Squeezing-induced nonreciprocal photon blockade in an optomechanical microresonator

Dong-Yang Wang; Dong-Yang Wang; Lei-Lei Yan; Shi-Lei Su; Shi-Lei Su; Cheng-Hua Bai; Hong-Fu Wang; Erjun Liang

doi:10.1364/OE.493208

1. Introduction

The optical nonreciprocal effect is an essential and intriguing phenomenon for fundamental studies and applied sciences [1–3]. To the best of our knowledge, a typical linear and time-independent system is time-reversal symmetric and obeys the Lorentz reciprocity [4,5], which means that the system’s optical transmission and statistic properties are isotropic. However, nonreciprocal optical devices can block the backward noise signal and unwanted feedback, which is indispensable in both classical and quantum optics research. The typical nonreciprocal optical devices include isolators [6], circulators [7], and directional amplifiers [8], which can break the time-reversal symmetric and be widely used in many fields, such as light propagation, quantum computation, quantum network, quantum information processing. So far, there are various schemes have been proposed to achieve optical nonreciprocity, such as parity-time symmetric nonlinear microresonators [9], spinning whispering gallery mode (WGM) [10,11], cavity magnonic interaction [12], effective gauge field [13], cross-Kerr nonlinearity [14], thermal motion of atoms [15,16], loss medium [4], and optomechanical interaction [17–19]. Most of the above schemes can be classified into three categories [20,21]: magnetic field breaking time-reversal symmetry, spatiotemporal modulation of the system, and nonlinearity.

The photon blockade (PB) phenomenon [22,23], the occupation of the first photon blocking the consequent photon injection, is a famous nonclassical antibunching effect and can be used to generate the single-photon source, which is crucially vital for quantum state preparation and quantum information processing of few photons [24,25]. Generally, there are two different mechanisms to generate the ideal PB effect, i.e., conventional photon blockade (CPB) and unconventional photon blockade (UPB), which respectively rely on the anharmonicity of eigenenergy spectrum [26,27] and the destructive quantum interference of different excitation paths [28–31]. The cavity optomechanical system is also an excellent platform for studying the PB due to its inherent nonlinear optomechanical interaction [32–35], which can result in the anharmonic eigenenergy spectrum. However, the required optomechanical coupling is too large for recent experiments and poses significant technological challenges. To this end, researchers have proposed some schemes to enhance the optomechanical coupling, e.g., the Josephson effect [36], mechanical amplification [37], and nonlinear medium [38]. Furthermore, the UPB in cavity optomechanical systems has also been studied via adding an auxiliary cavity [39–41] or atom [42,43] to construct another excitation path needed in the quantum interference mechanism. On the other hand, research on nonreciprocal PB effects has attracted widespread attention in recent years, where the nonreciprocity of PB is generated by the spinning-caused frequency shifting [44–47], nonlinearity [48–50], asymmetric system [51,52], and nonreciprocal coupling [53,54]. Recently, the nonreciprocal PB has been confirmed experimentally in a strongly coupled cavity quantum electrodynamics system [55].

In this paper, we will discuss how to achieve the nonreciprocal PB effect by quantum squeezing in a stationary whispering gallery mode optomechanical system. Here, the quantum squeezing is described by the nonlinear parametric process in the clockwise mode as proposed in Ref. [56]. First, we study the nonreciprocal conventional photon blockade (NCPB) according to the quantum squeezing transformation with a strong nonlinear parametric process. The quantum squeezing not only shifts the frequency but also enhances the single-photon optomechanical coupling strength in the clockwise mode, which results in the achievement of NCPB. Second, different from the above mechanism, we discuss the nonreciprocal unconventional photon blockade (NUPB) based on the destructive quantum interference mechanism when the nonlinear parametric process is weak, where the nonlinear parametric process is seen as a different excitation path. Although the nonlinear parametric process is weak, the resulting nonreciprocal ratio of PB is larger than the first scheme. Compared with the previous reports [44,48,50–52,54], our works have the following features: i) the nonreciprocal PB can be achieved whether the nonlinear parametric process is strong or weak, which gives an alternative scheme for experiments; ii) the system is stationary and does not require an excessive single-photon optomechanical coupling; iii) the proposal does not require a strong magnetic field or nonreciprocal coupling. Our proposal gives a new route to achieve nonreciprocal PB, which can also be expanded to engineer other nonreciprocal devices and would make sense in the chiral quantum network, unidirectional quantum information processing, and topological photonics.

The rest of the paper is organized as follows: In Sec. 2, we illustrate the proposed WGM optomechanical system, give its Hamiltonian under two different driving cases, and derive corresponding effective Hamiltonian for different PB mechanisms. In Sec. 3, we analytically and numerically calculate the second-order correlation function, discuss the generation of nonreciprocal PB, give the optimal parameter relation to optimize the nonreciprocity, compare the two mechanisms by calculating the nonreciprocal ratio. Finally, a conclusion is given in Sec. 4.

2. System and Hamiltonian

As depicted in Fig. 1, we consider a WGM optomechanical microresonator system, where the cavity boundary vibrates due to the optomechanical radiation pressure interaction, described as the mechanical mode $b$. There are two degenerate optical modes in the system, the clockwise and counterclockwise modes, respectively. Here, the considered microresonator is made of aluminum nitride or lithium niobate, which can generate the common $\chi ^{(2)}$ nonlinearity and support the parametric nonlinear optical process [57–59]. Then, the Hamiltonian of the proposed optomechanical microresonator system is written as ($\hbar =1$)

(1)$$H_{s}=\omega_{l}a_{l}^{{\dagger}}a_{l}+\omega_{r}a_{r}^{{\dagger}}a_{r}+\omega_{m}b^{{\dagger}}b-g\big(a_{l}^{{\dagger}}a_{l}+a_{r}^{{\dagger}}a_{r}\big)\left(b^{{\dagger}}+b\right)+\left[\frac{G}{2}e^{{-}i(\omega_{p}t-\theta)}a_{l}^{{\dagger} 2}+\mathrm{H.c.}\right],$$

where the first three terms represent the free Hamiltonian of clockwise mode, counterclockwise mode, and mechanical mode, respectively. $a_{l}\,(a_{l}^{\dagger })$, $a_{r}\,(a_{r}^{\dagger })$, and $b\,(b^{\dagger })$ are the annihilation (creation) operators of the clockwise mode, counterclockwise mode, and mechanical mode. The fourth term describes the nonlinear optomechanical interactions between the two propagating modes and the mechanical mode with single-photon optomechanical coupling strength $g$. The last term represents the nonlinear parametric process in the clockwise mode with nonlinear strength $G$, frequency $\omega _{p}$, and phase $\theta$. In this work, we would discuss the nonreciprocal PB effect [44] induced by the quantum squeezing.

Fig. 1. Schematic diagram of the WGM optomechanical microresonator system, where the forward (blue arrow) or backward (red arrow) driving is introduced by the optical fiber and evanescently coupled into the clockwise $a_{l}$ or counterclockwise $a_{r}$ mode of the microdisk. Both the propagating modes interact with the boundary vibrates mode $b$ via the optomechanical interaction.

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The considered optomechanical system can be driven from both sides, as shown in Fig. 1. The driving field with frequency $\omega _{l}$ input to the left (right) port excites the clockwise (counterclockwise) mode, which is called the forward (backward) driving case. The Hamiltonian of driving interaction is given by $H_{d}=E_{d}(a_{k}^{\dagger }e^{-i\omega _{d}t}+\mathrm {H.c.})$, where the driving amplitude $E_{d}=\sqrt {2\kappa P/(\hbar \omega _{d})}$ with the driving power $P$ and frequency $\omega _{d}$, $\kappa$ is the decay rate of optical propagating mode, and $k=l,r$ respectively represents the forward or backward driving case. Here, we have assumed the decay rate is the same for the clockwise and counterclockwise modes $\kappa _{l}=\kappa _{r}=\kappa$. The total Hamiltonian of the proposed system is $H=H_{s}+H_{d}$. In the rotating reference frame $V=\exp [-i\omega _{d}ta_{l}^{\dagger }a_{l}-i\omega _{d}ta_{r}^{\dagger }a_{r}]$ and setting $\omega _{p}=2\omega _{d}$, the system Hamiltonian without driving field is described by

(2)$$H^{\prime}=\Delta_{l}a_{l}^{{\dagger}}a_{l}+\Delta_{r}a_{r}^{{\dagger}}a_{r}+\omega_{m}b^{{\dagger}}b-g\big(a_{l}^{{\dagger}}a_{l}+a_{r}^{{\dagger}}a_{r}\big)\left(b^{{\dagger}}+b\right)+\left(\frac{G}{2}a_{l}^{{\dagger} 2}e^{i\theta}+\mathrm{H.c.}\right),$$

where $\Delta _{l,r}=\omega _{l,r}-\omega _{d}$ are detuning between driving field and propagating modes.

2.1 Effective Hamiltonian for nonreciprocal conventional photon blockade

For convenience, we can discuss the nonreciprocity of the system by dividing the system into two driving cases, i.e., the forward driving case and the backward driving case. The physical mechanism of the CPB in our proposal comes from the symmetry-breaking behavior of the system eigenenergy spectrum. Here, we derive an effective Hamiltonian to show the anharmonicity of the system by performing unitary transformations [60–62] $S=\exp [\frac {r}{2}e^{-i\theta }a_{l}^{2}-\frac {r}{2}e^{i\theta }a_{l}^{\dagger 2}]$ and $V_{\mathrm {md}}=\exp [(g\cosh 2ra_{l}^{s\dagger }a_{l}^{s}+g\sinh ^{2}r+ga_{r}^{\dagger }a_{r})(b^{\dagger }-b)/\omega _{m}]$ with $r=\frac {1}{4}\ln \frac {\Delta _{l}+G}{\Delta _{l}-G}$. The system effective Hamiltonian under the rotating-wave approximation reads

(3)$$H_{\mathrm{md}}=\Delta_{l}^{s}a_{l}^{s\dagger}a_{l}^{s}-\frac{g_{s}^{2}}{\omega_{m}}\left(a_{l}^{s\dagger}a_{l}^{s}\right)^{2}+\Delta_{r}a_{r}^{{\dagger}}a_{r}-\frac{g^{2}}{\omega_{m}}\left(a_{r}^{{\dagger}}a_{r}\right)^{2}+\omega_{m}b^{{\dagger}}b,$$

where the nonlinear optomechanical interaction is diagonalized in the mechanical displacement picture. $\Delta _{l}^{s}=\left (\Delta _{l}-G\right )e^{2r}-2g^{2}\cosh (2r)\sinh ^{2}r/\omega _{m}$ and $g_{s}=g\cosh (2r)$ are the frequency shift and the enhanced optomechanical coupling by the squeezing, respectively. Meanwhile, the system Hamiltonian is decoupled into three independent parts, so we can ignore those undriven terms in Eq. (3). And then the Hamiltonian of the forward driving case is rewritten as

(4)$$\mathcal{H}_{\mathrm{fw}}=\Delta_{l}^{s}a_{l}^{s\dagger}a_{l}^{s}-\frac{g_{s}^{2}}{\omega_{m}}\left(a_{l}^{s\dagger}a_{l}^{s}\right)^{2}+\left(E_{d}^{s}a_{l}^{s\dagger}+\mathrm{H.c.}\right),$$

where $g_{s}^{2}/\omega _{m}$ is the enhanced anharmonicity of system eigenenergy spectrum via quantum squeezing and $E_{d}^{s}=E_{d}(\cosh r-e^{i\theta }\sinh r)$. The Hamiltonian of the backward driving case is rewritten as

(5)$$\mathcal{H}_{\mathrm{bw}}=\Delta_{r}a_{r}^{{\dagger}}a_{r}-\frac{g^{2}}{\omega_{m}}\left(a_{r}^{{\dagger}}a_{r}\right)^{2}+\left(E_{d}a_{r}^{{\dagger}}+\mathrm{H.c.}\right),$$

which is consistent with the standard cavity optomechanical system [32,33]. The difference of the two driving cases would result in the nonreciprocity of the system.

Here, the dynamical master equation describing the system coupling to a normal vacuum reservoir in the original picture is written as

(6)$$\frac{\partial\rho}{\partial t}={-}i\left[H_{s}+H_{d},\rho\right]+\mathcal{L}[a_{l}]\rho+\mathcal{L}[a_{r}]\rho+\mathcal{L}[b]\rho+\mathcal{L}[b^{{\dagger}}]\rho,$$

where $\mathcal {L}[a_{l,r}]\rho =\kappa /2(2a_{l,r}\rho a_{l,r}^{\dagger }-a_{l,r}^{\dagger }a_{l,r}\rho -\rho a_{l,r}^{\dagger }a_{l,r})$, $\mathcal {L}[b]\rho =(n_{\mathrm {th}}+1)\gamma /2(2b\rho b^{\dagger }-b^{\dagger }b\rho -\rho b^{\dagger }b)$, and $\mathcal {L}[b^{\dagger }]\rho =n_{\mathrm {th}}\gamma /2(2b^{\dagger }\rho b-bb^{\dagger }\rho -\rho bb^{\dagger })$, $\kappa$ and $\gamma$ represent the decay and damping rate of the optical propagating and mechanical modes, respectively. Meanwhile, the clockwise and counterclockwise modes are coupled to the normal vacuum reservoir. However, the mechanical mode is coupled to the thermal reservoir with a mean thermal phonon number $n_{\mathrm {th}}=\{\exp [\hbar \omega _{m}/(k_{B}T)]-1\}^{-1}$, where $k_{B}$ is the Boltzmann constant and $T$ is temperature.

It is worth emphasizing that the clockwise mode needs to be transformed into the squeezing picture, and the Bogoliubov squeezing transformation inevitably introduces additional noise [56]. At this time, the effective master equation of the forward driving case takes the form $\partial \rho /\partial t=-i\left [\mathcal {H}_{\mathrm {fw}},\rho \right ]+\mathcal {L}[a_{l}^{s}]\rho +\mathfrak {L}[a_{l}^{s}]\rho$, where $\rho$ is the reduced system density matrix, $\mathcal {L}[a_{l}^{s}]\rho =\kappa /2(2a_{l}^{s}\rho a_{l}^{s\dagger }-a_{l}^{s\dagger }a_{l}^{s}\rho -\rho a_{l}^{s\dagger }a_{l}^{s})$ represents the decay of the squeezed clockwise mode $a_{l}^{s}$ with a decay rate $\kappa$, the last term $\mathfrak {L}[a_{l}^{s}]\rho =N_{p}\mathcal {L}[a_{l}^{s}]\rho +N_{p}\mathcal {L}[a_{l}^{s\dagger }]\rho -M_{p}\mathcal {L}^{\prime }[a_{l}^{s}]\rho -M_{p}^{\ast }\mathcal {L}^{\prime }[a_{l}^{s\dagger }]\rho$ describes the effective thermalization noise of the squeezed clockwise mode $a_{l}^{s}$ coming from the Bogoliubov squeezing transformation, where $\mathcal {L}^{\prime }[a_{l}^{s}]\rho =\kappa /2(2a_{l}^{s}\rho a_{l}^{s}-a_{l}^{s}a_{l}^{s}\rho -\rho a_{l}^{s}a_{l}^{s})$, $N_{p}=\sinh ^{2}r$, and $M_{p}=e^{-i\theta }\cosh r\sinh r$. The additional thermalization noise term $\mathfrak {L}[a_{l}^{s}]\rho$ induced by squeezing can thermalize the quantum system and destroy the study in the quantum regime. Thanks to the technology of input broadband squeezed vacuum field [56,60,63,64], the additional thermalization noise induced by squeezing can be eliminated when the input broadband squeezed vacuum field with squeezing parameter $r^{\prime }$ and phase $\theta ^{\prime }$ satisfies the relationships $r=r^{\prime }$ and $\theta -\theta ^{\prime }=(2n+1)\pi$, where $n\in Z$. The needed broadband squeezed vacuum field has been reported via an optical parametric amplifier [65]. Meanwhile, the squeezed clockwise mode $a_{l}^{s}$ is equivalently coupled to the effective vacuum reservoir without the squeezed-induced thermalization noise. Thus, the final quantum master equation is changed to

(7)$$\frac{\partial\rho}{\partial t}={-}i\left[\mathcal{H}_{\mathrm{fw}},\rho\right]+\mathcal{L}[a_{l}^{s}]\rho.$$

In the backward driving case, there is no input broadband squeezed vacuum field nor the nonlinear parametric process. So the system dynamics are not influenced by the squeezing transformation, and the external environment is a normal vacuum reservoir. And the effective quantum master equation for the backward driving case reads $\partial \rho /\partial t=-i\left [\mathcal {H}_{\mathrm {bw}},\rho \right ]+\mathcal {L}[a_{r}]\rho$, where $\rho$ is the reduced system density matrix, $\mathcal {L}[a_{r}]\rho =\kappa /2(2a_{r}\rho a_{r}^{\dagger }-a_{r}^{\dagger }a_{r}\rho -\rho a_{r}^{\dagger }a_{r})$ describes the decay of the counterclockwise mode.

2.2 Effective Hamiltonian for nonreciprocal unconventional photon blockade

The UPB comes from the destructive quantum interference between two different excitation paths. In our proposal, the nonlinear parametric process can be seen as a two-excitation path in the clockwise propagating mode, which means there may be an UPB phenomenon in the forward driving case. However, since the optomechanical coupling is tiny and there is no scatter coupling between the two propagating modes in our proposal, the counterclockwise mode is only excited by the backward driving, which results in the UPB not appearing in the backward driving case. This way, nonreciprocity emerges because of the different blockade mechanisms in the two driving cases.

Similarly, the effective Hamiltonian without the driving field is written as

(8)$$H_{\mathrm{md}}^{\prime}=\Delta_{l}a_{l}^{{\dagger}}a_{l}-\frac{g^{2}}{\omega_{m}}\left(a_{l}^{{\dagger}}a_{l}\right)^{2}+\frac{G}{2}\left(a_{l}^{{\dagger} 2}e^{i\theta}+a_{l}^{2}e^{{-}i\theta}\right)+\Delta_{r}a_{r}^{{\dagger}}a_{r}-\frac{g^{2}}{\omega_{m}}\left(a_{r}^{{\dagger}}a_{r}\right)^{2}+\omega_{m}b^{{\dagger}}b,$$

where the weak optomechanical coupling has been diagonalized via the unity transformation $V_{\mathrm {md}}^{\prime }=\exp [g(a_{l}^{\dagger }a_{l}+a_{r}^{\dagger }a_{r})(b^{\dagger }-b)/\omega _{m}]$. We can see that the system Hamiltonian is also decoupled since the optomechanical interaction has been eliminated. Thus, the effective Hamiltonian of the forward driving case is given by

(9)$$\mathbf{H}_{\mathrm{fw}}=\Delta_{l}a_{l}^{{\dagger}}a_{l}-\frac{g^{2}}{\omega_{m}}\left(a_{l}^{{\dagger}}a_{l}\right)^{2}+\frac{G}{2}\left(a_{l}^{{\dagger} 2}e^{i\theta}+a_{l}^{2}e^{{-}i\theta}\right)+E_{d}\left(a_{l}^{{\dagger}}+a_{l}\right).$$

The effective Hamiltonian of the backward case is

(10)$$\mathbf{H}_{\mathrm{bw}}=\Delta_{r}a_{r}^{{\dagger}}a_{r}-\frac{g^{2}}{\omega_{m}}\left(a_{r}^{{\dagger}}a_{r}\right)^{2}+E_{d}\left(a_{r}^{{\dagger}}+a_{r}\right).$$

The PB phenomenon in the counterclockwise mode belongs to the conventional blockade mechanism. However, the nonreciprocity in this section comes from the appearance of UPB in the clockwise mode. So we call this part the derivation of effective Hamiltonian for NUPB.

At this moment, the quantum master equation for the forward driving case is $\partial \rho /\partial t=-i\left [\mathbf {H}_{\mathrm {fw}},\rho \right ]+\mathcal {L}[a_{l}]\rho$, and for the backward driving case is $\partial \rho /\partial t=-i\left [\mathbf {H}_{\mathrm {bw}},\rho \right ]+\mathcal {L}[a_{r}]\rho$. Here, the noise term includes only the respective driven propagating modes in the above quantum master equation. That is because the effective Hamiltonian is decoupled, unaffected by the noise of other modes, which can be verified by solving the quantum master equation with the original Hamiltonian, which is given by

(11)$$\frac{\partial\rho}{\partial t}={-}i\left[H_{\mathrm{fw(bw)}},\rho\right]+\mathcal{L}[a_{l}]\rho+\mathcal{L}[a_{r}]\rho+\mathcal{L}[b]\rho+\mathcal{L}[b^{{\dagger}}]\rho.$$

The validity of the above calculation will be demonstrated via the following numerical simulations.

3. Nonreciprocal photon blockade

In the above discussion, we have reduced the initial three-mode system to a single-mode one by utilizing some unitary transformations, normal rotating-wave approximation, and weak optomechanical coupling conditions. Next, based on different PB mechanisms, we would investigate the nonreciprocal photon statistic property in the two reverse propagating modes. Moreover, the PB is characterized by calculating the photon correlation function analytically and numerically. The validity of those reduced Hamiltonians in Sec. 2. can be proved by comparing the analytical and numerical results.

3.1 Nonreciprocal conventional photon blockade

To explain the mechanism of NCPB, we respectively give the system eigenenergy spectrum of the two driving cases

(12)$$E_{\mathrm{fw}}(n_{l}^{s})=\Delta_{l}^{s}n_{l}^{s}-\frac{g_{s}^{2}}{\omega_{m}}n_{l}^{s2},{\kern 1cm} E_{\mathrm{bw}}(n_{r})=\Delta_{r}n_{r}-\frac{g^{2}}{\omega_{m}}n_{r}^{2},$$

where $n_{l}^{s}$ and $n_{r}$ are the photon number in modes $a_{l}^{s}$ and $a_{r}$, respectively. The result represents that the anharmonicities of the two driving cases are different, and the energy level shift occurs in the forward driving case due to the quantum squeezing. It is similar to the investigation [56], where the free energy is shifted, and the tunneling coupling is exponentially enhanced by the quantum squeezing.

Compared with the system eigenenergy spectrum of the two driving cases, we can find that the blockade effect of the two driving cases would be different because of their unequal anharmonic eigenenergy spectrum. The analytical calculation of the second-order correlation function can be obtained by solving the non-Hermitian Schrödinger equation, where the influence of the external environment is included by adding the system decay phenomenologically into the effective Hamiltonian. On the way, the non-Hermitian Schrödinger equation for the forward driving case can be written directly as

(13)$$i\frac{\partial|\psi(t)\rangle_{\mathrm{fw}}}{\partial t}=\left(\mathcal{H}_{\mathrm{fw}}-i\frac{\kappa}{2}a_{l}^{s\dagger}a_{l}^{s}\right)|\psi(t)\rangle_{\mathrm{fw}},$$

and for the backward driving case is

(14)$$i\frac{\partial|\psi(t)\rangle_{\mathrm{bw}}}{\partial t}=\left(\mathcal{H}_{\mathrm{bw}}-i\frac{\kappa}{2}a_{r}^{{\dagger}}a_{r}\right)|\psi(t)\rangle_{\mathrm{bw}},$$

where $|\psi (t)\rangle _{\mathrm {fw,bw}}=\sum _{n}C_{n}^{\mathrm {fw,bw}}(t)|n\rangle$ is the time-dependent optical state for the corresponding propagating mode. Here, $C_{n}^{\mathrm {fw,bw}}(t)$ is the probability amplitudes of $n$-photons state, and $n\in Z$ represents the photon number in the propagating mode.

Under the condition of weak driving $\{E_{d}^{s},\,E_{d}\}\ll \kappa$, the propagating mode is always low excitation. The dynamical evolution subspace of the system can be truncated by a small enough number to simplify the analytical calculation, e.g., $n\leqslant 2$. Based on the above assumption and taking the forward driving case as an example, we obtain a set of differential equations for probability amplitudes

(15)$$\begin{array}{l} i\frac{\partial C_{0}^{\mathrm{fw}}}{\partial t}=E_{d}^{s}C_{1}^{\mathrm{fw}},\\ i\frac{\partial C_{1}^{\mathrm{fw}}}{\partial t}=K_{s1}C_{1}^{\mathrm{fw}}+E_{d}^{s}C_{0}^{\mathrm{fw}}+\sqrt{2}E_{d}^{s}C_{2}^{\mathrm{fw}},\\ i\frac{\partial C_{2}^{\mathrm{fw}}}{\partial t}=2K_{s2}C_{2}^{\mathrm{fw}}+\sqrt{2}E_{d}^{s}C_{1}^{\mathrm{fw}}, \end{array}$$

where, for simplicity, we have made the following substitutions $K_{s1}=\Delta _{l}^{s}-i\kappa /2-g_{s}^{2}/\omega _{m}$ and $K_{s2}=\Delta _{l}^{s}-i\kappa /2-2g_{s}^{2}/\omega _{m}$. Because of the weak driving condition $E_{d}^{s}\ll \kappa$, the system would reach a steady state of dynamic equilibrium, where the time-independent probability amplitudes are calculated as

(16)$$C_{0}^{\mathrm{fw}}\simeq1,{\kern 1cm} C_{1}^{\mathrm{fw}}=\frac{E_{d}^{s}K_{s2}}{E_{d}^{s2}-K_{s1}K_{s2}},{\kern 1cm} C_{2}^{\mathrm{fw}}=\frac{-E_{d}^{s2}/\sqrt{2}}{E_{d}^{s2}-K_{s1}K_{s2}}.$$

According to the equal-time second-order correlation defined by $g_{l}^{(2)}(0)=\langle a_{l}^{s\dagger }a_{l}^{s\dagger }a_{l}^{s}a_{l}^{s}\rangle /\langle a_{l}^{s\dagger }a_{l}^{s}\rangle ^{2}$, we can obtain the simplified analytical solution $g_{l}^{(2)}(0)\simeq \left |K_{s1}/K_{s2}\right |^{2}$. Here, to simplify the final solution, we have taken the fact $C_{0}^{\mathrm {fw}}\gg C_{1}^{\mathrm {fw}}\gg C_{2}^{\mathrm {fw}}$ and the forward driving field is feeble $E_{d}^{s}\ll \kappa$.

Generally, the PB coming from the conventional blockade mechanism occurs at the single excitation resonance. However, when the anharmonicity caused by optomechanical interaction is small, the well-known conclusion no longer holds. Meanwhile, the location occurring PB is solved as $\Delta _{l}^{s}=3g_{s}^{2}/(2\omega _{m})-\sqrt {g_{s}^{4}/\omega _{m}^{2}+\kappa ^{2}}/2$, which can be simplified to the common single excitation resonance result $\Delta _{l}^{s}\simeq g_{s}^{2}/\omega _{m}$ only when $g_{s}^{2}/\omega _{m}\gg \kappa$, i.e.,

(17)$$\Delta_{l}=G+\frac{2g^{2}\cosh(2r)\sinh^{2}(r)+g^{2}\cosh^{2}(2r)}{\omega_{m}e^{2r}}.$$

Similar to the forward driving case, when we study the backward driving case, the equal-time second-order correlation of photons in the counterclockwise mode reads

(18)$$g_{r}^{(2)}(0)\simeq\left|\frac{\left(\Delta_{r}-i\frac{\kappa}{2}-\frac{g^{2}}{\omega_{m}}\right)}{\left(\Delta_{r}-i\frac{\kappa}{2}-\frac{2g^{2}}{\omega_{m}}\right)}\right|^{2},$$

where the detuning and single-photon optomechanical coupling strength have been changed to $\Delta _{r}$ and $g$ in the final result since there is no influence of the quantum squeezing transformation. Due to the optomechanical coupling being small in the counterclockwise mode, the optimal PB is located at

(19)$$\Delta_{r}=\frac{3g^{2}}{2\omega_{m}}-\frac{1}{2}\sqrt{\frac{g^{4}}{\omega_{m}^{2}}+\kappa^{2}},$$

which cannot be simplified to the common single excitation resonance location $\Delta _{r}=g^{2}/\omega _{m}$. Therefore, we can conclude that the nonreciprocity of PB in both reversed propagating modes is generated. The physical mechanism is that the quantum squeezing not only shifts the detuning but also enhances the single-photon optomechanical coupling strength in the clockwise mode. The physical mechanism is similar to the principle of nonreciprocal optical transmission induced by quantum squeezing [56].

The analytical and numerical results of correlation functions in both reversed propagating modes are shown in Fig. 2(a). We can see that the correlation functions of the two reversed driving cases are different in value and location, which represents the nonreciprocal photon blockade obtained. For the forward driving case, an ideal PB ($g_{l}^{(2)}(0)\simeq 0.07$) occurs at $\Delta _{l}/\kappa =25.48$, where the correlation function of photons is $g_{r}^{(2)}(0)=1$ in backward driving case. This indicates that an obvious NCPB is obtained at the location given by Eq. (17). However, in the backward driving case, the minimum correlation function value $g^{(2)}(0)\simeq 0.84$ is at $\Delta _{r}/\kappa =-0.367$ according to Eq. (19). But the PB phenomenon disappears in the forward driving case. At this point, the nonreciprocal effect is not very obvious. Compared with the results of both driving cases, the nonreciprocal photon correlation is obtained in the two propagating modes by quantum squeezing. Furthermore, the analytical and numerical results also validate previous analyses and interpretations.

Fig. 2. (a) The equal-time second-order correlation functions versus the driving field detuning. The blue and red solid lines represent the analytical solutions of the two driving cases. And the blue and red dashed lines come from the numerical simulation by solving the quantum master equation. Here, the system parameters are set as $\kappa =2\pi \,\mathrm {MHz}$, $\omega _{m}=100\kappa$, $\gamma =10^{-6}\omega _{m}$, $g=0.03\omega _{m}$, $G=25\kappa$, $\theta =\pi$, and $E_{d}=0.05\kappa$ [3,8,63]. (b) The nonreciprocal ratio of PB versus the driving field detuning and nonlinear strength. The inset represents the nonreciprocal ratio versus the squeezing parametric $r$.

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To quantify the nonreciprocity of photon correlation functions in the two reversed propagating modes, we define a nonreciprocal ratio of PB

(20)$$\eta={-}10\log_{10}\left[\frac{g_{l}^{(2)}(0)}{g_{r}^{(2)}(0)}\right]$$

and show results in Fig. 2(b). The maximal ratio appears on the brighter yellow line, which satisfies the relationship in Eq. (17). When the nonlinear strength $G=25\kappa$, the maximal ratio $\eta _{\mathrm {max}}=12\,\mathrm {dB}$. Moreover, the maximum value of the nonreciprocal ratio does not increase significantly as the nonlinear strength increases. That is because the enhancement of CPB requires tremendous nonlinear strength in optomechanical systems.

On the other hand, the above steady-state solutions can also be verified by solving the corresponding dynamical quantum master equations. Since the final steady state is independent of the system’s initial state, it can be assumed that it is initially in the vacuum state. The dynamical evolution of equal-time second-order correlation functions in both propagating modes is shown in Fig. 3. We can see that correlation functions gradually reach their stable value over time. Here, Fig. 3(a) represents the dynamical correlation functions in different propagating modes with the same detuning $\Delta _{l,r}=3g^{2}/(2\omega _{m})-\sqrt {g^{4}/\omega _{m}^{2}+\kappa ^{2}}/2$, which corresponds to the dip of the red line in Fig. 2. Conversely, Fig. 3(b) describes the dynamical evolution of correlation functions located at the dip of the blue line, where an ideal PB only happens in the clockwise mode.

Fig. 3. The dynamical evolution of equal-time second-order correlation functions versus the scaled time $\kappa t$. The blue solid and red dashed lines respectively represent the dynamical evolution of correlation functions for the forward and backward driving cases. Here, the system parameters are the same as in Fig. 2.

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So far, we have given the analytical expression and numerical simulation of the equal-time second-order correlation function, verified its dynamical evolution at different locations, shown the nonreciprocal ratio of PB, and explained that the reason for nonreciprocity comes from the frequency shift and enhanced optomechanical interaction. Moreover, the optimal PB (minimum correlation function) for the backward driving case does not occur at the single excitation resonance because of the tiny optomechanical interaction. Next, we investigate the possibility of NUPB.

3.2 Nonreciprocal unconventional photon blockade

In addition to the traditional cognition, the quantum squeezing shifting the frequency and enhancing the optomechanical interaction, we will give another scheme based on the unconventional blockade mechanism, where the second-order nonlinear parametric process is seen as an extra two-excitation path [66,67]. At this moment, the system does not need to be transformed into the squeezing picture because the nonlinear parametric process is too weak. The squeezing-induced thermalization noise also does not need to be eliminated. Therefore, this proposal does not need the input broadband squeezed vacuum field (green arrow in Fig. 1).

Equations (9,10) give the effective Hamiltonian of the two driving cases, respectively. For the forward driving case, the clockwise mode can be excited through two different paths, i.e., $|0\rangle \xrightarrow {E_{d}}|1\rangle \xrightarrow {\sqrt {2}E_{d}}|2\rangle$ and $|0\rangle \xrightarrow {Ge^{i\theta }/\sqrt {2}}|2\rangle$, where $\theta$ is the relative phase. To find the optimal condition of UPB in the clockwise mode, we can directly solve the non-Hermitian Schrödinger equation, which is given by

(21)$$i\frac{\partial|\psi(t)\rangle_{\mathrm{fw}}}{\partial t}=\left(\mathbf{H}_{\mathrm{fw}}-i\frac{\kappa}{2}a_{l}^{{\dagger}}a_{l}\right)|\psi(t)\rangle_{\mathrm{fw}}.$$

Similar to the above calculation, the differential equations for probability amplitudes are written as

(22)$$\begin{array}{l} i\frac{\partial C_{0}^{\mathrm{fw}}}{\partial t}=E_{d}C_{1}^{\mathrm{fw}}+\frac{G}{\sqrt{2}}e^{{-}i\theta}C_{2}^{\mathrm{fw}},\\ i\frac{\partial C_{1}^{\mathrm{fw}}}{\partial t}=K_{1}C_{1}^{\mathrm{fw}}+E_{d}C_{0}^{\mathrm{fw}}+\sqrt{2}E_{d}C_{2}^{\mathrm{fw}},\\ i\frac{\partial C_{2}^{\mathrm{fw}}}{\partial t}=2K_{2}C_{2}^{\mathrm{fw}}+\sqrt{2}E_{d}C_{1}^{\mathrm{fw}}+\frac{G}{\sqrt{2}}e^{i\theta}C_{0}^{\mathrm{fw}}, \end{array}$$

where we have used the renormalized parameters $K_{1}=\left (\Delta _{l}-i\kappa /2-g^{2}/\omega _{m}\right )$ and $K_{2}=\left (\Delta _{l}-i\kappa /2-2g^{2}/\omega _{m}\right )$. When the system reaches its steady state, the time-independent probability amplitudes are solved as

(23)$$C_{0}^{\mathrm{fw}}\simeq1,{\kern 1cm} C_{1}^{\mathrm{fw}}=\frac{E_{d}\left(2K_{2}-Ge^{i\theta}\right)}{2E_{d}^{2}-2K_{1}K_{2}},{\kern 1cm} C_{2}^{\mathrm{fw}}=\frac{\left(K_{1}Ge^{i\theta}-2E_{d}^{2}\right)}{2\sqrt{2}\left(E_{d}^{2}-K_{1}K_{2}\right)}.$$

At this time, the equal-time second-order correlation defined by $g_{a_{l}}^{(2)}(0)=\langle a_{l}^{\dagger }a_{l}^{\dagger }a_{l}a_{l}\rangle /\langle a_{l}^{\dagger }a_{l}\rangle ^{2}$ in the clockwise mode is approximately given by

(24)$$g_{a_{l}}^{(2)}(0)\simeq\frac{\left|2K_{1}K_{2}\left(K_{1}Ge^{i\theta}-2E_{d}^{2}\right)\right|^{2}}{\left|E_{d}\left(2K_{2}-Ge^{i\theta}\right)\right|^{4}}.$$

Based on the analytical result of the equal-time second-order correlation function, we can obtain the optimal UPB that occurs at

(25)$$\Delta_{l}=\frac{2E_{d}^{2}\cos\theta}{G}+\frac{g^{2}}{\omega_{m}},\,\,\,\mathrm{with}\,\,\, \theta=\arcsin\left(\frac{G\kappa}{4E_{d}^{2}}\right).$$

Different from the CPB in Sec. 3.1, the location of optimal PB coming from the unconventional blockade mechanism does not depend on the single excitation resonance condition given by Eq. (17), and it can be modulated via changing the second-order nonlinear parametric process. It is worth noting that the nonlinear strength should satisfy the relationship $G\leqslant 4E_{d}^{2}/\kappa$ in the unconventional blockade mechanism, e.g., $G\leqslant 0.01\kappa$ for a weak driving amplitude $E_{d}=0.05\kappa$, which is much smaller than it in the conventional blockade mechanism. That is because the required nonlinear strength in the CPB is large enough to enhance the effective optomechanical interaction, further producing a noticeable anharmonic eigenenergy spectrum. However, the nonlinear parametric process in UPB only needs to satisfy the condition of destructive quantum interference, which is on the order of input weak driving field and easier to implement in experiments.

On the other hand, the photon statistics property in the counterclockwise mode is invariant because it is not affected by the second-order nonlinear parametric process in the clockwise mode. So the PB behavior for the backward driving case is still described by Eq. (18). Furthermore, the nonreciprocal PB phenomena in both reversed propagating modes are characterized by calculating their equal-time second-order correlation functions, as shown in Fig. 4(a). For the forward driving case, we find that the correlation function has two dips located at $\Delta _{l}=3g^{2}/(2\omega _{m})-\sqrt {g^4/\omega _{m}^{2}+\kappa ^{2}}/2$ and $\Delta _{l}=2E_{d}^2\cos \theta /G+g^2/\omega _{m}$, which respectively occur an insignificant CPB and an ideal UPB. However, for the backward driving case, the PB phenomenon is also insignificant due to the too weak single-photon optomechanical coupling. As the optomechanical coupling strength increases, the PB phenomenon gradually changes obviously and occurs at $\Delta _{r}=g^{2}/\omega _{m}$ as the previous reports [32,33]. At this time, the nonreciprocal PB is obtained. Unlike the above discuss in Sec. 3.1, the nonreciprocity comes from the shifted frequency and enhanced coupling by the quantum squeezing transformation. Here, the nonreciprocity of photon statistics arises from different blockade mechanisms in the reversed propagating modes.

Fig. 4. (a) The equal-time second-order correlation functions versus the driving field detuning based on the unconventional blockade mechanism. The blue and red solid lines represent the analytical solutions of the two driving cases. The blue and red dashed lines come from the numerical simulation by solving the quantum master equation, respectively. (b) The nonreciprocal ratio of PB versus the driving field detuning and nonlinear strength. Here, the nonlinear strength of second-order parametric process is set as $G=10^{-3}\kappa$ and the other parameters are the same as in Fig. 2.

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The nonreciprocal ratio of PB in the two reversed propagating modes is given in Fig. 4(b). We can see that although the nonlinear strength $G$ is weak, the resulting nonreciprocal ratio is larger $\eta _{\mathrm {max}}>30\,\mathrm {dB}$ than it caused by the quantum squeezing transformation. Such as, the maximal ratio $\eta _{\mathrm {max}}=31\,\mathrm {dB}$ when the nonlinear strength $G=10^{-3}\kappa$. Moreover, the optimal nonlinear strength appearing maximal ratio satisfies the inverse proportional function relationship $G=2E_{d}^{2}\omega _{m}\cos \theta /(\Delta _{l,r}\omega _{m}-g^{2})$ in Fig. 4(b).

The dynamical evolution of equal-time second-order correlation functions in both propagating modes is shown in Fig. 5, where the two subfigures correspond to the dynamical evolution of correlation functions located at $\Delta _{l,r}=3g^{2}/(2\omega _{m})-\sqrt {g^{4}/\omega _{m}^{2}+\kappa ^{2}}/2$ and $\Delta _{l,r}=2E_{d}^{2}\cos \theta /G+g^{2}/\omega _{m}$, respectively. In Fig. 5(a), the two correlation functions in both propagating modes are weakly less than $1$ after a period of evolution, which represents the insignificant PB effect occurring in the two driving cases. The reason for the insignificant PB effect is that the nonlinear single-photon optomechanical coupling strength is too small in our proposal. However, an ideal PB $g^{(2)}(0)\sim 10^{-3}$ appears at $\Delta _{l,r}=2E_{d}^{2}\cos \theta /G+g^{2}/\omega _{m}$ even with the same optomechanical coupling for the forward driving case. The dynamical evolution of the correlation function in the clockwise mode reaches $10^{-3}$ over time, consistent with the above results in Fig. 4. Therefore, an ideal PB happens in the clockwise mode based on the UPB mechanism, where the effect of PB is better than it is in Fig. 3.

Fig. 5. The dynamical evolution of equal-time second-order correlation functions versus the scaled time $\kappa t$. The blue solid and red dashed lines respectively represent the dynamical evolution of correlation functions for the forward and backward driving cases. Here, the system parameters are the same as in Fig. 4.

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Compared with the nonreciprocal PB induced by the quantum squeezing transformation, the required nonlinear strength is much smaller based on destructive quantum interference. This conclusion can be clearly seen in Fig. 6, which shows the nonreciprocal ratio of PB changing with the strength of the nonlinear parametric process. We can see that the required nonlinear strength for the NUPB is roughly in the range of $10^{-3}\,\kappa \sim 10^{-2}\,\kappa$. However, it is required to be greater than $10\,\kappa$ to achieve an obvious NCPB. For example, in order to obtain a large nonreciprocal ratio $\eta >30\,\mathrm {dB}$, the required nonlinear strength should be over $300\,\kappa$, which is too large for current experiments. On the other hand, the nonreciprocal ratio of NUPB gradually decreases as the nonlinear strength increases, which is the exact opposite of the NCPB. It is worth noting that the nonlinear strength is not as small as possible. That is because the photon blockade effect will be far away from the cavity resonance. Here, we roughly estimate the minimum of the nonlinear strength according to the principle of $\Delta _{l}\leqslant \kappa$, and give the range of nonlinear strength $4E_{d}^{2}/\sqrt {4g^{4}/\omega _{m}^{2}-8\kappa g^{2}/\omega _{m}+5\kappa ^2}\leqslant G\leqslant 4E_{d}^{2}/\kappa$. Moreover, the broadband squeezing vacuum field is no longer needed in the unconventional blockade mechanism. So it is easier to implement experimentally. In the scheme based on the quantum squeezing transformation, the relative phase needs to satisfy the phase-matching condition $\theta -\theta ^{\prime }=(2n+1)\pi$ to eliminate the squeezing-induced thermalization. However, the phase in unconventional mechanism is $\theta =\arcsin [G\kappa /(4E_{d}^{2})]$, which is given in Eq. (25).

Fig. 6. The nonreciprocal ratio of PB versus the nonlinear strength for the two different mechanisms. The yellow region corresponds to the NUPB with a weak nonlinear parametric process. The magenta region is the NCPB with a strong nonlinear parametric process.

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4. Conclusions

In conclusion, we have discussed the nonreciprocal photon blockade based on the nonlinear parametric process in a WGM optomechanical microresonator system, where two completely different mechanisms are respectively given to achieve the nonreciprocal PB. First, the squeezing-based mechanism is clarified with a strong nonlinear parametric process. The frequency shift and enhanced optomechanical interaction caused by the Bogoliubov squeezing transformation cause the generation of nonreciprocity in the two reversed propagating modes. When the phase-matching condition is satisfied, the squeezing-induced thermalization noise can be eliminated via an extra broadband squeezed vacuum field. Second, we give another untraditional mechanism based on quantum interference, which can result in a larger nonreciprocal ratio even with a weak nonlinear parametric process. Here, the nonlinear parametric process is not used to generate squeezing but as a two-excitation path to constructing the destructive quantum interference, where the needed nonlinear strength is much smaller than that in the squeezing-based conventional physical mechanism. Moreover, we compare the nonreciprocal ratio of the two mechanisms in different regions of nonlinear strength, which gives an alternative scheme for different experimental conditions. Our proposal explores the feasibility of nonreciprocal PB occurring even with weak single-photon optomechanical coupling. It might have potential application in generating the single-photon isolator and nonreciprocal quantum information processing.

Funding

National Natural Science Foundation of China (12147149, 12204424, 12274376, 12074346, U21A20434); China Postdoctoral Science Foundation (2022M722889); Natural Science Foundation of Henan Province (212300410085).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Squeezing-induced nonreciprocal photon blockade in an optomechanical microresonator

Abstract

1. Introduction

2. System and Hamiltonian

2.1 Effective Hamiltonian for nonreciprocal conventional photon blockade

2.2 Effective Hamiltonian for nonreciprocal unconventional photon blockade

3. Nonreciprocal photon blockade

3.1 Nonreciprocal conventional photon blockade

3.2 Nonreciprocal unconventional photon blockade

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (25)

Optics Express