Quantum nonreciprocality in quadratic optomechanics

Xunwei Xu; Yanjun Zhao; Hui Wang; Hui Jing; Aixi Chen

doi:10.1364/PRJ.8.000143

Abstract

We propose to achieve nonreciprocal quantum control of photons in a quadratic optomechanical (QOM) system based on directional nonlinear interactions. We show that by optically pumping the QOM system in one side, the effective QOM coupling can be enhanced significantly in that side, but not for the other side. This, contrary to the intuitive picture, allows the emergence of a nonreciprocal photon blockade in such optomechanical devices with weak single-photon QOM coupling. Our proposal opens up the prospect of exploring and utilizing quantum nonreciprocal optomechanics, with applications ranging from single-photon nonreciprocal devices to on-chip chiral quantum engineering.

1. INTRODUCTION

Optomechanical systems can be used to test the foundations of physics, and have important applications ranging from gravitational wave detections to quantum information processing (for reviews, see Refs. [1–6]). Recently, optomechanically induced nonreciprocity has been proposed theoretically [7–18] and demonstrated experimentally [19–28]. Nonreciprocal devices [29], such as isolators and circulators, have drawn an immense amount of interest in the past few years, for their irreplaceable role in signal processing and communication. An optomechanical system opens a route for nonreciprocity with nonlinear interactions [7,8,19,20], inherent non-trivial topology [9,21–24], and synthetic magnetism [10–18,25–28].

In this paper, we show that nonreciprocity can be realized in a whispering-gallery-mode (WGM) optomechanical system with quadratic optomechanical (QOM) coupling for quadratic mechanical-position dependent change in the optical frequency. Specifically, by optically pumping the quadratic WGM optomechanical system in one side, the effective QOM coupling can be enhanced significantly in that side, but not for the other side. The directional nonlinear interactions can induce nonreciprocal photon transport. This new possibility, as far as we know, has not been revealed in previous works.

It has been shown that the QOM systems driven by a strong optical driving field can generate strong (second-order) nonlinear photon-phonon interaction, even under weak single-photon optomechanical coupling conditions [30,31]. Nevertheless, we show here that the WGM optomechanical system with QOM coupling can be used to generate directional nonlinear interactions, which opens up the prospect of exploring nonreciprocal photon transport based on nonlinearity without the limitations of dynamic reciprocity [32]. Moreover, we demonstrate that directional nonlinear interactions can not only induce classical nonreciprocity, but also achieve nonreciprocal quantum control of photons by manipulating the statistic of the nonreciprocal transport photons.

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Recently, a quantum nonreciprocal effect called nonreciprocal photon blockade (PB) was predicted theoretically [33] in a spinning Kerr resonator [34–37], that PB can emerge when the resonator is driven in one direction but not the other. Different from the nonreciprocal PB induced by the Fizeau–Sagnac drag [33], which relies on the condition of strong single-photon nonlinearity, here we show that the nonreciprocal PB can also be explored even with weak single-photon QOM coupling. Our proposal provides a feasible method to realize nonreciprocal photon blockade.

We note that the WGM optomechanical system has already been used for nonreciprocity with inherent non-trivial topology [9,21–24]. However, the WGM optomechanical systems with linear mechanical-position dependent change in the optical frequency (i.e., linear optomechanical coupling) can only enhance linear photon-phonon interaction, which cannot be used to observe nonreciprocal PB. Our work provides not only a new platform (quadratic WGM optomechanics) but a new approach (i.e., directional nonlinear interaction) to designing nonreciprocal single-photon devices for backaction-immune quantum communication and chiral quantum photonics.

2. MODEL AND HAMILTONIAN

We note that the cavity frequency of a WGM optomechanical system is almost linearly proportional to the mechanical position [38,39], even though the effects of QOM coupling have also been observed in experiments [40,41]. The QOM coupling can facilitate nondestructive measurements of the energy of the mechanical oscillator [42,43], which is instead impossible for a system with linear optomechanical coupling. We present a scheme to generate QOM interactions in the normal optical modes of a WGM optomechanical system by eliminating the linear optomechanical couplings [44–47].

We consider a near-field cavity optomechanical setup consisting of one mechanical resonator optomechanically coupling to two optical resonators ( $j = 1, 2$ ) via the optical evanescent field, with each optical resonator supporting two degenerate clockwise (CW) and counter-clockwise (CCW) travelling-wave WGMs, as shown in Figs. 1(a) and 1(b). The model can be described by the optomechanical interaction Hamiltonian $H_{om} = \sum_{j = 1, 2} \sum_{λ = cw, ccw} [ω_{0} + {(- 1)}^{j} g_{0} q] a_{j, λ}^{†} a_{j, λ} + J (a_{1, ccw} a_{2, cw}^{†} + a_{1, cw} a_{2, ccw}^{†} + H.c.) + \frac{1}{2} ω_{m} (q^{2} + p^{2}),$ (1)where $a_{j, λ}$ and $a_{j, λ}^{†}$ ( $j = 1, 2$ and $λ = cw, ccw$ ) are the annihilation and creation operators of the optical modes with frequency $ω_{0}$ , and $q$ and $p$ are the dimensionless displacement and momentum operators of the mechanical resonator with frequency $ω_{m}$ . As shown in Figs. 1(a) and 1(a), the mechanical resonator is placed at the middle of the gap of the two WGM resonators, so that the mechanical mode couples to the two WGM optical modes with linear optomechanical coupling of equal magnitude $g_{0}$ but opposite signs. Due to the tunneling coupling $J$ between the optical modes, there is an avoided crossing for the optical resonance frequency, $ω_{\pm} (q) = ω_{0} \pm \sqrt{J^{2} + {(g_{0} q)}^{2}}$ , as shown in Figs. 1(c) and 1(d). We assume that the optical mode $a_{j, λ}$ is coupled to a waveguide (port $j$ ) with strength $γ_{c}$ , and the damping rate of mechanical resonator $q$ is $γ_{m}$ . This theoretical model may also be realized with a membrane suspended on WGM microcavities [48] or photonic-crystal microcavities [49].

(a), (b) Schematic diagram for generating QOM coupling, where a mechanical nanostring oscillator is placed between two whispering gallery mode (WGM) resonators. (c), (d) Dispersion of the optical modes as a function of the displacement.

Figure 1.(a), (b) Schematic diagram for generating QOM coupling, where a mechanical nanostring oscillator is placed between two whispering gallery mode (WGM) resonators. (c), (d) Dispersion of the optical modes as a function of the displacement.

Following the approach in Refs. [44–47] (see Appendix A for more details), where $| J | ≫ ω_{m}$ is assumed such that $q$ can be treated as a quasi-static variable, the Hamiltonian can be diagonalized quasi-statically as $H_{om} \approx (ω_{+} + g q^{2}) (a_{L, +}^{†} a_{L, +} + a_{R, +}^{†} a_{R, +}) + (ω_{-} - g q^{2}) (a_{L, -}^{†} a_{L, -} + a_{R, -}^{†} a_{R, -}) + \frac{1}{2} ω_{m} (q^{2} + p^{2})$ (2)for normal modes $a_{R, \pm} \approx (a_{1, cw} \pm a_{2, ccw}) / \sqrt{2}$ and $a_{L, \pm} \approx (a_{1, ccw} \pm a_{2, cw}) / \sqrt{2}$ with eigenfrequencies $ω_{\pm} (q) \approx ω_{0} \pm (J + g q^{2})$ and single-photon QOM coupling $g \equiv g_{0}^{2} / (2 J)$ .

As already shown in the experiment [47], when the tunneling coupling between the optical modes is strong, i.e., $J > γ_{c}$ , the transmission spectrum of a laser probe through the coupled optical modes features resonance dips at the normal resonance frequencies $ω_{\pm} = ω_{0} \pm J$ , not at the bare optical resonance frequencies $ω_{0}$ . That is to say, the normal modes are coupled to the external waveguides and can be used to describe the input–output characteristic of the coupled optical modes system. Specifically, the normal modes $a_{L, \pm}$ and $a_{R, \pm}$ are coupled to both of the ports with strength $γ_{c} / 2$ , respectively, and the total damping rates are $γ_{c}$ .

We can choose either pair of degenerated quasi-static normal optical modes ( $a_{L, +}$ and $a_{R, +}$ , or $a_{L, -}$ and $a_{R, -}$ ), to generate strong nonlinear interaction for a few photons travelling in one direction, but not in the reverse direction. Without loss of generality, the two optical modes are denoted as $a_{L}$ and $a_{R}$ , with frequency $ω_{a} = ω_{+}$ or $ω_{-}$ , for photons travelling from port 1 to port 2 and the opposite direction, respectively. For example, to enhance the nonlinear optomechanical coupling between the optical mode $a_{L}$ and the mechanical resonator $b$ , the optical mode $a_{L}$ is pumped by a strong field with amplitude $Ω$ and frequency $ω_{L} \sim ω_{a} - 2 ω_{m}$ . In the rotating reference frame with the optical frequency $ω_{L}$ , the system can be described by $H_{om + driv} = (Δ_{a} + g q^{2}) (a_{L}^{†} a_{L} + a_{R}^{†} a_{R}) + \frac{1}{2} ω_{m} (q^{2} + p^{2}) + Ω a_{L}^{†} + Ω a_{L},$ (3)with detuning $Δ_{a} = ω_{a} - ω_{L}$ . Under the strong driving condition $Ω ≫ γ_{c}$ , we perform the displacement transformations: $a_{L} \to α_{L} + a_{L}$ , $a_{R} \to α_{R} + a_{R}$ , $q \to q_{s} + q$ , and $p \to p_{s} + p$ , where $α_{L}$ , $α_{R}$ , $q_{s}$ , and $p_{s}$ are the steady state values, and $a_{L}$ , $a_{R}$ , $q$ , and $p$ (on the right side of the arrow) are the quantum fluctuation operators. Adopting the standard linearization technique (see Appendix B), an effective Hamiltonian [30,31] for the quantum fluctuation operators is obtained as $H_{eff}^{'} = Δ_{a} (a_{L}^{†} a_{L} + a_{R}^{†} a_{R}) + ω_{m}^{'} b^{†} b + G a_{L}^{†} b^{2} + G^{*} a_{L} b^{† 2},$ (4)where $ω_{m}^{'} = ω_{m} + g {| α_{L} |}^{2}$ is the effective mechanical frequency, and $G = g α_{L} / 2$ is the effective (second-order) nonlinear coupling strength between the optical and mechanical modes. Without loss of generality, $G$ is assumed to be real in the following.

To investigate the system’s response behavior to a weak probe field, a weak field with amplitude $ε ≪ γ_{c}$ and frequency $ω_{p} \approx ω_{a}$ is input from one of the ports. In the rotating reference frame with the unitary operator $R^{'} (t) = \exp [i δ (a_{L}^{†} a_{L} + a_{R}^{†} a_{R} + b^{†} b / 2) t]$ , the total Hamiltonian is given by $H_{tot} = H_{eff} + H_{probe}$ , where $H_{eff} = Δ a_{L}^{†} a_{L} + Δ a_{R}^{†} a_{R} + Δ_{m} b^{†} b + G a_{L}^{†} b^{2} + G^{*} a_{L} b^{† 2},$ (5)with detunings $δ = ω_{p} - ω_{L}$ , $Δ = Δ_{a} - δ$ , and $Δ_{m} = ω_{m} + g {| α_{L} |}^{2} - δ / 2$ satisfying the condition $\max {| Δ |, | Δ_{m} |} ≪ ω_{m}$ . $H_{probe} = ε a_{ξ}^{†} + H.c.$ describes the driving of the probe field with subscript $ξ = L$ for weak field input from port 1 and $ξ = R$ for weak field input from port 2. The effective Hamiltonian $H_{eff}$ describes a directional nonlinear interaction between the optical and mechanical modes. If a probe field is input from port 1, the enhanced nonlinear interaction $G$ between $a_{L}$ and $b$ may suppress the excitation of the second photon under a single-photon resonant condition. In contrast, by driving from port 2, as there is no strong nonlinear interaction between $a_{R}$ and $b$ , the photons are free to cross the system.

According to the input–output relations [50], we have $a_{1, in} = ε / \sqrt{γ_{c} / 2}$ and $a_{2, out} = \sqrt{γ_{c} / 2} a_{L}$ ( $a_{2, in} = ε / \sqrt{γ_{c} / 2}$ and $a_{1, out} = \sqrt{γ_{c} / 2} a_{R}$ ), and then the transmission coefficient for the weak probe field can be defined by $T_{21} \equiv \frac{⟨ a_{2, out}^{†} a_{2, out} ⟩}{⟨ a_{1, in}^{†} a_{1, in} ⟩} = \frac{γ_{c}^{2}}{4 ε^{2}} ⟨ a_{L}^{†} a_{L} ⟩,$ (6)for photon transport from port 1 to port 2, and $T_{12} \equiv \frac{⟨ a_{1, out}^{†} a_{1, out} ⟩}{⟨ a_{2, in}^{†} a_{2, in} ⟩} = \frac{γ_{c}^{2}}{4 ε^{2}} ⟨ a_{R}^{†} a_{R} ⟩,$ (7)for photon transport from port 2 to port 1, where $n_{L} \equiv ⟨ a_{L}^{†} a_{L} ⟩$ and $n_{R} \equiv ⟨ a_{R}^{†} a_{R} ⟩$ are the mean photon numbers. The isolation for probe field transport from port 1 to port 2 is defined by $I \equiv T_{21} / T_{12}$ . Using the input–output relations, the statistic properties of the transmitted photons $a_{2, out}$ and $a_{1, out}$ can be described by the second-order correlation functions in the steady state ( $t \to \infty$ ) as follows: $g_{21}^{(2)} (τ) \equiv \frac{⟨ a_{2, out}^{†} (t) a_{2, out}^{†} (t + τ) a_{2, out} (t + τ) a_{2, out} (t) ⟩}{{⟨ a_{2, out}^{†} (t) a_{2, out} (t) ⟩}^{2}} = \frac{⟨ a_{L}^{†} (t) a_{L}^{†} (t + τ) a_{L} (t + τ) a_{L} (t) ⟩}{{⟨ a_{L}^{†} (t) a_{L} (t) ⟩}^{2}},$ (8)for photon transport from port 1 to port 2, and $g_{12}^{(2)} (τ) \equiv \frac{⟨ a_{1, out}^{†} (t) a_{1, out}^{†} (t + τ) a_{1, out} (t + τ) a_{1, out} (t) ⟩}{{⟨ a_{1, out}^{†} (t) a_{1, out} (t) ⟩}^{2}} = \frac{⟨ a_{R}^{†} (t) a_{R}^{†} (t + τ) a_{R} (t + τ) a_{R} (t) ⟩}{{⟨ a_{R}^{†} (t) a_{R} (t) ⟩}^{2}},$ (9)for photon transport from port 2 to port 1.

3. NONRECIPROCAL PB

The transmission coefficients and correlation functions can be obtained by numerically solving the master equation for the density matrix $ρ$ of the system [51] $\frac{\partial ρ}{\partial t} = - i [H_{tot}, ρ] + γ_{c} L [a_{L}] ρ + γ_{c} L [a_{R}] ρ + γ_{m} (n_{th} + 1) L [b] ρ + γ_{m} n_{th} L [b^{†}] ρ,$ (10)where $L [o] ρ = o ρ o^{†} - (o^{†} o ρ + ρ o^{†} o) / 2$ denotes a Lindbland term for an operator $o$ , and $n_{th}$ is the mean thermal phonon number, given by the Bose–Einstein statistics $n_{th} = {[\exp (ℏ ω_{m} / k_{B} T) - 1]}^{- 1}$ with the Boltzmann constant $k_{B}$ and the temperature $T$ of the reservoir at the thermal equilibrium.

In Fig. 2(a), we show the transmission coefficients $T_{21}$ for probe field transport from port 1 to port 2, and $T_{12}$ for probe field transport from port 2 to port 1 versus the detuning $Δ / G$ . For $T_{21}$ , there are two peaks at $Δ = \pm \sqrt{2} G$ and one dip at $Δ = 0$ . In contrast, there is only one peak at $Δ = 0$ for $T_{12}$ . Figure 2(b) shows the isolation $I = T_{21} / T_{12}$ for probe field transport from port 1 to port 2 as a function of the detuning $Δ / G$ . The isolation for the direction $1 \to 2$ is more than 17 dB at detuning $Δ = \pm \sqrt{2} G$ , and the isolation for the reverse direction $2 \to 1$ , i.e., $\tilde{I} \equiv T_{12} / T_{21}$ , is more than 70 dB at detuning $Δ = 0$ .

Figure 2.(a) The transmission coefficients $T_{21}$ (solid black curve) and $T_{12}$ (dashed red curve) as a function of the detuning $Δ / G$ . (b) The isolation as a function of the detuning $Δ / G$ . (c) The equal-time second-order correlation function $\log_{10} [g_{i j}^{(2)} (0)]$ ( $i j = 12, 21$ ) as a function of the detuning $Δ / G$ . (d) The second-order correlation function $\log_{10} [g_{21}^{(2)} (τ)]$ as a function of the normalized time delay $γ_{c} τ / (2 π)$ at detuning $Δ = \sqrt{2} G$ . The other parameters are $Δ_{m} = Δ / 2$ , $G = 3 γ_{c}$ , $ε = γ_{c} / 10$ , $γ_{m} = γ_{c} / 100$ , and $n_{th} = 0$ .

To explore the statistic properties of the transmitted photons, the equal-time second-order correlation function $\log_{10} [g_{i j}^{(2)} (0)]$ ( $i j = 12, 21$ ) is shown as a function of the detuning $Δ / G$ in Fig. 2(c). The photon transport from port 2 to port 1 is coherent in full frequency, i.e., $g_{12}^{(2)} (0) = 1$ . The photon transport from port 1 to port 2 exhibits strong antibunching effect, i.e., $g_{21}^{(2)} (0) ≪ 1$ , around the detunings $Δ = \pm \sqrt{2} G$ , and exhibits strong bunching effect, i.e., $g_{21}^{(2)} (0) ≫ 1$ , around the detunings $Δ = 0$ and $\pm 2 G$ . The time duration for nonreciprocal PB at $Δ = \pm \sqrt{2} G$ is on the order of $2 π / (10 γ_{c})$ , as shown in Fig. 2(d).

The peak for $T_{12} \approx 1$ at detuning $Δ = 0$ can be understood by the fact that when the probe field is injected from port 2, only the (linear) optical mode $a_{R}$ can be excited. Thus the maximum transmission coefficient is reached for the probe field in resonance with the optical mode $a_{R}$ , i.e., $Δ = 0$ , and the transmitted photons keep the statistic properties of the probe field (a coherent field), i.e., $g_{12}^{(2)} (0) = 1$ , as there is no nonlinear interaction in the optical path from port 2 to port 1.

In order to understand the origin of the peak for $T_{21} \approx 0.8$ around the detuning $Δ = \pm \sqrt{2} G$ and the dip for $T_{21} \approx 10^{- 7}$ at $Δ = 0$ , we show energy spectrum (see Appendix C) of the linearized QOM coupling between optical mode $a_{L}$ and mechanical resonator $b$ in Fig. 3. Under weak probe condition, the maximum transmission coefficient $T_{21} \approx 0.8$ is reached for the probe field in resonance with the transition $| 0_{0} ⟩ \to | 2_{\pm 1} ⟩$ , i.e., $Δ = \pm \sqrt{2} G$ , as shown in Fig. 3(a). However, the photon absorbed in the transition $| 0_{0} ⟩ \to | 2_{\pm 1} ⟩$ blocks the transition $| 2_{\pm 1} ⟩ \to | 4_{\pm 1} ⟩$ for large detuning, so we have $g_{12}^{(2)} (0) ≪ 1$ around the detuning $Δ = \pm \sqrt{2} G$ . The dip for $T_{21} \approx 10^{- 7}$ at $Δ = 0$ arises from the quantum interference between the transitions $| 2_{+ 1} ⟩ \to | 0_{0} ⟩$ and $| 2_{- 1} ⟩ \to | 0_{0} ⟩$ [Fig. 3(b)], in an equivalent picture as optomechanically induced transparency [52–54], or electromagnetically induced transparency in lambda-type three-level atoms [55,56]. Moreover, when $Δ = 0$ , the transition $| 0_{0} ⟩ \to | 2_{\pm 1} ⟩$ is suppressed, but the two-photon transition $| 0_{0} ⟩ \to | 4_{0} ⟩$ is resonant, which induces two-photon tunneling from port 1 to point 2, i.e., $g_{21}^{(2)} (0) ≫ 1$ . Similarly, $g_{21}^{(2)} (0) ≫ 1$ around $Δ = \pm 2 G$ is induced by the resonant transition $| 0_{0} ⟩ \to | 4_{\pm} ⟩$ .

Figure 3.Schematic energy spectrum of the linearized QOM coupling between optical mode $a_{L}$ and mechanical resonator $b$ , where $| 0_{0} ⟩ \equiv | 0, 0 ⟩$ , $| 1_{0} ⟩ \equiv | 0, 1 ⟩$ , $| 2_{\pm 1} ⟩ \equiv (| 1, 0 ⟩ \pm | 0, 2 ⟩) / \sqrt{2}$ , $| 3_{\pm 1} ⟩ \equiv (| 1, 1 ⟩ \pm | 0, 3 ⟩) / \sqrt{2}$ , $| 4_{0} ⟩ \equiv (- \sqrt{3} | 2, 0 ⟩ + | 0, 4 ⟩) / 2$ , $| 4_{\pm 1} ⟩ \equiv (| 2, 0 ⟩ \pm 2 | 1, 2 ⟩ + \sqrt{3} | 0, 4 ⟩) / (2 \sqrt{2})$ , and $| n, m ⟩$ represents the Fock state with $n$ photons in $a_{L}$ and $m$ phonons in $b$ .

We now discuss the effect of thermal phonons on the nonreciprocal PB. Figures 4(a) and 4(b) show the transmission coefficient $T_{21}$ and the equal-time second-order correlation function $\log_{10} [g_{21}^{(2)} (0)]$ versus the detuning $Δ / G$ with a different mean thermal phonon number ( $n_{th} = 0, 0.1, 1$ ). Thermal phonons have little influence on the transmission coefficient $T_{21}$ around $Δ = \pm \sqrt{2} G$ , but have great effect on the transmission coefficient $T_{21}$ around $Δ = 0$ . Thermal phonons have great effect on the second-order correlation function $\log_{10} [g_{21}^{(2)} (0)]$ around $Δ = 0$ and $\pm \sqrt{2} G$ .

Figure 4.(a) Transmission coefficient $T_{21}$ . (b) The equal-time second-order correlation function $\log_{10} [g_{21}^{(2)} (0)]$ versus the detuning $Δ / G$ with different mean thermal phonon number ( $n_{th} = 0, 0.1, 1$ ). (c) The isolation $T_{21} / T_{12}$ . (d) The equal-time second-order correlation function $\log_{10} [g_{21}^{(2)} (0)]$ versus the mean thermal phonon number $n_{th}$ with different detuning ( $Δ = 0, \sqrt{2} G, \sqrt{6} G$ ). The other parameters are the same as in Fig. 2.

The isolation $T_{21} / T_{12}$ and the second-order correlation function $\log_{10} [g_{21}^{(2)} (0)]$ versus the mean thermal phonon number $n_{th}$ are shown in Figs. 4(c) and 4(d) with detuning $Δ = 0, \sqrt{2} G, \sqrt{6} G$ . The isolation $T_{21} / T_{12}$ around $Δ = \sqrt{2} G$ is robust against the thermal phonons, but the antibunching effect of the transport photons becomes much weaker for greater thermal phonons. Both the isolation $T_{21} / T_{12}$ and second-order correlation function $\log_{10} [g_{21}^{(2)} (0)]$ around $Δ = 0$ are sensitive to the mean thermal phonon number $n_{th}$ , and this quality may be used in accurate temperature measurement at ultra-low temperature.

More interestingly, two peaks appear around $Δ = \pm \sqrt{6} G$ in the transmission coefficient $T_{21}$ , and the isolation $T_{21} / T_{12}$ can be improved with a larger thermal phonon number $n_{th}$ . This abnormal effect is induced by the phonon states, e.g., $| 1_{0} ⟩$ in Fig. 3(c). As the temperature increases, the population probability in phonon state $| 1_{0} ⟩$ increases, and the transitions of $| 1_{0} ⟩ \to | 3_{\pm 1} ⟩$ with resonance frequency $Δ = \pm \sqrt{6} G$ become remarkable gradually, which induces the increasing peaks of the transmission coefficient $T_{21}$ (or the isolation $T_{21} / T_{12}$ ) around $Δ = \pm \sqrt{6} G$ .

4. DISCUSSIONS AND CONCLUSIONS

Let us now discuss the experimental requirements for our proposal. Considering the parameters from a recent near-field WGM optomechanical experiment [39]: $g_{0} \sim 2 π \times (10 - 100) kHz$ , $γ_{c} \sim 2 π \times (100 - 1000) MHz$ , and $J \approx 2 π \times 1000 MHz$ for strong coupling, we have QOM coupling $g / 2 π \sim 10 Hz$ , which is too weak to realize $G = g α_{L} / 2 > γ_{c}$ . However, many schemes have been proposed to enhance the optomechanical coupling $g_{0}$ in recent years, such as by using the collective mechanical modes in scatter arrays [57], the squeezed cavity mode [58], or the Josephson effect for electromechanical systems [59]. The optomechanical coupling $g_{0}$ can be enhanced by several orders of magnitude and the strong coupling regime ( $g_{0} > γ_{c}$ ) can even be approached [57–59]. Furthermore, the frequency range of the SiN nanostring oscillators in the near-field cavity optomechanical setup is about 0.1–10 MHz [38–41], which is too low to observe nonreciprocal PB. One possible solution is replaying the SiN nanostring oscillator with an optomechanical crystal nanobeam, as optomechanical crystal nanobeams with GHz mechanical modes have been designed and demonstrated recently [60,61]. Moreover, the thermal phonons have important influence on the nonreciprocal PB, and the thermal effect can be suppressed for higher frequency mechanical mode.

In addition, there are two driving fields coupling to the system with frequencies $ω_{L} \sim ω_{a} - 2 ω_{m}$ and $ω_{p} \sim ω_{a}$ , and the strengths $Ω ≫ ε$ . In order to detect the photon correlation induced by the weak probe field, we should spectrally filter out the strong optical driving field at frequency $ω_{L}$ under the condition $ω_{a} - ω_{L} ≫ {γ_{c}, G}$ . A similar method has already been used to measure correlations of phonons in a recent experiment [62].

In summary, we have shown that the WGM optomechanical system with QOM coupling can be used to obtain directional nonlinear interactions and observe nonreciprocal PB. In addition, the thermal phonons have important influence on the nonreciprocal PB, especially on the statistic properties of the transport photons, and this quality may be used in temperature sensing at ultra-low temperature. Moreover, this work can be extended to study phonon manipulation in double-cavity optomechanics, e.g., nonreciprocal phonon blockades [63], nonreciprocal phonon lasers [64–70], nonreciprocal photon-phonon entanglements and quantum transfers. Directional nonlinear interaction can also be applied in the high-Q microtoroid resonator with phase-matched parametric amplification [71] or superconducting microwave circuit with Josephson parametric converters [72] to achieve nonreciprocal quantum control of photons. Our proposal provides a new routine towards the realization of on-chip quantum nonreciprocal devices, backaction-immune quantum measurement, and chiral quantum physics.

APPENDIX A: QUADRATIC OPTOMECHANICAL COUPLING

To realize quantum nonreciprocality in optomechanics, we consider a near-field cavity optomechanical setup consisting of one mechanical resonator optomechanical coupling to two optical resonators ( $j = 1, 2$ ) via the optical evanescent field, with each optical resonator supporting two degenerate clockwise (CW) and counter-clockwise (CCW) travelling-wave whispering-galley-modes (WGMs). The model can be described by the optomechanical interaction Hamiltonian $H_{om} = (\begin{matrix} a_{1, ccw}^{} & a_{2, cw}^{} \end{matrix}) (\begin{matrix} ω_{0} g_{0} q & J \\ J & ω_{0} + g_{0} q \end{matrix}) (\begin{matrix} a_{1, ccw} \\ a_{2, cw} \end{matrix}) + (\begin{matrix} a_{1, cw}^{} & a_{2, ccw}^{} \end{matrix}) (\begin{matrix} ω_{0} g_{0} q & J \\ J & ω_{0} + g_{0} q \end{matrix}) (\begin{matrix} a_{1, cw} \\ a_{2, ccw} \end{matrix}) + \frac{1}{2} ω_{m} (q^{2} + p^{2}),$ (A1)where $a_{j, λ}$ and $a_{j, λ}^{}$ ( $j = 1, 2$ and $λ = cw, ccw$ ) are the annihilation and creation operators of the optical modes with frequency $ω_{0}$ , $J$ is the tunneling amplitude between the optical modes, $q$ and $p$ are the dimensionless displacement and momentum operators of the mechanical resonator with frequency $ω_{m}$ , and $g_{0}$ is the linear optomechanical coupling strength between the mechanical resonator and optical modes. We assume that the optical mode $a_{j, λ}$ is coupled to a waveguide (port $j$ ) with strength $γ_{c}$ , and the damping rate of mechanical resonator $q$ is $γ_{m}$ .

The Hamiltonian in Eq. (A1) can be diagonalized as $H_{om} = (\begin{matrix} a_{L, +}^{} & a_{L,}^{} \end{matrix}) (\begin{matrix} ω_{+} (q) & 0 \\ 0 & ω_{} (q) \end{matrix}) (\begin{matrix} a_{L, +} \\ a_{L,} \end{matrix}) + (\begin{matrix} a_{R, +}^{} & a_{R,}^{} \end{matrix}) (\begin{matrix} ω_{+} (q) & 0 \\ 0 & ω_{} (q) \end{matrix}) (\begin{matrix} a_{R, +} \\ a_{R,} \end{matrix}) + \frac{1}{2} ω_{m} (q^{2} + p^{2}),$ (A2)in the normal modes basis, $a_{L, \pm} = \frac{1}{D_{\pm}} [J a_{1, ccw} + (g_{0} q \pm \sqrt{J^{2} + {(g_{0} q)}^{2}}) a_{2, cw}]$ (A3)and $a_{R, \pm} = \frac{1}{D_{\pm}} [J a_{1, cw} + (g_{0} q \pm \sqrt{J^{2} + {(g_{0} q)}^{2}}) a_{2, ccw}],$ (A4)with $D_{\pm}^{2} = J^{2} + {(g_{0} q \pm \sqrt{J^{2} + {(g_{0} q)}^{2}})}^{2},$ (A5)and eigenfrequencies $ω_{\pm} (q) = ω_{0} \pm \sqrt{J^{2} + {(g_{0} q)}^{2}} .$ (A6)

Moreover, $| J | g_{0} q$ is assumed [44–47] such that we can Taylor-expand the eigenfrequencies in Eq. (A6) as $ω_{\pm} (q) \approx ω_{\pm} \pm \frac{g_{0}^{2}}{2 J} q^{2},$ (A7)with frequencies $ω_{\pm} \equiv ω_{\pm} (0) = ω_{0} \pm J$ , and the quasi-static Hamiltonian in Eq. (A2) is given approximately by Eq. (2) in the main text with quadratic optomechanical coupling $g \equiv g_{0}^{2} / (2 J)$ , and quasi-static normal optical modes, $a_{L, \pm} \approx (a_{1, ccw} \pm a_{2, cw}) / \sqrt{2}$ and $a_{R, \pm} \approx (a_{1, cw} \pm a_{2, ccw}) / \sqrt{2}$ .

APPENDIX B: LINEARIZED OPTOMECHANICAL HAMILTONIAN

We consider a pair of degenerated quasi-static normal optical modes ( $a_{L, +}$ and $a_{R, +}$ , or $a_{L,}$ and $a_{R,}$ ), to generate strong nonlinear interaction for a few photons travelling in one direction, but not in the reverse direction. Without loss of generality, the two optical modes are denoted as $a_{L}$ and $a_{R}$ , with frequency $ω_{a} = ω_{+}$ or $ω_{}$ , for photons travelling from port 1 to port 2 and the opposite direction, respectively. For example, to enhance the nonlinear optomechanical coupling between the optical mode $a_{L}$ and the mechanical resonator, the optical mode $a_{L}$ is pumped by a strong field with amplitude $Ω$ and frequency $ω_{L} ～ ω_{a} 2 ω_{m}$ . In the rotating reference frame with the optical frequency $ω_{L}$ , the system can be described by Eq. (3) in the main text with detuning $Δ_{a} = ω_{a} ω_{L}$ .

Under strong driving condition $Ω γ_{c}$ , we perform the displacement transformations: $a_{L} \to α_{L} + a_{L}$ , $a_{R} \to α_{R} + a_{R}$ , $q \to q_{s} + q$ , and $p \to p_{s} + p$ , where $α_{L}$ , $α_{R}$ , $q_{s}$ , and $p_{s}$ are the steady state values, and $a_{L}$ , $a_{R}$ , $q$ , and $p$ (on the right side of the arrow) are the quantum fluctuation operators. The steady state values $α_{L}$ , $α_{R}$ , $q_{s}$ , and $p_{s}$ can be obtained by the equations of motions yielding $α_{L} = \frac{i 2 Ω}{γ_{c} + i 2 Δ_{a}}$ (B1)and $α_{R} = q_{s} = p_{s} = 0 .$ (B2)

The operators $q$ and $p$ for the mechanical resonator can be written in terms of phonon creation and annihilation operators as $q = (b^{} + b) / \sqrt{2}$ , $p = i (b^{} b) / \sqrt{2}$ , and the effective Hamiltonian for the quantum fluctuation operators reads $H_{eff}^{'} = Δ_{a} a_{L}^{} a_{L} + Δ_{a} a_{R}^{} a_{R} + ω_{m} b^{} b + \frac{g}{2} ({| α_{L} |}^{2} + a_{L}^{} a_{L} + a_{R}^{} a_{R}) {(b^{} + b)}^{2} + \frac{g}{2} (α_{L} a_{L}^{} + α_{L}^{*} a_{L}) {(b^{} + b)}^{2} .$ (B3)

We assume that the optical driving field is strong enough so we have ${| α_{L} |}^{2} a_{L}^{} a_{L} ～ a_{R}^{} a_{R}$ , and so that the term $g (a_{L}^{} a_{L} + a_{R}^{} a_{R}) {(b^{} + b)}^{2}$ in the above equation can be neglected safely. For $Δ_{a} ～ 2 ω_{m} | g α_{L} | / 2$ , the effective Hamiltonian can be further simplified by rotating-wave approximation and neglecting the high frequency terms [30,31], e.g., $b^{2}$ and $a b^{2}$ . Then, we get Eq. (4) in the main text, where $ω_{m}^{'} = ω_{m} + g {| α_{L} |}^{2}$ is the effective mechanical frequency, and $G = g α_{L} / 2$ is the effective nonlinear coupling strength between the optical and mechanical modes.

APPENDIX C: ENERGY SPECTRUM

In this appendix, we will show the energy spectrum of the effective Hamiltonian in the diagonal basis under the condition $Δ = 2 Δ_{m}$ , and similar results are shown in Refs. [30,31]. In the noncoupling basis, $| n, m$ represents the Fock state with $n$ photons in $a_{L}$ and $m$ phonons in $b$ . It is clear that the vacuum state $| 0_{0} \equiv | 0, 0$ (C1)and the single-phonon state $| 1_{0} \equiv | 0, 1,$ (C2)are decoupled from the other states. The degenerate states $| 1, 0$ and $| 0, 2$ are coupled with strengh $\sqrt{2} G$ , so the eigenstates are $| 2_{\pm 1} \equiv \frac{1}{\sqrt{2}} (| 1, 0 \pm | 0, 2),$ (C3)with energy splitting $Δ_{2, \pm} = \pm \sqrt{2} G .$ (C4)

The degenerate states $| 1, 1$ and $| 0, 3$ are coupled with strength $\sqrt{6} G$ , so the eigenstates are $| 3_{\pm 1} \equiv \frac{1}{\sqrt{2}} (| 1, 1 \pm | 0, 3),$ (C5)with energy splitting $Δ_{3, \pm} = \pm \sqrt{6} G .$ (C6)

The states $| 2, 0$ , $| 1, 2$ , and $| 0, 4$ are degenerate, where $| 2, 0$ and $| 1, 2$ are coupled with strength $2 G$ , and $| 0, 4$ and $| 1, 2$ are coupled with strength $2 \sqrt{3} G$ , so the eigenstates are $| 4_{0} \equiv \frac{1}{2} (\sqrt{3} | 2, 0 + | 0, 4),$ (C7) $| 4_{\pm 1} \equiv \frac{1}{2 \sqrt{2}} (| 2, 0 \pm 2 | 1, 2 + \sqrt{3} | 0, 4),$ (C8)with energy splitting $Δ_{4, 0} = 0,$ (C9) $Δ_{4, \pm} = \pm 4 G .$ (C10)

The schematic energy spectrum in the diagonal basis is given in Fig. 3.

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