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 (< 104 [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 106) 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 ωb1 (LC1) and ωb2 (LC2), and an optical cavity of frequency ωa1. 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.

 figure: Fig. 1

Fig. 1 The setup of both the cascaded and parallel schemes. In the cascade scheme, the optical cavity is driven by a laser of frequency ωL1 or ωL2 for the different periods, while in the parallel scheme, we apply both of these driving lasers at the same time.

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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.

 figure: Fig. 2

Fig. 2 A diagram of the process of the cascaded scheme, and a1, b1, b2 are the annihilation operators for the optical mode and two microwave modes, respectively. When 0 < t < T1 or T2 < t < T1 + T2, LC1 and the optical cavity exchange quantum states with each other, and during T1 < t < T2, the optical cavity mode and the LC2 mode get entangled.

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The detailed steps of this scheme are as following. We drive the optical cavity with different lasers in the different periods: (i) 0 < t < T1, using the laser of frequency ωL1; (ii) T1 < t < T2, using the laser of frequency ωL2; (iii) T2 < t < T1 + T2, using the laser of frequency ωL1, again. We assume that ωL1ωa1 = −ωb1, ωL2ωa1 = ωb2, 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 bi, i = 1, 2 in each period is:

HI,iC=gia1a1(bi+bi),
gi=ωa1n3r02dωbi2Ci,
where n, r0, and d are the refractive index, electro-optic coefficient and height of the medium respectively. Ci refers to the capacitance of the i th resonator.

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

HdC=i[E0a1eiωL1tE0*a1eiωL1t],
where E0 is the amplitude of the driving laser field. If we choose a rotating frame with respect to the optical mode a1 of frequency ωL1, the total Hamiltonian then becomes:
H1C=Δa1a1+ωb1b1b1+ωb2b2b2g1a1a1(b1+b1)+i(E0a1E0*a1),
where Δ = ωL1ωa1. 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 a1 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:
H1,effC=Δδa1δa1+ωb1b1b1+ωb2b2b2G1(δa1b1+δa1b1).
The effective coupling strength in the first period is G1 = ā1g1, 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 G2 and G1, respectively. For simplicity we introduce the non-dimension time: τ = G1t, τi = G1Ti, i = 1, 2 and the ratio of the effective coupling strengths r = G2/G1. Then through solving these Heisenberg equations for all three periods, we can get the final states of the system at τ = τ1 + τ2:
[δa1(τ1+τ2)b1(τ1+τ2)b2(τ1+τ2)]=M1M2M1[δa1(0)b1(0)b2(0)],
M1=[cos(τ1)isin(τ1)0isin(τ1)cos(τ1)0001],
M2=[cosh[r(τ2τ1)]0isinh[r(τ2τ1)]010isinh[r(τ2τ1)0cosh[r(τ2τ1)]].

Here we assume the evolution time for the first and third periods to be equal, so that we can use the same matrix M1 to describe the time evolution of the system during both periods. The matrix M2 characterizes the time evolution of the system in the second period. If cos(τ1) = 0, at τ = τ1 + τ2 the time evolution matrix in Eq. (6) becomes:

M1M2M1=[1000cosh[r(τ2τ1)]sinh[r(τ2τ1)]0sinh[r(τ2τ1)]cosh[r(τ2τ1)]].
Equation (9) shows that the optical mode decouples from the two microwave modes, and these two microwave modes form the Bogoliubov modes at the instant τ = τ1 + τ2. 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:

H=H0+HI+Hd,
H0=ωa1a1a1+i=12ωbibibi,
HI=j=12gja1a1(bj+bj),
Hd=j=12(1)jEj(a1eiωLjt+a1eiωLjt).
Here the driving lasers have real amplitudes Ej (j = 1, 2) and initial phases ϕ1=π2,ϕ2=π2.

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:

H=j=12gja1a1(bjeiωbjt+bjeiωbjt)+j=12(1)jEj[a1e(1)j1iΔjt+a1e(1)jiΔjt],
where Δ1 = ωa1ωL1, Δ2 = ωL2ωa1.

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

U=Texp{i0tj=12(1)jEj[a1e(1)j1iΔjt+a1e(1)jiΔjt]dt},
where T is the time-ordering operator. Unlike the cascade scheme, here we assume Ejj ≪ 1. Therefore, we can keep the leading term up to Ejj, instead of making the linearization approximation as before. Equation (14) then becomes,
HP=U(Hit)U=k=12gk{a1a1+j=12EjΔj[a(e(1)j+1iΔjt1)+H.c.]}×(bkeiωbkt+bkeiωbkt).
Through setting Δ1 = ωb1, Δ2 = ωb2 and using the rotating-wave approximation, HP reads:
HeffP=g1E1Δ1(a1b1+a1b1)g2E2Δ2(a1b2+a1b2).
Equation (17) yields the Langevin equation of this system,
ddτ[a1b1b2]=[k0iirik10ir0k2][a1b1b2]+[f0f1f2],
where τ and r are now defined by τ=g1E1Δ1t, r = (E2Δ1g2) /(E1Δ2g1). The non-dimensional decay rates and noise operators are defined by ki=ΓiΔ12E1g1, fi=FiΔ1g1E1, with {Γi, i = 1, 2, 3} and {Fi, i = 0, 1, 2} the real decay rates and noise operators for each mode in Eq. (18), respectively. According to Ref. [37], Fi(t)Fj(t)R=Γini,thδijδ(tt). Thus, it is straightforward that:
fi(τ)fj(τ)R=2kini,thδijδ(ττ),
where ni,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.
  1. If ki ≠ 0, i = 0, 1, 2, we can obtain the numerical solutions for the Langevin equations.
  2. If ki = 0, i = 0, 1, 2, Eq. (18) becomes a homogeneous equation. The time evolution of the system is:
    [a1(τ)b1(τ)b2(τ)]=MP[a1(0)b1(0)b2(0)],
    MP=[cos(1r2τ)isin(1r2τ)1r2irsin(1r2τ)1r2isin(1r2τ)1r2cos(1r2τ)r21r2[cos(1r2τ)1]r1r2irsin(1r2τ)1r2[1cos(1r2τ)]r1r21r2cos(1r2τ)1r2].

When 1r2τ=π, Eq. (21) becomes:

MP=[10001+r21r22r1r202r1r21+r21r2].
Equation (22) indicates that at the instant Tπ = π/(1 − r2)1/2, the optical mode decouples from the dynamics of the system. We assume cosh (ξ) = (1 + r2)/(1 − r2), sinh (ξ) = 2r /(1 − r2) and introduce the operator S=exp{ξ[b1(0)b2(0)b1(0)b2(0)]}. Then the annihilation operators of the two microwave modes can be expressed as b1(Tπ) = −Sb1(0)S, b2(Tπ)=Sb2(0)S, which indicates that the two microwave modes are prepared in the Bogoliubov modes at the instant Tπ. 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 ξ = tanh−1 [2r /(1 + r2)]. 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 × Tπ, 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 Tπ that the photons of the two microwave modes can be entangled completely.

 figure: Fig. 3

Fig. 3 The time evolution of the fluctuating photon number for different values of the non-dimensional decay rate ki. Na1, Nb1 and Nb2 are photon numbers for optical cavity, LC1 and LC2, respectively. (a) r = 0.5, k0 = k1 = k2 = 0, (b) r = 0.5, k0 = k1 = k2 = 0.1. The initial condition for both (a) and (b) is Na1(0) = n0,th = 0, Nb1(0) = Nb2(0) = n1,th = n2,th = 0.1. In (a) the ideal case, at each instant , the photon number of the optical mode goes to zero and those numbers of LC1 and LC2 become equal. From (b), we can find that even for the optical cavity, the photon number can not be zero at steady state.

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In order to investigate the entanglement properties of the microwave modes, we calculate the total variance V = 〈(Δu)2 + (Δv)2〉 of EPR-like operators u = x1 + x2, v = p1p2, with xi=(bi+bi)/2 and pi=i(bibi)/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 = 2e−2ξ. 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 Tπ. 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.

 figure: Fig. 4

Fig. 4 The total variance versus phase for different values of the parameters: (a) without dissipation, r = 1 − 10−3, n1,th = n2,th = nth, and nth = 0, 0.1, 1; (b) with dissipation, r = 1 − 10−3, nth = 0, k0 = k1 = k2 = k, and k = 0.001, 0.01, 0.1.

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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 ki ≪ 1) optical cavities are necessary in those cases. However, we can also prepare the target states with low-Q (satisfying ki ≫ 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 n 0,th ≈ 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 ωa2, which satisfies |ωa2/ωa1 − 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 k0 ≫ 1 ≫ kb, where k0 is the non-dimensional decay rate of the two optical cavities, while kb 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:

ddτ[a1(τ)a2(τ)b1(τ)b2(τ)]=[k00iir0k0irriir00iri00][a1(τ)a2(τ)b1(τ)b2(τ)]+[fa1(τ)fa2(τ)00].
Here the definition of all the non-dimensional variables has the same form as those in the parallel scheme. Under the assumption k0 ≫ 1, we can make the adiabatic approximations to the optical modes:
a1(τ)=ik0[b1(τ)+rb2(τ)]+fa1(τ)k0,
a2(τ)=ik0[rb1(τ)+b2(τ)]+fa2(τ)k0.
Inserting Eq. (24) and Eq. (25) into Eq. (23) yields the relationship between b1 and b2:
ddτ[b1(τ)b2(τ)]=1k0[r2100r21][b1(τ)b2(τ)]+ik0[fa1(τ)+rfa2(τ)rfa1(τ)fa2(τ)].

 figure: Fig. 5

Fig. 5 The setup for the dissipative dynamical scheme. The two optical cavities are filled with electro-optic media, which are modulated by both LC1 and LC2 via the electro-optic effect. LC1 is in the red-detuned regime with respect to the previous optical cavity and blue-detuned regime with respect to the second one. For LC2, the situation is just opposite.

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

τnew=(1r2)τk0,τnew=(1r2)τk0,
f˜ai(τnew)=fai(τnew)2k0,i=1,2,
S˜(τnew,ς)=exp[ς(f˜a1(τnew)f˜a2(τnew)f˜a1(τnew)f˜a2(τnew))],
where ς = arctan [r] is the new squeezing parameter of the optical modes. By the definition of ai, we can get f˜ai(τnew)f˜aj(τnew)R=δijδ(τnewτnew). With Eqs. (27)(29), the solution of Eq. (26) can be expressed as:
b1(τnew)=eτnewb1(0)+2i0τnewe(τnewτnew)S˜(τnew)f˜a1(τnew)S˜(τnew)dτnew,
b2(τnew)=eτnewb2(0)2i0τnewe(τnewτnew)S˜(τnew)f˜a2(τnew)S˜(τnew)dτnew.
From Eqs. (27)(31), we can find that the stable condition for this system is r < 1, under which the system will converge to the final state of the Bogoliubov modes as τnew → ∞. As in the parallel scheme, we calculate the total variance of EPR-like operators composed of b1 and b2 under the long-time limit, V=limτnew2(1r)1+r(1e2τnew)=2(1r)1+r. That is exactly the total variance of the ideal two-mode vacuum state 2e−2ς with the squeezing parameter ς = arctan[r]. Thus, under the assumption that k0 ≫ 1 ≫ kb, regardless of the initial condition, the two microwave modes will finally evolve to the two-mode squeezed vacuum state definitely.

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=i=12{e2(τnew+kiτ)(2nth,i+1)+(1r)2+(nth,i+1)k0ki1r2+k0ki×[1e2(τnew+kiτ)]}. 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.

 figure: Fig. 6

Fig. 6 Plot of the total variance V versus the scaled time τnew under different decay conditions, with the parameters r = 0.5, nb,th = 0.01, k0 = 10, together with the result for an ideal two-mode squeezed state. The effective decay rate of the subsystem formed by the two superconducting microwave resonators in Eq. (26) is γ=2(1r2)k0=0.15.

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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 C1 = C2 = 40 fF, ωb1 = 2π×3 GHz, and ωb2 = 2π×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 103 pm/V. The achievable value of such a product in other references varies from 102 to 103 [43,44]. Thus, it is reasonable to take n3r0 ≈300 pm/V. For the optical cavities, we can choose the resonance frequency of the optical cavity ωa1 ≈ 2π × 200 THz, the decay rate of the superconducting microwave resonators Γi ≈ 2π × 1 kHz, i = 1, 2, and the distance between the two planes of each capacitor d ≈ 10 μm. The pump power can be P ≈ 10 mW [41]. Thus, the coefficients gi given by Eq.(2) can reach g1 ≈ 2π × 15 kHz, and g2 ≈ 2π × 19 kHz. In the “overcoupled” case, we can also obtain the mean photon number of the optical cavity caused by the external pump cav,i through the following equation [16]

n˜cav,i=ΓΔi2+(Γ/2)2Pωa1.
Here Γ is the total loss rate of the optical cavity, which is dominated by the external loss rate of the cavity (Γ ≈ Γ0). For the Q-factor of the optical cavities exceeding 108, we have Γ0 ≈ 2π × 0.2 MHz. In our case we choose Δi = ωbi, and get cav,1 ≈ 400, cav,2 ≈ 144. Then the effective electro-optic coupling strength can reach n¯cav,1g12π×0.3 MHz, and n¯cav,2g22π×0.23 MHz. If we set the time for the “two-mode squeezing” interaction T2 ∼ 1.6 μs, the operation time for generating the target states will be Tc = π/G1 + T2 ∼ 3.2 μs in the cascaded scheme.

As for the parallel and dissipative dynamical schemes, we assume the pump power is relatively low, i.e. P = 10 μW. Other related parameters are chosen as: the distance d = 5 μm, the capacitance and inductance of the superconducting microwave resonators C1 = C2 = 4 fF, ωb1 = 2π × 8 GHz, ωb2 = 2π × 10 GHz, and Γ1 = Γ2 ≈ 2π × 1 kHz. In our case, Γ ≪ Δi. Then the amplitudes of the driving lasers in both schemes can be expressed as Ei=n¯cav,inewωbi, i = 1, 2. Therefore, the operation time for the parallel scheme to generate the target states is Tp=π/[g1n¯cav,1newn¯cav,2new]~4.2μs and the time for reaching the stationary state of the dissipative dynamical scheme Td=Γ/[2g12(n¯cav,1newn¯cav,2new)]~1.65μs 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 2π × 1 MHz. For example, if we take L = 63 μH and keep the other parameters the same as those in the cascaded scheme, the effective electro-optic coupling strength can reach 2π×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 (τ2τ1). Obviously, ζ will increase with the increase of τ2. However, the detrimental effect of decoherence also becomes great when τ2 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 nth,1 ≈ 0.3, and nth,2 ≈ 0.1. Through the numerical simulations shown in Fig. 7, we can find that the optimized scaled time τ2 is 2.43 or equally T2 ≈ 1.3 μs, 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 ki = Γi/2G1, 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 C1 = C2 = 1 fF, L1 = 360 nH, L2 = 350 nH, at the same temperature, the minimal variance drops to V ≈ 0.77.

 figure: Fig. 7

Fig. 7 Plot of the total variance V versus the scaled time τ2 under the above experimental parameters. We can see that the optimized scaled time is τ2 = 2.43, or equally T2 ≈ 1.3 μs.

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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 k0 ≈ 0.9, k1k2 ≈ 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:

V=2(1r)/(1+r)+2α(nth,1+nth,2+2)1+2α,
α=ki/γ,i=1,2,
γ=2(1r2)/k0..
Here we assume that the two superconducting resonators have the same decay rates Γ1 = Γ2, and γ is the effective decay rate of the system. If we choose Γ0 = 2π × 30 MHz, Γ1 = Γ2 ≈ 2π × 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.

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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).

References

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

2017 (1)

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

2016 (5)

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

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]

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

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]

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]

2015 (2)

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]

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]

2014 (3)

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]

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

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]

2013 (7)

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]

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]

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]

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

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]

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]

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

2012 (4)

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]

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]

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]

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]

2011 (4)

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]

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]

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]

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

2010 (3)

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]

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]

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

2009 (4)

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]

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]

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

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]

2008 (4)

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

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

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]

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]

2007 (2)

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]

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).

2004 (1)

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

2000 (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]

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]

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]

Abdi, M.

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]

Allman, M. S.

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]

An, J. H.

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]

Andrews, R. W.

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]

Anetsberger, G.

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]

Ankerhold, J.

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

Ansmann, M.

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]

Arcizet, O.

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]

Ashhab, S.

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]

Aspelmeyer, M.

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

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]

Ballester, D.

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]

Barad, Y.

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]

Barnea, T.

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

Barzanjeh, S.

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]

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]

Baust, A.

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]

Bernier, N. R.

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]

Bialczak, R. C.

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]

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]

Boisen, A.

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]

Bowen, W.

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]

Brawley, G.

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]

Carmichael, H. J.

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

Chen, A. X.

Chen, B.

Chen, G.

Chen, H. Y.

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).

Chen, P.

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).

Cheng, G. L.

Cheng, Z. Y.

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]

Cicak, K.

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]

Cirac, J. I.

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]

Cleland, A. N.

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]

Clerk, A. A.

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

Cui, Y.

Dambach, S.

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

Deppe, F.

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]

Di Candia, R.

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]

Donner, T.

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]

Duan, L. M.

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]

Eder, P.

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]

Eisert, J.

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

Fang, A. P.

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]

Feofanov, A. K.

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]

Ficek, Z.

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]

Frimmer, M.

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

Galland, C.

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

Gao, S. Y.

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]

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]

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]

Genes, C.

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]

Giedke, G.

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]

Gieseler, J.

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

Gong, Z. R.

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]

Gröblacher, S.

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]

Gross, R.

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]

Günter, P.

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

Haeberlein, M.

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]

Hammerer, K.

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]

Han, S. Y.

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]

He, Q. Y.

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]

Herzog, C.

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

Hoffmann, E.

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]

Hofheinz, M.

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]

Huang, P.

Ian, H.

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]

Ihmig, M.

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]

Ilchenko, V. S.

Inomata, K.

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]

Javerzac-Galy, C.

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]

Jennewein, T.

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]

Jiang, C.

Jiang, W. C.

Kampel, N. S.

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]

Kang, J. Y.

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).

Kippenberg, T. J.

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]

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

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]

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

Kubala, B.

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

Larsen, P. E.

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]

Law, C. K.

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

Lehnert, K.

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]

Lehnert, K. W.

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]

Lenander, M.

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]

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]

Li, D.

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]

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]

Li, F. L.

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]

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]

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Zeilinger, A.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
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Zhang, J.

Zhang, Q. M.

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).
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Zhao, J.

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).
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Zhao, L.

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).

Zheng, L. L.

Zheng, S. B.

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]

Zhong, L.

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]

Zhong, W. X.

Zoller, P.

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]

Zubairy, M. S.

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

Zuo, Y. H.

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).

Appl. Phys. Lett. (2)

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]

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).

J. Opt. Soc. Am. B (2)

Nat. Commun. (1)

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).
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Nat. Phys. (1)

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).
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Nature (4)

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).
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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).
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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).
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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).
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New J. Phys. (1)

S. Dambach, B. Kubala, and J. Ankerhold, “Generating entangled quantum microwaves in a josephson-photonics device,” New J. Phys. 19, 023027 (2017).
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Opt. Commun. (1)

C. Herzog, G. Poberaj, and P. Günter, “Electro-optic behavior of lithium niobate at cryogenic temperatures,” Opt. Commun. 281, 793–796 (2008).
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Opt. Express (3)

Phys. Rev. A (13)

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).
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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).
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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).
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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).
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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).
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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).
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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).
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J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
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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).
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M. Tsang, “Cavity quantum electro-optics,” Phys. Rev. A 81, 063837 (2010).
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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).
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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).
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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).
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Phys. Rev. Lett. (12)

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).
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L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
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V. C. Vivoli, T. Barnea, C. Galland, and N. Sangouard, “Proposal for an optomechanical bell test,” Phys. Rev. Lett. 116, 070405 (2016).
[Crossref] [PubMed]

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]

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]

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]

M. Frimmer, J. Gieseler, and L. Novotny, “Cooling mechanical oscillators by coherent control,” Phys. Rev. Lett. 117, 163601 (2016).
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A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
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S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109, 130503 (2012).
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L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
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Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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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).
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Science (1)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
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Other (1)

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

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Figures (7)

Fig. 1
Fig. 1 The setup of both the cascaded and parallel schemes. In the cascade scheme, the optical cavity is driven by a laser of frequency ωL1 or ωL2 for the different periods, while in the parallel scheme, we apply both of these driving lasers at the same time.
Fig. 2
Fig. 2 A diagram of the process of the cascaded scheme, and a1, b1, b2 are the annihilation operators for the optical mode and two microwave modes, respectively. When 0 < t < T1 or T2 < t < T1 + T2, LC1 and the optical cavity exchange quantum states with each other, and during T1 < t < T2, the optical cavity mode and the LC2 mode get entangled.
Fig. 3
Fig. 3 The time evolution of the fluctuating photon number for different values of the non-dimensional decay rate ki. Na1, Nb1 and Nb2 are photon numbers for optical cavity, LC1 and LC2, respectively. (a) r = 0.5, k0 = k1 = k2 = 0, (b) r = 0.5, k0 = k1 = k2 = 0.1. The initial condition for both (a) and (b) is Na1(0) = n0,th = 0, Nb1(0) = Nb2(0) = n1,th = n2,th = 0.1. In (a) the ideal case, at each instant , the photon number of the optical mode goes to zero and those numbers of LC1 and LC2 become equal. From (b), we can find that even for the optical cavity, the photon number can not be zero at steady state.
Fig. 4
Fig. 4 The total variance versus phase for different values of the parameters: (a) without dissipation, r = 1 − 10−3, n1,th = n2,th = nth, and nth = 0, 0.1, 1; (b) with dissipation, r = 1 − 10−3, nth = 0, k0 = k1 = k2 = k, and k = 0.001, 0.01, 0.1.
Fig. 5
Fig. 5 The setup for the dissipative dynamical scheme. The two optical cavities are filled with electro-optic media, which are modulated by both LC1 and LC2 via the electro-optic effect. LC1 is in the red-detuned regime with respect to the previous optical cavity and blue-detuned regime with respect to the second one. For LC2, the situation is just opposite.
Fig. 6
Fig. 6 Plot of the total variance V versus the scaled time τnew under different decay conditions, with the parameters r = 0.5, nb,th = 0.01, k0 = 10, together with the result for an ideal two-mode squeezed state. The effective decay rate of the subsystem formed by the two superconducting microwave resonators in Eq. (26) is γ = 2 ( 1 r 2 ) k 0 = 0.15.
Fig. 7
Fig. 7 Plot of the total variance V versus the scaled time τ2 under the above experimental parameters. We can see that the optimized scaled time is τ2 = 2.43, or equally T2 ≈ 1.3 μs.

Equations (35)

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H I , i C = g i a 1 a 1 ( b i + b i ) ,
g i = ω a 1 n 3 r 0 2 d ω b i 2 C i ,
H d C = i [ E 0 a 1 e i ω L 1 t E 0 * a 1 e i ω L 1 t ] ,
H 1 C = Δ a 1 a 1 + ω b 1 b 1 b 1 + ω b 2 b 2 b 2 g 1 a 1 a 1 ( b 1 + b 1 ) + i ( E 0 a 1 E 0 * a 1 ) ,
H 1 , eff C = Δ δ a 1 δ a 1 + ω b 1 b 1 b 1 + ω b 2 b 2 b 2 G 1 ( δ a 1 b 1 + δ a 1 b 1 ) .
[ δ a 1 ( τ 1 + τ 2 ) b 1 ( τ 1 + τ 2 ) b 2 ( τ 1 + τ 2 ) ] = M 1 M 2 M 1 [ δ a 1 ( 0 ) b 1 ( 0 ) b 2 ( 0 ) ] ,
M 1 = [ cos ( τ 1 ) i sin ( τ 1 ) 0 i sin ( τ 1 ) cos ( τ 1 ) 0 0 0 1 ] ,
M 2 = [ cosh [ r ( τ 2 τ 1 ) ] 0 i sinh [ r ( τ 2 τ 1 ) ] 0 1 0 i sinh [ r ( τ 2 τ 1 ) 0 cosh [ r ( τ 2 τ 1 ) ] ] .
M 1 M 2 M 1 = [ 1 0 0 0 cosh [ r ( τ 2 τ 1 ) ] sinh [ r ( τ 2 τ 1 ) ] 0 sinh [ r ( τ 2 τ 1 ) ] cosh [ r ( τ 2 τ 1 ) ] ] .
H = H 0 + H I + H d ,
H 0 = ω a 1 a 1 a 1 + i = 1 2 ω b i b i b i ,
H I = j = 1 2 g j a 1 a 1 ( b j + b j ) ,
H d = j = 1 2 ( 1 ) j E j ( a 1 e i ω L j t + a 1 e i ω L j t ) .
H = j = 1 2 g j a 1 a 1 ( b j e i ω b j t + b j e i ω b j t ) + j = 1 2 ( 1 ) j E j [ a 1 e ( 1 ) j 1 i Δ j t + a 1 e ( 1 ) j i Δ j t ] ,
U = T exp { i 0 t j = 1 2 ( 1 ) j E j [ a 1 e ( 1 ) j 1 i Δ j t + a 1 e ( 1 ) j i Δ j t ] d t } ,
H P = U ( H i t ) U = k = 1 2 g k { a 1 a 1 + j = 1 2 E j Δ j [ a ( e ( 1 ) j + 1 i Δ j t 1 ) + H . c . ] } × ( b k e i ω b k t + b k e i ω b k t ) .
H e f f P = g 1 E 1 Δ 1 ( a 1 b 1 + a 1 b 1 ) g 2 E 2 Δ 2 ( a 1 b 2 + a 1 b 2 ) .
d d τ [ a 1 b 1 b 2 ] = [ k 0 i i r i k 1 0 i r 0 k 2 ] [ a 1 b 1 b 2 ] + [ f 0 f 1 f 2 ] ,
f i ( τ ) f j ( τ ) R = 2 k i n i , t h δ i j δ ( τ τ ) ,
[ a 1 ( τ ) b 1 ( τ ) b 2 ( τ ) ] = M P [ a 1 ( 0 ) b 1 ( 0 ) b 2 ( 0 ) ] ,
M P = [ cos ( 1 r 2 τ ) i sin ( 1 r 2 τ ) 1 r 2 i r sin ( 1 r 2 τ ) 1 r 2 i sin ( 1 r 2 τ ) 1 r 2 cos ( 1 r 2 τ ) r 2 1 r 2 [ cos ( 1 r 2 τ ) 1 ] r 1 r 2 i r sin ( 1 r 2 τ ) 1 r 2 [ 1 cos ( 1 r 2 τ ) ] r 1 r 2 1 r 2 cos ( 1 r 2 τ ) 1 r 2 ] .
M P = [ 1 0 0 0 1 + r 2 1 r 2 2 r 1 r 2 0 2 r 1 r 2 1 + r 2 1 r 2 ] .
d d τ [ a 1 ( τ ) a 2 ( τ ) b 1 ( τ ) b 2 ( τ ) ] = [ k 0 0 i i r 0 k 0 i r r i i r 0 0 i r i 0 0 ] [ a 1 ( τ ) a 2 ( τ ) b 1 ( τ ) b 2 ( τ ) ] + [ f a 1 ( τ ) f a 2 ( τ ) 0 0 ] .
a 1 ( τ ) = i k 0 [ b 1 ( τ ) + r b 2 ( τ ) ] + f a 1 ( τ ) k 0 ,
a 2 ( τ ) = i k 0 [ r b 1 ( τ ) + b 2 ( τ ) ] + f a 2 ( τ ) k 0 .
d d τ [ b 1 ( τ ) b 2 ( τ ) ] = 1 k 0 [ r 2 1 0 0 r 2 1 ] [ b 1 ( τ ) b 2 ( τ ) ] + i k 0 [ f a 1 ( τ ) + r f a 2 ( τ ) r f a 1 ( τ ) f a 2 ( τ ) ] .
τ new = ( 1 r 2 ) τ k 0 , τ new = ( 1 r 2 ) τ k 0 ,
f ˜ a i ( τ new ) = f a i ( τ new ) 2 k 0 , i = 1 , 2 ,
S ˜ ( τ new , ς ) = exp [ ς ( f ˜ a 1 ( τ new ) f ˜ a 2 ( τ new ) f ˜ a 1 ( τ new ) f ˜ a 2 ( τ new ) ) ] ,
b 1 ( τ new ) = e τ new b 1 ( 0 ) + 2 i 0 τ new e ( τ new τ new ) S ˜ ( τ new ) f ˜ a 1 ( τ new ) S ˜ ( τ new ) d τ new ,
b 2 ( τ new ) = e τ new b 2 ( 0 ) 2 i 0 τ new e ( τ new τ new ) S ˜ ( τ new ) f ˜ a 2 ( τ new ) S ˜ ( τ new ) d τ new .
n ˜ cav , i = Γ Δ i 2 + ( Γ / 2 ) 2 P ω a 1 .
V = 2 ( 1 r ) / ( 1 + r ) + 2 α ( n t h , 1 + n t h , 2 + 2 ) 1 + 2 α ,
α = k i / γ , i = 1 , 2 ,
γ = 2 ( 1 r 2 ) / k 0 ..

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