Real-space mean-field theory of a spin-1 Bose gas in synthetic dimensions

The internal degrees of freedom provided by ultracold atoms provide a route for realizing higher dimensional physics in systems with limited spatial dimensions. Nonspatial degrees of freedom in these systems are dubbed “synthetic dimensions.” This connection is useful from an experimental standpoint but complicated by the fact that interactions alter the condensate ground state. Here we use the Gross-Pitaevskii equation to study the ground-state properties of a spin-1 Bose gas under the combined influence of an optical lattice, spatially varying spin-orbit coupling, and interactions at the mean-field level. The associated phases depend on the sign of the spin-dependent interaction parameter and the strength of the spin-orbit field. We find “charge”- and spin-density-wave phases which are directly related to helical spin order in real space and affect the behavior of edge currents in the synthetic dimension. We determine the resulting phase diagram as a function of the spin-orbit coupling and spin-dependent interaction strength, considering both attractive (ferromagnetic) and repulsive (polar) spin-dependent interactions, and we provide a direct comparison of our results with the noninteracting case. Our findings are applicable to current and future experiments, specifically with ${}^{87}\mathrm{Rb}, {}^7\mathrm{Li}, {}^{41}\mathrm{K}$, and ${}^{23}\mathrm{Na}$.

Figure 1

(a) The physical system consists of an optical lattice (black line; period ${\lambda }_{\mathrm{L}}/2$) with Raman lasers forming an effective helical magnetic field (green arrows; period ${\lambda }_{\mathrm{R}}/2$). (b) Single-particle dispersion relation for the spin-1 spin-orbit-coupled Bose gas in an optical lattice at $\mathrm{\Omega }=0.25E_{\mathrm{L}},c_2=0,c_0=0$. The six lowest energy bands are pictured, with the three lowest bands well split from the higher energy modes. (c) Synthetic dimension visualization. The hyperfine levels $m_F$ are viewed as an additional dimension with $2F+1$ sites. Each plaquette has a uniform flux $\mathrm{\Phi }\approx 2\pi k_{\mathrm{R}}/k_{\mathrm{L}}$. (d) The three lowest bands in the synthetic dimensions set up for $\mathrm{\Omega }=0.25E_{\mathrm{L}}$. At small $\mathrm{\Omega }$ and $\varepsilon =0$ the bottom band has three minima, with the lowest energy minimum at $k=0$. For $\varepsilon <0$ the bottom band has two degenerate minima, reflecting degeneracy in $m_F=\pm 1$.

Figure 2

Phase schematic for $\varepsilon =0$ and $\varepsilon =-E_{\mathrm{L}}$. (a) $\varepsilon =0$. For $\mathrm{\Omega }\lesssim 0.5E_{\mathrm{L}}$ the system exhibits charge-density-wave behavior and spin polarization along $S_x$ and is denoted ${\mathrm{CDW}}_{\mathrm{FM}}$. Increasing $\mathrm{\Omega }$ leads to a uniform-density phase with a helical spin texture. The period of the spin helix is determined by the Raman field. Positive $c_2$ values suppress density fluctuations. (b) $\varepsilon =-E_{\mathrm{L}}$. The BEC exhibits distinct charge-density-wave phases with different ordering wave vectors and different spin textures, denoted ${\mathrm{CDW}}_{\mathrm{SW}1}$ and ${\mathrm{CDW}}_{\mathrm{SW}2}$. A crossover occurs between the two with increasing $\mathrm{\Omega }$. At $\mathrm{\Omega }\approx 2.4E_{\mathrm{L}}$ there is a first-order transition to a uniform-density state with helical spin polarization.

Figure 3

(a) Schematic of the ${\mathrm{CDW}}_{\mathrm{FM}}$ phase. The BEC predominantly occupies the $m_F=0$ level. The total density is modulated at neighboring sites due to the Raman field. (b) Schematic of the ${\mathrm{CDW}}_{\mathrm{SW}1}$ phase. The edges are preferentially occupied and there is an overall density modulation. (c) Schematic of the ${\mathrm{CDW}}_{\mathrm{SW}2}$ phase. The bulk is more occupied than in (b), and the overall density modulation remains. (d, e) Fractional population as a function of $\mathrm{\Omega }$. (d) $c_2/c_0=-0.25,\varepsilon =0$. The system begins in a ${\mathrm{CDW}}_{\mathrm{FM}}$ ground state at $\mathrm{\Omega }\approx 0$ with $n_0=1/2$ (top line) and $n_{\pm 1}=1/4$ (bottom line) and moves to meet the single-particle occupation. The noninteracting case is indicated by dashed lines. (e) $c_2/c_0=0.25,\varepsilon =-E_{\mathrm{L}}$. The system starts with $n_{\pm 1}=1/2$ (top line) and $n_0=0$ (bottom line). As $\mathrm{\Omega }$ increases, it undergoes an edge to the bulk first-order transition at $\mathrm{\Omega }\approx 2.0E_{\mathrm{L}}$, which is weakened to a crossover in the limit $c_2=0$. As $\mathrm{\Omega }$ increases the bulk is preferentially occupied. Dotted lines indicate the case for $c_0\ne 0,c_2=0$.

Figure 4

Interplay of spin and density for $c_2<0$ at small $\mathrm{\Omega }$. (a) Lattice potential $V\left(x\right)$. (b) Local spin polarization and Raman field at each site for $+S_x$ polarization. The blue arrow shows the spin polarization, and the red arrow shows the local Raman field. (c) ${\mathrm{CDW}}_{\mathrm{FM}}$ phase with $+S_x$ polarization. The density increases at sites where $\mathbf{F}·{\mathbf{\Omega }}_{\mathrm{R}}<0$. (d) Local spin polarization and Raman field at each site for $-S_x$ polarization. The green arrow shows the spin polarization, and the red arrow shows the local Raman field. (e) ${\mathrm{CDW}}_{\mathrm{FM}}$ phase with $-S_x$ polarization. The density increases at the sites where $\mathbf{F}·{\mathbf{\Omega }}_{\mathrm{R}}<0$.

Figure 5

(a, b) GPE density computed for $c_2/c_0=-0.25$ and $\varepsilon =0$. (a) The density in real space in the ${\mathrm{CDW}}_{\mathrm{FM}}$ phase shows modulation between lattice sites. (b) The density in real space in the uniform-density phase. (c) $\left|n\right(2k_{\mathrm{R}}\left)\right|\ne 0$ signals a CDW phase. The noninteracting case (dashed line at 0) has density modulation only from the lattice, and only slight density modulation appears for $c_2=0,c_0>0$ (bottom, cyan line). An increasing spin-dependent interaction strength $|c_2|$ leads to greater overall density modulation until a crossover occurs to the uniform-density regime. The largest peak (red line) indicates the highest interaction strength tested.

Figure 6

Spin-wave order for $c_2/c_0=-0.7,\varepsilon =0$. Dashed lines show the $c_2,c_0=0$ case for reference. (a) $\langle S_x\left(k\right)\rangle$ is pinned to the density in the ${\mathrm{CDW}}_{\mathrm{FM}}$ phase, shown by $|\langle S_x\left(2k_{\mathrm{L}}\right)\rangle |\ne 0$ (yellow line). After the transition $\langle S_x\rangle$ is modulated primarily at the Raman wave vector $k=2k_{\mathrm{R}}$ (red line) and $|\langle S_x\left(2k_{\mathrm{L}}\right)\rangle |\to 0$. (b) $\langle S_y\left(k\right)\rangle$ is almost unaffected by the optical lattice but follows the Raman beam, shown by $|\langle S_y\left(2k_{\mathrm{L}}\right)\rangle |\ll |\langle S_y\left(2k_{\mathrm{R}}\right)\rangle |$. Right-axis labels correspond to the yellow line. (c) Amplitude of spin oscillations with increasing $\mathrm{\Omega }$. $S_x$ is indicated by the black line; $S_y$, by the blue line. We see that the ferromagnetic state crosses over to Raman polarized at $\mathrm{\Omega }\approx 0.5E_{\mathrm{L}}$. (d, e) Example real-space spin texture. (d) ${\mathrm{CDW}}_{\mathrm{FM}}$ phase. (e) Uniform-density phase with a helical spin texture. The legend is the same as in (c).

Figure 7

$j_S\left(\mathrm{\Omega }\right)$ computed for $c_2/c_0\le 0$ and $\varepsilon =0$. Ferromagnetic spin-dependent interactions first suppress and then slightly enhance the overall spin current compared to the noninteracting case (dashed line) in the regime where the density modulation is highest. For spin-independent interactions only ($c_2=0$; cyan line) the current is hardly changed.

Figure 8

(a, b) GPE density computed for $c_2/c_0=0.25$ and $\varepsilon =-E_{\mathrm{L}}$. Density in real space in the ${\mathrm{CDW}}_{\mathrm{SW}1}$ (a) and ${\mathrm{CDW}}_{\mathrm{SW}2}$ (b) phases. (c) $n\left(k\right)$ at $k=2k_R$ and (d) $4k_R$. The noninteracting case (black line) has no density modulation other than by the lattice, so it is 0 in this case. For $c_0\ne 0$ the density is modulated at two different wave vectors of the same order of magnitude, varying slightly with varying $c_2$. For $|n\left(4k_{\mathrm{R}}\right)|>|n\left(2k_{\mathrm{R}}\right)|$ we denote the ${\mathrm{CDW}}_{\mathrm{SW}1}$ phase, while for $|n\left(4k_{\mathrm{R}}\right)|<|n\left(2k_{\mathrm{R}}\right)|$ the system is in the ${\mathrm{CDW}}_{\mathrm{SW}2}$ phase. The crossover from ${\mathrm{CDW}}_{\mathrm{SW}1}$ to ${\mathrm{CDW}}_{\mathrm{SW}2}$ occurs for $\mathrm{\Omega }\approx 1.7E_{\mathrm{L}}$. The system undergoes a first-order transition to uniform density for $\mathrm{\Omega }\approx 2.4E_{\mathrm{L}}$, corresponding to the transition to the single-minimum regime. This first-order transition occurs at slightly higher $\mathrm{\Omega }$ values for increasing interaction strength. The red line indicates the highest interaction strength tested.

Figure 9

$j_S\left(\mathrm{\Omega }\right)$ computed for $c_2/c_0\ge 0$ and $\varepsilon =-E_L$. Repulsive spin-independent ($c_0$) interactions suppress the current compared to the noninteracting case, indicated by the dashed black line. The first-order transition causes a sharp increase in the current and is weakly dependent on $c_2$. It occurs at higher values of $\mathrm{\Omega }$ for higher values of $c_2/c_0$. The red line indicates the highest interaction strength tested. In the noninteracting case (dashed line), the discrete steps in the current are a finite-size effect due to the change in curvature of the lowest band. As $\varepsilon$ is tuned, the momentum $k$ where the band minimum occurs decreases in discrete steps from the original value of $k=\pm 2k_{\mathrm{R}}$ until the single-minimum regime at $k=0$ is reached. In the infinite system this curve would be smooth.

Figure 10

Spin-wave order for $c_2/c_0=1.0,\varepsilon =-E_{\mathrm{L}}$. Dashed lines show the $c_2,c_0=0$ case. (a, b) $\langle S_x\left(k\right)\rangle$ and $\langle S_y\left(k\right)\rangle$ are modulated at both the Raman and the lattice wave vectors for $\mathrm{\Omega }\lesssim 2.4E_{\mathrm{L}}$. After the first-order transition the spin comes unpinned from the lattice as evidenced by $|\langle S_{x,y}(k=2k_{\mathrm{L}})\rangle |=0$ (yellow lines) (c) The amplitude of spin oscillations increases with increasing $\mathrm{\Omega }$. $S_x$ is indicated by the black line; $S_y$, by the blue line. The noninteracting case initially occupies a single minimum and is polarized in $\langle S_z\rangle$ for $\mathrm{\Omega }=0$ (not shown). For $\mathrm{\Omega }\ne 0, \langle S_x\rangle$ and $\langle S_y\rangle$ increase continuously, with $\langle S_y\rangle$ suppressed in the interacting case. (d, e) Real-space spin textures for ${\mathrm{CDW}}_{\mathrm{SW}1}$ and ${\mathrm{CDW}}_{\mathrm{SW}2}$. The legend is the same as in (c).

Figure 11

Dependence of the ${\mathrm{CDW}}_{\mathrm{FM}}$ phase on the lattice depth. As $V_{\mathrm{L}}$ increases, the CDW increases in amplitude and then undergoes a first-order transition to a uniform-density phase. Condensation moves toward the Brillouin-zone edge, as shown by the increasing $\left|n\right(2k_{\mathrm{L}}\left)\right|$ (red line) even after the transition. Note that here $\left|n\right(k\left)\right|$ is normalized by $\left|n\right(k=0\left)\right|$.

Figure 12

Dependence of the ${\mathrm{CDW}}_{\mathrm{SW}1}$ and ${\mathrm{CDW}}_{\mathrm{SW}2}$ phases on the lattice depth. In both cases, condensation moves toward the Brillouin-zone edge, as shown by increasing $\left|n\right(2k_{\mathrm{L}}\left)\right|$ (top, red line; right axis), which is much larger in magnitude than the other order parameters. (a) ${\mathrm{CDW}}_{\mathrm{SW}1}$. As $V_{\mathrm{L}}$ increases, the CDW increases slightly before decreasing. (b) ${\mathrm{CDW}}_{\mathrm{SW}2}$. As $V_{\mathrm{L}}$ increases, the CDW decreases. Notably, $\left|n\right(2k_{\mathrm{L}}\left)\right|$ is the same order of magnitude as in the ${\mathrm{CDW}}_{\mathrm{FM}}$ case, while $\left|n\right(2k_{\mathrm{R}}\left)\right|$ (yellow line) and $\left|n\right(4k_{\mathrm{R}}\left)\right|$ (orange line) are much smaller. Note that in both (a) and (b) $\left|n\right(k\left)\right|$ is normalized by $\left|n\right(k=0\left)\right|$.