Nonreciprocal unconventional photon blockade in a driven dissipative cavity with parametric amplification

In this paper, we show that nonreciprocal unconventional photon blockade can be observed in a spinning cavity immersed in a degenerate optical parametric amplifier (OPA) driven by the laser driving with weak Kerr nonlinearities through the Fizeau drag. We analytically derive the optimal conditions for strong antibunching, which are in good agreement with those obtained by numerical simulations. Under the weak driving condition, we discuss the physical origins of the nonreciprocal unconventional photon blockade, which originates from the destructive interference between different paths from the ground state to two-photon states by driving the device from the left side. While the quantum interference paths are broken when the device is driven from the right side, which leads to the occurring of the photon bunching. Moreover, we extend the above results to the general non-Markovian regimes, in which the cavity couples with a thermal reservoir consisting of collection of infinite oscillators (bosonic photonic modes). We show nonreciprocal unconventional photon blockade exhibits a transition from the non-Markovian to Markovian regimes by controlling environmental spectral width regardless of the weakness of OPA gain and driving field.

Figure 1

A nonreciprocal photon blockade can be realized in a spinning cavity coupled with a degenerate optical parametric amplifier (OPA) [12, 58, 81, 83, 113, 114, 115, 116, 117, 118, 119, 120]. Spinning the resonator results in different Sagnac-Fizeau shifts ${\mathrm{\Delta }}_F$ for controlling CW or CCW optical mode experiencing different refractive indices with an angular velocity $\mathrm{\Omega }$. (a) Indicates driving the device from the left side (${\mathrm{\Delta }}_F>0$), which thereafter is denoted by $L$, while (b) indicates driving the cavity from the right side (${\mathrm{\Delta }}_F<0$), which thereafter is denoted by $R$. With the Sagnac-Fizeau shifts, photon blockade or photon-induced tunneling can be observed by driving the spinning resonator from one side, but not from the other side. (c) By driving the system from the left side, the direct excitation from state $|0\rangle$ to state $|2\rangle$ (red dashed arrow) will be forbidden by destructive quantum interference with the other paths drawn by blue arrows, which leads to photon antibunching. (d) Photon bunching occurs by driving the system from the right side, due to the lack of the complete destructive quantum interference between the indicated levels (drawn by crossed blue dashed arrows).

Figure 2

The second-order correlation function $g^{\left(2\right)}\left(0\right)$ versus the phase $\theta$ (The unit of $\theta$ is $\mathrm{rad}$) in two-photo pumping for different input directions. Here and hereafter, ${\mathrm{\Delta }}_l$, ${\mathrm{\Delta }}_F$, $G$, $F$, and $\lambda$ are rescaled in units of $\gamma$, and $t$ is then in units of $1/\gamma$. Hence, all parameters are dimensionless. (a) At $f=0.1504\gamma$, $\varphi =1.5498\mathrm{rad}$, single-photon blockade in Fig. 2 or photon bunching in Fig. 2 occurs at $\theta =3\mathrm{rad}$ by driving the device from the left or right side, with the same strength. At $f=0.1504\gamma$, $\varphi =2.0498\mathrm{rad}$, single-photon blockade in Fig. 2 or photon bunching in Fig. 2 occurs at $\theta =4\mathrm{rad}$ by driving the device from the left or right side, with the same strength. The other parameters chosen are ${\mathrm{\Delta }}_l=0$, $g=0.0045\gamma$, and $|{\mathrm{\Delta }}_F|=5\gamma$.

Figure 3

This figure indicates that the optimal value given by the analytical method is consistent with that given by numerical simulation (The unit of $\theta$ is $\mathrm{rad}$). The blue-square and red-circle lines denote $g^{\left(2\right)}\left(0\right)$ for the analytical expression given by Eq. (23) and full numerical simulation by solving the master Eq. (13), respectively. The parameters chosen are the same as those in Fig. 2.

Figure 4

The second-order correlation function $g^{\left(2\right)}\left(0\right)$ varies with the driving strength $f$ for different input directions. For the driving on the cavity from the left side, ${\mathrm{\Delta }}_F>0$, the figure shows the influence of the non-Markovian environment on the nonreciprocal unconventional photon blockade with the second-order correlation function $g^{\left(2\right)}\left(0\right)$ by numerically solving master Eq. (13) as a function of the driving strength $f$ in unconventional single-photon blockade regime. $\varphi$ takes its optimal value ${\varphi }_{\text{opt}}$ given by Eq. (19). From the figure, we point out that as the increase of the environmental spectrum width $\lambda$, the non-Markovian unconventional single-photon blockade effect gradually tends to the case of Markovian limit ($\lambda \to \infty$ or $\lambda \gg \gamma$). In contrast, for the driving on the cavity from the right side, ${\mathrm{\Delta }}_F<0$, the photon bunching occurs. Parameters chosen are $g=0.002\gamma$, ${\mathrm{\Delta }}_l=0$, $|{\mathrm{\Delta }}_F|=0.004\gamma$, $\theta =4\mathrm{rad}$. The chosen parameters satisfy ${({\mathrm{\Delta }}_F+{\mathrm{\Delta }}_l)}^2=4g^4$, which leads to the time-dependent coefficients of Eq. (25) to have analytical steady solutions.

Figure 5

The second-order correlation function $g^{\left(2\right)}\left(0\right)$ varies with the driving strength $f$ for different input directions. This figure shows the consistency of nonreciprocal unconventional single-photon blockade between non-Markovian limit with $\lambda =300\gamma$ given by Eq. (25) and Markovian approximation given by Eq. (13). The parameters chosen are the same as those in Fig. 4.

Figure 6

Ratio ${\mathrm{\Delta }}_F/\gamma$ of Fizeau drag to decay versus angular velocity (The unit of $\mathrm{\Omega }$ is Hz) of the cavity for ${\mathrm{\Delta }}_F/\gamma >0$ (blue line) and ${\mathrm{\Delta }}_F/\gamma <0$ (red-dashed line) cases. The optical wavelength is $\lambda =1.5\mu \mathrm{m}$, the radius of the cavity is $r=30\mu \mathrm{m}$, and the linear refractive index of the cavity is $n=1.4$.

Figure 7

Our model consists of spinning cavity coupled with a degenerate optical parametric amplifier (OPA) [12, 58, 81, 83, 113, 114, 115, 116, 117, 118, 119, 120] and single-photon laser driving. The figure represents the OPA process: A pump photon with frequency ${\omega }_p$ is down-converted into an identical pair of photons with frequency ${\omega }_c$ after passing through the ${\chi }^{\left(2\right)}$ nonlinear medium.