Bose-Einstein Condensates with Spin-Orbit Interaction

Motivated by recent experiments carried out by Spielman’s group at NIST, we study a general scheme for generating families of gauge fields, spanning the scalar, spin-orbit, and non-Abelian regimes. The NIST experiments, which impart momentum to bosons while changing their spin state, can in principle realize all these. In the spin-orbit regime, we show that a Bose gas is a spinor condensate made up of two non-orthogonal dressed spin states carrying different momenta. As a result, its density shows a stripe structure with a contrast proportional to the overlap of the dressed states, which can be made very pronounced by adjusting the experimental parameters.

Figure 1

Schematic of the experimental setup at NIST [3]. The Raman process consists of two lasers with wave vectors ${\mathbf{k}}_{\mathrm{op}}+q\widehat{x}$ and ${\mathbf{k}}_{\mathrm{op}}$, frequencies ${\omega }_{\mathrm{op}}+\omega$ and ${\omega }_{\mathrm{op}}$ impinging on the atomic cloud. Atoms excited by the laser will have their momenta increased by $q\widehat{x}$ while their spin projection along $\widehat{\mathbf{y}}$ is changed by 1, as shown in the energy diagram at top.

Figure 2

The energy levels $E_0(p)$ and $E_1(p)$ as a function of $k\equiv p+q/2$. The lower branch $E_0(p)$ has two degenerate minima at $k=\pm k_0$, where $k_0=(q/2)\cos \theta$. The energy difference between the lower and upper branch at $\pm k_0$ is ${ϵ}_q={\hslash }^2{q}^2/2\mathrm{m}$.

Figure 3

The phase diagram of pseudospin-$1/2$ Bose gas: Region $\mathbf{I}$ is a superposition of two dressed state with momentum $p_{+}$ and $p_{-}$, $\mathbf{I}\mathbf{I}$ and $\mathbf{I}\mathbf{I}\mathbf{I}$ are the single dressed states $p_{+}$ and $p_{-}$ respectively. $\alpha$, $\beta$, ${\alpha }_c$, and ${\beta }_c$ are defined in text.

Figure 4

The upper figure is the column density ${\tilde{n}}_1(x,y)=\int dz{n}_1(x,y,z)$. The lower frame is ${\tilde{n}}_1(x,0)$. The period of oscillation is $\pi /k_0$. The contrast of oscillation at the center is 70%. Our calculation is performed for ${}^{87}\mathrm{Rb}$ with $N=2.5\times {10}^5$ atoms, $q=1.56\times {10}^7{\mathrm{m}}^{-1}$, $\theta =\frac{1}{4}\pi$, $\hslash {\Omega }_R=h\times 7.1\mathrm{KHz}$, cloud size $R_{\mathrm{TF}}=20\mu \mathrm{m}$ [1]. The values $\alpha$ and $\beta$ used are given by $\alpha =\frac{1}{4}{\alpha }_c$ and $\beta =\frac{1}{4}{\beta }_c$. The length displayed is in units of the laser wavelength 804.3 nm in Ref. 1.