Ground-State Phase Diagram of a Spin-Orbital-Angular-Momentum Coupled Bose-Einstein Condensate

By inducing a Raman transition using a pair of Gaussian and Laguerre-Gaussian laser beams, we realize a ${}^{87}\mathrm{Rb}$ condensate whose orbital angular momentum (OAM) and its internal spin states are coupled. By varying the detuning and the coupling strength of the Raman transition, we experimentally map out the ground-state phase diagram of the system for the first time. The transitions between different phases feature a discontinuous jump of the OAM and the spin polarization, and hence are of first order. We demonstrate the hysteresis loop associated with such first-order phase transitions. The role of interatomic interaction is also elucidated. Our work paves the way to explore exotic quantum phases in the spin-orbital-angular-momentum coupled quantum gases.

Figure 1

Scheme of the SOAM coupling. (a) Experimental schematics. Two laser beams with different OAM ($l_1=-2$ and $l_2=0$) copropagate along the $z$ axis and interact with the Rb condensate. The magnetic field is along the $x$ axis. (b) Energy level diagram. Two $\lambda =790.02\mathrm{nm}$ lasers couple two spin states, $|\uparrow ⟩=|F=1,m_F=0⟩$ and $|\downarrow ⟩=|F=1,m_F=-1⟩$. $\delta$ is the two-photon detuning of the Raman transition. ${\omega }_q=2\pi \times 5.52\mathrm{kHz}$ is the quadratic Zeeman shift. (c) The single-particle phase diagram. Three quantum phases are denoted by the quasi-orbital-angular momentum $l_z=1$, 0, $-1$, respectively. $P_{1,2,3,4}$ represent four different paths that cross phase boundaries. The representative dispersion curve and the spin-dependent atomic density distribution for each phase are also shown.

Figure 2

Observation of the phase transitions. $P_1$, $P_2$, $P_3$, and $P_4$ denote the four paths indicated in Fig. 1. The left-hand panels show the density distributions of both spin states across the phase boundaries. The right-hand panels show the corresponding spin polarization $⟨{\sigma }_z⟩$. The black dashed (red solid) curves are results from the theoretical calculations for $T=0$ (finite temperature with $T/T_c=0.32$, where $T_c$ is the condensate critical temperature in the absence of SOAM coupling). The error bars indicate the standard deviation of experimental measurements. The shaded areas indicate the phase transition regions determined from the left-hand panel. $P_1$ is for ${\mathrm{\Omega }}_R/\hslash \omega =1604.5$, $P_2$ for ${\mathrm{\Omega }}_R/\hslash \omega =1134.5$, $P_3$ for $\delta /\hslash \omega =12.9$, and $P_4$ for $\delta /\hslash \omega =-12.9$.

Figure 3

Phase diagram of the SOAM coupled condensate. Three phases are denoted by the quasi-orbital-angular momentum $l_z=1,0,-1$, respectively. The solid curves denote the calculated phase boundaries including the many-body interaction. The experimental phase boundaries are determined from measurements shown in Fig. 2. The error bars indicate the experimental uncertainties in determining the boundaries of the phase transitions. The error bars of the red points come from the measurements along the paths $P_1$ and $P_3$, the error bars of the black points from the measurements along the path $P_2$, and the error bars of the blue points from the measurements along the path $P_4$. The background color denotes the spin polarization $⟨{\sigma }_z⟩=(N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })$.

Figure 4

Histogram of the spin polarization $⟨{\sigma }_z⟩$ at $\delta =0$ and ${\mathrm{\Omega }}_R/\hslash \omega =73.3$. The left-hand (right-hand) inset shows exemplary density distribution for $⟨{\sigma }_z⟩<0$ ($⟨{\sigma }_z⟩>0$). The image size is $600\times 600\mu {\mathrm{m}}^2$.

Figure 5

Observation of the hysteresis across the first-order phase transition. We fix ${\mathrm{\Omega }}_R/\hslash \omega =73.3$ while ramping $\delta$ at different rates. The spin polarization $⟨{\sigma }_z⟩$ is measured as a function of $\delta$. We initially prepare condensate in the $l_z=1$ ($l_z=-1$) phase at $\delta /\hslash \omega \approx 142$ ($\delta /\hslash \omega \approx -116$) and then ramp $\delta$ across the phase transition as shown with blue circles (black squares). The error bar indicates the standard deviation of experimental measurements. The ramping rates are 0.1 and $0.5\mathrm{kHz}/\mathrm{ms}$, respectively.