Spin-orbit angular momentum coupling in a spin-1 Bose-Einstein condensate

We propose a simple model with spin and orbit angular momentum coupling in a spin-1 Bose-Einstein condensate, where three internal atomic states are Raman coupled by a pair of copropagating Laguerre-Gaussian beams. The resulting Raman transition imposes a transfer of orbital angular momentum between photons and the condensate in a spin-dependent way. Focusing on a regime where the single-particle ground state is nearly threefold degenerate, we show that the weak interatomic interaction in the condensate produces a rich phase diagram, and that a many-body Rabi oscillation between two quantum phases can be induced by a sudden quench of the quadratic Zeeman shift. We carried out our calculations using both a variational method and a full numerical method, and found excellent agreement.

Figure 1

(a) A schematic picture showing two LG beams with opposite OAM copropagating along the $z$ axis shine on a condensate. (b) Atomic energy level structure. (c) The lowest-energy band ($n=1$) in the single-particle spectrum possesses different number of minima, which yield this phase diagram in the $\left|{\mathrm{\Omega }}_0\right|\text{-q}$ space. On the yellow dashed line, the single-particle ground state is threefold degenerate. (d) Solid lines represent the three degenerate ground-state wave functions $|{\psi }_m|=|u_m|$ ($m=-1$, 0, and 1). Here $\mathrm{\Omega }=-4$ and $q=-0.817$. The dashed lines represent the effective potential $V_m$ experienced by different spin states. The red numbers in the figure represent the OAM quantum number is the laboratory frame, which is equal to ${\tilde{l}}_z\mp 2ml$, and we take $l=1$ in all our calculations. All quantities plotted in the figures throughout the paper are expressed in a dimensionless unit system with $\hslash =M=\omega =1$.

Figure 2

The single-particle energy spectrum at ${\mathrm{\Omega }}_0=-4$. (a), (b), and (c) correspond to $q=-2, q=-0.817$, and $q=2$, respectively, where the energy spectrum for the $n=1$ band (connected with dots) exhibit two, three, and one minimum, respectively. (d) A detailed look at the $n=1$ band with three energy minima for three different values of $q$. At $q=-0.817$, all three minima are degenerate.

Figure 3

(a) Ground state phase diagram at ${\mathrm{\Omega }}_0=-4$ and $c_0=1$. Background color represents the magnitude of the total spin $\left|〈\mathit{S}〉\right|$. Broken lines and disk-shaped markers show the relative weights of $\alpha , \beta$, and $\gamma$ in $\mathrm{\Psi }$, where hollow circle has a $\pi$-phase difference with solid dots. (b) Typical spin density distributions in each phase. (c) Normalized local spin texture $\mathit{S}\left(\mathbf{r}\right)/\rho \left(\mathbf{r}\right)$ for different phases. The arrows represent transverse spin in the $xy$ plane and the background color represents longitudinal spin along the $z$ axis. Phase IVb (not shown) has similar transverse spin textures as Phase IVa but with $S_z\left(\mathbf{r}\right)=0$. (d) Top: Dependence of $\alpha$ (red dashed line), $\beta$ (black solid line), and $\gamma$ (blue dotted line) on $c_2$ at $q=-0.82$ corresponding to the white dashed vertical line in (a). Bottom: Dependence of $\left|〈\mathit{S}〉\right|$ on $c_2$ at $q=-0.82$.

Figure 4

(a) Time evolution of total transverse spin magnitude $\left|\langle S_{\perp }\rangle \right|$ (red solid curve with dots) and $\langle S_z^2\rangle$ (blue dashed curve with hollow circles) after the quadratic Zeeman shift $q$ is quenched from $-0.83$ to $-0.73$ at $t=0$. (b) Time evolution of the spin density profiles. The evolution is roughly periodic with a period of $T\approx 89$. Here $c_2=1$.