# Perfect Optical Nonreciprocity in a Double-Cavity Optomechanical System

###### Abstract

Nonreciprocal devices are indispensable for building quantum networks and ubiquitous in modern communication technology. Here, we use optomechanical interaction and linearly-coupled interaction to realize optical nonreciprocal transmission in a double-cavity optomechanical system. The scheme relies on the interference between the two interactions. We derive the essential conditions to realize perfect optical nonreciprocity in the system, and analyse the properties of optical nonreciprocal transmission and the output fields from mechanical mode. These results can be used to control optical transmission in quantum information processing.

###### pacs:

42.50.Ex, 42.50.Wk, 07.10.Cm## I Introduction

Nonreciprocal optical devices, such as isolators and circulators, allow transmission of signals exhibit different characteristics if source and observer are interchanged, see Fig. 1(a). They are essential to several applications in quantum signal processing and communication, as they can suppress spurious modes and unwanted signal Jalas2013 (). For example, they can protect devices from noise emanating from readout electronics in quantum superconducting circuits. To violate reciprocity and obtain asymmetric transmission, breaking time-reversal symmetry is required in any such device. Traditionally, nonreciprocal transmission has relied on applied magnetic bias fields to break time-reversal symmetry and Lorentz reciprocity Hogan (); Aplet (). While these conventional devices are typically bulky, incompatible with ultra-low loss superconducting circuits because they require sizable magnetic fields. Many recent alternative schemes have been proposed to replace conventional nonreciprocal devices, such as coupled-mode systems Ranzani2015 (); BingHe2018 (), reservoir engineering Metelmann2015 (), Brillouin scattering Poulton2012 (); Kim2015 (); Dong2015 (), angular momentum biasing Sounas2013 (); Estep2014 (); Sounas2014 () and spatiotemporal modulation of the refractive index Fang2012 (). These schemes are particularly promising because they can be integrated on-chip with existing superconducting technology.

In recent years, the rapidly growing field of cavity optomechanics Aspelmeyer2014 (), where optical fields and mechanical resonators are coupled through radiation pressure, has shown promising potential for applications in quantum information processing and communication. So far, many interesting quantum phenomena have been studied in this field, such as mechanical ground-state cooling Marquardt2007 (); Wilson-Rae2007 (); Bhattacharya2007 (); Chan2011 (); Teufel2011 (); BingHe2017 (), optomechanically induced transparency Huang2009 (); Weis2010 (); YCLiu2017 (); HXiong2018 (), entanglement Tian2013 (); Deng2015 (); Deng2016 (); Vitali2007 (); Yan2017 (); Yan2018 (), nonlinear effects Komar2013 (); Lemonde2013 (); Borkje2013 (); LuXY2013 (), and coherent perfect absorption Yan2014 (). Very recently, it has been realized that optomechanical coupling can lead to nonreciprocal transmission and optical isolation Manipatruni2009 (); Rabl2012 (); Habraken2012 (); ZqWang2015 (); Xu2015 (); Xu2016 (); Ruesink2016 (); Shen2016 (); Miri2017 (); Peterson2017 (); Tian2017 (); Sounas2017 (); Bernier2017 (); Barzanjeh2017 (); Fang2017 (); Ruesink2018 (); Malz2018 (). In most of these references, perfect optical nonreciprocity can be achieved under the conditions of equal damping rate (mechanical damping rate is equal to cavity damping rate) or nonreciprocal phase difference .

Here, we show that perfect optical nonreciprocity can be achieved under more general conditions, using the example of a double-cavity system in Fig. 1(b). With this simple model, we can easily capture the essential mechanisms about nonreciprocity, i.e., quantum interference of signal transmission between two possible paths corresponding to two interactions (optomechanical interaction and linearly-coupled interaction). From the expressions of output fields, we derive essential conditions to achieve perfect optical nonreciprocity, and find some interesting results. One of them is that mechanical decay rate has not any effect on the appearance of perfect optical nonreciprocity. It means that in the realistic parameter regime in cavity optomechanics (mechanical decay rate is much less than cavity damping rate), perfect optical nonreciprocity can still occur. Another interesting result is that perfect optical nonreciprocity can be achieved with any phase difference () as long as rotating wave approximation is a good approximation. We believe the results of this paper can be used to control optical transmission in modern communication technology.

## Ii System model and equations

We consider a double-cavity optomechanical system in which two cavities are coupled to a common mechanical resonator, see Fig. 1(b). The mechanical resonator with an eigen frequency and a decay rate is described by annihilation operator . The two optical modes with same frequency and decay rates and are described by annihilation operators and respectively. Two strong coupling fields (probe fields) with same frequency () and amplitudes and ( and ) are used to drive the double-cavity system from the left and right fixed mirror respectively. The linearly-coupled interaction between the two cavities is described by , and is the coupling strength. Then the total Hamiltonian in the rotating-wave frame of coupling frequency can be written as ()

(1) |

Here, is the detuning between cavity modes and coupling fields, is the detuning between probe fields and coupling fields, and is the single photon coupling constant between mechanical and optical modes.

The dynamics of the system is described by the quantum Langevin equations for the relevant operators of the mechanical and optical modes

(2) |

In the absence of probe fields , and with the factorization assumption , we can obtain the steady-state mean values

(3) |

with denoting the effective detunings between cavity modes and coupling fields. In the presence of both probe fields, however, we can write each operator as the sum of its mean value and its small fluctuation, i.e., , , to solve Eq. (2) when both coupling fields are sufficiently strong. Then keeping only the linear terms of fluctuation operators and moving into an interaction picture by introducing , , , we obtain the linearized quantum Langevin equations

(4) |

with and . The phase difference between effective optomechanical coupling and can be controlled by adjusting the coupling fields amplitudes and according to Eq. (3). It will be seen the phase difference is a critical factor to attain optical nonreciprocity. Without loss of generality, we take and as positive number (not negative to avoid introducing unimportant phase difference ).

If each coupling field drives one cavity mode at the mechanical red sideband , the system is operating in the resolved sideband regime , the mechanical resonator has a high mechanical quality factor , and the mechanical frequency is much lager than and , then Eq. (4) will be simplified to

(5) |

with . For simplicity, we set equal cavity damping rate and equal coupling in the following (actually, it can be proven that must equal if when the system exhibits perfect optical nonreciprocity).

We can solve Eq. (5) by assume with as follows

(6) |

with , , .

To study optical nonreciprocity, we must study the output optical fields and which can be obtained according to the input-output relation Walls ()

(7) |

With the same assumption as above, the output fields can be obtained as

(8) |

and .

## Iii perfect optical nonreciprocity

Perfect optical nonreciprocity can be achieved if transmission amplitudes () satisfy

(9a) |

or

(9b) |

It means that the input signal from one side can be completely transmitted to the other side, but not vice versa. What the Eq. (9a), (9b) represent is the two different directions of isolation. In this section, we just discuss the case of Eq. (9a) as the case of Eq. (9b) is similar. The subscript indicates there is not signal injected into the system from right/left side. We omit the subscripts because, in general, nonreciprocity is only related to one-way input, and write transmission amplitudes as for simplicity in the following.

According to Eq. (6), Eq. (8), the two optical output fields can be obtained as

(10) |

It can be seen from Eq. (10) that the two output fields are equal as ( is an integer). It means the photon transmission is reciprocal in this case. When , the system will exhibit a nonreciprocal response. It can be clearly seen from the numerator of Eq. (10) that the optical nonreciprocity comes from quantum interference between the optomechanical interaction and the linearly-coupled interaction .

With Eq. (10), we find perfect optical nonreciprocity Eq. (9a) can be achieved only when

(11) |

which can take positive real number only if

(12a) |

or

(12b) |

In the following, we will discuss perfect optical nonreciprocity in two cases, Eq. (12a), (12b), respectively.

### iii.1 Phase difference

With nonreciprocal phase difference , the two optical output fields Eq. (10) now become as

(13) |

According to Eq. (13), the perfect optical nonreciprocity Eq. (9a) can be achieved only when

(14) |

It is surprising that there is not any restriction on mechanical decay rate in Eq. (14). In other words, mechanical decay rate has not any effect on perfect optical nonreciprocity. It means that perfect optical nonreciprocity can still occur even in the case of as long as the conditions Eq. (14) is satisfied. This is important because, in general, mechanical decay rate is much less than cavity decay rate in cavity optomechanics. In addition, even with very weak optomechanical coupling , perfect optical nonreciprocity can still occur as according to Eq. (14). In Fig. 2(a)–2(d), we plot transmission amplitudes (red line) and (black line) vs normalized detuning with , for , , , respectively. It can be clearly seen from Fig. 2 that mechanical decay rate really does not affect the appearance of perfect optical nonreciprocity, but can strongly affect the width of transmission spectrum, especially for the case of . The two curves of transmission amplitudes and will tend to coincide except in the vicinity of resonance frequency () as , such as (see Fig. 2(d)). It means that the system can only exhibit optical nonreciprocity near resonance frequency in the case. By the way, the perfect optical nonreciprocity Eq. (9b) will occur if .

Now we examine the output fields (, and the mechanical mode ) for when the system exhibits perfect optical nonreciprocity. Besides the output fields Eq. (13), we obtain the other four output fields as

(15) |

and . With the coupling strength (), and detuning according to Eq. (14), it is not difficult to verify that

(16a) | ||||

(16b) |

According to Eq. (13) and Eq. (15), in Fig. 3(a)–3(f), we plot the transmission amplitudes of output fields and vs normalized detuning for different mechanical damping rate: (a), (b) ; (c), (d) ; (e), (f) . It can be seen from Fig. 3 that the input signal from left side can be completely transmitted to right side without any light output from the other two modes (see Fig. 3(a), 3(c), 3(e)), while the input signal from right side can not be transmitted to the left side, but can be transmitted out from the mechanical mode (see Fig. 3(b), 3(d), 3(f)). It can be seen from Eq. (16b) when the system exhibits perfect optical nonreciprocity (), the transmission amplitude unless . It means the intensity of output signal from the mechanical mode will be amplified (weakened) as () (see Fig. 3(b), 3(f)).

### iii.2 Equal damping rate

With equal damping rate , the two optical output fields Eq. (10) now become as

(17) |

From Eq. (17), we can obtain the conditions for perfect optical nonreciprocity as follows

(18) |

with negative sign and for Eq. (9a), positive sign and for Eq. (9b). It means that we can change the direction of isolation by adjusting the nonreciprocal phase difference or . In Fig. 4, we plot the normalized coupling strengths , () (blue line) and detuning (yellow line) vs phase difference according to Eq. (18). For the special case of , the coupling strength () takes the minimum value and detuning (see Fig. 4), and the transmission spectrums and take a symmetric form with respect to detuning (see Fig. 2(b)).

From Eq. (18), we can see that perfect optical nonreciprocity can occur with any phase () as long as () with the result that rotating wave approximation is a good approximation. It means the strongest quantum interference takes place at detuning in the case of . In Fig. 5(a)–5(d), we plot the transmission amplitudes (red line) and (black line) vs normalized detuning with () for , , , , respectively. It can be seen from Fig. 5, the transmission spectrums and will not take the symmetric form anymore as , and for , for . Moreover, the transmission amplitudes of output fields form mechanical mode remain unchanged, not like those in the previous section, we do not show them here.

## Iv conclusion

In summary, we have theoretically studied how to achieve perfect optical nonreciprocity in a double-cavity optomechanical system. In this paper, we focus on under what conditions the system can exhibit perfect optical nonreciprocity, and we have obtained them. From the condition expressions, we can draw three important conclusions: (1) When nonreciprocal phase difference , the mechanical damping rate has not any effect on the appearance of perfect optical nonreciprocity as long as Eq. (14) is satisfied; (2) When and (), the intensity of output fields from mechanical mode will be amplified (weakened); (3) The system can exhibit perfect optical nonreciprocity with any nonreciprocal phase difference if and Eq. (18) is satisfied. Our results can also be applied to other parametrically coupled three-mode bosonic systems, in addition to optomechanical systems.

###### Acknowledgements.

L. Yang is supported by National Natural Science Foundation of China (Grants No. 11804066), and the China Postdoctoral Science Foundation (Grant No. 2018M630337).## References

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