Parametric Interaction-Induced Asymmetric Behaviors in a Coupled-Cavities Quantum Electrodynamics System

Abstract

We investigate a quantum electrodynamics system consisting of two coupled single-mode cavities. The left cavity couples with a two-level atom, while the right cavity incorporates a second-order nonlinear medium, activated by a pumping field. In the absence of nonlinear medium, we show that the transmitted field intensity reveals only classical asymmetric behavior at the central frequency. However, the parametric interaction induced by the nonlinear medium leads to various quantum asymmetric behaviors at single photon excitation frequencies, including the squeezing, quantum statistics, and phase-space characteristics of the transmitted photons. These asymmetric behaviors arise from additional excitation pathways enabled by the parametric interaction-induced two-photon processes. We demonstrate these asymmetric behaviors through Klyshko’s figures of merit, the Wigner function, and the steady-state second-order correlation function of the transmitted photons. These results present promising applications for remote quantum-state manipulation and contribute significantly to the advancement of quantum networking.

1. Introduction

Optical asymmetric behavior, such as a difference in the transmission of light for the opposite directions of excitation, plays an important role in modern optical research [1,2,3,4]. In the field of optical communications, it can help improve the transmission efficiency and stability of signals [5,6]. Thus, optical asymmetric behavior has been widely used to construct key devices such as optical switches [7,8] and optical modulators [9,10] that play vital role in improving the performance of communication systems. In the field of quantum networks, optical asymmetric behavior is useful for realizing quantum information processing protocols such as quantum entanglement and quantum teleportation. It also provides a new method to manipulate quantum states for achieving more efficient and secure quantum communications [11,12,13,14]. Indeed, optical asymmetric behavior has played important roles in a wide range of applications in optical imaging [15], optical storage [16,17], and optical sensing [18,19], improving the resolution and contrast of optical imaging, achieving large-capacity and high-reliability optical storage systems, enhancing the sensitivity and selectivity of sensors.

Conventional approaches for realizing optical asymmetric behavior rely on magnetic fields [20,21,22], spatiotemporal modulation [23,24,25,26], or nonlinearity [27,28,29,30]. For example, by utilizing the influence of magnetic fields on the optical polarization state, the propagation dynamics and phase of light can be manipulated so that asymmetric transmission of light can be achieved [20,31]. The spatiotemporal phase modulation can lead to completely asymmetric reflections in the forward and backward directions. Thus, the back reflected field does not return to the same state as the forward incident field [23]. In addition, optical nonlinearities, such as the self- and cross-Kerr nonlinearities, can not only enhance the light-matter interaction but also precisely control the propagation and the state of light. When the forward- and backward-propagating light fields enter a device, e.g., a micro-ring resonator, the modulation coming from the self- and cross-Kerr nonlinearities will cause different resonance frequencies for the forward- and backward-circulating modes [29].

The cavity quantum electrodynamics (QED) system has emerged as an excellent platform for achieving the asymmetric transmission of photons due to its strong light-matter interaction and adjustable nonlinearity [32,33,34,35]. Particularly, the bistability in transmission and optical asymmetric behavior at the few-photon level have been theoretically and experimentally studied in an asymmetric cavity containing a single two-level atom [36,37]. Except for the asymmetric Fabry–Perot cavity, likewise behaviors can also be achieved in optomechanical systems [38,39,40,41]. Recently, the asymmetric behavior in quantum statistics has been reported in a spinning resonator [42,43], where the conversion between bunching and antibunching photons can be realized by changing the direction of the incident field. In addition, the asymmetric behavior in quantum phase transition has been studied in waveguide QED system by using the second-order harmonic oscillation [44]. Furthermore, likewise behavior in quantum interference has been predicted for an asymmetric cavity QED system where the unconventional photon blockade can only be achieved in a single driving direction [45,46].

In this paper, we examine a system of two coupled cavities, as described in the literature [47,48]. The left cavity is strongly coupled to a two-level atom, while the right cavity contains a pumped second-order nonlinear material, which leads to the parametric interaction [49]. We note that quantum squeezing induced by the parametric interaction results in various quantum phenomena, such as nonreciprocal transmission [50], nonreciprocal photon blockade [51], nonreciprocal entanglement [52], and so on [53]. Here, we demonstrate that, in the absence of the parametric interaction, classical asymmetric behavior can be observed by detecting the transmitted photon intensity when the probe field is injected from different directions. However, when a parametric drive is applied, various quantum asymmetric behaviors can be achieved, including the squeezing, quantum statistics, and phase-space properties of the transmitted photons. We show that such kind of quantum asymmetric behaviors can be analyzed by measuring Klyshko’s figures of merit, the Wigner function, and the steady-state second-order correlation function, which may have potential applications in remote quantum communication, quantum sensing, and quantum networking [54,55].

2. Model

As shown in Figure 1, two single-mode cavities with resonant frequencies ${\omega }_j(j=\mathrm{L},\mathrm{R})$ are coupled with a coupling strength V. In the left cavity, a two-level atom with resonant transition frequency ${\omega }_A$ is strongly coupled to the cavity mode with coupling strength J. The right cavity interacts with a ${\chi }^{\left(2\right)}$ nonlinear medium being transversely pumped with an external pump field (angular frequency ${\omega }_{\mathrm{d}}$). A weak probe field with angular frequency ${\omega }_p={\omega }_{\mathrm{d}}/2$ is injected from the left cavity (red solid arrow) or the right cavity (blue solid arrow), resulting in a driving-field Rabi frequency $\epsilon$. Thus, the Hamiltonian of the system in the rotating frame of the probe field frequency ${\omega }_p$ can be written as [48,56]

$$\begin{array}{ccc}\hfill H& =& {\Delta }_{\mathrm{L}}{a}_{\mathrm{L}}^{†}{a}_{\mathrm{L}}+{\Delta }_{\mathrm{R}}{a}_{\mathrm{R}}^{†}{a}_{\mathrm{R}}+{\Delta }_A{\sigma }_{+}{\sigma }_{-}+J(a_{\mathrm{L}}^{†}{\sigma }_{-}+a_{\mathrm{L}}{\sigma }_{+})\hfill \\ & & +V(a_{\mathrm{L}}^{†}{a}_{\mathrm{R}}+a_{\mathrm{R}}^{†}{a}_{\mathrm{L}})+G(a_{\mathrm{R}}^2+a_{\mathrm{R}}^{†2})+H_D,\hfill \end{array}$$

(1)

where ${\Delta }_{\mathrm{L}}={\omega }_{\mathrm{L}}-{\omega }_p$, ${\Delta }_{\mathrm{R}}={\omega }_{\mathrm{R}}-{\omega }_p$ and ${\Delta }_{\mathrm{A}}={\omega }_A-{\omega }_p$ are detunings of the probe field angular frequency from the left/right cavity mode frequencies and atomic transition frequency, respectively. Here, $a_j\left(a_j^{†}\right)$ is the annihilation (creation) operator for the left ($j=\mathrm{L}$) or right ($j=\mathrm{R}$) cavity. ${\sigma }_{\pm }$ denote the atomic raising and lowering operators. The nonlinear medium-induced parametric interaction strength is denoted by G. $H_D=\epsilon (a_j+a_j^{†})$ denotes the driving term where $j=$ L or R represents that the probe field drives the left or right cavity mode, respectively. For mathematical simplicity, we set ${\omega }_L={\omega }_R={\omega }_A\equiv \omega$ in the following derivation, yielding ${\Delta }_L={\Delta }_R={\Delta }_A\equiv \Delta$.

Before studying the effects of the parametric drive, let us first examine the properties of the system with $G=0$. Neglecting the weak probe field, the Hamiltonian of such a system can be written as $H_{G=0}=\omega (a_{\mathrm{L}}^{†}{a}_{\mathrm{L}}+a_{\mathrm{R}}^{†}{a}_{\mathrm{R}}+{\sigma }_{+}{\sigma }_{-})+J(a_{\mathrm{L}}^{†}{\sigma }_{-}+a_{\mathrm{L}}{\sigma }_{+})+V(a_{\mathrm{L}}^{†}{a}_{\mathrm{R}}+a_{\mathrm{R}}^{†}{a}_{\mathrm{L}})$. In the strong atom–cavity coupling regime, the properties of the system can be described by a set of dressed states [57]. In one photon space, the Hamiltonian $H_{G=0}$ can be expressed in terms of the basic states $|00e\rangle$, $|10g\rangle$ and $|01g\rangle$ [the first (second) number denotes the photon number in the left (right) cavity, while the third letter (e or g) indicates the atomic state], i.e.,

$$H^{\left(1\right)}\equiv \left[\begin{array}{ccc}\omega & J& 0\\ J& \omega & V\\ 0& V& \omega \end{array}\right].$$

(2)

The eigenvalues of this Hamiltonian are given by ${\lambda }_0^{\left(1\right)}=\omega$, ${\lambda }_{\pm }^{\left(1\right)}=\omega \pm \sqrt{J^2+V^2}$ with corresponding eigenstates:

$${\psi }_0^{\left(1\right)}={\displaystyle \frac{V}{\sqrt{J^2+V^2}}}|00e\rangle -{\displaystyle \frac{J}{\sqrt{J^2+V^2}}}|01g\rangle ,$$

(3)

and

$${\psi }_{\pm }^{\left(1\right)}=\sqrt{{\displaystyle \frac{1}{2}}}\left\{|10g\rangle \pm {\displaystyle \frac{J}{\sqrt{J^2+V^2}}}|00e\rangle \pm {\displaystyle \frac{V}{\sqrt{J^2+V^2}}}|01g\rangle \right\}.$$

(4)

Clearly, the state ${\psi }_0^{\left(1\right)}$ can not be excited when the probe field is injected from the left side of the left cavity since the state $|10g\rangle$ is not included.

Likewise, the Hamiltonian $H_{G=0}$ in two photon space can be expressed in terms of the basic states $|02g\rangle$, $|20g\rangle$, $|11g\rangle$, $|01e\rangle$, and $|10e\rangle$, yielding

$$H^{\left(2\right)}\equiv \left[\begin{array}{ccccc}2\omega & 0& \sqrt{2}V& 0& 0\\ 0& 2\omega & \sqrt{2}V& 0& \sqrt{2}J\\ \sqrt{2}V& \sqrt{2}V& 2\omega & J& 0\\ 0& 0& J& 2\omega & V\\ 0& \sqrt{2}J& 0& V& 2\omega \end{array}\right].$$

(5)

The eigenvalues of Equation (5) are given by ${\lambda }_0^{\left(2\right)}=2\omega$, ${\lambda }_{1\pm }^{\left(2\right)}=2\omega \mp {\displaystyle \frac{\sqrt{2}}{2}}$$\sqrt{3J^2+5V^2-\sqrt{{\Delta }_1}}$ and ${\lambda }_{2\pm }^{\left(2\right)}=2\omega \mp {\displaystyle \frac{\sqrt{2}}{2}}\sqrt{3J^2+5V^2+\sqrt{{\Delta }_1}}$, where ${\Delta }_1=J^4+30J^2{V}^2+9V^4$. The corresponding eigenstates are given by

$${\psi }_0^{\left(2\right)}=-{\displaystyle \frac{J^2-V^2}{\sqrt{2}JV}}|02g\rangle -{\displaystyle \frac{V}{\sqrt{2}J}}|20g\rangle +|01e\rangle ,$$

(6)

$$\begin{array}{ccc}\hfill {\psi }_{1\mp }^{\left(2\right)}& =& \mp {\displaystyle \frac{c_{1-}}{3J{\delta }_{-}}}|02g\rangle \mp {\displaystyle \frac{c_{2-}}{3J{\delta }_{-}}}|20g\rangle -{\displaystyle \frac{c_{3+}}{6JV}}|11g\rangle \pm {\displaystyle \frac{\sqrt{2}{c}_{4-}}{3V{\delta }_{-}}}|01e\rangle +|10e\rangle ,\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill {\psi }_{2\mp }^{\left(2\right)}& =& \mp {\displaystyle \frac{c_{1+}}{3J{\delta }_{+}}}|02g\rangle \mp {\displaystyle \frac{c_{2+}}{3J{\delta }_{+}}}|20g\rangle -{\displaystyle \frac{c_{3-}}{6JV}}|11g\rangle \pm {\displaystyle \frac{\sqrt{2}{c}_{4+}}{3V{\delta }_{+}}}|01e\rangle +|10e\rangle ,\hfill \end{array}$$

(8)

where ${\delta }_{\pm }=\sqrt{3J^2+5V^2\pm \sqrt{{\Delta }_1}}$, $c_{1\pm }=-J^2+3V^2\pm \sqrt{{\Delta }_1}$, $c_{2\pm }=5J^2+3V^2\pm \sqrt{{\Delta }_1}$, $c_{3\pm }=J^2-3V^2\pm \sqrt{{\Delta }_1}$ and $c_{4\pm }=-2V^2+2J^2-{\delta }_{\pm }^2/2$. It is clear to see that, in the presence of the parametric interaction, the right cavity can be excited via the two photon process, i.e., the transition from state $|00g\rangle$ to state $|02g\rangle$.

3. Classical Asymmetric Behavior

Case I. A probe photon is injected from the left side of the left cavity and detected on the right side of the right cavity. According to Equation (3), the transition ${\psi }_0^{\left(0\right)}\equiv |00g\rangle \stackrel{\mathrm{L}}{\leftrightarrow }{\psi }_0^{\left(1\right)}$ [see the red dashed arrow in Figure 2a] is forbidden when the probe field is injected from the left side of the left cavity. Therefore, only two excitation pathways [denoted by 1 ph in Figure 2a], i.e., ${\psi }_0^{\left(0\right)}\stackrel{\mathrm{L}}{\leftrightarrow }{\psi }_{\pm }^{\left(1\right)}$, are allowed in the single-photon space. This feature can be verified by numerically solving the master equation with dissipation, i.e.,

$$\begin{array}{ccc}\hfill \dot{\rho }& =& -i[H,\rho ]+{\kappa }_1(2a_L\rho a_L^{†}-a_L^{†}{a}_L\rho -\rho a_L^{†}{a}_L)\hfill \\ & & +{\kappa }_2(2a_R\rho a_R^{†}-a_R^{†}{a}_R\rho -\rho a_R^{†}{a}_R)+\gamma (2{\sigma }_{-}\rho {\sigma }_{+}-{\sigma }_{+}{\sigma }_{-}\rho -\rho {\sigma }_{+}{\sigma }_{-})\hfill \end{array}$$

(9)

where ${\kappa }_j,\gamma$ are decoherence rates for the left ($j=1$) or right ($j=2$) cavity decay and atomic spontaneous emission. Here, we employ the inverse-power method to obtain the steady state solution of the Equation (9), and truncate the Hilbert space of each cavity in our numerical calculations to a size of 30 to ensure numerical convergence. The system parameters are chosen as $J/2\pi =30$ MHz, ${\kappa }_1/2\pi ={\kappa }_2/2\pi =3.7$ MHz, $\gamma /2\pi =2.6$ MHz, $\epsilon /2\pi \approx 3.6$ MHz [37]. As shown in Figure 2b, only two peaks can be observed in the transmitted photon excitation spectrum $\langle a_{\mathrm{R}}^{†}{a}_{\mathrm{R}}\rangle$ at the frequencies of $\Delta =\pm \sqrt{J^2+V^2}$.

Case II. A probe photon is injected from the right side of the right cavity and detected on the left side of the left cavity. In this case, there exists three transition pathways in the single-photon space [see blue arrows in Figure 2a], i.e., ${\psi }_0^{\left(0\right)}\stackrel{\mathrm{R}}{\leftrightarrow }{\psi }_0^{\left(1\right)}$ and ${\psi }_0^{\left(0\right)}\stackrel{\mathrm{R}}{\leftrightarrow }{\psi }_{\pm }^{\left(1\right)}$, when the probe field is injected from the right side of the right cavity. As a result, one can observe three peaks in the transmitted photon excitation spectrum $\langle a_{\mathrm{L}}^{†}{a}_{\mathrm{L}}\rangle$ [see Figure 2c]. It is noted that the transmitted photon intensity at the central frequency can be reasonably observed for the case of $V<J$, but it drops quickly when the coupling strength $V\gg J$. Thus, under the condition of $G=0$ and $V<J$, the transmitted photon intensity changes greatly at $\Delta =0$ for different injection directions, yielding classical asymmetric propagating behavior.

To show the above features clearer, we select $V=0.1J$ as an example and plot the transmitted photon spectrum in Figure 3a. In this figure, the red solid curve represents the injection direction from the left side of the left cavity, while the blue dashed curve shows the injection from the right side of the right cavity. The other system parameters remain consistent with those used in Figure 2b. It is evident that, aside from the asymmetric behavior observed at $\Delta =0$, similar behaviors can also be detected at the one-photon (1 ph), two-photon (2 ph), and multiphoton excitation (3 ph) frequencies (i.e., different transmitted photon number). The corresponding excitation frequencies occur at $\Delta \approx \pm J$, $\pm J/\sqrt{2}$, and $\pm J/\sqrt{3}$, respectively. Additionally, we note that multiphoton excitations, such as two-photon and three-photon excitations, are more pronounced when the probe field is injected from the left side of the left cavity, resulting in a reverse behavior at one-photon excitation frequency. Therefore, the direction of this asymmetric propagating behavior in the transmitted photon intensity can be controlled by selecting a specific frequency for the probe field.

In Figure 3b, we set $\Delta =0$ and show the numerical result of the transmitted field intensity against the Rabi frequency of the probe field, $\epsilon /2\pi$. The red curves represent the injection direction from the left side of the left cavity, while the blue curves represent the injection direction from the right side of the right cavity. In the absence of the second order nonlinear medium (i.e., $G=0$, as shown by the dashed red and blue curves), transmitted photons can only be detected when the probe field is injected from the right side of the right cavity. It is important to note that as the intensity of the probe field increases, the amplitude of the transmitted field grows rapidly, demonstrating strong classical asymmetric propagating behavior.

However, when we include the parametric interaction caused by the nonlinear medium (for example, when $G/2\pi =0.2$ MHz, represented by the solid red and blue curves), the two-photon process introduces additional transition pathways. As a result, the classical asymmetric behavior disappears at high probe field intensities, and transmitted photons can be detected with nearly the same amplitudes regardless of whether the injection is from the left or right. The corresponding physical mechanics interpretation can be described as follows. When the probe field is injected from the left side of the left cavity, photons can NOT enter into the left cavity due to the atom–cavity coupling. Thus, the transmitted photons from the right side of the right cavity only depends on the parametric interaction strength G, yielding a constant mean photon number [see red solid curve in Figure 3b]. When the probe field is injected from the right side of the right cavity, photons can enter into the left cavity via the coupling between two cavities. Thus, the transmitted photon intensity from the left side of the left cavity will be slightly enhanced due to the parametric interaction [see blue solid curve in Figure 3b].

4. Quantum Asymmetric Behavior

Now, we will discuss how the parametric drive-induced two-photon excitation processes lead to various asymmetric behaviors in quantum statistics. First, let us study the squeezing property of the transmitted photons at the detuning $\Delta =0$ MHz. We set the parameter $\epsilon /2\pi \approx 3.5$ MHz and evaluate Klyshko’s figures of merit based on the steady state solution of Equation (9), defined as $K_n=\left[(n+1)P_{n-1}{P}_{n+1}\right]/\left(n{P}_n^2\right)$, to demonstrate the nonclassicality of the transmitted photon number distribution [58,59]. Other system parameters are the same as those used in Figure 2.

According to Klyshko’s criterion for nonclassicality, if any $K_n$ value is less than unity, the state of the transmitted photons is considered nonclassical. In Figure 4a, we observe that $K_n$ is less than unity for $n=2,4,$ and 6 when the injection direction is from the left side of the left cavity, confirming Klyshko’s criterion. In contrast, when the injection occurs from the right side of the right cavity, the nonclassicality of the transmitted photon state is notably diminished, as $K_n>1$ for all photon numbers n except for $n=2$ [see Figure 4b]. Furthermore, the nonclassical nature of the transmitted photons can also be verified by calculating the Wigner functions, as shown in the insets of panels (a) and (b). When the probe field is injected from the left side of the left cavity, the transmitted field exhibits slight squeezing.

The corresponding physical mechanics can be easily explained qualitatively. When photons are injected from the left side of the left cavity, they are blockaded by the atom before entering the right cavity. Thus, photons in the right cavity are originated from the parametric interaction-induced two photon excitation. As a result, the photons leaking from the right cavity will be squeezed [see Figure 4a]. Conversely, when coherent photons are injected from the right cavity, they will not be blockaded and enter into the left cavity via the coupling between two cavities. In this case, the transmitted photon from the left side of the left cavity is coherent as shown in Figure 4b. Note that the squeezing component induced by the parametric interaction is weaker than the coherent component induced by the probe field.

Next, we will investigate how the parametric drive influences the quantum statistical properties of the transmitted field. Specifically, we will calculate the steady-state second-order correlation function of the transmitted photons, expressed as $g^{\left(2\right)}\left(0\right)=\langle a_j^{†}{a}_j^{†}{a}_j{a}_j\rangle$/${\langle a_j^{†}{a}_j\rangle }^2$, where j can be either L or R, indicating photons transmitted from the left or right side, respectively. In Figure 5, we present the second-order correlation functions for the two injection directions: panel (a) shows results for the left side, while panel (b) reports results for the right side. At the frequencies near single photon resonant excitations [see Figure 5a], we observe that the second-order correlation function for the transmitted photons shifts from $g^{\left(2\right)}\left(0\right)<1$ to $g^{\left(2\right)}\left(0\right)>1$ as the parametric interaction strength G increases when the probe field is injected from the left side of the left cavity. The increase in $g^{\left(2\right)}\left(0\right)$ can be attributed to the squeezing of transmitted photons from the right side due to the parametric interaction occurring in the right cavity, where two photon excitations are dominant. For our calculations, we set $\epsilon /2\pi \approx 3.5$ MHz, while the other system parameters are consistent with those in Figure 2b. Conversely, when photons are injected from the right side of the right cavity, we find that the second-order correlation function remains unchanged and remains less than unity at the single photon resonance frequencies [see Figure 5b]. This phenomenon is attributed to the energy anharmonicity induced by the interaction between the atoms and photons in the left cavity, commonly known as the photon blockade phenomenon and yielding antibunched photons leaking from the left side of the left cavity. Therefore, it shows that asymmetric behavior in the quantum statistics of transmitted photons can be induced by increasing the parametric drive strength at frequencies near the single photon excitations.

Finally, let us discuss the implementation of the parametric interaction discussed in our model. In experiments, such kind of parametric interaction can be achieved by using anisotropic crystals with quadratic nonlinearity [60,61,62], or using three wave mixing and second harmonic generation in a dielectric medium with a quadratic intensity-dependent response [63].

5. Conclusions

In conclusion, we have studied the classical and quantum asymmetric behaviors in a two-coupled cavity quantum electrodynamics system. In this system, the left cavity interacts with a two-level atom, while the right cavity is filled with a second-order nonlinear material. The nonlinear material is driven by a pumping field, and a probe field is coherently injected either from the left side of the left cavity or the right side of the right cavity. When the condition $V\ll J$ is fulfilled and without the parametric drive, classical asymmetric behavior in the transmitted photon intensity occurs at the central resonant excitation frequency due to the interaction between the atom and the left cavity. However, the parametric drive introduces additional transition pathways through a two-photon process, effectively destroys the classical asymmetric propagating behavior at this central frequency. Instead, various quantum asymmetric behaviors can be detected near the single-photon resonant frequencies by analyzing Klyshko’s figures of merit, the Wigner function, and the steady-state second-order correlation function of the transmitted photons. As the strength of the parametric drive increases, photons leaking from the right side of the right cavity become squeezed due to parametric interactions when the probe field is injected from the left side of the left cavity. Conversely, photons leaking from the left side of the left cavity are purified by the atom and remain coherent when the probe field is injected from the right side of the right cavity. Additionally, we observe that the photons leaking from the left side exhibit antibunching due to energy inharmonicity caused by strong atom–cavity interactions, while the photons from the right side display bunching since the two-photon processes induced by the parametric drive dominate. These classical and quantum asymmetric behaviors in two coupled cavity QED system may lead to potential applications in remote quantum communication, quantum sensing, and quantum networking.