## Abstract

We study a double-cavity optomechanical system in which a movable mirror with perfect reflection is inserted between two fixed mirrors with partial transmission. This optomechanical system is driven from both fixed end mirrors in a symmetric scheme by two strong coupling fields and two weak probe fields. We find that three interesting phenomena: coherent perfect absorption (CPA), coherent perfect transmission (CPT), and coherent perfect synthesis (CPS) can be attained within different parameter regimes. That is, we can make two input probe fields totally absorbed by the movable mirror without yielding any energy output from either end mirror (CPA); make an input probe field transmitted from one end mirror to the other end mirror without suffering any energy loss in the two cavities (CPT); make two input probe fields synthesized into one output probe field after undergoing either a perfect transmission or a perfect reflection (CPS). These interesting phenomena originate from the efficient hybrid coupling of optical and mechanical modes and may be all-optically controlled to realize novel photonic devices in quantum information networks.

© 2014 Optical Society of America

## 1. Introduction

Cavity optomechanics has quickly developed over the past few years as a crossover field of nanophysics and quantum optics dealing with the hybrid interaction between light fields and mechanical motions. Relevant studies allow the high-sensitivity detection of tiny mass, force, and displacement by virtue of the radiation-pressure coupling between optical modes of cavities and mechanical degrees of freedom [1–4]. One standard and simplest optomechanical setup is a Fabry-Perot cavity with one end mirror being a micro- or nano-mechanical vibrating object [5–7]. Other optomechanical devices were also designed and tested such as silica toroidal optical microresonators with radiation pressure coupling [8–11] and typical optomechanical cavities confining a Bose-Einstein condensate (BEC) [12–17]. Three-mode or multimode optomechanical systems have been further studied to attain hybrid quantum entanglement [18], optomechanically induced light transparency [19], single-photon nonlinearities [20], and ground-state cooling of mechanical modes [21, 22].

Electromagnetically induced transparency (EIT) and normal mode splitting (NMS), in particular, have been predicted and observed in a few optomechanical systems where the output probe field depends on the incident control field via the nonlinear optomechanical interaction [23–27]. EIT in multi-mode optomechanical systems is an analogue of that in multi-level atomic systems [28]: a three-level atomic medium can be made transparent to a weak probe field as illuminated by a strong coupling field in the case of two-photon resonance. Then slowing down and even stopping light signals [29, 30] in the long-lived mechanical vibrations may be realized by the EIT technique. Recently, Agarwal and Huang studied a double-ended cavity optomechanical system with a movable in-between mirror of partial reflection inserted between two fixed end mirrors of equal transmission [31]. They showed that a phenomenon opposed to EIT, coherent perfect absorption (CPA), could be attained with realistic parameters when this system was driven by one strong coupling field and two weak probe fields. Such a double-cavity optomechanical system was first studied by Paternostro et al. in 2007 under the viewpoint of quantum correlations for achieving macroscopic entanglement [32].

In this paper, we study a slightly different double-cavity optomechanical system with an movable mirror of perfect reflection and driven by two coupling fields and two probe fields. We find that it is possible to attain coherent perfect transmission (CPT) and coherent perfect synthesis (CPS) in addition to CPA [31] as far as the probe field propagation is concerned. Here CPT means that a probe field input from one end mirror is observed from the other end mirror without suffering any loss after a perfect transmission through the double-cavity system; CPS means that two probe fields input from different end mirrors are synthesized into one output probe field at one end mirror by undergoing either a perfect transmission or a perfect reflection. Note that the realization of CPA [31] needs a large decay rate of the movable mirror while both CPT and CPS can be realized only when the movable mirror has a small decay rate. Actual probe frequencies of CPA, CPT, and CPS can be easily changed by controlling the effective optomechanical coupling rate, i.e., by modulating one or both coupling fields in amplitude. We expect that the all-optically controlled realization of CPA, CPT, and CPS could be explored to build new tunable photonic devices in quantum information networks.

## 2. Model and equations

We consider a double-cavity hybrid system with one movable mirror (membrane oscillator) of perfect reflection inserted between two fixed mirrors of partial transmission (see Fig. 1). The membrane oscillator has an eigen frequency *ω _{m}* and a decay rate

*γ*and thus exhibits a mechanical quality factor

_{m}*Q*=

*ω*. Two identical optical cavities of lengths

_{m}/γ_{m}*L*and frequencies

*ω*

_{0}are got when the membrane oscillator is at its equilibrium position in the absence of external excitation. We describe the two optical modes, respectively, by annihilation (creation) operators

*c*

_{1}( ${c}_{1}^{\u2020}$) and

*c*

_{2}( ${c}_{2}^{\u2020}$) while the only mechanical mode by

*b*(

*b*

^{†}). These annihilation and creation operators are restricted by the commutation relation $[{c}_{i},{c}_{i}^{\u2020}]=1$ (

*i*= 1, 2), [

*c*

_{1},

*c*

_{2}] = 0, and [

*b*,

*b*

^{†}] = 1. Two probe (coupling) fields are used to drive the double-cavity system from either left or right fixed mirrors with their amplitudes denoted by ${\epsilon}_{L}=\sqrt{2\kappa {\wp}_{L}/(\overline{h}{\omega}_{p})}$ and ${\epsilon}_{R}=\sqrt{2\kappa {\wp}_{R}/(\overline{h}{\omega}_{p})}$ ${\epsilon}_{cL}=\sqrt{2\kappa {\wp}_{cL}/(\overline{h}{\omega}_{c})}$ and ${\epsilon}_{cR}=\sqrt{2\kappa {\wp}_{cR}/(\overline{h}{\omega}_{c})}$). Here

*℘*,

_{L}*℘*,

_{R}*℘*, and

_{cL}*℘*are relevant field powers,

_{cR}*κ*is the common decay rate of both cavity modes, and

*ω*(

_{p}*ω*) is the probe (coupling) field frequency. Then the total Hamiltonian in the rotating-wave frame of coupling frequency

_{c}*ω*can be written as

_{c}*=*

_{c}*ω*

_{0}−

*ω*being the detuning between cavity modes and coupling fields,

_{c}*δ*=

*ω*−

_{p}*ω*being the detuning between probe fields and coupling fields,

_{c}*θ*being the relative phase between left-hand and right-hand probe fields, and ${g}_{0}=\frac{{\omega}_{0}}{L}\sqrt{\frac{\overline{h}}{2m{\omega}_{m}}}$ being the hybrid coupling constant between mechanical and optical modes.

With Eq. (1), it is easy to obtain the following quantum Langevin equations for relevant annihilation operators of mechanical and optical modes

*b*being the thermal noise on the movable mirror with zero mean value, ${c}_{1}^{\mathit{in}}$ ( ${c}_{2}^{\mathit{in}}$) is the input quantum vacuum noise from the left (right) cavity with zero mean value. In the absence of probe fields

_{in}*ε*and

_{L}*ε*, Eq.s (2) can be solved with the factorization assumption 〈

_{R}*bc*〉 = 〈

_{i}*b*〉 〈

*c*〉 to generate the steady-state mean values

_{i}*g*

_{0}|

*b*| is typically very small as compared to

_{s}*κ*and becomes even exactly zero in the case of |

*c*

_{1}

*| = |*

_{s}*c*

_{2}

*| (|*

_{s}*ε*| = |

_{cL}*ε*|).

_{cR}In the presence of both probe fields, however, we can write each operator as the sum of its mean value and its small fluctuation (*b* = *b _{s}* +

*δb*,

*c*

_{1}=

*c*

_{1}

*+*

_{s}*δc*

_{1},

*c*

_{2}=

*c*

_{2}

*+*

_{s}*δc*

_{2}) 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

*δb*→

*δbe*

^{−iωmt},

*δb*→

_{in}*δb*

_{in}e^{−iωmt},

*δc*

_{1}→

*δc*

_{1}

*e*

^{−iΔ1t}, $\delta {c}_{1}^{\mathit{in}}\to \delta {c}_{1}^{\mathit{in}}{e}^{-i{\mathrm{\Delta}}_{1}t}$,

*δc*

_{2}→

*δc*

_{2}

*e*

^{−iΔ2t}, $\delta {c}_{2}^{\mathit{in}}\to \delta {c}_{2}^{\mathit{in}}{e}^{-i{\mathrm{\Delta}}_{2}t}$, we obtain the linearized quantum Langevin equations

If each coupling field drives one cavity mode at the mechanical red sideband (Δ_{1} ≈ Δ_{2} ≈ *ω _{m}*), the hybrid system is operating in the resolved sideband regime (

*ω*>>

_{m}*κ*), the membrane oscillator has a high mechanical quality factor (

*ω*>>

_{m}*γ*), and the mechanical frequency

_{m}*ω*is much larger than

_{m}*g*

_{0}|

*c*

_{1}

*| and*

_{s}*g*

_{0}|

*c*

_{2}

*|, Eq.s (4) will be simplified to*

_{s}*x*=

*δ*−

*ω*≈

_{m}*ω*−

_{p}*ω*

_{0}being the detuning between probe fields and cavity modes.

Noting that the mean values of quantum and thermal noise terms are vanishing, we can examine the expectation values of small fluctuations arising from both weak probe fields by the following three coupled dynamic equations

*δs*〉 =

*δs*

_{+}

*e*

^{−}

*+*

^{ixt}*δs*

_{−}

*e*with

^{ixt}*s*=

*b*,

*c*

_{1},

*c*

_{2}. Then it is straightforward to obtain with relevant substitutions the following results

*G*=

*g*

_{0}

*c*

_{1}

*as the effective optomechanical coupling rate and |*

_{s}*c*

_{2}

_{s}/c_{1}

*|*

_{s}^{2}=

*n*

^{2}as the photon number ratio of two cavity modes. In deriving Eq.s (7), we have also assumed that

*c*

_{1}

_{s}_{,2}

*is real-valued without loss of generality.*

_{s}With Eq.s (7), it is possible to determine the output fields *ε _{outL}* and

*ε*leaving from both cavity mirrors with the following input-output relation [33]

_{outR}*ε*=

_{outL}*ε*

_{outL}_{+}

*e*

^{−}

*+*

^{ixt}*ε*

_{outL}_{−}

*e*and

^{ixt}*ε*=

_{outR}*ε*

_{outR}_{+}

*e*

^{−}

*+*

^{ixt}*ε*

_{outR}_{−}

*e*. Note that the output components

^{ixt}*ε*

_{outL}_{+}and

*ε*

_{outR}_{+}have the same Stokes frequency

*ω*as the input probe fields

_{p}*ε*and

_{L}*ε*while the output components

_{R}*ε*

_{outL}_{−}and

*ε*

_{outR}_{−}are generated at the anti-Stokes frequency 2

*ω*−

_{c}*ω*in a nonlinear wave-mixing process of optomechanical interaction. Then with Eq.s (8) we can obtain

_{p}## 3. Results and discussion

Now we will implement numerical calculations based on Eq.s (7) and Eq.s (9) to show that a few interesting phenomena such as CPA, CPT, and CPS could be attained in this double-cavity optomechanical system. Each of these phenomena corresponds to a distinct parameter regime accessible by devising the optical and mechanical devices or modulating the external driving laser fields in a proper way. In the following calculations, we will adopt realistic parameters as in ref. [34] with *L* = 25 mm, *m* = 145 ng, *κ* = 2*π* × 215 kHz, *ω _{m}* = 2

*π*× 947 kHz, and

*λ*= 2

*πc/ω*= 1064 nm for a relevant experimental setup.

_{c}#### 3.1. Coherent perfect absorption

First we consider how to achieve CPA by pursuing a parameter regime where we have *ε _{outL}*

_{+}=

*ε*

_{outR}_{+}= 0 with

*ε*≠ 0 and

_{L}*ε*≠ 0. That is, both input probe fields are perfectly absorbed by this double-cavity system without yielding any output from either fixed cavity mirror. We find by combining Eq.s (7) and Eq.s (9) that CPA can be realized only when conditions are simultaneously satisfied. These conditions

_{R}*n*could be different from 1.0 in our system.

The normalized output probe energy |*ε _{outL}*

_{+}|

^{2}/|

*ε*|

_{L}^{2}(|

*ε*

_{outR}_{+}|

^{2}/|

*ε*|

_{R}^{2}) is plotted as a function of the normalized input probe detuning

*x/κ*for different values of the effective optomechanical coupling rate

*G*in Fig. 2; for different values of the cavity photon number ratio

*n*in Fig. 3. It is clear that CPA can occur at zero, one resonant, or two detuned points depending on the values of

*G*[see Fig. 2]. Here CPA means that both input probe fields are fully absorbed by the mechanical mode without being reflected or transmitted out of this double-cavity system. This is the result of a perfect destructive interference between the left-going and right-going probe photons with their energy dissipated via the fast mechanical decay. In addition, we can tune the two detuned CPA points

*x*

_{±}by increasing the value of

*n*(right coupling field amplitude

*ε*) for a fixed value of

_{cR}*G*(left coupling field amplitude

*ε*) [see Fig. 3]. But the right probe field amplitude should be accordingly modified to guarantee the occurrence of CPA as required by

_{cL}*ε*=

_{R}*nε*in Eq.s (10). More discussions on CPA can be found in ref. [31].

_{L}#### 3.2. Coherent perfect transmission

Then we consider how to achieve CPT by pursuing a parameter regime where we have |*ε _{outL}*

_{+}/

*ε*| = 0 and |

_{L}*ε*

_{outR}_{+}/

*ε*| = 1 with

_{L}*ε*≠ 0 and

_{L}*ε*= 0. In this case, the left probe field will be observed from the right cavity mirror after a perfect transmission through this double-cavity system in the absence of the right probe field. Setting

_{R}*ε*= 0 and

_{R}*n*= 1, we find from Eq.s (7) and Eq.s (9) that CPT will occur at three points

*γ*→ 0. That is, a high-

_{m}*Q*mechanical mode is required to attain the perfect light transmission. Otherwise, the input probe field will experience a remarkable energy loss due to the fast mechanical decay.

In Fig. 4 and Fig. 5, we plot the normalized output probe field energy |*ε _{outL}*

_{+}/

*ε*|

_{L}^{2}and |

*ε*

_{outR}_{+}/

*ε*|

_{L}^{2}as a function of the normalized input probe detuning

*x/κ*for different values of the effective optomechanical coupling rate

*G*. It is clear that we have |

*ε*

_{outL}_{+}/

*ε*|

_{L}^{2}= 0 and |

*ε*

_{outR}_{+}/

*ε*|

_{L}^{2}= 1 at

*x*

_{0}= 0 with

*G*= 0.2

*κ*[see the black curves]; at

*x*

_{0}= 0 and

*x*

_{±}= ±1.37

*κ*with

*G*= 1.2

*κ*[see the red curves]; at

*x*

_{0}= 0 and

*x*

_{±}= ±2.03

*κ*with

*G*= 1.6

*κ*[see the blue curves]. These numerical results are exactly consistent with Eq.s (11) indicating that we can choose on demand the desired points where the left probe field is observed at the right cavity mirror after a perfect transmission. We stress here that CPT is a pure quantum optomechanical phenomenon because the 100% energy transfer from one cavity mode to the other cavity mode relies on the lossless tunnelling of a probe field through the membrane oscillator with the perfect reflection. Then a considerable energy loss can be successfully avoided during the probe field tunnelling process only when the membrane’s decay rate is sufficiently small. With the increasing of

*γ*, the energy transfer from one cavity mode to the other cavity mode will become evidently less than 100% for all probe field detunings.

_{m}#### 3.3. Coherent perfect synthesis

Finally we consider how to achieve CPS by pursuing a parameter regime where we have |*ε _{outL}*

_{+}/

*ε*| = 0 and |

_{L}*ε*

_{outR}_{+}/

*ε*|

_{L}^{2}= 2 or |

*ε*

_{outL}_{+}/

*ε*|

_{L}^{2}= 2 and |

*ε*

_{outR}_{+}/

*ε*| = 0 with

_{L}*ε*=

_{L}*ε*≠ 0. In this case, both input probe fields are expected to be observed (without suffering any energy loss) from either the right or the left cavity mirror. Setting

_{R}*ε*=

_{R}*ε*and

_{L}*n*= 1, we find from Eq.s (7) and Eq.s (9) that CPS will occur at two points

*θ*= 0.5

*π*and

*θ*= 1.5

*π*in the limit of

*γ*→ 0. Thus once again we should have a high-

_{m}*Q*mechanical mode to avoid the remarkable output energy loss via fast mechanical decay.

In Fig. 6 and Fig. 7, we plot the normalized output probe field energy |*ε _{outL}*

_{+}/

*ε*|

_{L}^{2}and |

*ε*

_{outR}_{+}/

*ε*|

_{R}^{2}as a function of the normalized input probe detuning

*x/κ*for different values of the probe field relative phase

*θ*. It is clear that we have |

*ε*

_{outL}_{+}/

*ε*|

_{L}^{2}= 0 and |

*ε*

_{outR}_{+}/

*ε*|

_{R}^{2}= 2 at

*x*

_{+}= +0.6

*κ*but |

*ε*

_{outL}_{+}/

*ε*|

_{L}^{2}= 2 and |

*ε*

_{outR}_{+}/

*ε*|

_{R}^{2}= 0 at

*x*

_{−}= −0.6

*κ*with

*θ*= 0.5

*π*[see the red curves]; |

*ε*

_{outL}_{+}/

*ε*|

_{L}^{2}= 0 and |

*ε*

_{outR}_{+}/

*ε*|

_{R}^{2}= 2 at

*x*

_{−}= −0.6

*κ*but |

*ε*

_{outL}_{+}/

*ε*|

_{L}^{2}= 2 and |

*ε*

_{outR}_{+}/

*ε*|

_{R}^{2}= 0 at

*x*

_{+}= +0.6

*κ*with

*θ*= 1.5

*π*[see the blue curves]. Then a simple relative phase modulation from

*θ*= 0.5

*π*to

*θ*= 1.5

*π*will result in the exchange of output probe field from left cavity mirror to right cavity mirror or vice versa. It is clear that in CPS one probe field experiences a perfect transmission while the other probe field experiences a perfect reflection. It is not difficult to find from Eq.s (7) and Eq.s (9) that CPS is once again the result of quantum interference between

*ε*and

_{L}*ε*, which is constructive at one cavity mirror but destructive at the other cavity mirror in our optomechanical system.

_{R}## 4. Conclusions

In summary, we have studied in theory a two-mode optical cavity with an in-between membrane oscillator driven at most by two strong coupling fields and two weak probe fields. Our analytical and numerical results show that three interesting phenomena on the probe field transmission (CPA, CPT, and CPS) could be attained in this optomechanical system. In CPA, both probe fields experience a complete absorption without yielding any energy output from the two cavity mirrors when the membrane oscillator has a very large decay rate. In CPT, one probe field travels through the optomechanical system without suffering any energy loss in the absence of the other probe field if the membrane oscillator has a very small decay rate. In CPS, one probe field is totally transmitted while the other one is totally reflected to generate a perfect synthesis at one cavity mirror when the membrane oscillator has a very small decay rate. Each phenomenon is realized at one, two, or three points of probe frequency depending mainly on the two coupling field amplitudes and the two probe field phases. A flexible control of CPA, CPT, and CPS should be important in quantum information networks, e.g., for achieving photonic switching, routing, swapping, and entanglement.

## Acknowledgments

This work is supported in part by the National Natural Science Foundation of China ( 61378094) and the National Basic Research Program of China ( 2011CB921603).

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