Quantum Zeno effect and nonclassicality in a $\mathcal{PT}$-symmetric system of coupled cavities

The interplay between the nonclassical features and the parity-time ($\mathcal{PT}$) symmetry (or its breaking) is studied here by considering a $\mathcal{PT}$-symmetric system consisting of two cavities with gain and loss. The conditions for $\mathcal{PT}$ invariance are obtained for this system. The behavior of the average photon number corresponding to the gain and loss modes for different initial states (e.g., vacuum, NOON, coherent, and thermal states) has also been obtained. With the help of the number operators, quantum Zeno and anti-Zeno effects are studied, and the observed behavior is compared in $\mathcal{PT}$-symmetric (PTS) and $\mathcal{PT}$-symmetry-broken (PTSB) regimes. It has been observed that the relative phase of the input coherent fields plays a key role in the occurrence of these effects. Further, some nonclassicality features are witnessed using criteria based on the number operator(s). Specifically, intermodal antibunching, sum, and difference squeezing are investigated for specific input states. It is found that the various nonclassical features, including the observed quantum Zeno and anti-Zeno effects, are suppressed when one goes from the PTS to PTSB regime. In other words, the dominance of the loss or gain rate in the field modes over the coupling strength between them diminishes the nonclassical features of the system.

Figure 1

The model. Two cavities bearing modes $a_1$ and $a_2$ coupled through coupling constant $g$ are also interacting with baths $B_1$ and $B_2$, respectively. The baths cause gain ${\gamma }_1$ and loss ${\gamma }_2$ in the first and second cavity, respectively.

Figure 2

The real part of the eigenvalues ${\lambda }_{\pm }$ is plotted as a function of the coupling strength $g$ and the gain (loss) rate $\gamma$. The points where the two eigenvalues coalesce are called exceptional points. In (a), the blue (solid) and red (dashed) curves correspond to ${\lambda }_{+}$ and ${\lambda }_{-}$, respectively. All the parameters are in units of ${\sec }^{(-1)}$ in this figure, while in the rest of the paper, all the quantities are dimensionless unless stated otherwise.

Figure 3

Average photon number $n_{a_1}=\langle a_1^{†}\left(t\right)a_1\left(t\right)\rangle$ (solid blue curve) and $n_{a_2}=\langle a_2^{†}\left(t\right)a_2\left(t\right)\rangle$ (dashed red curve) with respect to the dimensionless parameter $gt$ for PTS (left panel) and PTSB (right panel) cases. The value of $\gamma$ is $0.5g$ and $1.1g$ corresponding to the PTS and PTSB regimes, respectively. The input states are (a, b) vacuum state $|00\rangle$, (c, d) coherent state $|{\alpha }_1{\alpha }_2\rangle$ with ${\alpha }_k=r_k{e}^{i{\theta }_k}$ for $k=1,2$ and coherent state parameters $r_1=r_2=1,{\theta }_1={\theta }_2=\pi /4$, (e, f) NOON state $(|10\rangle +|01\rangle )/\sqrt{2}$, and (g, h) thermal state ${\rho }_0={(1-e^{\beta })}^{2}exp[-\beta (a_1^{†}{a}_1+a_2^{†}{a}_2)]$ with $\beta =\hslash \omega /kT$. Here we have chosen $\beta =1$.

Figure 4

The Zeno parameter as defined in Eq. (16), plotted with respect to time $t$ (sec) and loss/gain rate $\gamma \left({\sec }^{-1}\right), {\zeta }_{a_1}$ (blue surface) and ${\zeta }_{a_2}$ (red surface) with input state as vacuum (a), NOON state $(|10\rangle +|01\rangle )/\sqrt{2}$ (b), and thermal state (c). In all cases, the lossy mode ($a_2$) shows the QAZE while the gain mode ($a_1$) shows the QZE. Here we have chosen coupling strength $g=1$ so that $\gamma <1$ and $\gamma >1$ correspond to the PTS and PTSB regimes, respectively.

Figure 5

Zeno parameter, as defined in Eq. (16), is plotted with respect to time $t$ (sec), and the phase difference $\mathrm{\Delta }\theta$ (radians), with input state as coherent state $|r_1{e}^{i{\theta }_1},r_2{e}^{i{\theta }_2}\rangle$ for $r_1=r_2=1$. In (a), ${\theta }_1=\pi ,{\theta }_2=-\pi /4$. The blue and red surfaces correspond to ${\zeta }_{a_1}$ and ${\zeta }_{a_2}$, respectively. Here the coupling strength between the cavities $g=1$. (b) Variation with respect to the relative phase parameter $\mathrm{\Delta }\theta ={\theta }_1-{\theta }_2$. The color scheme is as follows: blue for ${\zeta }_{a_1}$, red for ${\zeta }_{a_2}$ with $\gamma =0.5g$, that is, PTS regime; green for ${\zeta }_{a_1}$ and gray for ${\zeta }_{a_2}$ with $\gamma =1.5g$, PTSB regime. The parameter $\mathrm{\Delta }\theta$ decides which of the two modes ($a_1$ or $a_2$) would show the QZE or QAZE. The maxima and minima in the plot occur at $\mathrm{\Delta }\theta =\pi /2,3\pi /2$.

Figure 6

Intermodal antibunching with input state as coherent state $|{\alpha }_1,{\alpha }_2\rangle$ (a) and NOON state $(|10\rangle +|01\rangle )/\sqrt{2}$ (b). In the former ${\alpha }_1=r_1{e}^{i{\theta }_1}$ and ${\alpha }_2=r_2{e}^{i{\theta }_2}$ with $r_1=1,r_2=2,{\theta }_1={\theta }_2=\pi /2$. The nonclassical behavior corresponds to $\mathcal{A}\left(a_1{a}_2\right)<0$. The behavior in the PTS regime ($\gamma <g$) is very different from the PTSB regime ($\gamma >g$).

Figure 7

Sum squeezing parameter $V(a_1,a_2)$ (a) and difference squeezing parameter $W(a_1,a_2)$ (b) as defined in Eqs. (20) and (21), plotted against dimensionless parameter $\gamma t$ with vacuum as the initial state. A state is sum (difference) squeezed if $V(a_1,a_2)<0[W(a_1,a_2)<0]$. Here, we used $\varphi =\pi /4$.