Extended Bose-Hubbard model with pair hopping induced by a quadratically coupled optomechanical system

We present a scheme to realize the (extended) Bose-Hubbard model in an $N$-coupled optomechanical system. By treating the cavities as intermediary and eliminating them adiabatically with the condition of large detuning or fast decay, we can obtain the effective Hamiltonian for the $N$ oscillators, with the regular terms in the Bose-Hubbard model, i.e., the pair tunnelings and the density-density interactions. Then we verify and provide the condition for our approximation with numerical results. Due to the existence of the pair tunnelings and the density-density interactions, we can investigate the density wave and supersolid phases in our model. Moreover, we also discuss the competition between the regular tunneling and the pair tunneling.

Figure 1

Schematic depiction of the $N$-coupled optomechanical system. The membranes ${\widehat{b}}_i$ are coupled to the microwave cavities ${\widehat{c}}_i$ and also quadratically coupled to the optical cavities ${\widehat{a}}_i$. $J$ and $K$ describe the tunnelings strength of the cavities in the neighboring sites. By supposing the large detuning (or fast decay), we can eliminate the cavity modes and obtain an effective Hamiltonian of the $N$ membranes.

Figure 2

Numerical verification of the validity of the effective Hamiltonian. Starting with the same initial state, we calculate the overlap $F\left(t\right)$ of the two output states corresponding to $H_{\mathrm{ori}}$ and $H_{\mathrm{eff}}$, respectively. Other parameters are set to be (a) $h=0,K/g=10,\mathrm{\Gamma }=0,\delta /g=20$, and (b) $g=0,J/h=10,\kappa =0,\mathrm{\Delta }/h=40$. From the data, we can see that with large detuning, the cavities can serve as intermediary with some fluctuation, which can be wiped out by the fast decay. The insets show the particle numbers $n_a={\widehat{a}}^{†}\widehat{a}$ and $n_c={\widehat{c}}^{†}\widehat{c}$ in time evolution.

Figure 3

The phase diagram in the $s\text{−}\mu$ space when $t=0$ and $s<0$. The red dashed lines are obtained by Eq. (16). The black solid lines are obtained by the Gutzwiller ansatz, and we find the ground state by evolving the wave function subject to the Hamiltonian (10) in imaginary time. The calculations are based on a six-site system. PSS denotes the pair supersolid phase as in this region ${\psi }_{\alpha }=0$ while ${\varphi }_{\alpha }\ne 0$ and ${\varphi }_A\ne {\varphi }_B$. The three lobes denote the density wave phase. $(n_A,n_B)=(1,1),(2,2),(3,3)$ are the Mott insulator phases.

Figure 4

(a) The black solid lines show the phase diagram when $s<0,t=0.2$, and $U=1$. When $-s$ is small, the system is in the SF phase. By increasing $-s$, the system may turn to the SS or DW phase. The three lobes of the DW phase are completely separated by the SS and SF phase. The dashed lines show the phase diagram when $s>0$, and other parameters are the same as the solid lines. On the left side of the (left) orange dashed line, it is the SF phase, and on the right side of the (right) blue dashed line is the MI phase. The region between the two dashed lines is the PSF phase, where we have $\psi =0$ and $\varphi \ne 0$. (b) $\lambda$ as a function of $\left|s\right|$ when $t=0.2$ and $U=1.0$. The inset shows the details of the curve when $s>0$.