Simulating the Lipkin-Meshkov-Glick model in a hybrid quantum system

We propose an efficient scheme for simulating the Lipkin-Meshkov-Glick (LMG) model with nitrogen-vacancy (NV) center ensembles in diamond magnetically coupled to superconducting coplanar waveguide cavities. With the assistance of external microwave driving fields, we show that the interaction of the NV spins can be easily controlled, and several types of the LMG model can be realized by tuning the different parameters. Under the thermal dynamical limit, the distinct nonequilibrium second-order quantum phase transition of the spin ensemble can be achieved at the critical point. Furthermore, we show that the spin squeezed state can be generated by tailoring the LMG Hamiltonian to possess the two-axis countertwisting form in this hybrid quantum system.

Figure 1

The scheme diagrams. (a) Two superconducting coplanar waveguide cavities are coupled together (the coupling coefficient is $\epsilon$), an NV center ensemble is set in the center of one cavity, and this ensemble is magnetically coupled to this cavity. (b) Level diagram of the NV center ground triplet state and the feasible four Raman transition channels. The blue solid arrows indicate the coupling to the two normal modes of the quantized cavities fields with the frequencies ${\nu }_a$ and ${\nu }_b$. The red dashed arrows show the four additional classical microwave fields with frequencies ${\omega }_1$, ${\omega }_2$, ${\omega }_3$, and ${\omega }_4$, and they are used to implement Raman transitions between the two excited spin states $|+\rangle$ and $|-\rangle$.

Figure 2

The average values of the first-order spin components and the quadratic spin components vs the dimensionless variables $\lambda /\gamma$ and ${\mathrm{\Gamma }}_b/\gamma$, where the parameters are chosen as ${\gamma }_{\text{dep}}=0.2\gamma$ and $h=\gamma$. The average values of $\langle {\widehat{J}}_i\rangle /j$ and $\langle {\widehat{J}}_i^2\rangle /j^2$ vs $\lambda /\gamma$ with ${\mathrm{\Gamma }}_b=0.2\gamma$ (a, b) and ${\mathrm{\Gamma }}_b=0.8\gamma$ (c, d). (e, f) The three-dimensional plots of $\langle {\widehat{J}}_i\rangle /j$ and $\langle {\widehat{J}}_i^2\rangle /j^2$ vs ($\lambda /\gamma$, ${\mathrm{\Gamma }}_b/\gamma$).

Figure 3

The average values of the first-order spin components and the quadratic spin components vs the dimensionless variables $\lambda /\gamma$ and ${\gamma }_{\text{dep}}/\gamma$, where the parameters are chosen as ${\mathrm{\Gamma }}_b=0.5\gamma$ and $h=\gamma$. The average values of $\langle {\widehat{J}}_i\rangle /j$ and $\langle {\widehat{J}}_i^2\rangle /j^2$ vs $\lambda /\gamma$ with ${\gamma }_{\text{dep}}=0.2\gamma$ (a, b) and ${\gamma }_{\text{dep}}=0.4\gamma$ (c, d). (e, f) The three-dimensional plots of $\langle {\widehat{J}}_i\rangle /j$ and $\langle {\widehat{J}}_i^2\rangle /j^2$ vs ($\lambda /\gamma$, ${\gamma }_{\text{dep}}/\gamma$).

Figure 4

Time evolution of ${\xi }^2$ with $H_T$ for $N\simeq {10}^{12}$, where the parameters are chosen as $\lambda /\gamma \simeq 1$ and $2h/\gamma \simeq 0$; the different effective dissipations are ${\mathrm{\Gamma }}_a={\mathrm{\Gamma }}_b=\mathrm{\Gamma }=0.1\gamma ,0.01\gamma ,\text{and}0.001\gamma$; and the dephasing rates are ${\gamma }_{\text{dep}}=0.02\gamma ,0.03\gamma ,\text{and}0.04\gamma$, respectively. (a) The evolution for a shorter time $0\le t\le 1.5/\gamma$. (b) The evolution for a longer time $0\le t\le 3/\gamma$.