Collective spin systems in dispersive optical cavity QED: Quantum phase transitions and entanglement

We propose a cavity QED setup which implements a dissipative Lipkin-Meshkov-Glick model—an interacting collective spin system. By varying the external model parameters the system can be made to undergo both first- and second-order quantum phase transitions, which are signified by dramatic changes in cavity output field properties, such as the probe laser transmission spectrum. The steady-state entanglement between pairs of atoms is shown to peak at the critical points and can be experimentally determined by suitable measurements on the cavity output field. The entanglement dynamics also exhibits pronounced variations in the vicinities of the phase transitions.

Figure 1

Atomic level and excitation scheme for the general model.

Figure 2

(a) Schematic of potential ring cavity system and setup for measurement of the output transmission spectrum of a weak probe laser field of amplitude ${\mathcal{E}}_p$ and frequency ${\nu }_p$. (b) Possible atomic level scheme as described in the text.

Figure 3

Semiclassical (solid line) and finite-$N$ steady-state second-order moments for $h=1$, ${\Gamma }_a=0.01$, ${\Gamma }_b=0.2$, and $N=25$ (dotted line), 50 (short dashed line), 100 (long dashed line).

Figure 4

Eigenvalues of the linearized equations of motion, ${\mu }_{\pm }$, as given by Eqs. (34, 35), for $h=1$ and ${\Gamma }_b=0.2$. The right-hand column gives a magnified view of the region around ${\lambda }_{\text{c}}=1.01$.

Figure 5

Transmission spectrum in the linearized regime, for $\lambda =0.3,0.93,0.992,1.000625(={\lambda }_{\text{c}}),\text{1}.00\text{5},\text{1}.0\text{5},\text{1}.\text{5}$, with microscopic parameters ${\kappa }_a=0.3$, ${\delta }_a=15$, and ${\lambda }_b=0.87$, ${\kappa }_b=15$, giving ${\Gamma }_b=0.05$. We set $h=1$ as usual. Note that ${\lambda }_a$ is chosen to give the indicated $\lambda$ for the given choice of ${\kappa }_a$ and ${\delta }_a$, viz. Eq. (12a) and recalling that $\lambda =2{\Lambda }_a$, while ${\Gamma }_a$ varies according to Eq. (12b), with ${\Gamma }_a=0.01$ when $\lambda ={\lambda }_{\mathrm{c}}$.

Figure 6

Entanglement measure $\text{max}\left\{0,C_{\phi }\right\}$ for $N=100$, $h=1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$.

Figure 7

(Color online) Steady-state spin $Q$-function, $Q_{\text{s}}\left(\eta \right)$, on the Bloch sphere for (a) $\lambda =0.5$, (b) $\lambda =1.01$, (c) $\lambda =1.1$, and (d) $\lambda =2$, with $N=50$, $h=1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$. Note that dark blue corresponds to the minimum value of zero of $Q_{\text{s}}\left(\eta \right)$ while dark red indicates the maximum value of $Q_{\text{s}}\left(\eta \right)$.

Figure 8

Rescaled concurrence $C_{\text{R}}$ for $N=100$ (dashed line) and in the thermodynamic limit (solid line) with $h=1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$.

Figure 9

Entanglement measure $\text{max}\left\{0,C_{\phi }\right\}$ in the thermodynamic limit for $h=1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$. Above the critical point, the system is linearized about only one of the two semiclassical steady-state amplitudes; hence the less-sensitive dependence of $\text{max}\left\{0,C_{\phi }\right\}$ on $\phi$ for $\lambda >{\lambda }_{\text{c}}$ as compared with the finite-$N$ results.

Figure 10

Rescaled concurrence $C_{\text{R}}\left(t\right)$ as a function of $\lambda$ and $t$ for $N=100$, $h=1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$.

Figure 11

Rescaled concurrence $C_{\text{R}}\left(t\right)$ as a function of $\lambda$ and $t$ in the thermodynamic limit, with $h=1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$. Above the critical point the dynamics is linearized around one of the two possible semiclassical steady-state amplitudes.

Figure 12

Semiclassical (solid line) and finite-$N$ steady-state solutions for $\lambda =1$, ${\Gamma }_a=0.01$, ${\Gamma }_b=0.2$, and $N=25$ (dotted line), 50 (short dashed line), 100 (long dashed line). Note the inset in the bottom panel is a magnified plot of the semiclassical solution of $⟨J_y^2⟩/j^2$.

Figure 13

Eigenvalues of the linearized equations of motion, ${\mu }_{\pm }$, as given by Eqs. (34, 35), for $\lambda =1$, and ${\Gamma }_b=0.2$. The right-hand column gives a magnified view of the region around $h_{\text{c}}=0.0101$.

Figure 14

Transmission spectra in the linearized regime, for $h=-0.6,-0.1,-0.01,6.25\times {10}^{-4}(=h_{\text{c}}),0.0\text{5},0.\text{3}$, with microscopic parameters ${\kappa }_a=0.3$, ${\delta }_a=15$, ${\lambda }_a=2.7$, ${\lambda }_b=0.87$, and ${\kappa }_b=15$, giving $\lambda =1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.05$.

Figure 15

Entanglement measure $\text{max}\left\{0,C_{\phi }\right\}$ as a function of $h$ and $\phi$ for $N=100$, $\lambda =1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$.

Figure 16

(Color online) Steady-state spin $Q$ function, $Q_{\text{s}}\left(\eta \right)$, on the Bloch sphere for (a) $h=-0.5$, (b) $h=-0.01$, (c) $h=2.5\times {10}^{-3}$, (d) $h=5\times {10}^{-3}$, (e) $h=0.015$, and (f) $h=0.15$, with $N=50$, $\lambda =1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$. Note that dark blue corresponds to the minimum value of zero of $Q_{\text{s}}\left(\eta \right)$ while dark red indicates the maximum value of $Q_{\text{s}}\left(\eta \right)$.

Figure 17

Rescaled concurrence $C_{\text{R}}$ versus $h$ for $N=100$ (dashed line) and in the thermodynamic limit (solid line) with $\lambda =1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$.

Figure 18

Rescaled concurrence $C_{\text{R}}\left(t\right)$ for $N=100$, with $\lambda =1$, ${\Gamma }_a=0.01$, and ${\Gamma }_b=0.2$.