Surviving Entanglement in Optic-Microwave Conversion by Electro-Optomechanical System

In recent development of quantum technologies, a frequency conversion of quantum signals has been studied widely. We investigate the optic-microwave entanglement that is generated by applying an electro-optomechanical frequency conversion scheme to one mode in an optical two-mode squeezed vacuum state. We quantify entanglement of the converted two-mode Gaussian state, where surviving entanglement of the state is analyzed with respect to the parameters of the electro-optomechanical system. Furthermore, we show that there exists an upper bound for the entanglement that survives after the conversion of highly entangled optical states. Our study provides a theoretical platform for a practical quantum illumination system.


I. INTRODUCTION
Entanglement is the essential source of a quantum advantage in quantum-enhanced sensing [1] and quantum illumination [2][3][4]. In optics, we can generate two-mode entangled states with mean photon number of 7.4 for each mode under the current technology [5]. Although optical entanglement was demonstrated in a long-distance quantum communication over clear sky [6,7], it is very challenging to distribute entanglement in free space due to its heavy scattering. To overcome the issue, it is natural to consider microwave regime which has lower attenuation than optical regime in the atmosphere. For example, 10 GHz microwave signal has very low attenuation, and is used in radar system to detect a remote target. However, the microwave is in a low energy level so that thermal occupancy is a dominant issue.
To compensate both frequency regimes, we take both advantages of optical and microwave regimes, by taking optic-microwave bi-directional conversion. First, we prepare an optical entangled photon that is composed of signal and idler modes. Second, we convert the optical photon of the signal mode into microwave one. Third, we send the microwave photon into a target in the atmosphere while we keep the optical photon of the idler mode ideally. At this moment, we focus on how much entanglement the optic-microwave two-mode state contains. A frequency conversion of electromagnetic fields requires nonlinear interaction, which has been implemented in ferroelectric crystal [8][9][10][11], magnon [12,13], a rare-earth-doped crystal [14,15], and an electro-optomechanical system [16,17]. Up to now, the highest conversion efficiency was achieved with the electro-optomechanical system that consists of a microwave cavity(MC), an optical cavity(OC), and a mechanical resonator(MR). The MR is coupled with the OC and the MC simultaneously and mediates a coherent conversion between them [18].
In this article, we analyze an optic-microwave entangled state generated by using the electro-optomechanical system. We start with an optical two-mode squeezed vacuum(TMSV) state which is conventionally exploited in continuousvariable quantum information processing. Then, the signal mode of the TMSV state is converted to microwave photons via the electro-optomechanical system while the idler mode is retained ideally. After the quantum frequency conversion, we analyze entanglement between the converted microwave signal mode and the optical idler mode. We find that entanglement can survive in the converted two-mode state when we employ a realistic system with feasible parameters [17].
This article is organized as follows. In Sec. II and III, we describe the electro-optomechanical system and its opticmicrowave conversion process. In Sec. IV, we derive an entanglement formula between the converted signal mode and the idler mode, and analyze the amount of surviving entanglement compared with the entanglement of an initial optical TMSV state. Then, we investigate a relation between the quantum frequency conversion and the amount of the surviving entanglement. Finally, it is concluded in Sec. V.

FIG. 1. (a)
A schematic diagram of the optic-microwave entangled state generation. One of the modes of an optical two-mode squeezed vacuum state is converted to microwave while the other mode is ideally retained. (b) A schematic diagram of the optic-microwave conversion using electro-optomechanical system. The yellow lines denote a capacitor. An optical cavity and a microwave cavity are connected by a mechanical resonator. Fig. 1(a) shows a schematic diagram of the optic-microwave entangled state generation. The signal mode of an optical TMSV state is converted to microwave, while the idler mode is ideally retained. Fig. 1(b) presents a schematic diagram of the electro-optomechanical system for optic-microwave conversion. The MR with resonance frequency ω m is connected to both the OC with the resonance frequency ω o and the MC with ω e . The MR is coupled to the MC by the capacitance formed between them, and to the OC by modulating the optical pathway.

II. ELECTRO-OPTOMECHANICAL SYSTEM
In Fig. 1(b), the driving pump frequency of the OC is set as ω d,o = ω o − ∆ d,o , and the frequency of the MC driving pump is ω d,e = ω e − ∆ d,e , where ∆ d,o and ∆ d,e denote optical and microwave pump detuning frequencies from the cavity resonance frequencies, respectively. Then the Hamiltonian of the entire system is written aŝ where (â,â † ), (b,b † ), and (ĉ,ĉ † ) show the annihilation and creation operators of the OC, MC, and MR, respectively [4]. The subscripts o and e denote OC and MC. g j is the coupling strength, E j is the field strength of the input driving pump. The first three terms in Eq. (1) denote the free Hamiltonian of the OC, MC, and MR. The fourth(fifth) term shows an interaction between MR and OC(MC). The last terms represent driving pumps of the OC and MC.
To analyze the optic-microwave conversion, we need to solve an input-output relation of the entire system. To construct the input-output relation, we simplify the Hamiltonian of Eq. (1) by linearizing the Hamiltonian [4,18] and by using the rotating wave approximation [19][20][21]. We define intracavity quantum noise operatorsα andβ aŝ is the mean photon number in the cavity induced by the driving pumps and N j 1. The effective cavity detuning is obtained from the following equation: and Γ j = γ j + γ j is a cavity decay rate which is the sum of an input loss rate γ j and an intrinsic loss rate γ j . Then, the interaction Hamiltonian with respect to the free Hamiltonian can be linearized in the following equation: where G j = g j N j . The interaction Hamiltonian can be simplified with the appropriate setting of the effective cavity detuning. If ω m = ∆ o = ∆ e , i.e., both the optical and microwave pumps are red-detuned, then the interaction Hamiltonian can be simplified toĤ with the rotating wave approximation. In this Hamiltonian, both OC-MR and MC-MR relations are a beam-splitter interaction. This means that one phonon in the MR is created when one optical photon or one microwave photon is annihilated, and vice versa. Thus, one optical photon can be converted to one microwave photon through the MR.

III. OPTIC-MICROWAVE CONVERSION
The dynamics of the electro-optomechanical system with noise and damping can be described by using quantum Langevin equation(QLE) [22]. The nonlinear QLEs of the system can be written as follows: where γ m is a damping rate of the MR, and the subscript "in" denotes an input field, "out" does an output field, and "loss" does intrinsic loss. With the following cavity input-output relations [4,18]: and we can derive an input-output relation between optical wave and microwave as: where ω is the frequency difference of the output field from the resonance frequency. The coefficients of Eq. (7) are given as where In our analysis, we take the electro-optomechanical system in the recent demonstration [17]. The following values are exploited in our simulation: the coupling strengths are g o /2π = 6.6 Hz and g e /2π = 3.8 Hz; the OC and MC input loss rates are γ o /2π = 1.1 MHz and γ e /2π = 2.3 MHz; the intrinsic loss rates of OC and MC are γ o /2π = 1 MHz and γ e /2π = 0.2 MHz; the MR intrinsic loss rate γ m /2π = 11 Hz; the resonance frequencies of OC, MC, and MR are ω o /2π = 282 THz, ω e /2π = 6 GHz, and ω m /2π = 1.4732 MHz, respectively; the temperature of the electro-optomechanical system T = 35 mK; and N j = 1.7 × 10 8 , where j ∈ {o, e}.

A. Entanglement of converted two-mode Gaussian state
The TMSV state is an entangled state that is conventionally exploited in continuous-variable quantum information processing. It is generated by using two single-mode squeezed vacuum states and a 50:50 beam splitter [23] or by using a spontaneous parametric down-conversion crystal [24,25]. A TMSV state can be written in the photon number basis as follows: where N S is the mean photon number of each mode, and the subscripts S and I under the ket vectors denote the signal and idler modes of the TMSV state. As shown in Fig. 1 (a), we analyze the quantum state that the signal mode of an optical TMSV state is converted to microwave frequency, and we will call this state as converted two-mode Gaussian(CTMG) state. Since thermal noise is produced by the electro-optomechanical system during the conversion, the CTMG state is not a TMSV state. Entanglement of the two-mode Gaussian state can be quantified by using various methods, such as the logarithmic negativity(LN) [26], quantum discord [27,28], and the entanglement of formation [29]. Here, we quantify entanglement by using the LN for a two-mode Gaussian state [30,31]. To quantify entanglement, we first determine the covariance matrix(CM) of our system. Here, we consider a TMSV state with a thermal bath, where their first-order moments are zero. The element of CM can be obtained from where with the microwave-converted signal mode operatorβ S . The CM can be written with 2 × 2 block matrices A, B, and C as and then the logarithmic negativity can be evaluated from the symplectic eigenvalues under partial transposition. If symplectic eigenvalues of a partially transposed two-mode Gaussian state are not smaller than 1/2, the two-mode Gaussian state is separable, and if there is a symplectic eigenvalue that is smaller than 1/2, the two-mode Gaussian state is entangled [32]. The symplecic eigenvalues under partial transposition can be obtained from the following equation [33]: where Det[X] denotes the determinant of matrix X. From two positive roots of this equation ξ ± , the LN is defined as since ξ + always satisfies ξ + ≥ 1/2 [33]. In order to obtain the LN of the CTMG state, we calculate the expectation value written in Eq. (11). Since we consider loss and noise in our QLE, we take a thermal bath as loss and noise in the conversion. Then, our initial state becomesρ whereρ Th is the thermal bath. To calculate LN of the CTMG state, we use the microwave-converted signal mode operator that is the microwave output modeβ S =β out of Eq. (7). The expectation values can be calculated by using the input-output relation in Eq. (7). For example, The block matrices A, B, and C can be obtained from these expectation values. In the optical input mode of the electro-optomechanical system, if we ignore the stationary cavity field, there are the signal mode of the TMSV state and a thermal noise, with α † inα in = N S + N o . Since the microwave input mode and the loss modes are not related with the input TMSV state, the other expectation values are given by β † inβ in = N e , α † lossα loss = N o , β † lossβ loss = N e , and ĉ † lossĉ loss = N m , where N j = [exp( ω j /k B T ) − 1] −1 is the mean photon number of the thermal noise induced from the electro-optomechanical system and T is a temperature of the electro-optomechanical system. In the phasesensitive cross correlation of Eq. (17c), onlyα in mode has a correlation with the idler mode, so the other terms become zero. The input optical wave is the signal mode of the TMSV state with thermal noise which is not related with the idler mode, and thus, α inâI = N S (N S + 1).
A full description of ξ − of the CTMG state can be derived analytically. We show ξ − at ω = 0 as follows: where Then, we can calculate the amount of surviving entanglement by comparing LN of the initial TMSV state and that of the CTMG state. Fig. 2(a) shows LN of the initial TMSV state(blue dashed line) and that of the CTMG state(black line) as a function of the mean photon number(N S ) of the signal mode in the TMSV state. The entanglement of the initial TMSV state partially survives after the optic-microwave conversion of the signal mode. The LN of the CTMG state has nonzero values and increases up to 0.304. Fig. 2(b) shows the entanglement surviving ratio after the optic-microwave conversion as a function of N S . Under a fixed conversion efficiency, the entanglement survival of the CTMG state depends on the system of the frequency conversion as well as the injected input mean photon number. With our electro-optomechanical system, the entanglement surviving ratio between LN of the converted state and that of the initial TMSV state is maximized as approximately 0.178 at N S = 0.157. Due to the induced thermal noise, the entanglement of the initial state does not well survive in the very low limit of N S .
The LN of the TMSV state in Eq. (21) can be infinitely large with increasing N S , but the LN of the CTMG state is upper bounded by 0.304. We define this upper bound as the entanglement surviving capacity of the electrooptomechanical system, since the upper bound is calculated from the system parameters. The capacity is defined as the LN of the CTMG state in the limit of N S → ∞: where k i with i ∈ {1, 2, 3, 4, 5} is given in appendix. The entanglement surviving capacity P converges if k 2 = (k 5 ) 2 satisfies, and this is always true in our system. By using the Taylor expansion, the capacity becomes If we consider the ideal situation, i.e., the noiseless conversion at T → 0, the capacity becomes simpler as shown in the following equation: where R(0) is the frequency conversion efficiency of the electro-optomechanical system at ω = 0 that will be described in the next section. From Eq. (24), the ideal entanglement surviving capacity without noise is related with the conversion efficiency of the system.

B. Conversion efficiency and entanglement survival
In this section, we analyze the conversion efficiency of the electro-optomechanical system and the surviving entanglement of the CTMG state. From Eq. (7), we define an optic-microwave conversion efficiency as R(ω) ≡ |C 1 (ω)| 2 which is the probability that an input optical photon is converted to a microwave photon [9]. As shown in Eq. (8a), C 1 (ω) does not depend on the number of input photons or temperature of the conversion system, so it is a characteristic of the electro-optomechanical system. The conversion efficiency of our system at ω = 0 approaches to R(0) ≈ 0.328, so that less than a half of input optical photons are converted to microwave photons.
Note that our conversion efficiency is not the same as that used in the existing experiments [16,17]. In those demonstrations, the ratio between input amplitude and converted amplitude is defined as a conversion efficiency, which is corresponding to β † Sβ S / â † Sâ S in our calculation. As shown in Eq. (17a), the mean photon number of the converted signal is affected by thermal noise. Due to the thermal noise, their conversion efficiency varies with the mean photon number of input and temperature of the electro-optomechanical system. In our system, β † Sβ S / â † Sâ S = 0.482 with a single-photon level N S = 1 and β † Sβ S / â † Sâ S = 0.636 with N S = 0.5. Because of the added thermal noise, β † Sβ S / â † Sâ S is larger than R(0) ≈ 0.328, which is our conversion efficiency. A full equation of our conversion efficiency can be written down. Here, we describe the conversion efficiency at ω = 0 as: If we can control the cavity input loss rates while the other parameters are fixed, the conversion efficiency is maximized at the following values: where the cavity input loss rates are related with the coupling rates and the intrinsic loss rates. In our electrooptomechanical system, the maximum conversion efficiency 0.962 is obtained at γ o = 82.4 MHz and γ e = 27.3 MHz. Fig. 3(a) shows R(0) as a function of the optical and microwave cavity input loss rates. The other parameters are the same as those described in the Sec. III. As previously studied [16,17], the conversion efficiency maximizes when G 2 o /Γ o γ m and G 2 e /Γ e γ m are equivalent, whereas it gets smaller when the values are far apart. Fig. 3(b) shows LN of the CTMG state with respect to the cavity input loss rates at ω = 0 when the initial TMSV state satisfies N S = 1. Note that the unit of the cavity input loss rates in Fig. 3(a) is GHz but that in Fig. 3(b) is MHz. Even though we showed that the maximum entanglement surviving capacity is related with the conversion efficiency of the system in Eq. (24), the two plots show that the conversion efficiency is not directly proportional to surviving entanglement of the output state with the noise. As previously obtained, the cavity input loss rates for the maximum conversion efficiency are γ o = 82.4 MHz and γ e = 27.3 MHz. However, in Fig. 3(b), the maximum LN of the CTMG state is obtained at γ o = 1.63 MHz and γ e = 1.02 MHz, and the conversion efficiency with these input loss rates is 0.550. These results implicate that the higher conversion efficiency does not guarantee the better survival of entanglement.

V. CONCLUSION
In this article, we analyzed entanglement of an optic-microwave entangled state that is generated by applying the quantum frequency conversion with an electro-optomechanical system to one mode in an optical two-mode squeezed vacuum(TMSV) state. We investigated entanglement of the initial TMSV state and the converted two-mode Gaussian(CTMG) state by the logarithmic negativity(LN). It was found that the ratio of surviving entanglement is affected by the mean photon number of the initial TMSV state even if the conversion efficiency of the electro-optomechanical system is fixed. In addition, we showed that the entanglement of the CTMG state has an upper bound which is a characteristic of the frequency conversion system. Finally, we found that the higher conversion efficiency does not guarantee the larger entanglement of the CTMG state, even though the conversion efficiency is related with the entanglement surviving capacity at the zero temperature limit. This result implies that parameters of the conversion system should be controlled for enhancing the entanglement survival rather than the conversion efficiency in order to generate a highly entangled optic-microwave two-mode state.
From the results, we theoretically confirm that entanglement of the initial state can survive after the quantum frequency conversion, and the CTMG state can be used in a long-distance target detection and quantum network that exploit optic-microwave entanglement.