Bose-Einstein condensate in Bloch bands with an off-diagonal periodic potential

We report on the Bose-Einstein condensate in the Bloch bands with off-diagonal periodic potential (ODPP), which simultaneously plays the role of spin-orbit coupling and Zeeman field. This model can be realized using two independent Raman couplings in the same three-level system, in which the time-reversal symmetry ensures the energy degeneracy between the two states with opposite momenta. We find that these two Raman couplings can be used to tune the spin polarization in momentum space, thus greatly modifying the effective scatterings over the Bloch bands. We observe a transition from the Bloch plane wave phase with condensate at one wave vector to the Bloch stripe phase with condensate at the two Bloch states with opposite wave vectors. These two phases will exhibit different spin textures and density modulations in real space, which are totally different from that in free space. In momentum space, multiple peaks differing by some reciprocal lattice vectors can be observed in both phases, reflecting the periodic structure of the ODPP. A three-band effective model is proposed to understand these observations. This ODPP will never approach the tight-binding limit, thus can provide an alternative platform in the investigations of various physics, such as collective excitations, polaron, and topological superlfuids, over the Bloch bands.

Figure 1

(a) Setup for the ODPP. The laser $L_1$ with frequency ${\omega }_1$ is linearly polarized along the $z$ direction, and the two lasers $L_{2/3}$ with identical frequency ${\omega }_2$ are orthogonally polarized in the $x-y$ plane. (b) These three laser beams form two independent Raman coupling in the same three-level system. (c) Momentum transfer in these two Raman couplings with ${\mathbf{k}}_i\equiv {\mathbf{k}}_{L_i}-{\mathbf{k}}_{L_3}$ for $i=1$, 2. (d) The plane wave $\mathbf{k}$ with spin $\sigma$ is coupled the two other plane wave vectors $\mathbf{k}\pm \mathbf{Q}$ with opposite spin $\overline{\sigma }$ by ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$, respectively.

Figure 2

(a) Energy spectrum with ${\mathrm{\Omega }}_1=0.2, {\mathrm{\Omega }}_2=0$ when $k_x=0$, here the ${\mathrm{\Omega }}_i$ have defined in unit of the recoil energy $E_R$. The blue dashed lines are obtained with $\tilde{\mathcal{H}}$, in which $k_y$ has been shift to match the bands obtained using Bloch theorem. (b) Spectrum with ${\mathrm{\Omega }}_1=0.2, {\mathrm{\Omega }}_2=0.1$ when $k_x=0$. The inset shows the detail of the double minima. (c) Double minima spectrum in two dimension when ${\mathrm{\Omega }}_1=0.6$ and ${\mathrm{\Omega }}_2=0.2$. (d) Single minimum at $(0,\pi )$ with ${\mathrm{\Omega }}_1=1.2$ and ${\mathrm{\Omega }}_2=0.2$. The arrows in (c) and (d) represent the spin polarization in ${\sigma }_x-{\sigma }_z$ space in momentum spacing, indicating of spin-momentum locking from two-dimensional SOC.

Figure 3

(a) Single-particle phase diagram in the parameter space ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. The color represents the distance between the two minima, thus the white color represents one local minimum. (b) The phase boundary between BPW ($g_1>g_{12}^c$) and BST ($g_{12}<g_{12}^c$) influenced by the two Raman couplings for the three horizon lines in (a), with ${\mathrm{\Omega }}_2=(0,0.8,0.9,1.0,1.2)$. The system collapses when $g_{12}/g<-1$. (c, d) Ground-state energy $\mathrm{\Delta }{E}_g=E_g\left(g_{12}\right)-E_g\left(g_{12}^c\right)$ and spin polarization across the phase boundaries for three typical parameters.

Figure 4

Properties of the BPW phase. (a) and (d) Wave function in momentum space for spin up and spin down components, respectively. (b) and (e) The corresponding wave functions for these two components in real space. Spin polarization in ${\sigma }_x-{\sigma }_y$ space and ${\sigma }_x-{\sigma }_z$ space are shown in (c) and (f). These results are obtained from GPE simulation with parameters: ${\mathrm{\Omega }}_1=0.4, {\mathrm{\Omega }}_2=0.8, g_{12}/g=0.1$, and ${\mathbf{k}}_0\approx (0.30,0.15)$.

Figure 5

Properties of the BST phase with parameters ${\mathrm{\Omega }}_1=0.4, {\mathrm{\Omega }}_2=0.8, g_{12}/g=-0.1, {\mathbf{k}}_0\approx (0.30,0.15)$. The other parameters are the same as that in Fig. 4.

Figure 6

(a) Two lasers, polarize along the $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ directions, couple the ground state ${|m\rangle }_g$ to the excite states ${|m-1\rangle }_e$ and ${|m+1\rangle }_e$. Here the ground state ${|m\rangle }_g$ denotes hyperfine state $5^2{\mathrm{S}}_{1/2}$ with $F=1,m_F=m$, and the excited state ${|m\rangle }_e$ denotes $5^2{\mathrm{P}}_{1/2}$ with $F=1,m_F=m$ [62]. The solid and dashed lines denote the two lasers, respectively. (b) Two independent Raman couplings in the same three-level system. Lasers $L_1$ and $L_2$, which polarize in the $\widehat{\mathbf{x}}-\widehat{\mathbf{y}}$ plane, couple $|\downarrow \rangle$ to $|e\rangle$, while Laser $L_3$ linearly polarizes along the $\widehat{\mathbf{z}}$ direction, couples $|\uparrow \rangle$ to $|e\rangle$.

Figure 7

Properties of the BPW when ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=1.0$ and $g_{12}/g=-0.3$. (a) and (e) show the wave function in momentum space for spin up and spin down components, respectively. The corresponding wave functions in real space are given in (b) and (f). (c) and (g) show the densities of spin up and spin down components in the line of $x=0$. (d) and (h) plot the spin polarization in ${\sigma }_x-{\sigma }_y$ and ${\sigma }_x-{\sigma }_z$ space. In (a), two major peaks are visible at $-{\mathbf{k}}_0\pm Q\widehat{y}$, while in (e), three peaks are visible at $-{\mathbf{k}}_0$ and $-{\mathbf{k}}_0\pm Q2\widehat{y}$.

Figure 8

Properties of the BST phase when ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=1.0$ and $g_{12}/g=-0.7$. (a) and (e) show the wave function in momentum space for spin up and spin down components, respectively. The corresponding wave function in real space are presented in (b) and (f). (c) and (g) show the densities of spin up and spin down components in the line of $x=0$. (d) and (h) plot the spin polarization in ${\sigma }_x-{\sigma }_y$ and ${\sigma }_x-{\sigma }_z$ space. In (a), five peaks can be found at ${\mathbf{k}}_0,-{\mathbf{k}}_0\pm Q\widehat{\mathbf{y}},{\mathbf{k}}_0\pm 2Q\widehat{\mathbf{y}}$. The peaks at $\mathbf{k}={\mathbf{k}}_0\pm 2Q\widehat{\mathbf{y}}$ is about one tenth in amplitude of the peaks at ${\mathbf{k}}_0$. In (e), there are also five peaks at $-{\mathbf{k}}_0,{\mathbf{k}}_0\pm Q\widehat{\mathbf{y}},-{\mathbf{k}}_0\pm 2Q\widehat{\mathbf{y}}$, just in the opposite position of the peaks in (a).