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arXiv:1702.03298 · Physical Review A

Phase-space curvature in spin-orbit-coupled ultracold atomic systems

J. Armaitis, Julius Ruseckas, Egidijus Anisimovas

Published Apr 17, 2017• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.95.043616

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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.
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.
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.
This dissertation contains a collection of topics on the spin-orbit coupling of cold atoms and semiconductors. We first consider the effects of an optically induced spin-orbit interaction on a system of Bosonic atoms. The spin-orbit term couples the emergent pseudo-spin- 1/2 degree of freedom to momentum. The single particle energy spectrum has a low energy band with two minima at non-zero momentum. At low temperatures, a many-body system will condense into these minima. In the presence of vanishing interactions, the ground state wavefunction is found to be N00N state of "left" and "right"-moving state. An experimentally observable signature of the condensate is predicted that can be observed using time-of-flight imaging. We then predict a bulk manifestation of the spin-Hall effect in an inhomoge- neous spin-orbit-coupled system. The phenomena is predicted in the framework of the spin diffusion equations generalized to include arbitrary Rashba and linear and cubic Dresselhaus terms. This framework shows that a bulk spin-density wave with a wavevector oriented perpendicular to an applied electric field will induce a charge- density wave characterized by a π/2-phase shift and a non-monotonic time-varying amplitude. The optimal values of spin-orbit coupling for observation of the effect are determined. We propose a scheme for using atom interferometry to measure weak time- dependent accelerations. This proposal uses an ensemble of dilute trapped bosons with two pseudo-spin states coupled to a synthetic magnetic field, but with opposite effective charges. The synthetic field acts to couple spin to momentum continuously, which continuously imparts the acceleration on the phase of the internal states. We use time reversal pulses to reduce noise. The sensitivity of such a system is estimated to be S ∼ 10−7 m/s^2/√Hz Finally, we consider the semiclassical dynamics of a trapped spin-orbit coupled system. We find non-linear dynamics parameterized by the value of the initial displacement of the trap, and the anisotropy of the spin-orbit energy spectrum. We show that the dynamics can give a Berry's phase, and propose an experiment to measure this phase. We then propose a generalization of the 4-level scheme that allows for spin-orbit coupling described by a vector potential that is proportional to the angular momentum operator.
The recent experimental realization of synthetic Rashba spin-orbit coupling (SOC) paves a new avenue for exploring topological phases in ultracold atoms. The unequivocal characterization of such topological physics requires a simple scheme for measuring the Berry phase originating from the SOC. Here we propose a scheme to realize momentum space Aharonov-Bohm interferometry in a Rashba spin-orbit–coupled Bose-Einstein condensate through a sudden change of the in-plane Zeeman field. We find that the π Berry phase for the Dirac point of the Rashba SOC is directly revealed by a robust dark interference fringe in the momentum space. An external perpendicular Zeeman field opens a band gap at the Dirac point, which reduces the Berry phase along the Rashba ring, leading to lower brightness of the interference fringe. We develop a variational model with semiclassical equations of motion of essential dynamical quantities for describing the interference process, yielding real and momentum space trajectories and geometric phases agreeing with the real-time simulation of the Gross-Pitaevskii equation. Our study may pave the way for the experimental detection of Berry phases in ultracold atomic systems and further exploration of momentum space interference dynamics.
We introduce a theoretical approach to determine the spin structure of harmonically trapped atoms with two-body zero-range interactions subject to an equal mixture of Rashba and Dresselhaus spin-orbit coupling created through Raman coupling of atomic hyperfine states. The spin structure of bosonic and fermionic two-particle systems with finite and infinite two-body interaction strength ggg is calculated. Taking advantage of the fact that the NNN-boson and NNN-fermion systems with infinitely large coupling strength ggg are analytically solvable for vanishing spin-orbit coupling strength kso{k}_{\mathrm{so}}kso​ and vanishing Raman coupling strength $\mathrm{\ensuremath{\Omega}}$, we develop an effective spin model that is accurate to second order in $\mathrm{\ensuremath{\Omega}}$ for any kso{k}_{\mathrm{so}}kso​ and infinite ggg. The three- and four-particle systems are considered explicitly. It is shown that the effective spin Hamiltonian, which contains a Heisenberg exchange term and an anisotropic Dzyaloshinskii-Moriya exchange term, describes the transitions that these systems undergo with the change of kso{k}_{\mathrm{so}}kso​ as a competition between independent spin dynamics and nearest-neighbor spin interactions.
We present an analysis of two-dimensional (2D) matter-wave solitons, governed by the pseudospinor system of Gross-Pitaevskii equations with self- and cross attraction, which includes the spin-orbit coupling (SOC) in the general Rashba-Dresselhaus form, and, separately, the Rashba coupling and the Zeeman splitting. Families of semivortex (SV) and mixed-mode (MM) solitons are constructed, which exist and are stable in free space, as the SOC terms prevent the onset of the critical collapse and create the otherwise missing ground states in the form of the solitons. The Dresselhaus SOC produces a destructive effect on the vortex solitons, while the Zeeman term tends to convert the MM states into the SV ones, which eventually suffer delocalization. Existence domains and stability boundaries are identified for the soliton families. For physically relevant parameters of the SOC system, the number of atoms in the 2D solitons is limited by ∼1.5×10^{4}. The results are obtained by means of combined analytical and numerical methods.