Topological energy bands have important geometrical properties described by the Berry curvature. We show that the Berry curvature changes the hydrodynamic equations of motion for a trapped Bose-Einstein condensate, and causes significant modifications to the collective mode frequencies. We illustrate our results for the case of two-dimensional Rashba spin-orbit coupling in a Zeeman field. Using an operator approach, we derive the effects of Berry curvature on the dipole mode in very general settings. We show that the sizes of these effects can be large and readily detected in experiment. Collective modes therefore provide a sensitive way to measure geometrical properties of energy bands.
Topological energy bands have important geometrical properties described by the Berry curvature. We show that the Berry curvature changes the hydrodynamic equations of motion for a trapped Bose-Einstein condensate, and causes significant modifications to the collective mode frequencies. We illustrate our results for the case of two-dimensional Rashba spin-orbit coupling in a Zeeman field. Using an operator approach, we derive the effects of Berry curvature on the dipole mode in very general settings. We show that the sizes of these effects can be large and readily detected in experiment. Collective modes therefore provide a sensitive way to measure geometrical properties of energy bands.
Topological phenomena arise in a wide range of systems, with fascinating physical consequences. There is great interest in finding new ways to measure such consequences in ultracold atomic gas experiments. These experiments have significant advantages over the solid-state as ultracold atoms are controllable, tuneable and clean. They can also be used to investigate properties which are inaccessible in other quantum systems. We explore some of the novel features of topological energy bands and topological solitons in ultracold gases. Topological energy bands have important geometrical properties described by the Berry curvature. Bands with nonzero Berry curvature arise in key areas of current research, such as optical lattices with more than one band; strong artificial magnetic fields and 2D spin-orbit coupling. Topological solitons are also relevant to cutting-edge experiments as they can be created and studied with high temporal and spatial resolution. In this thesis, we investigate the consequences of Berry curvature for the semiclassical dynamics of a wavepacket and the collective modes of an ultracold gas. We also study theoretically the dynamics of skyrmion-antiskyrmion pairs in a Bose Einstein condensate. Firstly, we propose a general method by which experiments can map the Berry curvature across the Brillouin zone, and thereby determine the topological properties of the energy bands of optical lattices. The Berry curvature modifies the semiclassical dynamics and hence the trajectory of a wavepacket undergoing Bloch oscillations. Our general protocol allows a clean measurement from the semiclassical dynamics of the Berry curvature over the Brillouin zone. We discuss how this protocol may be implemented and explore the semiclassical dynamics for three relevant systems. We discuss general experimental considerations for observing Berry curvature effects before reviewing some of the progress in the field since the publication of our work. Secondly, we show that the Berry curvature changes the hydrodynamic equations of motion for a trapped Bose-Einstein condensate, and causes significant modifications to the collective mode frequencies. We illustrate our results for the case of two-dimensional Rashba spin-orbit coupling in a Zeeman field, where we also apply both a sum rule and an operator approach to the dipole mode. Extending the operator method, we derive the effects of Berry curvature on the dipole mode in very general settings. We show that the sizes of these effects can be large and readily detected in experiment. Collective modes therefore provide a sensitive way to measure geometrical properties of topological energy bands. Lastly, we study theoretically the dynamics of two-component Bose-Einstein condensates in two dimensions, which admit topological excitations related to the skyrmions of nuclear physics. We explore a branch of uniformly propagating solitary waves, which, at high momentum, can be viewed as skyrmion-antiskyrmion pairs. We study these solitary waves for a range of interaction regimes and show that, for experimentally relevant cases, there is a transition to spatially extended spin-wave states at low momentum. We explain how this can be understood by analogy to the two-dimensional ferromagnet and discuss how such solitary waves can be generated and studied in experiment.
We consider a system with spin-orbit coupling and derive equations of motion which include the effects of Berry curvatures. We apply these equations to investigate the dynamics of particles with equal Rashba-Dresselhaus spin-orbit coupling in one dimension. In our derivation, the adiabatic transformation is performed first and leads to quantum Heisenberg equations of motion for momentum and position operators. These equations explicitly contain position-space, momentum-space, and phase-space Berry curvature terms. Subsequently, we perform the semiclassical approximation and obtain the semiclassical equations of motion. Taking the low-Berry-curvature limit results in equations that can be directly compared to previous results for the motion of wave packets. Finally, we show that in the semiclassical regime, the effective mass of the equal Rashba-Dresselhaus spin-orbit-coupled system can be viewed as a direct effect of the phase-space Berry curvature.
We studied the collective modes of a Bose-Einstein condensate (BEC) with spin-orbit coupling. We developed the hydrodynamic equations for spin-orbit coupled BECs and used them to study collective modes in the plane-wave phase and large Rabi coupling regime for both a uniform BEC and a BEC in a harmonic trap. In the homogeneous situation, we obtained energy spectra of elementary excitations and found that the spin-orbit coupling can increase the effective mass of the atoms, which will suppress the sound velocity. The spin-orbit coupling can also change the spin mixing, which will modify the interaction energy, and may lead to an enhancement of sound velocity. The competition between these two effects gives the behavior of sound velocity. In a harmonic trap, we found that the dipole mode and the breathing mode are coupled together in the plane-wave phase, and these two modes have a $\ensuremath{\pi}/2$ phase difference, because the spin-orbit coupling and the interaction are not invariant under spin rotation. However, in the large Rabi coupling regime, the dipole mode and the breathing mode are decoupled due to the symmetry restriction.
We propose a general method by which experiments on ultracold gases can be used to determine the topological properties of the energy bands of optical lattices, as represented by the map of the Berry curvature across the Brillouin zone. The Berry curvature modifies the semiclassical dynamics and hence the trajectory of a wave packet undergoing Bloch oscillations. However, in two dimensions these trajectories may be complicated Lissajous-like figures, making it difficult to extract the effects of Berry curvature in general. We propose how this can be done using a ``time-reversal'' protocol. This compares the velocity of a wave packet under positive and negative external force, and allows a clean measurement of the Berry curvature over the Brillouin zone. We discuss how this protocol may be implemented and explore the semiclassical dynamics for three specific systems: the asymmetric hexagonal lattice and two ``optical flux'' lattices in which the Chern number is nonzero. Finally, we discuss general experimental considerations for observing Berry curvature effects in ultracold gases.
The Berry curvature is a geometrical property of an energy band which acts as a momentum space magnetic field in the effective Hamiltonian describing single-particle quantum dynamics. We show how this perspective may be exploited to study systems directly relevant to ultracold gases and photonics. Given the exchanged roles of momentum and position, we demonstrate that the global topology of momentum space is crucially important. We propose an experiment to study the Harper-Hofstadter Hamiltonian with a harmonic trap that will illustrate the advantages of this approach and that will also constitute the first realization of magnetism on a torus.
We analytically and numerically study nonlinear dynamics in Bose–Einstein condensates (BECs) induced either by a harmonic modulation of the interaction or by the geometry of the trapping potential. To analytically describe BEC dynamics, we use a perturbative expansion based on the Poincaré–Lindstedt analysis of a Gaussian variational ansatz, whereas in the numerical approach we use numerical solutions of both a variational system of equations and the full time-dependent Gross–Pitaevskii equation. The harmonic modulation of the atomic s-wave scattering length of a BEC of 7 Li was achieved recently via Feshbach resonance, and such a modulation leads to a number of nonlinear effects, which we describe within our approach: mode coupling, higher harmonics generation and significant shifts in the frequencies of collective modes. In addition to the strength of atomic interactions, the geometry of the trapping potential is another key factor for the dynamics of the condensate, as well as for its collective modes. The asymmetry of the confining potential leads to important nonlinear effects, including resonances in the frequencies of collective modes of the condensate. We study in detail such geometric resonances and derive explicit analytic results for frequency shifts for the case of an axially symmetric condensate with two- and three-body interactions. Analytically obtained results are verified by extensive numerical simulations.