Quantum optical diode with semiconductor microcavities

The semiconductor diode, which acts as an electrical rectifier and allows unidirectional electronic transports, is the key to information processing in integrated circuits. Analogously, an optical rectifier (or diode) working at specific target wavelengths has recently become a highly sought-after device in optical communication and signal processing. In this paper, we propose a scheme to realize an optical diode for photonic transport at the level of few photons. The system consists of two spatially overlapping single-mode semiconductor microcavities coupled via ${\chi }^{\left(2\right)}$ nonlinearities. The photon blockade is predicted to take place in this system. These photon blockade effects can be achieved by tuning the frequency of the input laser field (driving field). Based on those blockades, we analytically derive the one- and two-photon current in terms of a zero and a finite time-delayed two-order correlation function. The results suggest that the system can serve as a one- and two-photon quantum optical diode which allows transmission of photons in one direction much more efficiently than in the other.

Figure 1

(a) Illustration of the setup. The two cavities driven by external fields (pumping) are coupled by ${\chi }^2$ nonlinearities. The pumping on the cavity $a$ is resonant with the field inside cavity $a$, while the pumping on cavity $b$ is at $2{\omega }_a+\sqrt{2}\Omega$ or $2{\omega }_a-\sqrt{2}\Omega$. (b) Level diagram for the two coupled cavities with ${\omega }_b=2{\omega }_a$. States are labeled by $|mn\rangle$ with the first (second) number denoting the photon number in cavity $a$ (cavity $b$). The coupling $\Omega$ splits the degeneracy of states $|mn\rangle$ and $|m-2n+1\rangle$ or $|m+2n-1\rangle$. The arrows show the frequency of the driven field.

Figure 2

Rescaled average photon number (black squares) and the equal-time second-order correlation function (blue circles) as a function of detunings. The photon number is rescaled by $F^2$ to fit the correlation function. (a) and (b) are for the case with cavity $a$ pumped, and (c) and (d) for cavity $b$ pumped. Once one cavity is pumped, another is left free. Dotted lines are analytical results [see Eqs. (14, 15, 16, 17)]. Red solid lines show the numerical results. Here and hereafter, $\Omega ,F,{\Delta }_a,{\Delta }_b$, and ${\kappa }_a$ are rescaled in units of $\kappa$, and $t$ is then in units of $1/\kappa$. Hence all parameters are dimensionless. In all plots, we chose $\Omega /\kappa =4,F/\kappa =0.2,{\omega }_a/\kappa =1$, ${\omega }_b/\kappa =2$, and ${\kappa }_a/\kappa =1$. A–F mark the maximum and minimum, where the corresponding detunings are A: ${\Delta }_a/\Omega =0$; B: $-\frac{1}{\sqrt{2}}$; C: $\frac{1}{\sqrt{2}}$; D: ${\Delta }_b/\Omega =-\sqrt{2}$; E: $\sqrt{2}$; and F: 0. The tiny deviation of the analytical results from the numerical ones is caused by the approximation $F/\kappa \ll 1$ used in Eqs. (14, 15, 16, 17). A–F in the bottom panels illustrate the transitions that may lead to the features seen at the points A–F. Suppression of the steady-state population of the level $|01\rangle$ is indicated by a red $X$. Further explanation of the features can be found in the main text.

Figure 3

Single-photon rectification in the semiconductor microcavities coupled via ${\chi }^2$ nonlinearities. The equal-time second-order correlation function when the system is pumped on the cavity $a$ [see analytical expression given by Eq. (17)] (log scale). (a) is for cavity $a$ and (b) is for cavity $b$. We set $x={\Delta }_b+2{\Delta }_a$, with detunings given by $\Delta ={\Delta }_b-2{\Delta }_a$ corresponding to ${\Delta }_a=(x-\Delta )/4$ and ${\Delta }_b=(x+\Delta )/2$. Parameters chosen are ${\kappa }_a=0.1\kappa ,F=0.1\kappa ,x=30\kappa .$ The very narrow region around $\Delta =30\kappa$, i.e., ${\Delta }_a\approx 0$, shows the complete single-photon blockade, the corresponding transition is shown in (c), and the single-photon current to right cavity $b$ is illustrated in (d).

Figure 4

Two-photon rectification in the semiconductor microcavities coupled by ${\chi }^2$ nonlinearities. The equal-time second-order correlation function [see the analytical expression given by Eq. (15), log scale] of (a) cavity $a$ and (b) cavity $b$. This figure is plotted with the right cavity $b$ pumped. We set $x={\Delta }_b+2{\Delta }_a$ with detunings defined by $\Delta ={\Delta }_b-2{\Delta }_a$. Parameters chosen are ${\kappa }_a=0.1\kappa ,F=0.1\kappa ,$ and $x=30\kappa .$ At the ellipse, ${\Delta }^2+8{\Omega }^2=x^2$, i.e., two-photon resonant transition from $\left|0\right.〉\to \left|2_{\pm }\right.〉$ when $2{\omega }_L={\omega }_{2\pm }$, the two-photon blockade happens in the semiconductor microcavities. The corresponding transition is illustrated in (c). This causes the two-photon current to the left cavity mode $a$; see (d). Note that the two-photon current is symmetric for the symmetric coupling $\Omega$ in (a) and (b); this is a direct reflection of Eq. (15).

Figure 5

(a) The rectifying factor, correlation function, and total current as a function of frequency of the external laser field $x$. The blue dashed line and red bold line denote the rectification factor for the analytical expression given by Eq. (21) and the fully numerical simulation in (a), respectively. (b) The equal-time second-order correlation function $g^{\left(2\right)}$ [see the analytical expression given by Eqs. (14, 15, 16, 17)] of the output photon in both directions. (c) The total current, $N\left(k\right)=N_a\left(k\right)+N_b\left(k\right)$ [see the analytical expression given by Eqs. (14, 15, 16, 17)]. Parameters chosen are ${\kappa }_a=0.1\kappa ,F=0.1\kappa ,$ and $\Omega =10\kappa .$ We assume that $\Delta =30\kappa$ with ${\Delta }_a=(x-\Delta )/4$ and ${\Delta }_b=(x+\Delta )/2$.

Figure 6

Finite temporal evolution of the second-order correlation function for selected driving strength $F$ with cavity $a$ being pumped. The blue dashed line and red bold line denote the analytical expression [given by Eqs. (30) and (31)] and the exact numerical simulation, respectively. Parameters chosen are ${\kappa }_a=1.6\kappa ,F=0.72\kappa ,\Omega =10\kappa ,{\Delta }_a=0.48\kappa ,$ and ${\Delta }_b=0.4\kappa$ for (a) and (c), and $F=2\kappa$ for (b) and (d).