Preparation of entangled states of microwave photons in a hybrid system via electro-optic effect

We propose to realize the two-mode continuous-variable entanglement of microwave photons in an electro-optic system, consisting of two superconducting microwave resonators and one or two optical cavities filled with certain electro-optic medium. 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"beam-splitter"and"two-mode squeezing"interactions, rather than their amplitudes.

Unlike optical photons, it is hard to entangle microwave photons through nonlinear optical methods with optical crystals. It is thus appealing to propose alternative approaches to generate entangled microwave photons. Many different approaches have been explored in different systems. For instance, some theoretical works have presented the dissipation-based approach in electromechanical systems [18], the coherent-control-based approach through excitations of cavity Bogoliubov modes in optomechanical systems [19] and electro-mechanical systems [18], as well as the schemes utilizing solid-state superconducting circuits [20][21][22][23][24][25][26].
In previous works, it has been demonstrated that the electro-optic coupling has the same form as optomechanical [27] and electro-mechanical couplings [18]. So all previous considered effects can in principle be observed in electro-optic systems [28]. But there are several challenges of optomechanical and electro-mechanical systems to cool down their nano-/micro-mechanical oscillators. First, due to the low frequencies of mechanical oscillators, the required environment temperature is ultra low. Moreover, there's a limitation of physical cooling as demonstrated in [29] that mechanical oscillators can not be cooled down to their ground states under the influence of cavity frequency noise. Besides cooling processes, the quality factors of high-frequency mechanical oscillators are relatively low. However, we can avoid those drawbacks of micro mechanical oscillators through electro-optic systems. Auxiliary modes in electro-optic systems are optical modes, which can be regard as vacuum states at experimental temperatures. In addition, with the well-developed fabrication of optical cavities and superconducting circuits, it is convenient to get optical cavities with desired quality factors as well as low-dissipation superconducting microwave resonators to prepare high-quality entanglement states.
In this work, inspired by optomechanical systems [29-40] and other hybrid quantum systems [18,27,28,41], we propose an electro-optic system comprising two separated superconducting microwave resonators and one or two auxiliary optical cavities, which are filled with certain electro-optic medium. With this system, we provide three schemes to entangle these two microwave resonators via electro-optic effect: (i) cascaded scheme; (ii) parallel scheme; (iii) dissipative dynamical scheme. The underlying physics for both cascaded and parallel schemes is the coherent control over their systems to realize the Bogoliubov modes consisting of the two microwave modes, while the last scheme is based on quantum reservoir engineering, which exploits the dissipations of two optical cavities as useful resources to entangle microwave photons. For each scheme, we've worked out the analytic solutions and numerical simulations. In Section 3, we also talk about the experimental feasibility of all schemes. Especially, Eq.(40)-Eq.(42) shows that the temperature dependence of the entanglement degree for the dissipative dynamical scheme is steerable. It's modulated by the decay rate of both optical cavities and superconducting microwave resonators, as well as the ratio of effective coupling strengths. Therefore, high quality entanglement can be realized through choosing cavities and resonators of optimized quality factors.

The cascaded scheme
As shown in Fig. 1, the hybrid quantum system we considered composes of two microwave resonators with frequency ω b1 (LC1) and ω b2 (LC2), and an optical cavity of frequency ω a1 , inside which a kind of electro-optic medium(EOM) such as KDP is filled. These two resonators are coupled to the optical cavity through electro-optic effect, but have no direct interaction with each other.
It is known that the effect of "beam-splitter" interaction is to exchange quantum states between two modes, and that of "two-mode squeezing" interaction is to get both modes entangled. Therefore, one straightforward approach to entangle the microwave photons in these separated resonators is to drive the optical cavity of this system with suitably detuned lasers in a cascaded way as shown in Fig. 2: (i) to set LC1 in the red-detuned regime; (ii) to set LC2 in the blue-detuned regime; (iii) to set LC1 in the red-detuned again. Then at the final moment, the Bogoliubov modes composed of the two resonator modes only will be excited.
The detailed steps of this scheme are as the following. We drive the optical cavity with different lasers in different periods: (i) 0 < t < T 1 , using the laser of frequency ω L1 ; (ii) T 1 < t < T 2 , using the laser of frequency ω L2 ; (iii) T 2 < t < T 1 + T 2 , we use the laser of frequency ω L1 , again. We assume ω L1 − ω a1 = −ω b1 , ω L2 − ω a1 = ω b2 to guarantee that the microwave resonators are in the suitable detuned regimes. Through the rotating-wave approximation, in each period there is only one microwave mode interacting with the optical mode. In other words, we set LC1 to be in the red-detuned regime and LC2 to be in the blue-detuned regime when they interact with the optical cavity, and both of them are isolated when they are far detuned from the optical cavity. As shown in [28], when we only consider one mode of the optical cavity, the interaction Fig. 1. The setup of both cascaded and parallel schemes. In the cascade scheme, the optical cavity is driven by a laser of frequency ω L1 or ω L2 for different periods, while in the parallel scheme, we impose both of these driving lasers at the same time.
Hamiltonian in each period is: where n, r 0 , and d are the refractive index, electro-optic coefficient and height of the medium respectively. C i refers to the capacitance of the i th resonator.
In the first period, the driven term in the total Hamiltonian is that: where E 0 is the complex amplitude of the driving laser. If we choose a rotating frame with frequency ω L1 respect to the optical mode a 1 , the total Hamiltonian then becomes: where ∆ = ω L1 − ω a1 . In Eq.(4), we have assumed that the driving laser is strong enough. Therefore, it is a good approximation to linearize the above Hamiltonian through replacing the Fig. 2. A diagram of the process of the cascaded scheme, and a 1 , b 1 , b 2 are annihilation operators of optical mode and two microwave modes, respectively. When 0 < t < T 1 or T 2 < t < T 1 + T 2 , LC1 and the optical cavity exchange quantum states with each other, and during T 1 < t < T 2 , modes of optical cavity and LC2 get entangled.
optical annihilate operator a 1 by the sum of its stable mean valueā and its fluctuation term δa. The interaction term between the optical mode and LC2 mode has been eliminated by the rotating wave approximation. Employing Heisenberg equation, the zero order and linear terms are eliminated, so we only consider the quadratic terms. Then Eq.(4) becomes: The effective coupling strength in the first period isā 1 g 1 , which can also be modulated by the power of the driving lasers. For simplicity we introduce the non-dimension time: Then in the interaction picture, the Heisenberg equations for all three modes can be written as: The solution of these equations is straightforward: Similarly, in the second period we have: where r =ā 2 g 2 /ā 1 g 1 is the ratio of the coupling strength between the blue-detuned and red-detuned type interactions, and ∆τ 1 = τ − τ 1 . As for the third period: where ∆τ 2 = τ − τ 2 . Now we can get the final state of the system at τ If cos(τ 1 ) = 0, at τ = τ 1 + τ 2 the evolution matrix in Eq.(10) becomes: Eq.(13) shows at the instant τ = τ 1 + τ 2 , the optical mode decouples with the two LC modes, and the two LC modes form the Bogoliubov modes. Moreover, in the case of initial vacuum states, the Bogoliubov modes turn to two-mode squeezed vacuum state [18].

Parallel scheme
In the parallel scheme, we want to set LC1 in the red-detuned regime, meanwhile LC2 is in the blue-detuned regime. Therefore, we need to apply two driving lasers of suitable frequencies simultaneously. The total Hamiltonian can be expressed as: Here we apply driving signals with real amplitudes E j ( j = 1, 2) and initial phases φ 1 = − π 2 , φ 2 = π Then we can use similar approach as used in [42] to simplify the Hamiltonian of our system. In the interaction picture, the total Hamiltonian becomes: where To engineer our desired coupling, we need an unitary transformation with the following defined unitary operator: where T is the time-ordering operator. Unlike the cascade scheme requiring extreme strong lasers, the parallel scheme requires lasers with relatively weak intensities: E j /∆ j << 1 such that we can keep the leading term of E j /∆ j . Eq.(18) then becomes, Through setting ∆ 1 = ω b1 , ∆ 2 = ω b2 and using the rotating-wave approximation, H P turns to: H P e f f = − g 1 Eq.(21) yields the Langevin equation of this system, d dτ where τ and r are now defined by τ = g 1 We solve the Heisenberg equation and Langevin equation for the non-dissipative and dissipative cases, respectively. (i)If k i = 0, i = 0, 1, 2, Eq.(22) converts into a homogeneous equation. The time evolution of the system is: (ii) If k i 0, i = 0, 1, 2, the time evolution of this system can be solved using the theory of linear differential equations. We assume that: The eigenvalues and corresponding column eigenvectors of matrix c are {λ i , ì u i , i = 1, 2, 3}, and the time evolution of such system can be written as: It is not necessary for us to get exact analytic solutions in this case, so we can get numerical solutions from the expressions above.
As for the non-dissipative case, when √ 1 − r 2 τ = π, Eq.(25) becomes: Eq.(29) indicates that at the instant T π = π/ √ 1 − r 2 , the optical mode decouples from the dynamics of the system. We assume cosh(ξ) = (1 + r 2 )/(1 − r 2 ), sinh(ξ) = 2r/(1 − r 2 ) and introduce the operator 0)S † , indicating the two superconducting microwave resonators are prepared in the Bogoliubov modes. If the initial states of the superconducting microwave resonators are vacuum states, they will be prepared in the two-mode squeezed state with the squeezing parameter ξ = tanh −1 [2r/(1 + r 2 )] at the moment T π . Fig.3 shows the time evolution of the photon numbers. For the non-dissipative case shown in Fig.3(a), at the instant T π the photon number of the optical cavity drops to 0 and the photon numbers of the superconducting microwave resonators become equal. This is in accordance with the conclusion that at that moment the optical mode decouples from the dynamics of the system and the microwave modes get entangled. From Fig.3(b), we can see that the periodic fluctuations of photon numbers are impeded by the dissipations. As a result, the photon number of the optical cavity can't decrease to 0, reflecting that some photons of microwave modes still interact with the remaining photons of optical modes. Therefore, when the effect of dissipations is notable, there is no such instant as T π that the photons of the two microwave modes can be entangled completely In order to investigate the entanglement of modes b 1 and b 2 , we need the total variance V = (∆u) 2 + (∆v) 2 of EPR-like operators u = In (a) the ideal case, at each instant nπ, the photon number of optical mode goes to zero and those numbers of LC1 and LC2 become equal, which is in accordance with the decoupling of optical mode and the entanglement of two superconducting microwave resonators. From (b), we can find that even for optical cavity, the photon number can not be zero at steady state. [18]. The two-mode Gaussian state is entangled if and only if V < 2 [18,44]. Especially, for the two-mode squeezed vacuum state, V = 2e −2ξ . We explore the effects of the dissipation and the initial thermal conditions to the entanglement of this system as illustrated in Fig.4. In Fig.4(a), we change the initial thermal condition of each mode, and find that at the neighborhood of T π the differences of all curves disappear. We've already known that at the instant T π , the two microwave modes are entangled. Therefore, the entanglement of this system is insensitive to the initial thermal conditions. But in Fig.4 (b), when we modulate the decay rates of all modes, the total variances vary greatly with them. Thus, the low-decay condition should be satisfied in order to get better entanglements.

Dissipative dynamical scheme
The system in the parallel scheme is very sensitive to dissipations as shown in Fig.4, and so does the cascaded scheme. This indicates in previous schemes, we need to attenuate the dissipation as much as possible. Thus, in many cases, high-Q cavities or resonators are necessary. But we can also prepare our target states with low-Q optical cavities("bad cavities"). In the dissipative dynamical scheme, large decay rates of the optical modes are required due to the fact that the dissipative effects of the optical modes here are treated as a useful resources. Such schemes in other hybrid quantum systems have been explored previously [18]. However, in our system, what we use is the optical thermal noise, where the mean photon number at thermal equilibrium is n 0,th ≈ 0. Therefore, through the electro-optic system we can get more ideal two-mode squeezed vacuum states at the same temperature, compared with the optomechanical systems. To realize our scheme, it is necessary to put LC1 and LC2 in both red-and -blue detuned regimes at the same time. One approach is to add another optical cavity of frequency ω a2 satisfying |ω a2 /ω a1 − 1| << 1 paralleled to the previous one as shown in Fig.5. Through modulating the parameters in Eq.(2), the coupling strengths of the "beam-splitter and "two-mode squeezing" interactions in the second optical cavity can keep the same as the first one. The ideal situation for this scheme is k 0 >> 1 >> k 1 , where k 0 stands for the non-dimensional decay rate of two optical cavities while k 1 denotes the non-dimensional decay rate of two microwave resonators. Therefore, we can ignore the dissipation of two microwave modes, Then the Langevin equation of this system becomes: Here the definition of all the non-dimensional variables has the same form as that in the parallel scheme. In the ideal situation, we can make the adiabatic approximations to the optical modes: Inserting Eq.(31) and Eq.(32) into Eq.(30) yields the relationship between b 1 and b † 2 : We introduce some new variables and operators to simplify our expressions.
where ς = arctan [r] is the squeezing parameter of optical modes, and by the definition off ai , we can get f ai (x)f † a j (x ) R = δ i j δ(x − x ). With Eq.(34)-(36), the solution of Eq.(33) can be expressed as: From Eq.(34)-(38), we can find the stable condition for this system is r < 1, under which as x → ∞ the system will converge to the final state of the Bogoliubov modes composed of optical modes. As in the parallel scheme, we calculate the total variance of EPR-like operators composed of b 1 and b † 2 in the stable case, V = lim 1+r . That is exactly the total variance of the ideal two-mode vacuum state 2e −2ς with squeezing parameter ς = arctan[r]. Thus, under the assumption that k 0 1 k 1 , no mater what the initial condition is, the two microwave modes will finally evolve to the two-mode squeezed vacuum state, definitely.
Let's consider this scheme in a more general case, where we only make adiabatic approximations to optical modes. Then such total variance becomes V = Σ 2 i=1 e −2(x+k i τ) (2n th,i + 1) + The effective decay rate in Eq.(33) varies inversely with the decay rate of optical modes, and that explains why we need the optical cavity with a large decay rate. In Fig.6, we present the time evolution of the total variance under different decay rates of microwave modes as well as the result of 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 know that when the decay rates of microwave modes are much smaller than the effective decay rate, at steady state we can prepare nearly ideal two-mode vacuum squeezed states.

Experimental Feasibility
We now talk about the experimental parameters. For the cascaded scheme, it is feasible to take the capacitances and inductances of LC resonators as C 1 = C 2 =40fF, L 1 =70nH, L 2 =25nH. Similar to that electro-optic system reported by Mankei Tsang [28], we can take the electro-optic coefficient n 3 r 0 ≈300pm/V, the resonance frequency of optical cavity ω a1 ≈ 2π × 200THz, the decay rate of superconducting microwave resonators Γ i ≈ 2π × 1KHz, i = 1, 2, and assume the distance between two planes of each capacitor d ≈ 10µm. The pump power is able to reach P = 10mW [45]. Thus, the coefficients g i given by Eq.(2) can reach g 1 ≈ 2π × 15KHz, g 2 ≈ 2π × 19KHz, and ω b1 ≈ 2π × 3GHz, ω b2 ≈ 2π × 5GHz. In the "overcoupled" case, we can also work out the mean photon number of optical cavity caused by external pumpn cav,i through the following equation [29]n cav,i = where Γ is the total loss rate of optical cavity, which is dominated by external loss rate of the cavity. For the Q-factor of optical cavities can reach 10 8 , we have Γ 0 ≈ 2π × 0.3MHz. In our case we choose ∆ i = ω bi , and getn cav,1 ≈ 400,n cav,2 ≈ 144. Then the effective electro-optic coupling strength can reach √n cav,1 g 1 ≈ 2π × 0.3MHz, √n cav,2 g 2 ≈ 2π × 0.23MHz. If we set the time for "two-mode squeezed" interaction is T 2 ∼ 1.6µs, the operation time for generating target states will be T c = π/(ā 1 g 1 ) + T 2 ∼ 3.2µs in the cascaded scheme.
As for parallel and dissipative dynamical schemes, we assume the pump power is relatively low, i.e. P = 10µW, the distance d = 5µm, the capacitances and inductances of superconducting microwave resonators C 1 = C 2 = 4fF, L 1 = 700nH, L 2 = 250nH, Γ 1 = Γ 2 ≈ 2π × 1KHz, and use the optical cavity with resonance frequency 2π × 1500THz(λ ≈ 200nm) to ensure the validity of the expansion applied in both schemes. In our case, Γ ∆ i . Then the amplitudes of driving lasers in both schemes can be expressed as E i = √n cav,inew ω bi , i = 1, 2. Therefore, the operation time for the parallel scheme to generate target states is T p = π/ g 1 √n cav,1new −n cav,2new ∼ 3.8µs, and the time for reaching stationary state of the dissipative dynamical scheme T d = Γ/ 2g 2 1 n cav,1new −n cav,2new ∼ 1.3µs. These times are much shorter than the photon lifetime in superconducting microwave resonators.
Further more, if it is allowed to realize large inductance, the effective electro-optic coupling strength will exceed 2π × 1MHz. For example, we take L = 63µH and keep other parameters the same as those in the cascaded scheme, its effective electro-optic coupling strength will reach 2π×1.6MHz and the optical loss rate can be ignored for simplicity.
We are also concerned about the entanglement properties of systems in each scheme. From Eq.(13), we can see the squeezed parameter of the cascaded scheme in ideal case is determined by ζ = r (τ 2 − τ 1 ). Obviously, ζ will increase with the increase of τ 2 . But effects of dissipation and decoherence also grow when τ 2 increases. Therefore, people need to strike a balance between both aspects. We assume the temperature is approximately 100mK. In the previous experimental parameters in the cascaded scheme, the thermal photon numbers are n th,1 ≈ 0.3, n th,2 ≈ 0.1. Through numerical simulation shown in Fig.7, we can find the optimized scaled time τ 2 is 2.43 or equally T 2 ≈ 1.3µs, and the minimum of total variance is approximately 1.56. We find that the total variance is insensitive to the environment temperature when it is below 1K, but greatly relies on scaled decay rates of all modes k i = Γ i /(2ā 1 g 1 ) , i = 0, 1, 2. Thus, we can reset those related parameters to improve the quality of target states. For example, when we change capacities and inductances to C 1 = C 2 = 1fF, L 1 = 360nH, L 2 = 350nH, at the same temperature, the minimal variance drops to V ≈ 0.77. And we can see that the optimized scaled time is τ 2 = 2.43, or equally T 2 ≈ 1.3µs.
In the parallel scheme, the total variance is greatly affected by not only scaled decay rates, but also the environment temperature. With those experimental parameters used for discussing operation times of this scheme, scaled decay rates are k 0 ≈ 0.9, k 1 ≈ k 2 ≈ 0.003. At the temperature T = 100mK, the lowest variance is 0.66, but at the temperature T = 1K, 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 its stable variance as following: V = 2 (1 − r) /(1 + r) + 2α n th,1 + n th,2 + 2 1 + 2α , Where we've assumed two superconducting resonators have same the decay rate Γ 1 = Γ 2 , and γ is the effective decay rate of the system. We choose Γ 0 = 2π × 30MHz, Γ 1 = Γ 2 ≈ 2π × 1.44KHz and keep other parameters same as the parallel scheme. At the same temperature T = 100mK, the total variance can decrease to 0.3.

Conclusions
To conclude, we propose three schemes to generate the entanglement of microwave photons with an electro-optic system, in which two superconducting microwave resonators are coupled by one or two optical cavities through electro-optic effect. The first two schemes are based on coherent control over the system to realize Bogoliubov modes consisting of two microwave modes while the last scheme is based on dissipative dynamics engineering, which exploits the thermal noises of two optical cavities as useful resources to entangle microwave modes. Compared to previous works, our electro-optic system can generate more ideal two-mode squeezed states in principle. These schemes based on the electro-optic system may have novel applications in quantum information processing.