) Hall regime. We also find that the entanglement spectrum exhibits an anomalous square-root dispersion relation, which leads to a subleading logarithmic term in the entanglement entropy. All of these are confirmed by numerical calculations based on the Bogoliubov theory with the lowest-Landau-level approximation. Finally, we investigate the effects of quantum fluctuations on the phase diagrams by calculating the correction to the ground-state energy due to zero-point fluctuations in the Bogoliubov theory. We find that the boundaries between rhombic-, square-, and rectangular-lattice phases shift appreciably with a decrease in the filling factor.
We study collective modes of vortex lattices in two-component Bose–Einstein condensates subject to synthetic magnetic fields in mutually parallel or antiparallel directions. By means of the Bogoliubov theory with the lowest-Landau-level approximation, we numerically calculate the excitation spectra for a rich variety of vortex lattices that appear commonly for parallel and antiparallel synthetic fields. We find that in all of these cases there appear two distinct modes with linear and quadratic dispersion relations at low energies, which exhibit anisotropy reflecting the symmetry of each lattice structure. Remarkably, the low-energy spectra for the two types of fields are found to be related to each other by simple rescaling when vortices in different components overlap owing to an intercomponent attraction. These results are consistent with an effective field theory analysis. However, the rescaling relations break down for interlaced vortex lattices appearing with an intercomponent repulsion, indicating a nontrivial effect of an intercomponent vortex displacement beyond the effective field theory. We also find that high-energy parts of the excitation bands exhibit line or point nodes as a consequence of a fractional translation symmetry present in some of the lattice structures.
Bose-Einstein condensates (BECs) in ultracold atomic gas have been studied intensively, since the state of atoms is well controlled by lasers and external electromagnetic fields.Synthetic gauge fields can be induced in a charge-neutral system by mechanical rotation of the system or by using lasers.In a BEC under a synthetic gauge fields, a quantized vortex or a vortex lattice, where multiple quantized vortices are periodically located, are observed.In binary BECs composed of bosonic atoms in two hyperfine states, parallel and antiparallel synthetic gauge fields are induced by the former (mechanical rotation) and the latter (laser) techniques, respectively.In the mean-field theory, vortex lattices appear with various configurations which depend on the ratio of the intercomponent interaction to the intracomponent one.While the ground-state phase diagrams for parallel and antiparallel fields are the same in the mean-field regime, two phase diagrams are significantly different in the quantum Hall regime, where a magnetic field is strong.In the first part of this thesis, we study the excitation spectra, correlation functions and the quantum depletion of vortex lattices by means of the effective-field theory and the Bogoliubov theory under the lowest-Landau-level (LLL) approximation.We obtain the effective-field theory of vortex lattices in binary BECs by introducing the missing elastic energy term from the symmetry consideration.By renormalizing the coupling constants, we find the effective-field theory for strong magnetic fields so that we can compare it with the Bogoliubov theory.We obtain the analytical expression of the dispersion relations for parallel and antiparallel fields by diagonalizing the effective Hamiltonian and the rescaling relations between them.From the rescaling relations, we clarify that the different excited states for parallel and antiparallel fields are indeed related to each other.By calculating the correlation function, we show that a one-particle density matrix exhibits a quasi-longrange order and that the fraction of depletion increases logarithmically with respect to the number of fluxes.By means of the Bogoliubov theory, we numerically obtain energy spectra with quadratic and linear dispersion relations for parallel and antiparallel fields and confirm the rescaling relations for all the phases.We find that the quantum depletion depends crucially on the direction of synthetic gauge fields and diverges logarithmically with respect to the number of fluxes.By calculating the correction to the ground-state energy due to zero-point fluctuations by the Bogoliubov theory, we show that the transition points shift appreciably due to quantum fluctuations.Furthermore, we show that the variations of an inner angle and an aspect ratio with respect to the ratio of interactions are altered from those obtained by the mean-field theory for rhombic and rectangular lattices, respectively.We find that the quantum fluctuation changes the ground-state phase diagram and leads to the distinction between the phase diagrams for parallel and antiparallel fields which are unique to binary BECs.Moreover, we relate the quantum depletion to shifts of the transition points since the phase diagram is largely modified when the quantum depletion which characterizes the quantum fluctuation is also large.In the second part of the present thesis, we study the intercomponent entanglement spectrum (ES) and entanglement entropy (EE) of vortex lattices in binary BECs by exploiting the formalisms developed in the first part.By means of the effective-field theory, we show that the ES exhibits a square-root dispersion relation for all the phases and that the rescaling relation holds between the ESs for parallel and antiparallel fields, similarly to the energy spectra.We confirm these predictions through a numerical calculation by List of PublicationsThis Thesis is based on the following publications. Collective modes of vortex lattices in two-component Bose-Einstein condensates under synthetic gauge fields
We consider a rapidly rotating two-component Bose-Einstein condensate (BEC) containing a vortex lattice. We calculate the dispersion relation for small oscillations of vortex positions (Tkachenko modes) in the mean-field quantum Hall regime, taking into account the coupling of these modes with density excitations. Using an analytic form for the density of the vortex lattice, we numerically calculate the elastic constants for different lattice geometries. We also apply this method to calculate the elastic constant for the single-component triangular lattice. For a two-component BEC, there are two kinds of Tkachenko modes, which we call acoustic and optical in analogy with phonons. For all lattice types, acoustic Tkachenko mode frequencies have quadratic wave-number dependence at long wavelengths, while the optical Tkachenko modes have linear dependence. For triangular lattices the dispersion of the Tkachenko modes are isotropic, while for other lattice types the dispersion relations show directional dependence consistent with the symmetry of the lattice. Depending on the intercomponent interaction there are five distinct lattice types, and four structural phase transitions between them. Two of these transitions are second order and are accompanied by the softening of an acoustic Tkachenko mode. The remaining two transitions are first order and while one of them is accompanied by the softening of an optical mode, the other does not have any dramatic effect on the Tkachenko spectrum. We also find an instability of the vortex lattice when the intercomponent repulsion becomes stronger than the repulsion within components.
We study the entanglement entropy and spectrum between components in binary Bose-Einstein condensates in d spatial dimensions. We employ effective field theory to show that the entanglement spectrum exhibits an anomalous square-root dispersion relation in the presence of intercomponent tunneling (Rabi coupling) and a gapped dispersion relation in its absence. These spectral features are associated with the emergence of long-range interactions in terms of the superfluid velocity and the particle density in the entanglement Hamiltonian. Our results demonstrate that unusual long-range interactions can be emulated in a subsystem of multicomponent BECs that have only short-range interactions. We also find that for a finite Rabi coupling the entanglement entropy exhibits a volume-law scaling with subleading logarithmic corrections originating from the Nambu-Goldstone mode and the symmetry restoration for a finite volume.
We review the topic of quantized vortices in multicomponent Bose–Einstein condensates of dilute atomic gases, with an emphasis on the two-component condensates. First, we review the fundamental structure, stability and dynamics of a single vortex state in a slowly rotating two-component condensates. To understand recent experimental results, we use the coupled Gross–Pitaevskii equations and the generalized nonlinear sigma model. An axisymmetric vortex state, which was observed by the JILA group, can be regarded as a topologically trivial skyrmion in the pseudospin representation. The internal, coherent coupling between the two components breaks the axisymmetry of the vortex state, resulting in a stable vortex molecule (a meron pair). We also mention unconventional vortex states and monopole excitations in a spin-1 Bose–Einstein condensate. Next, we discuss a rich variety of vortex states realized in rapidly rotating two-component Bose–Einstein condensates. We introduce a phase diagram with axes of rotation frequency and the intercomponent coupling strength. This phase diagram reveals unconventional vortex states such as a square lattice, a double-core lattice, vortex stripes and vortex sheets, all of which are in an experimentally accessible parameter regime. The coherent coupling leads to an effective attractive interaction between two components, providing not only a promising candidate to tune the intercomponent interaction to study the rich vortex phases but also a new regime to explore vortex states consisting of vortex molecules characterized by anisotropic vorticity. A recent experiment by the JILA group vindicated the formation of a square vortex lattice in this system.
We study the ground-state phase diagram of two-dimensional two-component (or pseudospin-21) Bose gases in a high synthetic magnetic field in the space of the total filling factor and the ratio of the intercomponent coupling ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ to the intracomponent one g>0. Using exact diagonalization, we find that when the intercomponent coupling is attractive (${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}<0$), the product states of a pair of nearly uncorrelated quantum Hall states are remarkably robust and persist even when $|{g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}|$ is close to g. This contrasts with the case of an intercomponent repulsion, where a variety of spin-singlet quantum Hall states with high intercomponent entanglement emerge for ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}\ensuremath{\approx}g$. We interpret this marked dependence on the sign of ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ in light of pseudopotentials on a sphere, and also explain recent numerical results in two-component Bose gases in mutually antiparallel magnetic fields where a qualitatively opposite dependence on the sign of ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ is found. Our results thus unveil an intriguing connection between multicomponent quantum Hall systems and quantum spin Hall systems in minimal setups.