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

Dao-Quan Zhu; Peng-Bo Li

doi:10.1364/OE.25.028305

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 ω_b₁ (LC1) and ω_b₂ (LC2), and an optical cavity of frequency ω_a₁. 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.

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 ω_L₁ or ω_L₂ 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.

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 ω_L₁; (ii) T₁ < t < T₂, using the laser of frequency ω_L₂; (iii) T₂ < t < T₁ + T₂, using the laser of frequency ω_L₁, again. We assume that ω_L₁ − ω_a₁ = −ω_b₁, ω_L₂ − ω_a₁ = ω_b₂, in order to guarantee that there is only one microwave mode interacting with the optical mode in each period.

As shown in Ref. [15], the interaction Hamiltonian between the optical mode a 1 and the microwave modes b_i, i = 1, 2 in each period is:

H_{I, i}^{C} = - ℏ g_{i} a_{1}^{†} a_{1} (b_{i} + b_{i}^{†}),

g_{i} = \frac{ω_{a 1} n^{3} r_{0}}{2 d} \sqrt{\frac{ℏ {ω_{b}}_{i}}{2 C_{i}},}

where n, r₀, 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 driving term in the total Hamiltonian is that:

H_{d}^{C} = i ℏ [E_{0} a_{1}^{†} e^{- i ω_{L 1} t} - E_{0}^{*} a_{1} e^{i ω_{L 1} t}],

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 ω_L₁, the total Hamiltonian then becomes:

\begin{matrix} H_{1}^{C} = - ℏ Δ a_{1}^{†} a_{1} + ℏ ω_{b 1} b_{1}^{†} b_{1} + ℏ ω_{b 2} b_{2}^{†} b_{2} \\ - ℏ g_{1} a_{1}^{†} a_{1} (b_{1}^{†} + b_{1}) + i ℏ (E_{0} a_{1}^{†} - E_{0}^{*} a_{1}), \end{matrix}

where Δ = ω_L₁ − ω_a₁. 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:

\begin{matrix} H_{1, eff}^{C} = - ℏ Δ δ a_{1}^{†} δ a_{1} + ℏ ω_{b 1} b_{1}^{†} b_{1} + ℏ ω_{b 2} b_{2}^{†} b_{2} \\ - ℏ G_{1} (δ a_{1}^{†} b_{1} + δ a_{1} b_{1}^{†}) . \end{matrix}

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, τ_i = G₁T_i, 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:

[\begin{matrix} δ a_{1} (τ_{1} + τ_{2}) \\ b_{1} (τ_{1} + τ_{2}) \\ b_{2}^{†} (τ_{1} + τ_{2}) \end{matrix}] = M_{1} M_{2} M_{1} [\begin{matrix} δ a_{1} (0) \\ b_{1} (0) \\ b_{2}^{†} (0) \end{matrix}],

M_{1} = [\begin{matrix} cos (τ_{1}) & i sin (τ_{1}) & 0 \\ i sin (τ_{1}) & cos (τ_{1}) & 0 \\ 0 & 0 & 1 \end{matrix}],

M_{2} = [\begin{matrix} 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})] \end{matrix}] .

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:

M_{1} M_{2} M_{1} = [\begin{matrix} - 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})] \end{matrix}] .

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 = H_{0} + H_{I} + {H^{'}}_{d},

H_{0} = ℏ {ω_{a}}_{1} a_{1}^{†} a_{1} + \sum_{i = 1}^{2} ℏ ω_{b i} b_{i}^{†} b_{i},

H_{I} = - \sum_{j = 1}^{2} ℏ g_{j} a_{1}^{†} a_{1} (b_{j} + b_{j}^{†}),

{H^{'}}_{d} = ℏ \sum_{j = 1}^{2} {(- 1)}^{j} E_{j} (a_{1}^{†} e^{- i ω_{L j} t} + a_{1} e^{i ω_{L j} t}) .

Here the driving lasers have real amplitudes E_j (j = 1, 2) and initial phases

ϕ_{1} = - \frac{π}{2}

,

ϕ_{2} = \frac{π}{2}

.

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

\begin{matrix} H^{'} = - \sum_{j = 1}^{2} ℏ g_{j} a_{1}^{†} a_{1} (b_{j} e^{- i ω_{b_{j}} t} + b_{j}^{†} e^{- i ω_{b_{j}} t}) \\ + ℏ \sum_{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}], \end{matrix}

where Δ₁ = ω_a₁ − ω_L₁, Δ₂ = ω_L₂ − ω_a₁.

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

U = T exp {- i \int_{0}^{t} \sum_{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},

where T is the time-ordering operator. Unlike the cascade scheme, here we assume E_j /Δ_j ≪ 1. Therefore, we can keep the leading term up to E_j /Δ_j, instead of making the linearization approximation as before. Equation (14) then becomes,

\begin{array}{l} H^{P} = U^{†} (H^{'} - i ℏ \frac{\partial}{\partial t}) U \\ = - \sum_{k = 1}^{2} ℏ g_{k} {a_{1}^{†} a_{1} + \sum_{j = 1}^{2} \frac{E_{j}}{Δ_{j}} [a^{†} {(e}^{{(- 1)}^{j + 1} i Δ_{j} t} - 1) + H . c .]} \\ \times (b_{k} e^{- i ω_{b k} t} + b_{k}^{†} e^{- i ω_{b k} t}) . \end{array}

Through setting Δ₁ = ω_b₁, Δ₂ = ω_b₂ and using the rotating-wave approximation, H^P reads:

H_{e f f}^{P} = - ℏ g_{1} \frac{E_{1}}{Δ_{1}} (a_{1}^{†} b_{1} + a_{1} b_{1}^{†}) - ℏ g_{2} \frac{E_{2}}{Δ_{2}} (a_{1}^{†} b_{2}^{†} + a_{1} b_{2}) .

Equation (17) yields the Langevin equation of this system,

\frac{d}{d τ} [\begin{array}{l} a_{1} \\ b_{1} \\ b_{2}^{†} \end{array}] = [\begin{array}{l} - k_{0} & i & i r \\ i & - k_{1} & 0 \\ - i r & 0 & - k_{2} \end{array}] [\begin{array}{l} a_{1} \\ b_{1} \\ b_{2}^{†} \end{array}] + [\begin{array}{l} f_{0} \\ f_{1} \\ f_{2}^{†} \end{array}],

where τ and r are now defined by

τ = g_{1} \frac{E_{1}}{Δ_{1}} t

, r = (E₂Δ₁g₂) /(E₁Δ₂g₁). The non-dimensional decay rates and noise operators are defined by

k_{i} = \frac{Γ_{i} Δ_{1}}{2 E_{1} g_{1}}

,

f_{i} = \frac{F_{i} Δ_{1}}{g_{1} E_{1}}

, with {Γ_i, i = 1, 2, 3} and {F_i, i = 0, 1, 2} the real decay rates and noise operators for each mode in Eq. (18), respectively. According to Ref. [37],

{〈 F_{i}^{†} (t) F_{j} (t^{'}) 〉}_{R} = Γ_{i} n_{i, t h} δ_{i j} δ (t - t^{'})

. Thus, it is straightforward that:

{〈 f_{i}^{†} (τ) f_{j} (τ^{'}) 〉}_{R} = 2 k_{i} n_{i, t h} δ_{i j} δ (τ - τ^{'}),

where n_i,th, i = 1, 2 refers to the thermal photon number of the i th superconducting microwave resonator. We solve the Heisenberg or Langevin equations for the non-dissipative or dissipative cases, respectively.

If k_i ≠ 0, i = 0, 1, 2, we can obtain the numerical solutions for the Langevin equations.
If k_i = 0, i = 0, 1, 2, Eq. (18) becomes a homogeneous equation. The time evolution of the system is: $[\begin{array}{l} a_{1} (τ) \\ b_{1} (τ) \\ b_{2}^{†} (τ) \end{array}] = M^{P} [\begin{array}{l} a_{1} (0) \\ b_{1} (0) \\ b_{2}^{†} (0) \end{array}],$ $M^{P} = [\begin{array}{l} cos (\sqrt{1 - r^{2}} τ) & i \frac{sin (\sqrt{1 - r^{2}} τ)}{\sqrt{1 - r^{2}}} & i r \frac{sin (\sqrt{1 - r^{2}} τ)}{\sqrt{1 - r^{2}}} \\ i \frac{sin (\sqrt{1 - r^{2}} τ)}{\sqrt{1 - r^{2}}} & \frac{cos (\sqrt{1 - r^{2}} τ) - r^{2}}{1 - r^{2}} & \frac{[cos (\sqrt{1 - r^{2}} τ) - 1] r}{1 - r^{2}} \\ - i r \frac{sin (\sqrt{1 - r^{2}} τ)}{\sqrt{1 - r^{2}}} & \frac{[1 - cos (\sqrt{1 - r^{2}} τ)] r}{1 - r^{2}} & \frac{1 - r^{2} cos (\sqrt{1 - r^{2}} τ)}{1 - r^{2}} \end{array}] .$

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

M^{P} = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - \frac{1 + r^{2}}{1 - r^{2}} & - \frac{2 r}{1 - r^{2}} \\ 0 & \frac{2 r}{1 - r^{2}} & \frac{1 + r^{2}}{1 - r^{2}} \end{matrix}] .

Equation (22) indicates that at the instant T_π = 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 [b_{1} (0) b_{2} (0) - b_{1}^{†} (0) b_{2}^{†} (0)]}

. Then the annihilation operators of the two microwave modes can be expressed as b₁(T_π) = −Sb₁(0)S^†,

b_{2}^{†} (T_{π}) = S b_{2}^{†} (0) S^{†}

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

Fig. 3 The time evolution of the fluctuating photon number for different values of the non-dimensional decay rate k_i. N_a₁, N_b₁ and N_b₂ 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 N_a₁(0) = n₀_,th = 0, N_b₁(0) = N_b₂(0) = n₁_,th = n₂_,th = 0.1. In (a) the ideal case, at each instant nπ, 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 = p₁ − p₂, with $x_{i} = (b_{i} + b_{i}^{†}) / \sqrt{2}$ and $p_{i} = - i (b_{i} - b_{i}^{†}) / \sqrt{2}$ , i = 1, 2 [5]. According to Ref. [5, 38], the two-mode Gaussian state is entangled if and only if V < 2. Especially, for the two-mode squeezed vacuum state (squeezing parameter ξ), V = 2e⁻²^ξ. In Fig. 4 we show the influence of the dissipation and the initial thermal conditions in the entanglement of this system. As illustrated in Fig. 4 (a), the difference between the curves disappears near T_π. 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.

Fig. 4 The total variance versus phase for different values of the parameters: (a) without dissipation, r = 1 − 10⁻³, n₁_,th = n₂_,th = n_th, and n_th = 0, 0.1, 1; (b) with dissipation, r = 1 − 10⁻³, n_th = 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 k_i ≪ 1) optical cavities are necessary in those cases. However, we can also prepare the target states with low-Q (satisfying k_i ≫ 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 ω_a₂, which satisfies |ω_a₂/ω_a₁ − 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 ≫ k_b, where k₀ is the non-dimensional decay rate of the two optical cavities, while k_b 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:

\frac{d}{d τ} [\begin{matrix} a_{1} (τ) \\ a_{2}^{†} (τ) \\ b_{1} (τ) \\ b_{2}^{†} (τ) \end{matrix}] = [\begin{matrix} - k_{0} & 0 & i & i r \\ 0 & - k_{0} & - i r & - r \\ i & i r & 0 & 0 \\ - i r & - i & 0 & 0 \end{matrix}] [\begin{matrix} a_{1} (τ) \\ a_{2}^{†} (τ) \\ b_{1} (τ) \\ b_{2}^{†} (τ) \end{matrix}] + [\begin{matrix} f_{a 1} (τ) \\ f_{a 2}^{†} (τ) \\ 0 \\ 0 \end{matrix}] .

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:

a_{1} (τ) = \frac{i}{k_{0}} [b_{1} (τ) + r b_{2}^{†} (τ)] + \frac{{f_{a}}_{1} (τ)}{k_{0}},

a_{2}^{†} (τ) = - \frac{i}{k_{0}} [r b_{1} (τ) + b_{2}^{†} (τ)] + \frac{f_{a 2}^{†} (τ)}{k_{0}} .

Inserting Eq. (24) and Eq. (25) into Eq. (23) yields the relationship between b₁ and

b_{2}^{†}

:

\begin{matrix} \frac{d}{d τ} [\begin{matrix} b_{1} (τ) \\ b_{2}^{†} (τ) \end{matrix}] = \frac{1}{k_{0}} [\begin{matrix} r^{2} - 1 & 0 \\ 0 & r^{2} - 1 \end{matrix}] [\begin{matrix} b_{1} (τ) \\ b_{2}^{†} (τ) \end{matrix}] \\ + \frac{i}{k_{0}} [\begin{matrix} f_{a 1} (τ) + r f_{a 2}^{†} (τ) \\ - r f_{a 1} (τ) - f_{a 2}^{†} (τ) \end{matrix}] . \end{matrix}

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.

τ_{new} = \frac{(1 - r^{2}) τ}{k_{0}}, {τ^{'}}_{new} = \frac{(1 - r^{2}) τ^{'}}{k_{0}},

{\tilde{f}}_{a i} (τ_{new}) = \frac{f_{a i} (τ_{new})}{\sqrt{2 k_{0}}}, i = 1, 2,

\tilde{S} (τ_{new,} ς) = exp [ς ({\tilde{f}}_{a 1} (τ_{new}) {\tilde{f}}_{a 2} (τ_{new}) - {\tilde{f}}_{a 1}^{†} (τ_{new}) {\tilde{f}}_{a 2}^{†} (τ_{new}))],

where ς = arctan [r] is the new squeezing parameter of the optical modes. By the definition of f̃_ai, we can get

{〈 {\tilde{f}}_{a i} (τ_{new}) {\tilde{f}}_{a j}^{†} ({τ^{'}}_{new}) 〉}_{R} = δ_{i j} δ (τ_{new} - {τ^{'}}_{new})

. With Eqs. (27)–(29), the solution of Eq. (26) can be expressed as:

b_{1} (τ_{new}) = e^{- τ_{new}} b_{1} (0) + \sqrt{2 i} \int_{0}^{τ_{new}} e^{- (τ_{new} - {τ^{'}}_{new})} \tilde{S} ({τ^{'}}_{new}) {\tilde{f}}_{a 1} ({τ^{'}}_{new}) {\tilde{S}}^{†} ({τ^{'}}_{new}) d {τ^{'}}_{new},

b_{2}^{†} (τ_{new}) = e^{- τ_{new}} b_{2}^{†} (0) - \sqrt{2 i} \int_{0}^{τ_{new}} e^{- (τ_{new} - {τ^{'}}_{new})} \tilde{S} ({τ^{'}}_{new}) {\tilde{f}}_{a 2}^{†} ({τ^{'}}_{new}) {\tilde{S}}^{†} ({τ^{'}}_{new}) d {τ^{'}}_{new} .

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 τ_new → ∞. As in the parallel scheme, we calculate the total variance of EPR-like operators composed of b₁ and

b_{2}^{†}

under the long-time limit,

V = lim_{τ_{new} \to \infty} \frac{2 (1 - r)}{1 + r} (1 - e^{- 2 τ_{new}}) = \frac{2 (1 - r)}{1 + r}

. That is exactly the total variance of the ideal two-mode vacuum state 2e⁻²^ς with the squeezing parameter ς = arctan[r]. Thus, under the assumption that k₀ ≫ 1 ≫ k_b, 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 $V = \sum_{i = 1}^{2} {e^{- 2 (τ_{new} + k_{i} τ)} (2 n_{t h, i} + 1) + \frac{{(1 - r)}^{2} + (n_{t h, i} + 1) k_{0} k_{i}}{1 - r^{2} + k_{0} k_{i}} \times [1 - e^{- 2 (τ_{new} + k_{i} τ)}]}$ . 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.

Fig. 6 Plot of the total variance V versus the scaled time τ_new under different decay conditions, with the parameters r = 0.5, n_b,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 $γ = \frac{2 (1 - r^{2})}{k_{0}} = 0.15$ .

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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, ω_b₁ = 2π×3 GHz, and ω_b₂ = 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 ω_a₁ ≈ 2π × 200 THz, the decay rate of the superconducting microwave resonators Γ_i ≈ 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 g_i 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 n̄_cav,i through the following equation [16]

{\tilde{n}}_{cav, i} = \frac{Γ}{Δ_{i}^{2} + {(Γ / 2)}^{2}} \frac{P}{ℏ ω_{a 1}} .

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 Δ_i = ω_bi, and get n̄_cav,1 ≈ 400, n̄_cav,2 ≈ 144. Then the effective electro-optic coupling strength can reach

\sqrt{{\bar{n}}_{cav, 1}} g_{1} \approx 2 π \times 0.3

MHz, and

\sqrt{{\bar{n}}_{cav, 2}} g_{2} \approx 2 π \times 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 T_c = 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, ω_b₁ = 2π × 8 GHz, ω_b₂ = 2π × 10 GHz, and Γ₁ = Γ₂ ≈ 2π × 1 kHz. In our case, Γ ≪ Δ_i. Then the amplitudes of the driving lasers in both schemes can be expressed as $E_{i} = \sqrt{{\bar{n}}_{cav, i new}} ω_{b i}$ , i = 1, 2. Therefore, the operation time for the parallel scheme to generate the target states is $T_{p} = π / [g_{1} \sqrt{{\bar{n}}_{cav, 1 new} - {\bar{n}}_{cav, 2 new}}] ~ 4.2 μ s$ and the time for reaching the stationary state of the dissipative dynamical scheme $T_{d} = Γ / [2 g_{1}^{2} ({\bar{n}}_{cav, 1 new} - {\bar{n}}_{cav, 2 new})] ~ 1.65 μ s$ times are much shorter than the photon lifetime in the superconducting microwave resonators.

Furthermore, if a large inductance is allowed, the effective electro-optic coupling strength will exceed 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 (τ₂ − τ₁). 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 n_th,₁ ≈ 0.3, and n_th,₂ ≈ 0.1. Through the numerical simulations shown in Fig. 7, we can find that the optimized scaled time 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 k_i = Γ_i/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.

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, k₁ ≈ k₂ ≈ 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 = \frac{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} ..

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

Abstract

1. Introduction

2. The Model and schemes

2.1. The cascaded scheme

2.2. Parallel scheme

2.3. Dissipative dynamical scheme

3. Experimental feasibility

4. Conclusions

Funding

References and links

Cited By

Figures (7)

Equations (35)

Optics Express