Interacting spin-orbit-coupled spin-1 Bose-Einstein condensates

The recent experimental realization of spin-orbit (SO) coupling for spin-1 ultracold atoms opens an interesting avenue for exploring SO-coupling-related physics in large-spin systems, which is generally unattainable in electronic materials. In this paper, we study the effects of interactions between atoms on the ground states and collective excitations of SO-coupled spin-1 Bose-Einstein condensates (BECs) in the presence of a spin-tensor potential. We find that ferromagnetic interaction between atoms can induce a stripe phase exhibiting in-phase or out-of-phase modulating patterns between spin-tensor and zero-spin-component density waves. We characterize the phase transitions between different phases using the spin-tensor density as well as the collective dipole motion of the BEC. We show that there exists a double maxon-roton structure in the Bogoliubov-excitation spectrum, attributed to the three band minima of the SO-coupled spin-1 BEC.

Figure 1

Left: The scheme to generate SO coupling in a spin-1 BEC with three Raman lasers. Right: Raman transitions between three hyperfine states with detuning ${\mathrm{\Omega }}_2$, which controls the spin-tensor potential.

Figure 2

(a) Ground-state phase diagram in the ${\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2$ plane for $(c_0,c_2)=(5,-0.1)$. (b) Closeup of the framed region in panel (a). Regions I, II, and IIIa/IIIb represent uniform, plane-wave, and stripe phases, respectively. The solid, dashed, and dot-dashed curves show first-order-transition, second-order-transition, and crossover boundaries, respectively. The star (diamond) sign denotes the triple (tricritical) point. The insets show schematic single-particle band structures and the BEC's occupation in the momentum space. (c) Phase diagram in the $|c_2|-{\mathrm{\Omega }}_2$ plane for $c_0=5$ and ${\mathrm{\Omega }}_1=0.1$. (d) Phase diagram in the ${\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2$ plane for $(c_0,c_2)=(0.5,-0.01)$.

Figure 3

(a) $\langle F_z^2\rangle$ (solid curve), $\langle F_z\rangle$ (dashed), and $P=10|C_{+}^2{C}_{-}^2|$ (dot-dashed) vs ${\mathrm{\Omega }}_2$ along path ${\mathrm{\Omega }}_1=0.2$ in Fig. 2. The inset shows $\frac{\partial \langle F_z^2\rangle }{\partial {\mathrm{\Omega }}_2}$. (b) [(c)] Dipole oscillation periods $T$ along path ${\mathrm{\Omega }}_1=2$ (${\mathrm{\Omega }}_2=-3$) across a first-order (second-order) transition. Filled circles (solid curves) are obtained from the GPE (effective-mass method). (d) $T$ across the stripe phase along path ${\mathrm{\Omega }}_1=1.3$ (with thick-dashed fitting curves). Phases are labeled in the corresponding regions separated by dashed lines.

Figure 4

(a), (b) The lowest (solid curve) and second-lowest (dashed curve) roton gaps vs ${\mathrm{\Omega }}_2$ along paths ${\mathrm{\Omega }}_1=2$ in Fig. 2 and $0.2$ in Fig. 2, respectively. Regions of different roton structures (see text) are labeled and separated by dashed lines. The corresponding ground-state phases are labeled above the panels. Curves in the insets show schematic Bogoliubov-excitation spectra ($x$ and $y$ axes represent momentum and energy, respectively) in the corresponding regions. The roton modes are circled. There are no roton excitations in the shaded region of the stripe phase [53].