## Abstract

We propose to realize the two-mode continuous-variable entanglement of microwave photons in an electro-optic system, consisting of superconducting microwave resonators and optical cavities that are filled with certain electro-optic media. The cascaded and parallel schemes realize such entanglement via coherent control on the dynamics of the system, while the dissipative dynamical scheme utilizes the reservoir-engineering approach and exploits the optical dissipation as a useful resource. We show that, for all the schemes, the amount of entanglement is determined by the ratio of the effective coupling strengths of the “beam-splitter” and “two-mode squeezing” interactions, instead of their amplitudes.

© 2017 Optical Society of America

## 1. Introduction

Entanglement is one of the most fascinating aspects of quantum theory. Entangled photons have already been applied to many fields, such as the test of fundamental law of quantum mechanics [1,2], quantum cryptography [3] and other related applications. With the development of quantum information science, microwave radiation has been widely used to couple different quantum systems to form hybrid quantum devices, due to its frequency range covering many types of qubits [4]. Therefore, it is very appealing to generate entangled microwave photons.

Unlike optical photons, it is very difficult to generate entangled microwave photons through the nonlinear optical method with optical crystals. It is thus appealing to propose alternative methods to generate entangled photons in the microwave range. Many different approaches have been explored in different systems: the dissipation-based approach in electro-mechanical systems [5], the coherent-control-based approach in optomechanical systems [6] and electro-mechanical systems [5], as well as the schemes utilizing solid-state superconducting circuits [7–13].

In the previous work, it has been demonstrated that the electro-optic coupling has the same form as the optomechanical [14] and electro-mechanical coupling [5]. So, in principle all the previously considered effects can be observed in electro-optic systems [15]. However, there are several drawbacks that need to be overcome in optomechanical and electro-mechanical systems. One outstanding problem is ground state cooling of the mechanical oscillators. Due to the relatively low frequency of mechanical oscillators (around the microwave range), simple physical cooling can’t cool them down to their ground states [16]. Thus, it requires some other techniques, such as sideband cooling [17–19] and other cooling methods [20]. Besides ground state cooling, the quality factors for high-frequency (*>* 1 GHz) mechanical oscillators are relatively low (*<* 10^{4} [21,22]). However, we can avoid those drawbacks of mechanical oscillators by the use of electro-optic systems. In electro-optic systems, we adopt optical modes as auxiliary modes, whose ground states can be taken as the vacuum states at room temperature. In addition, with the well-developed fabrication technology for optical cavities and superconducting circuits, it is much easier to get optical cavities and superconducting microwave resonators with desired quality factors (larger than 10^{6}) to prepare entanglement states [23,24].

In this work, we propose an electro-optic system with two separated superconducting microwave resonators and one or two auxiliary optical cavities. The optical cavities are filled with certain electro-optic media, while the two resonators are coupled to the optical cavity through the electro-optic effect. With this system, we provide three schemes to entangle these two microwave resonators via the electro-optic effect: (i) the cascaded scheme; (ii) the parallel scheme; (iii) the dissipative dynamical scheme. The underlying physics for both the cascaded and parallel schemes is the coherent control over the system to generate the Bogoliubov modes of the two microwave cavity modes. The last scheme is based on the quantum reservoir engineering approach, which exploits the dissipation of the two optical cavities as a useful resource to entangle the microwave cavity photons. For these schemes, we have obtained the analytic solutions and the numerical simulations. In Section 3, we also discuss the experimental . Especially, Eq. (33) – Eq. (35) show that the temperature dependence of the entanglement degree for the dissipative dynamical scheme is steerable. It is modulated by the decay rates of all the modes and the ratio of the effective coupling strengths. Therefore, high quality entanglement can be realized through suitably choosing the cavities with optimized quality factors.

## 2. The Model and schemes

#### 2.1. The cascaded scheme

As shown in Fig. 1, the hybrid quantum system we considered composes of two microwave resonators with frequencies *ω _{b}*

_{1}(LC1) and

*ω*

_{b}_{2}(LC2), and an optical cavity of frequency

*ω*

_{a}_{1}. The optical cavity is filled with a kind of electro-optic medium (EOM) such as KDP. The two resonators are coupled to the optical cavity through the electro-optic effect, but have no direct interaction with each other.

It is known that we can transfer quantum states between two boson modes through the “beam-splitter” interactions, and can entangle different boson modes by the “two-mode squeezing” interactions [16,25–35]. Therefore, to make both microwave modes entangled, one straightforward idea is to transfer the quantum state of one microwave mode to the auxiliary mode first, and then entangle the auxiliary mode with the other microwave mode. Finally one transfers the state of the auxiliary mode back to the first microwave mode, immediately. We can realize such a proposal by driving the optical cavity with suitably detuned lasers in a cascaded way as shown in Fig. 2: (i) setting LC1 in the red-detuned regime; (ii) setting LC2 in the blue-detuned regime; (iii) setting LC1 in the red-detuned again. Then at the final moment, the Bogoliubov modes composed of the two microwave modes will be excited.

The detailed steps of this scheme are as following. We drive the optical cavity with different lasers in the different periods: (i) 0 *< t < T*_{1}, using the laser of frequency *ω _{L}*

_{1}; (ii)

*T*

_{1}

*< t < T*

_{2}, using the laser of frequency

*ω*

_{L}_{2}; (iii)

*T*

_{2}

*< t < T*

_{1}+

*T*

_{2}, using the laser of frequency

*ω*

_{L}_{1}, again. We assume that

*ω*

_{L}_{1}−

*ω*

_{a}_{1}= −

*ω*

_{b}_{1},

*ω*

_{L}_{2}−

*ω*

_{a}_{1}=

*ω*

_{b}_{2}, in order to guarantee that there is only one microwave mode interacting with the optical mode in each period.

As shown in Ref. [15], the interaction Hamiltonian between the optical mode *a* 1 and the microwave modes *b _{i}*,

*i*= 1, 2 in each period is:

*n*,

*r*

_{0}, and

*d*are the refractive index, electro-optic coefficient and height of the medium respectively.

*C*refers to the capacitance of the

_{i}*i*th resonator.

In the first period, the driving term in the total Hamiltonian is that:

*E*

_{0}is the amplitude of the driving laser field. If we choose a rotating frame with respect to the optical mode

*a*

_{1}of frequency

*ω*

_{L}_{1}, the total Hamiltonian then becomes:

*ω*

_{L}_{1}−

*ω*

_{a}_{1}. In Eq. (4), taking into account the effect of the driving laser, it is a good approximation to linearize the above Hamiltonian through replacing the optical annihilation operator

*a*

_{1}with the sum of its stable mean value

*ā*and its fluctuation term

*δa*[16,25]. Meanwhile the interaction term between the optical mode and LC2 mode has been eliminated by the rotating wave approximation. By employing Heisenberg equation, the zero order and linear terms are eliminated, and we only consider the quadratic terms. Then Eq. (4) becomes:

*G*

_{1}=

*ā*

_{1}

*g*

_{1}, which can be amplified by the driving lasers. The above discussions can also apply to the case in the second and third periods, whose effective coupling strengths are

*G*

_{2}and

*G*

_{1}, respectively. For simplicity we introduce the non-dimension time:

*τ*=

*G*

_{1}

*t*,

*τ*=

_{i}*G*

_{1}

*T*,

_{i}*i*= 1, 2 and the ratio of the effective coupling strengths

*r*=

*G*

_{2}/

*G*

_{1}. Then through solving these Heisenberg equations for all three periods, we can get the final states of the system at

*τ*=

*τ*

_{1}+

*τ*

_{2}:

Here we assume the evolution time for the first and third periods to be equal, so that we can use the same matrix *M*_{1} to describe the time evolution of the system during both periods. The matrix *M*_{2} characterizes the time evolution of the system in the second period. If cos(*τ*_{1}) = 0, at *τ* = *τ*_{1} + *τ*_{2} the time evolution matrix in Eq. (6) becomes:

*τ*=

*τ*

_{1}+

*τ*

_{2}. Moreover, when the two microwave modes are in vacuum states initially, they will be prepared in a two-mode squeezed vacuum state [5].

#### 2.2. Parallel scheme

In the parallel scheme, we keep LC1 in the red-detuned regime, and LC2 in the blue-detuned regime. Therefore, we need to apply two driving lasers with suitable frequencies simultaneously. The total Hamiltonian can be expressed as:

*E*(

_{j}*j*= 1, 2) and initial phases ${\varphi}_{1}=-\frac{\pi}{2}$,${\varphi}_{2}=\frac{\pi}{2}$.

Then we can use the similar approach as used in [36] to simplify the Hamiltonian of our system. In the interaction picture, the total Hamiltonian becomes:

_{1}=

*ω*

_{a}_{1}−

*ω*

_{L}_{1}, Δ

_{2}=

*ω*

_{L}_{2}−

*ω*

_{a}_{1}.

To engineer the desired coupling, we apply an unitary transformation:

*T*is the time-ordering operator. Unlike the cascade scheme, here we assume

*E*/Δ

_{j}*≪ 1. Therefore, we can keep the leading term up to*

_{j}*E*/Δ

_{j}*, instead of making the linearization approximation as before. Equation (14) then becomes,*

_{j}_{1}=

*ω*

_{b}_{1}, Δ

_{2}=

*ω*

_{b}_{2}and using the rotating-wave approximation,

*H*reads:

^{P}*τ*and

*r*are now defined by $\tau ={g}_{1}\frac{{E}_{1}}{{\mathrm{\Delta}}_{1}}t$,

*r*= (

*E*

_{2}Δ

_{1}

*g*

_{2}) /(

*E*

_{1}Δ

_{2}

*g*

_{1}). The non-dimensional decay rates and noise operators are defined by ${k}_{i}=\frac{{\mathrm{\Gamma}}_{i}{\mathrm{\Delta}}_{1}}{2{E}_{1}{g}_{1}}$, ${f}_{i}=\frac{{F}_{i}{\mathrm{\Delta}}_{1}}{{g}_{1}{E}_{1}}$, with {Γ

*,*

_{i}*i*= 1, 2, 3} and {

*F*,

_{i}*i*= 0, 1, 2} the real decay rates and noise operators for each mode in Eq. (18), respectively. According to Ref. [37], ${\u3008{F}_{i}^{\u2020}(t){F}_{j}({t}^{\prime})\u3009}_{R}={\mathrm{\Gamma}}_{i}{n}_{i,th}{\delta}_{ij}\delta (t-{t}^{\prime})$. Thus, it is straightforward that:

*n*,

_{i,th}*i*= 1, 2 refers to the thermal photon number of the

*i*th superconducting microwave resonator. We solve the Heisenberg or Langevin equations for the non-dissipative or dissipative cases, respectively.

- If
*k*≠ 0,_{i}*i*= 0, 1, 2, we can obtain the numerical solutions for the Langevin equations. - If
*k*= 0,_{i}*i*= 0, 1, 2, Eq. (18) becomes a homogeneous equation. The time evolution of the system is:$$\left[\begin{array}{l}{a}_{1}(\tau )\hfill \\ {b}_{1}(\tau )\hfill \\ {b}_{2}^{\u2020}(\tau )\hfill \end{array}\right]={M}^{P}\left[\begin{array}{l}{a}_{1}(0)\hfill \\ {b}_{1}(0)\hfill \\ {b}_{2}^{\u2020}(0)\hfill \end{array}\right],$$$${M}^{P}=\left[\begin{array}{lll}\text{cos}\left(\sqrt{1-{r}^{2}}\tau \right)\hfill & i\frac{\text{sin}\left(\sqrt{1-{r}^{2}}\tau \right)}{\sqrt{1-{r}^{2}}}\hfill & ir\frac{\text{sin}\left(\sqrt{1-{r}^{2}}\tau \right)}{\sqrt{1-{r}^{2}}}\hfill \\ i\frac{\text{sin}\left(\sqrt{1-{r}^{2}}\tau \right)}{\sqrt{1-{r}^{2}}}\hfill & \frac{\text{cos}\left(\sqrt{1-{r}^{2}}\tau \right)-{r}^{2}}{1-{r}^{2}}\hfill & \frac{\left[\text{cos}\left(\sqrt{1-{r}^{2}}\tau \right)-1\right]r}{1-{r}^{2}}\hfill \\ -ir\frac{\text{sin}\left(\sqrt{1-{r}^{2}}\tau \right)}{\sqrt{1-{r}^{2}}}\hfill & \frac{\left[\text{1}-\text{cos}\left(\sqrt{1-{r}^{2}}\tau \right)\right]r}{1-{r}^{2}}\hfill & \frac{1-{r}^{2}\text{cos}\left(\sqrt{1-{r}^{2}}\tau \right)}{1-{r}^{2}}\hfill \end{array}\right].$$

When $\sqrt{1-{r}^{2}}\tau =\pi $, Eq. (21) becomes:

*T*=

_{π}*π*/(1 −

*r*

^{2})

^{1/2}, the optical mode decouples from the dynamics of the system. We assume cosh (

*ξ*) = (1 +

*r*

^{2})/(1 −

*r*

^{2}), sinh (

*ξ*) = 2

*r*/(1 −

*r*

^{2}) and introduce the operator $S=\text{exp}\left\{\xi \left[{b}_{1}(0){b}_{2}(0)-{b}_{1}^{\u2020}\left(0\right){b}_{2}^{\u2020}\left(0\right)\right]\right\}$. Then the annihilation operators of the two microwave modes can be expressed as

*b*

_{1}(

*T*) = −

_{π}*Sb*

_{1}(0)

*S*

^{†}, ${b}_{2}^{\u2020}\left({T}_{\pi}\right)=S{b}_{2}^{\u2020}\left(0\right){S}^{\u2020}$, which indicates that the two microwave modes are prepared in the Bogoliubov modes at the instant

*T*. If the initial states of the microwave modes are the vacuum state, they will be prepared in the two-mode squeezed state with the squeezing parameter

_{π}*ξ*= tanh

^{−1}[2

*r*/(1 +

*r*

^{2})]. That means the degree of squeezing and the amount of entanglement are determined by the ratio

*r*. Figure 3 shows the time evolution of the photon numbers for each mode. For the non-dissipative case shown in Fig. 3 (a), the photon number of the optical cavity drops to 0 and the photon numbers of the superconducting microwave resonators become equal at each instant

*τ*=

*N*×

*T*,

_{π}*N*= 1, 2, 3, . . . . This is in accordance with the conclusion that at that moment the optical mode decouples from the dynamics of the system and the two microwave modes get entangled. However, from Fig. 3 (b) we can see that the periodic fluctuations of the photon numbers are impeded by the dissipations. As a result, the photon number of the optical cavity can’t decrease to 0, and there are still interactions between the optical mode and the microwave modes. Therefore, when the dissipations have great effect, there is no such instant as

*T*that the photons of the two microwave modes can be entangled completely.

_{π}In order to investigate the entanglement properties of the microwave modes, we calculate the total variance *V* = 〈(Δ*u*)^{2} + (Δ*v*)^{2}〉 of EPR-like operators *u* = *x*_{1} + *x*_{2}, *v* = *p*_{1} − *p*_{2}, with ${x}_{i}=\left({b}_{i}+{b}_{i}^{\u2020}\right)/\sqrt{2}$ and ${p}_{i}=-i\left({b}_{i}-{b}_{i}^{\u2020}\right)/\sqrt{2}$, *i* = 1, 2 [5]. According to Ref. [5, 38], the two-mode Gaussian state is entangled if and only if *V <* 2. Especially, for the two-mode squeezed vacuum state (squeezing parameter *ξ*), *V* = 2*e*^{−2}* ^{ξ}*. In Fig. 4 we show the influence of the dissipation and the initial thermal conditions in the entanglement of this system. As illustrated in Fig. 4 (a), the difference between the curves disappears near

*T*. Therefore, such entanglement is insensitive to the initial thermal conditions. However, as shown in Fig. 4 (b), the total variance varies significantly with the decay rates of the system. Thus, the low-dissipation condition should be satisfied in order to get better entanglement.

_{π}#### 2.3. Dissipative dynamical scheme

The parallel scheme and the cascaded scheme are both sensitive to the dissipations as shown in Fig. 4. To overcome this problem, we need to reduce the effect of the dissipation as much as possible in the previous schemes. Thus, high-Q (to ensure the non-dimensional decay rates *k _{i}* ≪ 1) optical cavities are necessary in those cases. However, we can also prepare the target states with low-Q (satisfying

*k*≫ 1) optical cavities (“bad cavities”). In the dissipative dynamical scheme, large decay rates of the optical modes are required due to the fact that the dissipation of the optical modes here are treated as a useful resource. Such schemes have been explored previously in other hybrid quantum systems [5,39]. However, different from the previous works, here we use the optical noise in this system, which leads to the mean thermal photon number

_{i}*n*

_{0}

*≈ 0. Obviously, in the schemes that utilize mechanical noises, the mean thermal phonon number of the mechanical oscillators is much larger than that in our system at the same temperature. Therefore, through the electro-optic system, we can get more ideal two-mode squeezed vacuum states at the same temperature, compared to the optomechanical systems. To realize our scheme, it is necessary to keep LC1 and LC2 in both red- and -blue detuned regimes at the same time. One approach is to add another optical cavity of frequency*

_{,th}*ω*

_{a}_{2}, which satisfies |

*ω*

_{a}_{2}/

*ω*

_{a}_{1}− 1| ≪ 1 as shown in Fig. 5. Through modulating the parameters in Eq. (2), the coupling strengths of the “beam-splitter” and “two-mode squeezing” interactions between the microwave modes and the second optical mode can keep the same form. The ideal situation for this scheme is

*k*

_{0}≫ 1 ≫

*k*, where

_{b}*k*

_{0}is the non-dimensional decay rate of the two optical cavities, while

*k*denotes the non-dimensional decay rate of the two microwave resonators. Therefore, we can ignore the dissipations of the two microwave modes. Then the Langevin equation of this system becomes:

_{b}*k*

_{0}≫ 1, we can make the adiabatic approximations to the optical modes:

*b*

_{1}and ${b}_{2}^{\u2020}$:

We introduce some new variables and operators to simplify our expressions.

*ς*= arctan [

*r*] is the new squeezing parameter of the optical modes. By the definition of

*f̃*, we can get ${\u3008{\tilde{f}}_{ai}\left({\tau}_{\text{new}}\right){\tilde{f}}_{aj}^{\u2020}\left({{\tau}^{\prime}}_{\text{new}}\right)\u3009}_{R}={\delta}_{ij}\delta \left({\tau}_{\text{new}}-{{\tau}^{\prime}}_{\text{new}}\right)$. With Eqs. (27)–(29), the solution of Eq. (26) can be expressed as:

_{ai}*r <*1, under which the system will converge to the final state of the Bogoliubov modes as

*τ*

_{new}→ ∞. As in the parallel scheme, we calculate the total variance of EPR-like operators composed of

*b*

_{1}and ${b}_{2}^{\u2020}$ under the long-time limit, $V=\underset{{\tau}_{\text{new}}\to \infty}{\text{lim}}\frac{2\left(1-r\right)}{1+r}\left(1-{e}^{-2{\tau}_{\text{new}}}\right)=\frac{2(1-r)}{1+r}$. That is exactly the total variance of the ideal two-mode vacuum state 2

*e*

^{−2}

*with the squeezing parameter*

^{ς}*ς*= arctan[

*r*]. Thus, under the assumption that

*k*

_{0}≫ 1 ≫

*k*, regardless of the initial condition, the two microwave modes will finally evolve to the two-mode squeezed vacuum state definitely.

_{b}We turn to considering this scheme in a more general case, where we only make adiabatic approximations to the optical modes. Then the total variance becomes $V={\sum}_{i=1}^{2}\left\{{e}^{-2\left({\tau}_{\text{new}}+{k}_{i}\tau \right)}\left(2{n}_{th,i}+1\right)+\frac{{\left(1-r\right)}^{2}+({n}_{th,i}+1){k}_{0}{k}_{i}}{1-{r}^{2}+{k}_{0}{k}_{i}}\times \left[1-{e}^{-2\left({\tau}_{\text{new}}+{k}_{i}\tau \right)}\right]\right\}$. The effective decay rate in Eq. (26) varies inversely with the decay rate of the optical modes, which explains why we need the optical cavities with large decay rates. In Fig. 6, we present the time evolution of the total variance under different decay rates of the microwave modes together with the result for an ideal two-mode squeezed vacuum state. The initial conditions are chosen as the ground states for the optical cavities and the thermal states for the two microwave modes. From this figure, we can conclude that if the decay rates of the microwave modes are much smaller than the effective decay rate of the system in Eq. (26), we can prepare nearly ideal two-mode squeezed vacuum states when the system becomes stable.

## 3. Experimental feasibility

We now discuss the experimental parameters. Currently available experimental setups are promising platforms for realizing these schemes [40,41]. For the cascade scheme, it is feasible to take the capacitance and frequencies of the superconducting resonators as *C*_{1} = *C*_{2} = 40 fF, *ω _{b}*

_{1}= 2

*π*×3 GHz, and

*ω*

_{b}_{2}= 2

*π*×5 GHz. According to the study on the electro-optic coefficients of lithium niobate at cryogenic temperature [42], the largest product of the cubic refractive index and the electro-optic coefficient can reach 10

^{3}pm/V. The achievable value of such a product in other references varies from 10

^{2}to 10

^{3}[43,44]. Thus, it is reasonable to take

*n*

^{3}

*r*

_{0}≈300 pm/V. For the optical cavities, we can choose the resonance frequency of the optical cavity

*ω*

_{a}_{1}≈ 2

*π*× 200 THz, the decay rate of the superconducting microwave resonators Γ

*≈ 2*

_{i}*π*× 1 kHz,

*i*= 1, 2, and the distance between the two planes of each capacitor

*d*≈ 10

*μ*m. The pump power can be

*P*≈ 10 mW [41]. Thus, the coefficients

*g*given by Eq.(2) can reach

_{i}*g*

_{1}≈ 2

*π*× 15 kHz, and

*g*

_{2}≈ 2

*π*× 19 kHz. In the “overcoupled” case, we can also obtain the mean photon number of the optical cavity caused by the external pump

*n̄*

_{cav,i}through the following equation [16]

_{0}). For the Q-factor of the optical cavities exceeding 10

^{8}, we have Γ

_{0}≈ 2

*π*× 0.2 MHz. In our case we choose Δ

*=*

_{i}*ω*, and get

_{bi}*n̄*

_{cav,1}≈ 400,

*n̄*

_{cav,2}≈ 144. Then the effective electro-optic coupling strength can reach $\sqrt{{\overline{n}}_{\text{cav},\text{1}}}{g}_{1}\approx 2\pi \times 0.3$ MHz, and $\sqrt{{\overline{n}}_{\text{cav},\text{2}}}{g}_{2}\approx 2\pi \times 0.23$ MHz. If we set the time for the “two-mode squeezing” interaction

*T*

_{2}∼ 1.6

*μ*s, the operation time for generating the target states will be

*T*=

_{c}*π*/

*G*

_{1}+

*T*

_{2}∼ 3.2

*μ*s in the cascaded scheme.

As for the parallel and dissipative dynamical schemes, we assume the pump power is relatively low, i.e. *P* = 10 *μ*W. Other related parameters are chosen as: the distance *d* = 5 *μ*m, the capacitance and inductance of the superconducting microwave resonators *C*_{1} = *C*_{2} = 4 fF, *ω _{b}*

_{1}= 2

*π*× 8 GHz,

*ω*

_{b}_{2}= 2

*π*× 10 GHz, and Γ

_{1}= Γ

_{2}≈ 2

*π*× 1 kHz. In our case, Γ ≪ Δ

*. Then the amplitudes of the driving lasers in both schemes can be expressed as ${E}_{i}=\sqrt{{\overline{n}}_{\text{cav},\hspace{0.17em}i\hspace{0.17em}\text{new}}}{\omega}_{bi}$,*

_{i}*i*= 1, 2. Therefore, the operation time for the parallel scheme to generate the target states is ${T}_{p}=\pi /\left[{g}_{1}\sqrt{{\overline{n}}_{\text{cav},1\text{new}}-{\overline{n}}_{\text{cav},2\text{new}}}\right]~4.2\mu \text{s}$ and the time for reaching the stationary state of the dissipative dynamical scheme ${T}_{d}=\mathrm{\Gamma}/\left[2{g}_{1}^{2}\left({\overline{n}}_{\text{cav},1\text{new}}-{\overline{n}}_{\text{cav},2\text{new}}\right)\right]~1.65\mu \text{s}$ times are much shorter than the photon lifetime in the superconducting microwave resonators.

Furthermore, if a large inductance is allowed, the effective electro-optic coupling strength will exceed 2*π* × 1 MHz. For example, if we take *L* = 63 *μ*H and keep the other parameters the same as those in the cascaded scheme, the effective electro-optic coupling strength can reach 2*π*×1.6 MHz and the optical loss rate can be ignored.

We consider the entanglement properties of systems in each scheme. From Eq.(9), we can see that the squeezed parameter of the cascaded scheme for the ideal case is determined by *ζ* = *r* (*τ*_{2} − *τ*_{1}). Obviously, *ζ* will increase with the increase of *τ*_{2}. However, the detrimental effect of decoherence also becomes great when *τ*_{2} increases. Therefore, it needs to get a balance between both aspects. We assume the temperature is approximately 100 mK. For the parameters in the cascaded scheme, the thermal photon numbers are *n _{th,}*

_{1}≈ 0.3, and

*n*

_{th,}_{2}≈ 0.1. Through the numerical simulations shown in Fig. 7, we can find that the optimized scaled time

*τ*

_{2}is 2.43 or equally

*T*

_{2}≈ 1.3

*μ*s, and the minimum total variance is approximately 1.56. We also find that the total variance is insensitive to the environment temperature when it is below 1 K, but greatly relies on the scaled decay rates of all the modes

*k*= Γ

_{i}*/2*

_{i}*G*

_{1},

*i*= 0, 1, 2. Thus, we can suitably choose those related parameters to improve the quality of the target states. For example, when we change the capacitance and inductance to

*C*

_{1}=

*C*

_{2}= 1 fF,

*L*

_{1}= 360 nH,

*L*

_{2}= 350 nH, at the same temperature, the minimal variance drops to

*V*≈ 0.77.

In the parallel scheme, the total variance is greatly affected not only by the scaled decay rates, but also by the environment temperature. With those experimental parameters when discussing the operation time of this scheme, the scaled decay rates are *k*_{0} ≈ 0.9, *k*_{1} ≈ *k*_{2} ≈ 0.003. At the temperature *T* = 100 mK, the lowest variance is 0.66, but at the temperature *T* = 1 K, the entanglement will be destroyed.

In order to discuss the total variance of the dissipative dynamical scheme, it is useful to simplify the expression of the final variance as following:

Here we assume that the two superconducting resonators have the same decay rates Γ_{1}= Γ

_{2}, and

*γ*is the effective decay rate of the system. If we choose Γ

_{0}= 2

*π*× 30 MHz, Γ

_{1}= Γ

_{2}≈ 2

*π*× 1.44 kHz and keep the other parameters the same as those for the parallel scheme, at the temperature

*T*= 100 mK, the total variance decreases to 0.3.

## 4. Conclusions

To conclude, we have proposed three schemes to generate entangled microwave photons with an electro-optic system, in which two superconducting microwave resonators are coupled by one or two optical cavities through the electro-optic effect. The first two schemes are based on the coherent control over the system, while the last scheme is based on the dissipative dynamics engineering approach, which exploits the dissipation of the two optical cavities as a useful resource to entangle the microwave modes. These schemes based on the electro-optic system may have interesting applications in quantum information processing.

## Funding

NSFC (11474227 and 11774285); Fundamental Research Funds for the Central Universities.

## References and links

**1. **H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. **93**, 020401 (2004). [CrossRef] [PubMed]

**2. **V. C. Vivoli, T. Barnea, C. Galland, and N. Sangouard, “Proposal for an optomechanical bell test,” Phys. Rev. Lett. **116**, 070405 (2016). [CrossRef] [PubMed]

**3. **T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. **84**, 4729–4732 (2000). [CrossRef] [PubMed]

**4. **Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. **85**, 623–653 (2013). [CrossRef]

**5. **P. B. Li, S. Y. Gao, and F. L. Li, “Robust continuous-variable entanglement of microwave photons with cavity electromechanics,” Phys. Rev. A **88**, 043802 (2013). [CrossRef]

**6. **L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. **110**, 233602 (2013). [CrossRef] [PubMed]

**7. **S. L. Ma, Z. Li, A. P. Fang, P. B. Li, S. Y. Gao, and F. L. Li, “Controllable generation of two-mode-entangled states in two-resonator circuit QED with a single gap-tunable superconducting qubit,” Phys. Rev. A **90**, 062342 (2014). [CrossRef]

**8. **C. P. Yang, Q. P. Su, S. B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A **87**, 022320 (2013). [CrossRef]

**9. **H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. **106**, 060401 (2011). [CrossRef] [PubMed]

**10. **S. Dambach, B. Kubala, and J. Ankerhold, “Generating entangled quantum microwaves in a josephson-photonics device,” New J. Phys. **19**, 023027 (2017). [CrossRef]

**11. **G. L. Cheng, A. X. Chen, and W. X. Zhong, “Tripartite entanglement of microwave radiation via nonlinear parametric interactions enhanced by quantum interference in superconducting quantum circuits,” J. Opt. Soc. Am. B **30**, 2875–2881 (2013). [CrossRef]

**12. **E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. **109**, 250502 (2012). [CrossRef]

**13. **X. Y. Lü, L. L. Zheng, P. Huang, J. Li, and X. X. Yang, “Adiabatic passage scheme for entanglement between two distant microwave cavities interacting with single-molecule magnets,” J. Opt. Soc. Am. B **26**, 1162–1168 (2009). [CrossRef]

**14. **A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “On fundamental quantum noises of whispering gallery mode electro-optic modulators,” Opt. Express **15**, 17401–17409 (2007). [CrossRef] [PubMed]

**15. **M. Tsang, “Cavity quantum electro-optics,” Phys. Rev. A **81**, 063837 (2010). [CrossRef]

**16. **M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. **86**, 1391–1452 (2014). [CrossRef]

**17. **J. Teufel, T. Donner, D. Li, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature **475**, 359–363 (2011). [CrossRef] [PubMed]

**18. **A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. **4**, 415–419 (2008). [CrossRef]

**19. **R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. **116**, 063601 (2016). [CrossRef] [PubMed]

**20. **M. Frimmer, J. Gieseler, and L. Novotny, “Cooling mechanical oscillators by coherent control,” Phys. Rev. Lett. **117**, 163601 (2016). [CrossRef] [PubMed]

**21. **A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature **464**, 697–703 (2010). [CrossRef]

**22. **W. C. Jiang, X. Lu, J. Zhang, and Q. Lin, “High-frequency silicon optomechanical oscillator with an ultralow threshold,” Opt. Express **20**, 15991–15996 (2012). [CrossRef] [PubMed]

**23. **G. Brawley, M. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. **7**, 10988 (2016). [CrossRef] [PubMed]

**24. **C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A **94**, 053815 (2016). [CrossRef]

**25. **Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A **89**, 022332 (2014). [CrossRef]

**26. **J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A **83**, 033820 (2011). [CrossRef]

**27. **C. J. Yang, J. H. An, W. L. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A **92**, 062311 (2015). [CrossRef]

**28. **C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express **21**, 12165–12173 (2013). [CrossRef] [PubMed]

**29. **Z. Li, S. L. Ma, and F. L. Li, “Generation of broadband two-mode squeezed light in cascaded double-cavity optomechanical systems,” Phys. Rev. A **92**, 023856 (2015). [CrossRef]

**30. **C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A **78**, 032316 (2008). [CrossRef]

**31. **S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A **84**, 042342 (2011). [CrossRef]

**32. **A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. **103**, 213603 (2009). [CrossRef]

**33. **S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. **109**, 130503 (2012). [CrossRef] [PubMed]

**34. **Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A **80**, 065801 (2009). [CrossRef]

**35. **T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science **321**, 1172–1176 (2008). [CrossRef] [PubMed]

**36. **P. B. Li, S. Y. Gao, and F. L. Li, “Engineering two-mode entangled states between two superconducting resonators by dissipation,” Phys. Rev. A **86**, 012318 (2012). [CrossRef]

**37. **M. O. Scully and M. S. Zubairy, *Quantum optics* (Cambridge University, Cambridge, 1997). [CrossRef]

**38. **L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. **84**, 2722–2725 (2000). [CrossRef] [PubMed]

**39. **Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. **110**, 253601 (2013). [CrossRef] [PubMed]

**40. **J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature **471**, 204–208 (2010). [CrossRef]

**41. **S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature **460**, 724–727 (2009). [CrossRef] [PubMed]

**42. **C. Herzog, G. Poberaj, and P. Günter, “Electro-optic behavior of lithium niobate at cryogenic temperatures,” Opt. Commun. **281**, 793–796 (2008). [CrossRef]

**43. **Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park, and Q. M. Zhang, “Composition, temperature, and crystal orientation dependence of the linear electro-optic properties of Pb(Zn1/3Nb2/3)O-3-PbTiO3 single crystals,” Appl. Phys. Lett. **77**, 1247–1249 (2000). [CrossRef]

**44. **P. Chen, X. G. Tu, S. P. Li, J. C. Li, W. Lin, H. Y. Chen, D. Y. Liu, J. Y. Kang, Y. H. Zuo, and L. Zhao, “Enhanced Pockels effect in GaN/AlxGa1-xN superlattice measured by polarization-maintaining fiber Mach-Zehnder interferometer,” Appl. Phys. Lett. **91**, 1447 (2007).