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arXiv:1506.01172 · Physical Review A

Bose-Einstein-condensed cold atoms in a lattice system with k -space Berry curvature

Xiaohui Li, Ting-Pong Choy, Tai Kai Ng

Published Mar 27, 2017• 3 citations
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Abstract

In this paper we study the properties of cold bosons in two-dimensional optical lattices where Bose condensation occurs at a momentum point $\mathbf{k}$ with nonzero $k$-space Berry curvature. By combining results from both analytic and numerical approaches, the bulk and edge excitation spectra of the system are derived. We show that the boson system carries nonuniversal, nontopological temperature-dependent equilibrium angular momentum and edge current at low temperatures. The differences between boson systems with real- and $k$-space Berry curvature are pointed out. In particular, using a simple variational wave-function approach, we show that for bosons in a harmonic trap, the stability of the Bose-condensed state towards vortex formation is rather different in the two cases because of the different angular momentum scalings in the ground states.

DOI
10.1103/physreva.95.033628

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In this paper we study the properties of cold bosons in two-dimensional optical lattices where Bose condensation occurs at a momentum point k\mathbf{k}k with nonzero kkk-space Berry curvature. By combining results from both analytic and numerical approaches, the bulk and edge excitation spectra of the system are derived. We show that the boson system carries nonuniversal, nontopological temperature-dependent equilibrium angular momentum and edge current at low temperatures. The differences between boson systems with real- and kkk-space Berry curvature are pointed out. In particular, using a simple variational wave-function approach, we show that for bosons in a harmonic trap, the stability of the Bose-condensed state towards vortex formation is rather different in the two cases because of the different angular momentum scalings in the ground states.
We investigate the low-lying excitations of a stack of weakly coupled two-dimensional Bose-Einstein condensates, which is formed by a one-dimensional optical lattice. In particular, we calculate the dispersion relations of the monopole and quadrupole modes, both for the ground state as well as for the case in which the system contains a vortex along the direction of the lasers creating the optical lattice. Our variational approach enables us to determine analytically the dispersion relations for an arbitrary number of atoms in every two-dimensional condensate and for an arbitrary momentum. We also discuss the feasibility of observing our results experimentally.
This review explores the dynamics and the low-energy excitation spectra of Bose-Einstein condensates (BECs) of interacting bosons in external potential traps putting particular emphasis on the emerging many-body effects beyond mean-field descriptions. To do so, methods have to be used that, in principle, can provide numerically exact results for both the dynamics and the excitation spectra in a systematic manner. Numerically exact results for the dynamics are presented employing the well-established multicongurational time-dependent Hartree for bosons (MCTDHB) method. The respective excitation spectra are calculated utilizing the more recently introduced linear-response theory atop it (LR-MCTDHB). The latter theory gives rise to an, in general, non-hermitian eigenvalue problem. The theory and its newly developed implementation are described in detail and benchmarked towards the exactly-solvable harmonic-interaction model. Several applications to BECs in one- and two-dimensional potential traps are discussed. With respect to dynamics, it is shown that both the out-of-equilibrium tunneling dynamics and the dynamics of trapped vortices are of many-body nature. Furthermore, many-body effects in the excitation spectra are presented for BECs in different trap geometries. It is demonstrated that even for essentially-condensed systems, the spectrum of the lowest-in-energy excitations computed at the many-body level can differ substantially from the standard mean-field description. In general, it is shown that bosons carrying angular momentum are more sensitive to many-body effects than bosons without. These effects are present in both the dynamics and the excitation spectrum.
We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein condensate in an optical lattice. Since the Bogoliubov equation for a bosonic system is non-Hermitian, complex eigenvalues often appear and induce dynamical instability. As a function of momentum, the onset of appearance and disappearance of complex eigenvalues is an exceptional point (EP), which is a point where the Hamiltonian is not diagonalizable and hence the Berry connection and curvature are ill-defined, preventing defining topological invariants. In this paper, we propose a systematic procedure to avoid EPs in the Brillouin zone by introducing an imaginary part of the momentum. We then define the Berry phase for a one-dimensional bosonic Bogoliubov system. Extending the argument for Hermitian systems, the Berry phase for an inversion-symmetric system is shown to be Z2{\mathbb{Z}}_{2}Z2​. As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The Z2{\mathbb{Z}}_{2}Z2​ invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.
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In this paper we develop an analytic expression for the critical temperature for a gas of ideal bosons in a combined harmonic lattice potential, relevant to current experiments using optical lattices. We give corrections to the critical temperature arising from effective mass modifications of the low energy spectrum, finite size effects, and excited band states. We compute the critical temperature using numerical methods and compare to our analytic result. We study condensation in an optical lattice over a wide parameter regime and demonstrate that the critical temperature can be increased or reduced relative to the purely harmonic case by adjusting the harmonic trap frequency. We show that a simple numerical procedure based on a piecewise analytic density of states provides an accurate prediction for the critical temperature.
We analyze the vortex solution space of the ( 2 + 1 ) -dimensional nonlinear Dirac equation for bosons in a honeycomb optical lattice at length scales much larger than the lattice spacing. Dirac point relativistic covariance combined with s-wave scattering for bosons leads to a large number of vortex solutions characterized by different functional forms for the internal spin and overall phase of the order parameter. We present a detailed derivation of these solutions which include skyrmions, half-quantum vortices, Mermin–Ho and Anderson–Toulouse vortices for vortex winding ℓ = 1. For ℓ ⩾ 2 we obtain topological as well as non-topological solutions defined by the asymptotic radial dependence. For arbitrary values of ℓ the non-topological solutions include bright ring-vortices which explicitly demonstrate the confining effects of the Dirac operator. We arrive at solutions through an asymptotic Bessel series, algebraic closed-forms, and using standard numerical shooting methods. By including a harmonic potential to simulate a finite trap we compute the discrete spectra associated with radially quantized modes. We demonstrate the continuous spectral mapping between the vortex and free particle limits for all of our solutions.