Spin-orbit-coupled Bose-Einstein condensates

Abstract

Spin-orbit (SO) coupling--the interaction between a quantum particle's spin and its momentum--is ubiquitous in physical systems. In condensed matter systems, SO coupling is crucial for the spin-Hall effect and topological insulators; it contributes to the electronic properties of materials such as GaAs, and is important for spintronic devices. Quantum many-body systems of ultracold atoms can be precisely controlled experimentally, and would therefore seem to provide an ideal platform on which to study SO coupling. Although an atom's intrinsic SO coupling affects its electronic structure, it does not lead to coupling between the spin and the centre-of-mass motion of the atom. Here, we engineer SO coupling (with equal Rashba and Dresselhaus strengths) in a neutral atomic Bose-Einstein condensate by dressing two atomic spin states with a pair of lasers. Such coupling has not been realized previously for ultracold atomic gases, or indeed any bosonic system. Furthermore, in the presence of the laser coupling, the interactions between the two dressed atomic spin states are modified, driving a quantum phase transition from a spatially spin-mixed state (lasers off) to a phase-separated state (above a critical laser intensity). We develop a many-body theory that provides quantitative agreement with the observed location of the transition. The engineered SO coupling--equally applicable for bosons and fermions--sets the stage for the realization of topological insulators in fermionic neutral atom systems.

Figure 1 |. Scheme for creating SO coupling.

a, Level diagram. Two $\lambda \approx 804.1\text{nm}$ lasers (thick lines) coupled states $|F=1,m_F=0\rangle =|\uparrow \rangle$ and $|F=1,m_F=-1\rangle =|\downarrow \rangle$, differing in energy by a $ħ{\omega }_{\text{Z}}$ Zeeman shift. The lasers, with frequency difference $\text{Δ}{\omega }_{\text{L}}/2\pi =\left({\omega }_{\text{Z}}+\delta /ħ\right)/2\pi$, were detuned $\delta$ from the Raman resonance. $|m_F=0\rangle$ and $|m_F=+1\rangle$ had a $ħ\left({\omega }_{\text{Z}}-{\omega }_q\right)$ energy difference; because $ħ{\omega }_q=3.8E_{\text{L}}$ is large, $|m_F=+1\rangle$ can be neglected. b, Computed dispersion. Eigenenergies at $\delta =0$ for $\Omega =0$ (grey) to $5E_{\text{L}}$. When $\Omega <4E_{\text{L}}$ the two minima correspond to the dressed spin states $|{\uparrow }^{\prime }\rangle$ and $|{\downarrow }^{\prime }\rangle$. c, Measured minima. Quasimomentum $q_{\uparrow ,\downarrow }$ of $|{\uparrow }^{\prime },{\downarrow }^{\prime }\rangle$ versus $\Omega$ at $\delta =0$, corresponding to the minima of $E_{-}(q)$. Each point is averaged over about ten experiments; the uncertainties are their standard deviation. d, Spin-momentum decomposition. Data for sudden laser turn-off: $\delta \approx 0,\Omega =2E_{\text{L}}$ (top image pair), and $\Omega =6E_{\text{L}}$ (bottom image pair). For $\Omega =2E_{\text{L}},|{\uparrow }^{\prime }\rangle$ consists of $|\uparrow ,{\kappa }_x\approx 0\rangle$ and $|\downarrow ,{\kappa }_x\approx -2k_L\rangle$, and $|{\downarrow }^{\prime }\rangle$ consists of $|\uparrow ,{\kappa }_x\approx 2k_L\rangle$ and $|\downarrow ,{\kappa }_x\approx 0\rangle$.

Figure 2 |. Phases of a SO-coupled BEC.

a, b, Mean field phase diagrams for infinite homogeneous SO-coupled ⁸⁷Rb BECs (1.5-kHz chemical potential). The background colours indicate atom fraction in $|\uparrow \rangle$ and $|\downarrow \rangle$. Between the dashed lines there are two dressed spin states, $|{\uparrow }^{\prime }\rangle$ and $|{\downarrow }^{\prime }\rangle$. a, Single-particle phase diagram in the $\Omega -\delta$ plane. b, Phase diagram (enlargement of the grey rectangle in a), as modified by interactions. The dots represent a metastable region where the fraction of atoms $f_{{\uparrow }^{\prime },{\downarrow }^{\prime }}$ remains largely unchanged for $t_{\text{h}}=3\text{s}$. c, Miscible-to-immiscible transition. Phase line for mixtures of dressed spins and images after TOF (with populations $N_{\uparrow }\approx N_{\downarrow }$, mapped from $|{\uparrow }^{\prime }\rangle$ and $|{\downarrow }^{\prime }\rangle$ showing the transition from phase-mixed to phase-separated within the ‘metastable window’ of detuning.

Figure 3 |. Population relaxation.

a, Condensate fraction $f_{{\downarrow }^{\prime }}$ in $|{\downarrow }^{\prime }\rangle$ at $\Omega =0.6E_{\text{I}}$ versus detuning $\delta$ at $t_{\text{h}}=0.1$, 0.5 and 3 s showing $w_{\delta }$ decrease with increasing $t_{\text{h}}$. The solid curves are fits to the error function from which we obtained the width $w_{\delta }$. b, Metastable detuning width. Width $w_{\delta }$ versus $\Omega$ at $t_{\text{h}}=0.1$, 0.5 and 3 s; the data fits well to $a\left[b+{\left(\Omega /E_{\text{L}}\right)}^{-2}\right]$ (dashed curves).

Figure 4 |. Miscible to immiscible phase transition.

Phase separation $s$ versus $\Omega$ with $t_{\text{h}}=3\text{s}$; the solid curve is a fit to the function described in the text. The power-law component of the fit has an exponent $a=0.75\pm 0.07$; this is not a critical exponent, but instead results from the decreasing size of the domain wall between the regions of $|{\uparrow }^{\prime }\rangle$ and $|{\downarrow }^{\prime }\rangle$ as $\Omega$ increases. Each point represents an average over 15 to 50 realizations and the uncertainties are the standard deviation. Inset, phase separation $s$ versus $t_{\text{h}}$ with $\Omega =0.6E_{\text{L}}$ fitted to an exponential showing the rapid 0.14(3)-s timescale for phase separation.