Symmetry breaking and phase transitions in Bose-Einstein condensates with spin–orbital-angular-momentum coupling

Theoretical study is presented for a spinor Bose-Einstein condensate, whose two components are coupled by copropagating Raman beams with different orbital angular momenta. The investigation is focused on the behavior of the ground state of this condensate, depending on the atom-light coupling strength. By analyzing the ground state, we have identified a number of quantum phases, which reflect the symmetries of the effective Hamiltonian and are characterized by the specific structure of the wave function. In addition to the well-known stripe, polarized, and zero-momentum phases, our results show that the system can support phases whose wave functions contain a complex vortex molecule. Such a molecule plays an important role in the continuous phase transitions of the system. The predicted behavior of vortex-molecule phases can be examined in cold-atom experiments using currently existing techniques.

Figure 1

(a) Schematic representation of a disk-shaped BEC interacting with copropagating LG laser beams with OAM $l_1$ and $l_2$. (b) A sketch of the level diagram illustrating the angular momentum transfer in the two-photon Raman transition between atomic states $|\uparrow \rangle$ and $|\downarrow \rangle$.

Figure 2

Amplitudes and phases of the ground-state wave functions for up (two left columns) and down (two right columns) components at Raman coupling strengths $\mathrm{\Omega }=36$ (I), $\mathrm{\Omega }=45$ (II), $\mathrm{\Omega }=120$ (III), $\mathrm{\Omega }=150$ (IV), and $\mathrm{\Omega }=220$ (V). Vortices in each component are marked with circles on the phase plots. Arrows indicate directions of the phase winding. The length unit is set by $a_{\mathrm{ho}}=\sqrt{\hslash /M{\omega }_{\rho }}$.

Figure 3

Phase diagram with respect to the coupling strength $\mathrm{\Omega }$. Panel (a) shows the derivative of ground-state energy $\partial \epsilon /\partial \mathrm{\Omega }$. Panel (b) shows the expectation of total angular momentum $\langle {\widehat{J}}_z\rangle$ and its standard deviation $\mathrm{\Delta }{J}_z$. Panel (c) shows expectation values of the spin operators $\langle {\widehat{\sigma }}_z\rangle$ and $\langle {\widehat{\sigma }}_{2l}\rangle$. We show here only the non-negative branch of $\langle {\widehat{J}}_z\rangle$ and $\langle {\widehat{\sigma }}_z\rangle$, which corresponds to one of the two degenerate ground states. Phase transitions are detected at critical $\mathrm{\Omega }$ values ${\mathrm{\Omega }}_{c1}=36, {\mathrm{\Omega }}_{c2}=59, {\mathrm{\Omega }}_{c3}=140$, and ${\mathrm{\Omega }}_{c4}=213$.

Figure 4

Amplitudes of two wave-function components (two left columns) and corresponding azimuthal distributions (rightmost column) at the distance ${\rho }_0$ from the center (marked with a dashed circle). Four rows correspond to different values of the coupling strength: $\mathrm{\Omega }=36$ (a), $\mathrm{\Omega }=37$ (b), $\mathrm{\Omega }=45$ (c), $\mathrm{\Omega }=61$ (d). Row (a) corresponds to phase I, (b) and (c) to phase II, (d) to phase III. Vorex cores are marked with circles. The value of ${\rho }_0$ is chosen to pass through the vortex cores in the row (b). Characteristic size of two-vortex molecule is labeled as $d_{\mathrm{VV}}$. The logarithmic scale is applied in two left columns for a better visibility. The length unit is set by $a_{\mathrm{ho}}=\sqrt{\hslash /M{\omega }_{\rho }}$.

Figure 5

Vortex molecule size $d_{\mathrm{VV}}$, in units of harmonic length $a_{\mathrm{ho}}=\sqrt{\hslash /M{\omega }_{\rho }}$, as a function of coupling strength $\mathrm{\Omega }$.

Figure 6

Vortex-antivortex molecule size $d_{\mathrm{VA}}$, in units of harmonic length $a_{\mathrm{ho}}=\sqrt{\hslash /M{\omega }_{\rho }}$, as a function of Raman coupling strength $\mathrm{\Omega }$. Blue dots represent the (excited) stationary vortex-antivortex-molecule state in the region $\mathrm{\Omega }<{\mathrm{\Omega }}_{c3}$.