Nonreciprocal photon blockade in a two-mode cavity with a second-order nonlinearity

It is shown that the Fizeau drag can cause nonreciprocity. We propose the use of a nanostructured ring cavity made of ${\chi }^{\left(2\right)}$ nonlinear materials to achieve a nonreciprocal photon blockade (PB) through the Fizeau drag. Under the weak driving condition, we discuss the PB phenomena for the fundamental mode and the second-harmonic mode. We find that for the fundamental mode the PB happens when we drive the system from one side but disappears when we drive the system from the other side. However, there is no such phenomenon for the second-harmonic mode. Remarkably, the PB phenomenon occurs with a reasonably small optical nonlinearity, thus bringing the system parameters closer to the reasonably achievable realm by the current technology.

Figure 1

A rotating optical cavity with a ${\chi }^{\left(2\right)}$ nonlinear material. The cavity is rotating at a fixed angular velocity $\mathrm{\Omega }$ and an external classical light is coupled into and out of the cavity through an optical fiber. (a) ${\mathrm{\Delta }}_F>0$ when we drive the device from its left side. (b) ${\mathrm{\Delta }}_F<0$ when we drive the device from its right side.

Figure 2

The schematic energy spectrum of the nonlinear cavity based on the ${\chi }^{\left(2\right)}$ material, with the noncoupling Fock state (left) and energy eigenstates (right) when the angular velocity $\mathrm{\Omega }=0$. The red arrows show the frequency of the driving laser.

Figure 3

The schematic energy-level transitions when driving the system from the left side (left) and the right side (right). The angular velocity $\mathrm{\Omega }$ satisfies ${\mathrm{\Delta }}_F=\pm \frac{\sqrt{2}}{4}g$. The red arrows show the frequency of the driving laser.

Figure 4

(a) The second-order correlation function for the fundamental mode on a logarithmic scale ${\text{log}}_{10}{g}_{aa}^{\left(2\right)}\left(0\right)$ as a function of the detuning $\mathrm{\Delta }$ of the fundamental mode from the driving laser frequency and the hopping interaction $g$ between the two modes. (b) Contour plot of (a). (c) The second-order correlation function for the second-harmonic mode on a logarithmic scale ${\text{log}}_{10}{g}_{bb}^{\left(2\right)}\left(0\right)$ as a function of the detuning $\mathrm{\Delta }$ and the hopping interaction $g$. (d) Contour plot of (c). Other parameters are ${\kappa }_2={\kappa }_1,F=0.05{\kappa }_1$, and the fixed angular velocity is zero.

Figure 5

The second-order correlation function for the second-harmonic mode $g_{bb}^{\left(2\right)}\left(0\right)$ vs the hopping interaction $g$ between the two modes when the detuning $\mathrm{\Delta }=0$. Other parameters are the same as in Fig. 4.

Figure 6

The second-order correlation function for the second-harmonic mode on a logarithmic scale ${\text{log}}_{10}{g}_{bb}^{\left(2\right)}\left(0\right)$ is drawn as a function of the cavity loss rate of the second-harmonic mode ${\kappa }_2$ and the hopping interaction $g$ between the two modes. The detuning $\mathrm{\Delta }$ and the fixed angular velocity have been set as zero. The driving strength is chosen as $0.05{\kappa }_1$. The red dashed line represents the optimal condition for the PB in the second-harmonic mode according to the analytical calculation.

Figure 7

The second-order correlation function for the fundamental mode $g_{aa}^{\left(2\right)}\left(0\right)$ (a) and that for the second-harmonic mode $g_{bb}^{\left(2\right)}\left(0\right)$ (b) are drawn as a function of the detuning $\mathrm{\Delta }$ when the fixed angular velocity meets $|{\mathrm{\Delta }}_F|=\frac{\sqrt{2}}{4}g$. The blue solid (red dotted) curve represents driving the system from the left (right) side, and the orange dashed line refers to $\mathrm{\Delta }=-{\mathrm{\Delta }}_F$ when we drive the system from the left side. Other parameters are given by ${\kappa }_2={\kappa }_1$, $F=0.05{\kappa }_1$, and $g=5{\kappa }_1$ in (a) and $g=0.867{\kappa }_1$ in (b).

Figure 8

The mean photon number $\langle n_a\rangle$ of the fundamental mode and $\langle n_b\rangle$ of the second-harmonic mode are drawn as a function of the detuning $\mathrm{\Delta }$ when the fixed angular velocity meets $|{\mathrm{\Delta }}_F|=\frac{\sqrt{2}}{4}g$. The blue solid (red dotted) curve represents driving the system from the left (right) side, and the orange dashed line refers to $\mathrm{\Delta }=-|{\mathrm{\Delta }}_F|$. Other parameters are ${\kappa }_2={\kappa }_1$, $F=0.05{\kappa }_1$, and $g=5{\kappa }_1$ in (a) and $g=0.867{\kappa }_1$ in (b).