### Quantum nonreciprocality in quadratic optomechanics Download： 597次

## 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

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.

Recently, a quantum nonreciprocal effect called nonreciprocal photon blockade (PB) was predicted theoretically [33] in a spinning Kerr resonator [34–

We note that the WGM optomechanical system has already been used for nonreciprocity with inherent non-trivial topology [9,21–

## 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–

We consider a near-field cavity optomechanical setup consisting of one mechanical resonator optomechanically coupling to two optical resonators (

#### Fig. 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–

As already shown in the experiment [47], when the tunneling coupling between the optical modes is strong, i.e.,

We can choose either pair of degenerated quasi-static normal optical modes (

To investigate the system’s response behavior to a weak probe field, a weak field with amplitude

According to the input–output relations [50], we have

## 3. NONRECIPROCAL PB

The transmission coefficients and correlation functions can be obtained by numerically solving the master equation for the density matrix

In Fig.

#### Fig. 2. (a) The transmission coefficients ${T}_{21}$ (solid black curve) and ${T}_{12}$ (dashed red curve) as a function of the detuning $\mathrm{\Delta}/G$ . (b) The isolation as a function of the detuning $\mathrm{\Delta}/G$ . (c) The equal-time second-order correlation function ${\mathrm{log}}_{10}[{g}_{ij}^{(2)}(0)]$ ($ij=12,21$ ) as a function of the detuning $\mathrm{\Delta}/G$ . (d) The second-order correlation function ${\mathrm{log}}_{10}[{g}_{21}^{(2)}(\tau )]$ as a function of the normalized time delay ${\gamma}_{c}\tau /(2\pi )$ at detuning $\mathrm{\Delta}=\sqrt{2}G$ . The other parameters are ${\mathrm{\Delta}}_{m}=\mathrm{\Delta}/2$ , $G=3{\gamma}_{c}$ , $\epsilon ={\gamma}_{c}/10$ , ${\gamma}_{m}={\gamma}_{c}/100$ , and ${n}_{\mathrm{th}}=0$ .

To explore the statistic properties of the transmitted photons, the equal-time second-order correlation function

The peak for

In order to understand the origin of the peak for

#### Fig. 3. Schematic energy spectrum of the linearized QOM coupling between optical mode ${a}_{L}$ and mechanical resonator $b$ , where $|{0}_{0}\u27e9\equiv |0,0\u27e9$ , $|{1}_{0}\u27e9\equiv |0,1\u27e9$ , $|{2}_{\pm 1}\u27e9\equiv (|1,0\u27e9\pm |0,2\u27e9)/\sqrt{2}$ , $|{3}_{\pm 1}\u27e9\equiv (|1,1\u27e9\pm |0,3\u27e9)/\sqrt{2}$ , $|{4}_{0}\u27e9\equiv (-\sqrt{3}|2,0\u27e9+|0,4\u27e9)/2$ , $|{4}_{\pm 1}\u27e9\equiv (|2,0\u27e9\pm 2|1,2\u27e9+\sqrt{3}|0,4\u27e9)/(2\sqrt{2})$ , and $|n,m\u27e9$ 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

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

The isolation

More interestingly, two peaks appear around

## 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]:

In addition, there are two driving fields coupling to the system with frequencies

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*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.

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##### Article Outline

Xunwei Xu, Yanjun Zhao, Hui Wang, Hui Jing, Aixi Chen. Quantum nonreciprocality in quadratic optomechanics[J]. Photonics Research, 2020, 8(2): 02000143.