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arXiv:1504.03786 · Physical Review A

Spin diffusion in ultracold spin-orbit-coupled K40 gas

Tao Yu, M. W. Wu

Published Jul 6, 2015• 6 citations
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Abstract

We investigate the steady-state spin diffusion for ultracold spin-orbit-coupled $^{40}\mathrm{K}$ gas by the kinetic spin Bloch equation approach both analytically and numerically. Four configurations, i.e., the spin diffusions along two specific directions with the spin polarization perpendicular (transverse configuration) and parallel (longitudinal configuration) to the effective Zeeman field are studied. It is found that the behaviors of the steady-state spin diffusion for the four configurations are very different, which are determined by three characteristic lengths: the mean free path ${l}_{\ensuremath{\tau}}$, the Zeeman oscillation length ${l}_{\mathrm{\ensuremath{\Omega}}}$, and the spin-orbit coupling oscillation length ${l}_{\ensuremath{\alpha}}$. It is analytically revealed and numerically confirmed that by tuning the scattering strength, the system can be divided into five regimes: I, weak scattering regime (${l}_{\ensuremath{\tau}}\ensuremath{\gtrsim}{l}_{\mathrm{\ensuremath{\Omega}}},{l}_{\ensuremath{\alpha}}$); II, Zeeman field-dominated moderate scattering regime (${l}_{\mathrm{\ensuremath{\Omega}}}\ensuremath{\ll}{l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\ensuremath{\alpha}}$); III, spin-orbit coupling-dominated moderate scattering regime (${l}_{\ensuremath{\alpha}}\ensuremath{\ll}{l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\mathrm{\ensuremath{\Omega}}}$); IV, relatively strong scattering regime (${l}_{\ensuremath{\tau}}^{c}\ensuremath{\ll}{l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\mathrm{\ensuremath{\Omega}}},{l}_{\ensuremath{\alpha}}$); V, strong scattering regime (${l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\mathrm{\ensuremath{\Omega}}},{l}_{\ensuremath{\alpha}},{l}_{\ensuremath{\tau}}^{c}$), with ${l}_{\ensuremath{\tau}}^{c}$ representing the crossover length between the relatively strong and strong scattering regimes. In different regimes, the behaviors of the spacial evolution of the steady-state spin polarization are very rich, showing different dependencies on the scattering strength, Zeeman field and spin-orbit coupling strength. The rich behaviors of the spin diffusions in different regimes are hard to understand in the framework of the simple drift-diffusion model or the direct inhomogeneous broadening picture in the literature. However, almost all these rich behaviors can be well understood by means of our modified drift-diffusion model and/or modified inhomogeneous broadening picture. Specifically, several anomalous features of the spin diffusion are revealed, which are in contrast to those obtained from both the simple drift-diffusion model and the direct inhomogeneous broadening picture.

DOI
10.1103/physreva.92.013607

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We investigate the steady-state spin diffusion for ultracold spin-orbit-coupled 40K^{40}\mathrm{K}40K gas by the kinetic spin Bloch equation approach both analytically and numerically. Four configurations, i.e., the spin diffusions along two specific directions with the spin polarization perpendicular (transverse configuration) and parallel (longitudinal configuration) to the effective Zeeman field are studied. It is found that the behaviors of the steady-state spin diffusion for the four configurations are very different, which are determined by three characteristic lengths: the mean free path l\ensuremathτ{l}_{\ensuremath{\tau}}l\ensuremathτ​, the Zeeman oscillation length l\ensuremathΩ{l}_{\mathrm{\ensuremath{\Omega}}}l\ensuremathΩ​, and the spin-orbit coupling oscillation length l\ensuremathα{l}_{\ensuremath{\alpha}}l\ensuremathα​. It is analytically revealed and numerically confirmed that by tuning the scattering strength, the system can be divided into five regimes: I, weak scattering regime (l\ensuremathτ\ensuremath≳l\ensuremathΩ,l\ensuremathα{l}_{\ensuremath{\tau}}\ensuremath{\gtrsim}{l}_{\mathrm{\ensuremath{\Omega}}},{l}_{\ensuremath{\alpha}}l\ensuremathτ​\ensuremath≳l\ensuremathΩ​,l\ensuremathα​); II, Zeeman field-dominated moderate scattering regime (l\ensuremathΩ\ensuremath≪l\ensuremathτ\ensuremath≪l\ensuremathα{l}_{\mathrm{\ensuremath{\Omega}}}\ensuremath{\ll}{l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\ensuremath{\alpha}}l\ensuremathΩ​\ensuremath≪l\ensuremathτ​\ensuremath≪l\ensuremathα​); III, spin-orbit coupling-dominated moderate scattering regime (l\ensuremathα\ensuremath≪l\ensuremathτ\ensuremath≪l\ensuremathΩ{l}_{\ensuremath{\alpha}}\ensuremath{\ll}{l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\mathrm{\ensuremath{\Omega}}}l\ensuremathα​\ensuremath≪l\ensuremathτ​\ensuremath≪l\ensuremathΩ​); IV, relatively strong scattering regime (l\ensuremathτc\ensuremath≪l\ensuremathτ\ensuremath≪l\ensuremathΩ,l\ensuremathα{l}_{\ensuremath{\tau}}^{c}\ensuremath{\ll}{l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\mathrm{\ensuremath{\Omega}}},{l}_{\ensuremath{\alpha}}l\ensuremathτc​\ensuremath≪l\ensuremathτ​\ensuremath≪l\ensuremathΩ​,l\ensuremathα​); V, strong scattering regime (l\ensuremathτ\ensuremath≪l\ensuremathΩ,l\ensuremathα,l\ensuremathτc{l}_{\ensuremath{\tau}}\ensuremath{\ll}{l}_{\mathrm{\ensuremath{\Omega}}},{l}_{\ensuremath{\alpha}},{l}_{\ensuremath{\tau}}^{c}l\ensuremathτ​\ensuremath≪l\ensuremathΩ​,l\ensuremathα​,l\ensuremathτc​), with l\ensuremathτc{l}_{\ensuremath{\tau}}^{c}l\ensuremathτc​ representing the crossover length between the relatively strong and strong scattering regimes. In different regimes, the behaviors of the spacial evolution of the steady-state spin polarization are very rich, showing different dependencies on the scattering strength, Zeeman field and spin-orbit coupling strength. The rich behaviors of the spin diffusions in different regimes are hard to understand in the framework of the simple drift-diffusion model or the direct inhomogeneous broadening picture in the literature. However, almost all these rich behaviors can be well understood by means of our modified drift-diffusion model and/or modified inhomogeneous broadening picture. Specifically, several anomalous features of the spin diffusion are revealed, which are in contrast to those obtained from both the simple drift-diffusion model and the direct inhomogeneous broadening picture.
We compute spin diffusion in a dilute Fermi gas at arbitrary temperature, polarization, and strong interaction in the normal phase using kinetic theory. While the longitudinal spin diffusivity ${D}_{\ensuremath{\parallel}}$ depends weakly on polarization and diverges for small temperatures, the transverse spin diffusivity ${D}_{\ensuremath{\perp}}$ has a strong polarization dependence and approaches a finite value for $T\ensuremath{\rightarrow}0$ in the Fermi liquid phase. For a 3D unitary Fermi gas at infinite scattering length, the diffusivities reach a minimum near the quantum limit of diffusion $\ensuremath{\hbar}/m$ in the quantum degenerate regime and are strongly suppressed by medium scattering, and we discuss the importance of the spin-rotation effect. In two dimensions, ${D}_{\ensuremath{\perp}}$ attains a minimum at strong coupling $\ensuremath{-}1\ensuremath{\lesssim}\mathrm{ln}({k}_{F}{a}_{2D})\ensuremath{\lesssim}1$ and reaches ${D}_{\ensuremath{\perp}}\ensuremath{\sim}0.2...\phantom{\rule{0.16em}{0ex}}0.3\phantom{\rule{0.16em}{0ex}}\ensuremath{\hbar}/m$ at large polarization. These values are consistent with recent measurements of two-dimensional ultracold atomic gases in the strong coupling regime.
We report the anomalous D'yakonov-Perel' spin relaxation in ultracold spin-orbit-coupled 40{}^{40}40K gas when the coupling between $|9/2,9/2\ensuremath{\rangle}$ and $|9/2,7/2\ensuremath{\rangle}$ states, acting as an effective Zeeman magnetic field, is much stronger than the spin-orbit coupled field. Both the transverse and longitudinal spin relaxations are investigated with small and large spin polarizations. It is found that with small spin polarization, the transverse (longitudinal) spin relaxation is divided into four (two) regimes: the normal weak scattering regime, the anomalous D'yakonov-Perel'-like regime, the anomalous Elliott-Yafet-like regime, and the normal strong scattering regime (the anomalous Elliott-Yafet-like regime and the normal strong scattering regime). With large spin polarization, we find that the Hartree-Fock self-energy, which acts as an effective magnetic field, can markedly suppress the transverse spin relaxation in both weak and strong scattering limits. Moreover, by noting that as both the momentum relaxation time and the Hartree-Fock effective magnetic field vary with the scattering length in cold atoms, the anomalous D'yakonov-Perel'-like regime is suppressed and the transverse spin relaxation is hence divided into three regimes in the scattering length dependence: the normal weak scattering regime, the anomalous Elliott-Yafet-like regime, and the strong scattering regime. On the other hand, the longitudinal spin relaxation is again divided into the anomalous Elliott-Yafet-like and normal strong scattering regimes. Furthermore, it is found that the longitudinal spin relaxation can be either enhanced or suppressed by the Hartree-Fock effective magnetic field if the spin polarization is parallel or antiparallel to the effective Zeeman magnetic field.
In recent experiments, spin-orbit-coupled (SOC) bosonic gases in an optical lattice have been successfully prepared into any Bloch band [Hamner et al., Phys. Rev. Lett. 114, 070401 (2015)], which promises a viable contender in the competitive field of simulating gauge-related phenomena. However, the ground-state phase diagram of such systems in the superfluid regime is still lacking. Here we present a detailed study of the phase diagram in an optically trapped Bose gas with equal-weight Rashba and Dresselhaus SO coupling. We identify four different quantum phases, which include three normal phases and a mixed phase, by considering the wave vector k1{k}_{1}k1​, the longitudinal $\ensuremath{\langle}{\ensuremath{\sigma}}_{z}\ensuremath{\rangle}$, and the transverse $\ensuremath{\langle}{\ensuremath{\sigma}}_{x}\ensuremath{\rangle}$ spin polarizations as three order parameters. The ground state of normal phases is a Bloch wave with a single wave vector k1{k}_{1}k1​, which can position in arbitrary regions in the Brillouin zone. By contrast, the ground state of the mixed phase is a superposition of two Bloch waves with opposite k1{k}_{1}k1​, which, remarkably, may lack periodicity even though the system's Hamiltonian is periodic. This mixed phase in the lattice setting can be seen as the counterpart of the stripe phase associated with the uniform SOC gas. Furthermore, due to the lattice-renormalized SOC, the phase diagram of the model system becomes significantly different from the uniform case when the lattice strength grows. Finally, a scheme for experimentally probing the mixed phase using Bragg spectroscopy is proposed.
We study ultracold bosons in three dimensions with an anisotropic Rashba-Dresselhaus spin-orbit coupling. We first carry out the exact summation of ladder diagrams for the two-boson ttt matrix at zero energy. Then, with the ttt matrix as the effective interaction, we find the ground-state phase diagrams of bosons in mean field as a function of the spin-orbit coupling, the anisotropy, and the scattering lengths between particles in the same and in different pseudospin states. The resulting phase diagrams have much richer structures than those obtained using mean-field couplings, exhibiting three different phases: a plane-wave condensate, a striped condensate, and an unstable phase. The differences between the present approach using the ttt matrix compared to using mean-field couplings is significant for large scattering lengths, large spin-orbit-coupling strength, or small anisotropy.
Ultra-cold trapped atoms, with their high degree of tunability, provide ideal model systems to study physical phenomena with applications in many different fields of research. This thesis describes studies on spin transport phenomenon in a trapped ultra-cold spin-polarized 87Rb gas at temperatures above quantum degeneracy. This work is focused on the less studied regime of cross-over between classical and quantum transport. Diffusion is a fundamental dissipative process that tends to relax any system towards a state of minimum inhomogeneity. In this work we study longitudinal spin diffusion as a special case of spin transport. The system studied here consists of two anti-parallel longitudinal spin domains separated by a helical domain-wall. We report that the diffusion process manifests a significant deviation from classical diffusion due to purely quantum mechanical modifications. The two-domain spin textures are prepared using optical and microwave pulse techniques, and the dynamics of the spin structure is studied as it relaxes towards the final equilibrium state. Generally, there is a wide range of parameter space that could be studied. In this work we focused our studies on the effects related to the degree of coherence in the domainwall as well as the effective magnetic field acting on the spins. By controllably tuning these experimental parameters, we studied in detail how the spatiotemporal behaviour of the diffusion dynamics is modified. Our results show that the longitudinal spin diffusion time scales depend sensitively on the domain-wall degree of coherence. External magnetic field gradients also alter the dynamics noticeably, manifesting a significant dependence on the sign of the applied field gradients.
The dissertation explores the effects of synthetic spin-orbit coupling on the behaviour of quantum gases in several different contexts. We first study realistic methods to create vortices in spin-orbit-coupled (SOC) Bose-Einstein condensates (BEC). We propose two different methods to induce thermodynamically stable static vortex configurations: (1) to rotate both the Raman lasers and the anisotropic trap; and (2) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. We solve the Gross-Pitaevskii equation for several experimentally relevant regimes and find new interesting effects such as spatial separation of left- and right-moving spin-orbit-coupled condensates, and the appearance of unusual vortex arrangements. Next we consider cold atoms in an optical lattice with synthetic SOC in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model and derive the low-energy spin Hamiltonian which is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interaction. We find that the Hamiltonian supports a rich classical phase diagram with collinear, spiral and vortex phases. Next we study the time evolution of the magnetization in a Rashba spin-orbit-coupled Fermi gas, starting from a fully-polarized initial state. We model the dynamics using a Boltzmann equation, which we solve in the Hartree-Fock approximation. The resulting non-linear system of equations gives rise to three distinct dynamical regimes controlled by the ratio of interaction and spin-orbit-coupling strength λ: for small λ, the magnetization decays to zero. For intermediate λ, it displays undamped oscillations about zero and for large λ, a partially magnetized state is dynamically stabilized. Motivated by an interesting stripe phase which appears in BEC with SOC [Li et al., Phys. Rev. Lett. 108, 225301 (2011)], we study the finite-temperature phase diagram of a pseudospin-1/2 Bose gas with contact interactions. We show that strong inter-spin interactions can lead to the appearance of magnetically ordered phases at temperatures above the superfluid transition. For the case of inter-spin attraction, we also discuss the possibility of a bosonic analogue of the Cooper-paired phase, however this state is not energetically favourable. We extend our calculations to a spin-orbit-coupled Bose gas to investigate the possibility of stripe ordering in the normal phase. However, within our approximations, we do not find an instability towards stripe formation. Finally, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This Hamiltonian can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011)], or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha and C. Chin, Nature Physics 9, 769 (2013)]. By numerically comparing energies of various trial wave-functions, we show that, at low densities, the ground state is strongly correlated, in contrast to a typical mean-field BEC. The trial wave-function with the lowest energy is of Jastrow-type and it describes a state with finite, but strongly reduced, condensate fraction.