Unconventional Bose-Einstein condensation in a system with two species of bosons in the $p$-orbital bands in an optical lattice

In the context of Gross-Pitaevskii theory, we investigate the unconventional Bose-Einstein condensations in the two-species mixture with $p$-wave symmetry in the second band of a bipartite optical lattice. An imaginary-time propagation method is developed to numerically determine the $p$-orbital condensation. Different from the single-species case, the two-species boson mixture exhibits two nonequivalent complex condensates in the intraspecies-interaction-dominating regime, exhibiting the vortex-antivortex lattice configuration in the charge and spin channels, respectively. When the interspecies interaction is tuned across the SU(2) invariant point, the system undergoes a quantum phase transition toward a checkerboardlike spin-density wave state with a real-valued condensate wave function. The influence of lattice asymmetry on the quantum phase transition is addressed. Finally, we present a phase-sensitive measurement scheme for experimentally detecting the unconventional Bose-Einstein condensation in our model.

Figure 1

(a) Double-well optical lattice with the experiment parameters $V_0=6.2E_r, \eta =0.95, \theta =95.4^{\circ }, \epsilon =0.81$, and $\alpha =\pi /5$. The white dashed line illustrates the half wavelength of laser $a_0$ and $\sqrt{2}{a}_0$ is the lattice constant. The $A$ and $B$ sublattice sites are shown in (a). (b) Energy spectra of the second band, whose energy minima are located at ${\mathbf{K}}_1=(\pi /2a_0,\pi /2a_0)$ and ${\mathbf{K}}_2=(-\pi /2a_0,\pi /2a_0)$. (c) and (d) Density profiles (top panels) and phase profiles (bottom panels) for noninteracting gas for (c) ${\mathbf{K}}_1$ and (d) ${\mathbf{K}}_2$.

Figure 2

Spatial distributions of the density, phase, and spin texture of the condensate wave functions corresponding to various states in the symmetric lattice: (a) and (b) state I, (c) and (d) state II, and (e) and (f) the checkerboard state. The intraspecies interactions are ${\tilde{g}}_{AA}={\tilde{g}}_{BB}=0.025E_r$ with $E_r={\hslash }^2{k}_l^2/2M$; the interspecies ones are (a) and (b) ${\tilde{g}}_{AB}=0.25{\tilde{g}}_{AA}$ and (c) and (d) ${\tilde{g}}_{AB}=1.1{\tilde{g}}_{AA}$. (a), (c), and (e) Density and phase distributions are shown in the top and bottom panels for each species, respectively. (b), (d) and (f) Spin texture configurations with arrows indicating the orientation of spins and color bars representing the values of $S_z$. The parameters used are $V_0=6.2E_r, \eta =0.95, \theta =95.4^{\circ }, \epsilon =0.81$, and $\alpha ={\alpha }_0={cos}^{-1}\epsilon \approx 35.9^{\circ }$.

Figure 3

Plot of $E/N_{\mathrm{tot}}$ vs ${\tilde{g}}_{AB}/{\tilde{g}}_{AA}$ in the symmetric lattice with $\alpha ={\alpha }_0$. The red dots, blue triangles, and green squares are for different values of ${\tilde{g}}_{AA}=0.015E_r, 0.025E_r$, and $0.05E_r$, respectively. The dashed horizontal line represents the energy for $\gamma >1$ without including domain walls, i.e., solving the GP equation assuming fully polarization. The parameter values are the same as those in Fig. 2.

Figure 4

Condensate fraction of ${\psi }_{{\mathbf{K}}_1}, {cos}^2\delta$, as a function of $\gamma ={\tilde{g}}_{AB}/{\tilde{g}}_{AA}$. The red dots, blue triangles, and green squares indicate ${\tilde{g}}_{AA}/E_r=0.015, 0.025$, and $0.05$, respectively, and the dotted lines depict the fraction of spin-polarized state with the same total particle numbers for each interaction strength. The parameter values of the optical lattice here are the same as for Fig. 2 except for $\alpha =\pi /5>{\alpha }_0$.

Figure 5

(a) Density and phase and (b) spin texture of ${\tilde{g}}_{AA}={\tilde{g}}_{BB}=0.015E_r$, and $\gamma =0.1$. The parameter values of the optical lattice here are the same as for Fig. 1 except for $\alpha =\pi /5$.