## Abstract

Based on photon-phonon nonlinear interaction, a scheme of controllable photon-phonon converters is proposed at single-quantum level in a composed quadratically coupled optomechanical system. With the assistance of the mechanical oscillator, the Kerr nonlinear effect between photon and phonon is enhanced so that the single-photon state can be converted into the phonon state with high fidelity even under the current experimental condition that the single-photon coupling rate is much smaller than mechanical frequency (*g* ≪ *ω _{m}*). The state transfer protocols and their transfer fidelity are discussed analytically and numerically. A multi-path photon-phonon converter is designed by combining the optomechanical system with low frequency resonators, which can be controlled by experimentally adjustable parameters. This work provides us a potential platform for quantum state transfer and quantum information.

© 2017 Optical Society of America

## 1. Introduction

The radiation pressure in optomechanical system provides us nonlinear interaction between optical cavity mode and microcosmic or macroscopic mechanical mode [1–4]. In addition to some promising applications in fundamental physics research [5, 6], macroscopic mechanical oscillators cooling [7–9], weak force sensing [10–13] and quantum information processing [14–16], optomechanical interaction is naturally of the outstanding characteristic to transduce a state from high frequency object into another one with low frequency. It has been shown that the quantum-state can be converted from light to macroscopic oscillators [17, 18] via optomechanical systems in optical regime. In electro-opto-mechanical system, electrical and optical quantum states can be stored and transferred into mechanical resonators [19, 20], thus the system is able to serve as a microwave quantum-illumination device [21]. On the other hand, many proposal of convertors, transferring few photon state to different frequency electromagnetic wave state based on quantum nonlinearity [22–24], have been proposed, including four-wave mixing converter [24], single-photon frequency conversion in a Sagnac interferometer [23]. In order to employ the cavity optomechanical system in quantum information processing, single photon-phonon conversion is an important manipulation. There is a great challenge in converter when huge frequency difference exists between input state and output state such as optical mode and mechanical mode. Under current experimental parameters region, how to enhance the effective nonlinearity and how to employ their nonlinearity to perform quantum information processing deserve our investigation.

In this paper, we put forward a scheme to enhance the cross-Kerr nonlinearity in a quadratically coupled optomechanical system. Considering the realization, we use an auxiliary mechanical oscillators to enhance the quantum nonlinear effects in the system, and thus achieve an ultra-strong cross-Kerr nonlinearity (*g _{eff}/ω_{eff}* ≫ 1). By combining single bit operations in optical mode and mechanical mode, we can implement a photon-phonon converter at single-quantum level. Then we construct a multi-path photon-phonon converter by extending the dimension of system, which can be controlled by experimentally adjustable parameters.

## 2. Model

We consider an optomechanical system where a membrane in the middle of a Fabry-Pérot cavity is coupled with another mechanical oscillator, shown in Fig. 1. The membrane quadratically couples to the cavity field, and the interaction between two oscillators can be realized by a resonator interacting with a transmission line resonator through the medium of capacitance [25], or by using the geometrically interconnecting [26]. The Hamiltonian of the system is

*a*

^{†}(

*a*), ${b}_{1}^{\u2020}({b}_{1})$ and ${b}_{2}^{\u2020}({b}_{2})$ are the creation (annihilation) operators of the F-P cavity, the mechanical membrane and the target mechanical oscillator, respectively.

*ω*,

_{c}*ω*

_{m1}and

*ω*

_{m2}are the resonant frequency of them. The second term in

*H*describes the quadratic optomechanical coupling between the cavity and the mechanical membrane with strength

_{sys}*g*. The last term in

*H*represent the free energy and the phonon tunneling coupling between two oscillators with strength

_{sys}*V*[27, 28].

*H*denotes that the cavity is driven by a laser with frequency

_{d}*ω*. By eliminating the rapid evolution mode

_{d}*b*

_{1}due to large frequency

*ω*

_{m1}, we obtain the effective interaction between the optomechanical cavity and the oscillator mode

*b*

_{2}, under the condition {

*ω*

_{m1},

*ω*

_{m2}} ≫ {

*V*,

*g*} (Details are in Appendix). The effective Hamiltonian is

_{1}denotes the mechanically modulating detuning of the cavity with a driving frequency

*ω*.

_{d}*ω*means the effective frequency of the mechanical oscillator, and

_{eff}*g*represent nonlinear coupling strength. In the process of deriving effective Hamiltonian, we also obtain the effective and dampling rates ${\gamma}_{\mathit{eff}}={\gamma}_{2}+\frac{{V}^{2}({\gamma}_{1}-{\gamma}_{2})}{2{\left|A\right|}^{2}}$. We can clearly see the existence of the cross-Kerr nonlinear term between cavity mode and mechanical mode ${g}_{\mathit{eff}}{a}^{\u2020}a{b}_{2}^{\u2020}{b}_{2}$ which can provide a way to preform manipulation between photons and phonons. The effective coefficient

_{eff}*g*play the key factor in the quantum control schemes based on cross-Kerr nonlinearity [14, 29]. Usually, the cross-Kerr nonlinearity is very weak. Fortunately, in our scheme, the coefficients

_{eff}*g*, Δ′ and

_{eff}*ω*are adjustable.

_{eff}As is shown in Fig. 2(a), within certain region of *V*, *ω _{eff}* can be reduced, even approximation a minimal value zero, i.e., |

*ω*| ≈ 0 at the specific value of $V\approx \sqrt{{\omega}_{m1}({\omega}_{m1}-{\omega}_{m2})}$. Meanwhile, the effective coupling rate

_{eff}*g*increases with the mechanical coupling rate

_{eff}*V*raising, shown in Fig. 2(b). The ratio between the effective coupling rate and the effective mechanical frequency |

*g*| is shown in Fig. 2(c). There is a discontinuity point tending to infinity at a specific value of mechanical coupling rate $V\approx \sqrt{{\omega}_{m1}({\omega}_{m1}-{\omega}_{m2})}$. The coefficients

_{eff}/ω_{eff}*ω*,

_{eff}*g*and

_{eff}*γ*are affected by the frequency difference between two oscillators are shown in Fig. 3(d), 3(e) and 3(f). The modulated

_{eff}*ω*frequency also can be decreased by adjusting Δ

_{eff}_{ωm}[shown in (d)], and

*g*decrease with increasing of Δ

_{eff}_{ωm}[see (e)]. The ratio |

*g*| can be enlarged greatly at certain region, shown in (f). Although {

_{eff}/ω_{eff}*g*,

*V*} ≪ {

*ω*

_{m1},

*ω*

_{m2}}, the Δ

_{ωm}can be small so that effective Kerr nonlinearity is enlarged. For usual optomechanical system, the nonlinearity is

*g*

^{2}/

*ω*

_{m1}, while our modulated nonlinearity coefficient is

*g*(see Eq. (7)). Obviously, the nonlinearity is enlarged greatly, i.e., $\frac{2g{V}^{2}}{{\left|A\right|}^{2}}\gg {g}^{2}/{\omega}_{m1}$. Thus we can enlarge the ratio |

_{eff}*g*| ≫ 1 by adjusting the coupling rate

_{eff}/ω_{eff}*V*and the frequency difference Δ

_{ωm}. So it is possible for us to achieve an ultra-strong cross-Kerr nonlinearity in the system.

## 3. Photon-phonon control phase-flip gate

Now we show that the composed optomechanical system can work as a photon-phonon control phase-flip gate (CPFG) analytically and numerically. We use the ground- (excited-) state of photon and phonon to denote the logical states |0〉 (|1〉) of signal mode and control mode, respectively. The unknown signal qubit inputs via the optical cavity. The arbitrary initial state of the system can be describes as

*α*|

^{2}+ |

*β*|

^{2}+ |

*γ*|

^{2}+ |

*δ*|

^{2}= 1. |0〉

*(|1〉*

_{c}*) means no (one) photon in the cavity while |0〉*

_{c}*(|1〉*

_{m}*) stands for the ground state (the first excited state) of the oscillator. When we can accomplish the optomechanical CPFG, the target state should be*

_{m}*ψ*〉 =

_{f}*e*

^{−iHefft}|

*ψ*〉, which can be described as

*θ*

_{01}=

*ω*,

_{eff}t*θ*

_{10}= Δ′

*t*,

*θ*

_{11}= (

*ω*+ Δ′ −

_{eff}*g*)

_{eff}*t*. We define the fidelity

*F*= |〈

_{cp}*ψ*|Φ〉| between the final state and the target state, thus

_{f}*F*= 1, which means the CPFG is realized. Then the ratio of the parameters should meet the relation

_{cp}*ω*: Δ′:

_{eff}*g*=

_{eff}*n*

_{1}:

*n*

_{2}: (

*n*

_{1}+

*n*

_{2}−

*n*

_{3}− 1/2).

Including the dissipation of the system, we can directly employ the master equation of the system to reconsider CPFG. The master equation is as

*H*is the original Hamiltonian Eq. (1),

*κ*,

*γ*and

_{j}*n*are the decay rates of the cavity, mechanical resonator and the thermal occupancy of the mechanical bath respectively.

_{thj}*𝒟*[

*o*]

*ρ*=

*oρo*

^{†}−

*o*

^{†}

*oρ*/2 −

*ρo*

^{†}

*o*/2 is the Lindblad dissipation superoperator. Under this condition, ${F}_{\mathit{cp}}=\sqrt{\u3008\mathrm{\Phi}\left|\rho \right|\mathrm{\Phi}\u3009}$.

We plot the fidelity in both analytical and numerical method in Fig. 3(a). One can observe that the CPFG can be realized in a specific time when the additional phase equals to (2*n*+1)*π*, *n* ∈ ℜ. Comparing the analytical and numerical solution, we find that the two lines almost coincide. Thus we can safely conclude that the effective Hamiltonian and the analytical solution are correct, except some point with low fidelity due to the approximation in analytical solution. The Kerr nonlinear term
${g}_{\mathit{eff}}{a}^{\u2020}a{b}_{2}^{\u2020}{b}_{2}$ in Eq. (4) is the key point to realize the CPF gate. Choosing different values of *V/ω*_{m1} = 1.35 × 10^{−2}, 2.58 × 10^{−2}, 3.16 × 10^{−2}, we plot the fidelity in (b),(c) and (d) respectively. The corresponding ratio
$\left|\frac{{g}_{\mathit{eff}}}{{\omega}_{\mathit{eff}}}\right|=0.01,\hspace{0.17em}0.05,\hspace{0.17em}0.1$. The maximal fidelity with ratio are *F _{cpmax}* ≈ 0.83, 0.94, 0.97, respectively. It is clearly that the larger values of the ratio
$\left|\frac{{g}_{\mathit{eff}}}{{\omega}_{\mathit{eff}}}\right|$ the higher fidelity of the CPFG. In our scheme, with the assistance of mechanical oscillator we can enlarge the ratio
$\left|\frac{{g}_{\mathit{eff}}}{{\omega}_{\mathit{eff}}}\right|$. Therefore, the current scheme provides a realizable method to perform controlled-phase gate between photons and phonons under weak coupling regime.

## 4. Single-quantum photon-phonon convertor

Employing optomechanical interaction, we can build a link between photons and phonons. Now, we show that our system can perform a photon-phonon convertor at the single-photon level using the cross-Kerr nonlinearity effect. As shown in Fig. 4, the quantum circuit denotes a basic process to realize photon-phonon convertor. The single-qubit code in photons can be realized by using a velocity-selected circular Rydberg atom in the cavity [30], pre-excited single atoms in the cavity [31] and parametric down-conversion in a distributed microcavity [32]. We can produce signal state directly in optomechanical system [30, 31] or introduce it into the system using low losing fiber [33], monochromatic waveguide [34]. And the single optical state can be easily operated by linear optical device [35]. The ground- and single- phonon state can be manipulated by film bulk acoustic resonator [26,36,37]. In Fig. 4, the c-phase gate is realized by the optomechanical system with fidelity *F _{max}* = 0.97 which we have mentioned in section III. Here we code information as follows: |0〉

_{1}means no photon in the cavity while |1〉

_{1}means only one photon in the cavity. |0〉

_{2}means that mechanical osillator is in its gound state and |1〉

_{2}means in excited state. As shown in Fig. 4, to transfer an arbitrary optical state

*α*|0〉

_{1}+

*β*|1〉

_{1}to the mechanical oscillator through the convertor, we input the coded state into the cavity while the mechanical oscillator should be cooled into its ground state |0〉

_{2}. After a Hadamard gate operator manipulate in |0〉

_{2}, the state becomes $\frac{1}{\sqrt{2}}\left(\alpha {|0\u3009}_{1}+\beta {|1\u3009}_{1}\right)\left({|0\u3009}_{2}+{|1\u3009}_{2}\right)$. The system undergoing a CPFG process, the quantum state becomes $\frac{1}{\sqrt{2}}\left[\alpha {|0\u3009}_{1}\left({|0\u3009}_{2}+{|1\u3009}_{2}\right)+\beta {|1\u3009}_{2}({|0\u3009}_{2}-{|1\u3009}_{2}\right]$. Then, ultimating two Hadamard gate operation on both photon and phonon state, we get the system state, $\frac{1}{\sqrt{2}}\left[\alpha \left({|0\u3009}_{1}+{|1\u3009}_{1}\right){|0\u3009}_{2}+\beta \left({|0\u3009}_{1}-{|1\u3009}_{1}\right){|1\u3009}_{2}\right]$, which can be rewritten as

*α*|0〉

_{2}+

*β*|1〉

_{2}. If the photon counting is one, the mechanical oscillator will collapse to the state

*α*|0〉

_{2}−

*β*|1〉

_{2}, then we just need to performing a

*σ*operation. Thus, finally, we can transfer optical state.

_{z}In a recent experiment [38], single photon-phonon correlation using linearized optomechanical interaction *G*(*a*^{†}*b* + *ab*^{†}), can generate a single stokes scattering with probability *p* ≈ 3%. By using this scheme, a single photon is transfer into a single phonon with the efficiency that approximate equals to 3.7%. Differently, we aim to transfer a information encoded in a photon state (*α*|0〉 + *β*|1〉) into a phonon carrier. We employ an enhanced Kerr nonlinearity *g _{eff}a*

^{†}

*ab*

^{†}

*b*. Theoretically speaking, our scheme, the phonon-to-photon information conversion might be achieved with a 100% efficiency and a high fidelity considering the dissipation.

## 5. Controllable multi-path photon-phonon converter

Now we expand our system to a more general model. As shown in Fig. 5, there is an array of quadratically coupled optomechanics, cavity-*k* (*k* > 1) coupled to the cavity-1 with strength *J*_{k−1}. Each membranae of the optomechanical cavity coupled to a low frequency oscillator with the strength *V _{j}*. The Hamiltonian of the system can be writte as

*a*and

_{j}*ω*denote the cavity photon operator and frequency, respectively.

_{j}*ω*,

_{mj}*b*and

_{j}*ω*,

_{Aj}*b*describe the membranae mode and target oscillator mode, respectively. The fourth term denotes the optomechanical interaction. The fifth and sixth term denote the interaction between the mechanical modes and optical modes, respectively.

_{Aj}Using the same processing in section II, we get the effective Hamiltonian under the condition *ω _{mj}* ≫ {

*g*,

_{j}*V*,

_{j}*J*}.

_{j}*=*

_{j}*ω*−

_{j}*ω*, here

_{jL}*A*=

_{j}*i*(

*ω*−

_{mj}*ω*) + (

_{Aj}*γ*−

_{mj}*γ*)/2 and

_{Aj}*ω*denotes the driving frequency of cavity-

_{jL}*j*.

If we set cavity-1 as an input port of the multi-path convert system, the input photon state can be transmitted from cavity-1 to cavity-*k* due to the Beam Splitter (BS) interaction. According to the analysis in former section, we can convert the single-photon state from optical mode to mechanical mode by using cross-Kerr nonlinearity. Thus, composing the two manipulations, we can convert the arbitrary input single-photon state from cavity-1 to any other target oscillator through cavity-*k*. According to the mode shows in Fig. 5, only when the CPFG is achieved and the optical state is transferred into cavity-*j* at the time, can we convert the input photon state to the *j*-th phonon state. In order to evaluate the quality of the conversion, we define the conversion fidelity which is defined as

*F*denotes the fidelity of the CPFG between

_{Gj}*j*-th optical mode and mechanical mode,

*ρ*is the density operation of them.

_{j}*ψ*is the final state after a perfect CPFG operator.

_{f}*F*denotes the fidelity between input state

_{Sj}*ψ*

_{0}and the state in cavity-

*j*

*ψ*

_{0}.

Take *n* = 2 as an example, we show the controllable photon-phonon conversion process using *H _{eff}*. As shown in Fig. 6(a), we plot the conversion fidelity of the output port without dispassion. It shows that the photon-phonon conversion both for port 1 and 2 can be periodically realized at a specific time simultaneously. When the dissipation is included, the fidelity of the conversion for both ports decreased (see (b)). Although the parameters are the same for both of the subsystem, the fidelity

*F*

_{C2}is slightly lower than

*F*

_{C1}= 1, because the signal state is directly input from cavity 1, so it is directly conversed into port 1, while for port 2, it is needed firstly hopping from

*a*

_{1}into

*a*

_{2}and then conversed into port 2. Thus, it is reasonable to understand the slightly decrease of fidelity

*F*

_{C2}. By individually adjusting

*V*and

*ω*(

_{mi}*i*= 1, 2) as we have discussed above, we can change the ratio

*g*. The conversion fidelity for individual different values

_{eff}/ω_{eff}*g*(i=1,2) are shown in Fig. 6(c) and 6(d). It is obvious that the periodic time is determined by the value of

_{ei}/ω_{ei}*g*. Thus, on the one hand, we can enlarge the the ratios

_{ei}/ω_{ei}*g*to compensate the dissipation of the system, on the other hand, we can control the time of the conversion reached.

_{ei}/ω_{ei}## 6. Discussion and conclusion

As we have shown, the *F _{C}* of cavity-2 cannot reach 1 just like cavity-1 even without dispassion, because the rest systems can be seen as an environment, which introduces an effective dissipation rate to the sub-system (such as cavity-

*j*) that we focus on. Thus, the number of output ports in the multi-path converter is limited to some extent if we require high fidelity of the converted state. To increase the fidelity of converter, on the one hand, we can enlarge the ratio of |

*g*| to accelerate the evolution, which will reduce the cumulative effects of system dissipation (shown in Fig. 3). On the other hand, we can directly improve the quality factor of cavity to reduce the effective dissipation rate. In our scheme, the optomechanical oscillator can be realized by a suspended film bulk acoustic resonator with frequency in the order of 10

_{eff}/ω_{eff}^{9}

*Hz*[37]. The target oscillator can be realized by nanoobjects with frequency in the order of 10

^{6}

*Hz*[26, 39]. The interaction between them can be controlled by piezoelectric ceramics and LC circuits [25,37]. By changing the voltage

*V*

_{0}and capacitance

*C*

_{0}experimentally, we can control V because of

*V*∝

*C*

_{0}

*V*

_{0}. A two-dimensional electron gas and Schottky-contacted gold electrodes can be used to output the phonon signal we need [40].

We propose a scheme to realize single photon-phonon converter by combining the single-bit operation with the cross-Kerr nonlinear effect between photons and phonons in quadratically coupled optomechanical system. Considering the realization, we use the weak coupling optomechanical oscillator coupled to a target oscillator to accelerate the evolution of system so as to improve the fidelity of CPFG, we can achieves *F*_{cpmax} ≈ 0.97 in consideration of dissipation, as is shown in Fig. 3. By controlling the adjustable parameters, we are able to achieve an ultra-strong cross-Kerr nonlinearity (*g _{eff}/ω_{eff}* ≫ 1), which makes sense for the realization of single quantum photon-phonon converter to a large extent. Moreover, we also extend the converter protocol to a controllable multiple outputs scheme through a dimension extension. By choosing the detection time of the output signal of cavity-1, we can transduce an unknown input single optical state into a mechanical state of an arbitrary output port we want with high fidelity. This protocol provides us with a possibility to perform photon-phonon multipath conversion and operation.

## Appendix: derivation of the effective Hamiltonian

Under the weak diving condition, we tempt ignore the function of the Hamiltonian *H _{d}* so as to simplify the process of the deriving the effective Hamiltonian. Considering

*ω*

_{m1}≫

*g*,

*b*

_{1}

^{2}and

*b*

_{1}

^{†2}can be ignored by rotating wave approximation, the Hamiltonian can be rewritten as

_{1}=

*ω*−

_{c}*ω*−

_{d}*g*denotes the mechanically modulating detuning of the cavity with a driving frequency

*ω*.

_{d}*κ*,

*γ*

_{1}and

*γ*

_{2}represent the decay rates of mode

*a*,

*b*

_{1}and

*b*

_{2}. The

*a*is the sum of coherent amplitudes

_{in}*ā*and vacuum noise operator

_{in}*ξ*.

*b*

_{in,1}and

*b*

_{in,2}are the noise operators associated with the mechanical dissipations. Defining photon number operator

*N*=

_{a}*a*

^{†}

*a*and phtonon number operator ${N}_{b}={b}_{1}^{\u2020}{b}_{1}$, we can rewrite the nonlinear quantum Langevin equations as

*κ*

_{1}so that we can treat

*N*as constant in integration (21b), in other words, when the dissipation rate of mechanical oscillators is weak enough and the dynamic time of the oscillators is much smaller than that for cavity field, we can regard

_{a}*N*as independent of time

_{a}*t*in the phonon evolution period. Under the weak driving condition, the occupation number of the system is rather small. If

*ω*

_{m2}≫ {

*g*,

*V*}, we can employ iteration method to solve (22c) and obtain the zero order solution (in terms of V) as

*γ*

_{1}≫

*γ*

_{2}(the term containing

*e*

^{−γ1t}is a fast decaying term and can be neglected), we have

*A*=

*i*(

*ω*

_{m1}−

*ω*

_{m2})+(

*γ*

_{1}−

*γ*

_{2})/2. When |

*ω*

_{m1}−

*ω*

_{m2}| is much larger than 2

*gN*, we have

_{a}*b*

_{1}(

*t*) back to (21a) and (21c), we finally obtain

## Funding

National Natural Science Foundation of China (NSFC) (11474044, 11547134).

## Acknowledgments

We would like to thank Mr. Wen-Lin Li and Xun Li for helpful discussions.

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