Microwave Quantum Illumination with Optical Memory and
Single-Mode Phase-Conjugate Receiver

Quantum illumination (QI) is a protocol that detects a low-reflectivity target using an entangled state within a fixed range, which outperforms classical illumination (CI) under conditions of high background noise and a low signal mean photon number [1, 2, 3, 4]. Initially QI was proposed with single-photon-level entangled state and low background noise [5], and then it was developed with gaussian states and high background noise [6]. Two-mode squeezed vacuum (TMSV) state presents a nearly optimal entangled state in QI [7, 8, 9], whereas coherent state is theoretically a benchmark state. QI with TMSV state can attain 6 dB quantum advantage over coherent state [6], which requires collective measurement in a receiver [10, 11]. Since the collective measurement has technical difficulties such as implementing feedforward process, beam splitter array, nonlinear interaction, memory, and so on, it is more feasible to consider local joint measurement on each pair of the returned signal mode and the idler mode. By local joint measurement, QI with TMSV state can achieve the sub-optimal 3 dB advantage [12, 13, 14]. Until now, several QI experiments with local measurement were implemented in microwave range [15, 16, 17, 18] as well as in optical range [19, 20, 21, 22, 23], where an optical (or a Josephson) parametric amplifier receiver could only obtain quantum advantage in experiment [19, 18].

For the local joint measurement in QI, it is necessary to have a quantum memory to store the idler mode in order to be interfered with the reflected signal mode. There was a research from Reichert et al. [24] that quantum memory cannot be circumvented in the local joint measurement through a delay operation on a classical signal. Thus, quantum memory is inevitable for QI. The point is that the loss of the quantum memory in the idler mode highly affects the performance of QI. Recently, Kim et al. [25] demonstrated that the quantum memory loss of the idler mode dominates the performance of QI. When the memory loss is more than 50%, QI cannot outperform CI under local joint measurement. Note that the upper bound of the memory loss is 75% under collective measurement. In optical range, the quantum memory is simulated with a delay line in the idler mode [26]. The current state-of-the-art efficiency of the optical delay line is about 0.14 dB/km, which therefore acts as a bottleneck on the realization of the long-ranged QI technology [27].

Although current technologies are more feasible in optical ranges, quantum advantage is more achievable in microwave range that is surrounded with high background noise. There were some trial experiments [28, 29, 30, 31, 32] to convert from microwave to optical wave and vice versa. Alternatively, to take both benefits of optical and microwave ranges, Barzanjeh et al. [33] proposed microwave QI by employing two electro-optomechanical (EOM) devices as a generation of input entangled microwave-optic fields at a transmitter and a phase-conjugated EOM system at a receiver. In this way, microwave QI can utilize the microwave signal and optical idler, having both benefits in achieving the quantum advantage: a large mean photon number of background noise in microwave signal frequency and the feasibility of optical components in optical idler frequency.

Since the originally proposed microwave QI assumed an ideal optical quantum memory [33], it is important to consider a lossy optical memory in an idler mode, resulting in the minimum boundary of quantum advantage in microwave QI. Here, we propose a testbed of microwave QI with a lossy optical memory. First, we observe the memory-loss effect on quantum advantage, providing an optimal input state to achieve the best quantum advantage regarding memory loss. Second, we propose a new receiver that is capable of operating in a lossy condition. We show that our receiver can approximately achieve the maximum quantum advantage for local joint measurement, even with implementing an on-off detection. Additionally, we show that the performance of the receiver is robust against the memory loss.

Our paper is organized as follows. Section II outlines our model of microwave QI with lossy optical memory under some conditions. Section III discusses the optimal conditions for the transmitter to attain the maximum quantum advantage for local joint measurement. Section IV introduces a novel receiver performing effectively despite memory loss and an on-off detection, along with an analysis of its performance. Section V draws the conclusion.

We consider microwave quantum illumination with quantum memory in Fig. 1. (a), where an EOM transmitter generates entangled states that consist of pairs of microwave signal and optical idler. EOM transmitter and converter consist of microwave and optical cavities which are coupled through a mechanical resonator, where the microwave and optical pumps are injected into the corresponding cavities as shown in Fig. 1. (b) and (c). In the absence of a target, the signal passes through the air and only a background thermal noise arrives at the EOM converter. When the target is present, the signal hits the target and arrives at the EOM converter along with the background thermal noise. The reflected signal carries the target information that is proportional to the target reflectivity $\kappa$. Upon the EOM converter, the reflected signal mode is frequency-converted to match the optical frequency of the idler. Meanwhile, the idler is sent to the quantum memory to preserve its state until joint measurement, where the quantum memory is represented by a delay line that is simply modeled as an effective transmission efficiency [34] $\eta=e^{-\gamma_{I}t}$ ($\gamma_{I}$: a dampinig rate, $t$: delay time). Once both the converted signal mode and the stored idler mode are recombined, they are jointly measured in a receiver to detect a target. We assume that the receiver performs the local joint measurement.

In Fig. 1, a target is modeled as a beam splitter with a reflectivity $\kappa$, so that the transformation by the target interaction is given by $\hat{b}_{S}=\sqrt{\kappa}\hat{a}_{S}+\sqrt{1-\kappa}\hat{a}_{B}$, where $\hat{a}_{S(I)}$, $\hat{a}_{B}$, and $\hat{b}_{S}$ are the annihilation operators of signal(idler), background thermal noise, and returned signal modes, respectively. Since the thermal noise is independent of the target reflectivity after interacting with the target, the mean photon number of the thermal noise is defined as $N_{B}/(1-\kappa)$ and then the returned signal mode presents the mean photon number of $\langle\hat{b}_{S}^{\dagger}\hat{b}_{S}\rangle=\kappa N_{S}+N_{B}$. The best quantum advantage is obtained in the range of $\kappa,N_{S}\ll 1\ll N_{B}$, which validates microwave signal under high background thermal noise. Note that it is negligible for thermal noise in the memory since the idler is in optical regime.

QI system discriminates whether $\kappa>0$ (target presence) or $\kappa=0$ (target absence) by measuring an observable $\hat{O}$ on the joint returned signal and idler modes. The performance is described with a detection error probability. We focus on verifying the target presence by comparing mean of $M$ observables $\{O^{(k)}:1\leq k\leq M\}$ with a decision threshold $R_{th}$, which decides $\kappa>0$ if $\bar{O}:=\frac{1}{M}\sum_{k}O^{(k)}>R_{th}$ and $\kappa=0$ if $\bar{O}\leq R_{th}$. Then, the detection error probability $P_{err}$ with $\hat{O}$ is given by

which decreases exponentially with the number of modes $M$, so that $P_{err}(\hat{O})\sim\exp(-{\rm SNR(\hat{O})}M)$ [35], where the signal-to-ratio (SNR) is defined as

Under local measurement with the observable, the maximum performance of the QI is defined as the maximum error exponent $\gamma_{\rm QI}:=\max_{\hat{O}}{\rm SNR}(\hat{O})$.

Although a TMSV state is a nearly optimal input state for QI, the microwave QI of Ref. [33] generates a two-mode squeezed thermal state that is respresented as

where $\hat{S}_{SI}(r)=\exp[r(\hat{a}_{S}\hat{a}_{I}-\hat{a}_{S}^{\dagger}\hat{a}_{I}^{\dagger})]$ and $r$ is a squeezing parameter of the two-mode squeezing operator. $\rho_{th}(\nu)$ is a thermal state with a symplectic eigenvalue $\nu$, where $\nu_{S}$ and $\nu_{I}$ represent the symplectic eigenvalues of the signal and idler modes. Since the symplectic eigenvalues are the eigenvalues of the symplectic decomposition of the covariance matrix of the joint signal-idler state, it is given by $\nu_{i}=2N_{th,i}+1$ for $i\in\{S,I\}$, where $N_{th,i}$ is the mean photon number of $\rho_{th}(\nu_{i})$. This implies that a two-mode squeezed thermal state with high symplectic eigenvalues is generated by squeezing a thermal state with high temperature, whereas the TMSV state gives the minimum symplectic eigenvalue $\nu_{S}=\nu_{I}=1$. In the same scenario, it is reasonable to consider a displaced thermal state as a classical benchmark for microwave QI [36], but here we take a strict benchmark with a coherent state.

In order to evaluate the quantum advantage, we employ the maximum error exponent $\gamma_{\rm QI}$ for QI with an optimal input state that is compared with $\gamma_{\rm CI}$ for CI with a coherent state. Under the same signal mean photon number, let $\mathcal{F}=\gamma_{\rm QI}/\gamma_{\rm CI}$ be a figure of quantum advantage. Then $\mathcal{F}>1$ implies the quantum advantage. At $N_{S}\ll 1\ll N_{B}$, the quantum advantage can approach $\mathcal{F}=2$ with local measurement, which represents the sub-optimal 3 dB quantum advantage.

In this section, we present the optimal condition of an input state to achieve the best quantum advantage under memory loss. The input state has two different parameter domains as follows: First, the input state is a function of symplectic eigenvalue and two-mode squeezing parameter. Second, the input state is a function of cooperativities, where it is generated from the EOM transmitter.

Any QI setup does not always achieve the maximum quantum advantage, such that it is necessary to investigate the performance of QI with a practical receiver. Here, we employ $\mathcal{F}_{\rm receiver}={\rm SNR}(\hat{O}_{\rm receiver})/\gamma_{\rm CI}$ as the figure of quantum advantage, with an observable of the receiver $\hat{O}_{\rm receiver}$.

In the microwave QI of Ref. [33], a phase-conjugate (PC) receiver has been suggested by addressing the EOM device as a frequency-converting phase conjugator. PC receiver interferes the frequency-converted signal mode and stored idler mode by a 50:50 beam splitter and measures the photon number difference of the output modes with photon number detectors. Theoretically, the PC receiver achieves the quantum advantage $\mathcal{F}_{\rm PC}$ up to $3$ dB [12]. However, experimentally, it has not been completely realized while digitally being achieved with no quantum advantage [17]. The obstacle to implementing the PC receiver comes from the necessity of a photon number resolving (PNR) detector with high resolution, since the PC receiver setup assumes to fully count the frequency converted background thermal noise. Actually, it is a challenge to implement a high-resolution PNR detector.

We propose a single-mode PC (SMPC) receiver that consists of an EOM device, a low-reflectivity beam splitter, and a finite-resolution PNR detector, as shown in Fig. 6. The reflected signal mode $\hat{b}_{S}$ goes through the EOM converter, resulting in its frequency conversion into the optical frequency. The frequency-converted signal mode $\hat{c}_{S}$ and the stored idler mode $\hat{c}_{I}$ are interfered at the beam splitter with low reflectivity $R\ll 1$. Given the output mode $\hat{c}=\sqrt{R}\hat{c}_{S}+\sqrt{1-R}\hat{c}_{I}$, the PNR detector measures the photon number of the mode up to its resolution $K$. The corresponding observable of the PNR detector is given by $\hat{O}_{\rm SMPC}=\min(\hat{c}^{\dagger}\hat{c},K)$.

By choosing appropriate cooperativities of the EOM converter and the reflectivity $R=\sqrt{N_{c,I}}/N_{c,S}$ with $N_{c,i}:=\langle\hat{c}_{i}^{\dagger}\hat{c}_{i}\rangle$ for $i\in\{S,I\}$, the SMPC receiver can fully achieve 3 dB quantum advantage, i.e. $\mathcal{F}_{\rm SMPC}=2$. Note that $N_{c,S}$ is assumed to be invariant whether $\kappa=0$ or $\kappa>0$. We emphasize that the SMPC receiver can achieve the possible maximum quantum advantage regardless of $K$, which is proved in Appendix C. Suppose the quantum-advantage regime of $\Gamma_{o},\Gamma_{w}\gg 1$, which validates to assume $\nu_{S}=\nu_{I}=\nu$ from $N_{S}\simeq N_{I}$. Fig. 7. (a) shows $\mathcal{F}_{\rm SMPC}$ corresponding to $K=1$ and $\infty$ assuming the input state as a TMSV state. As shown in the figure, even in the case of $K=1$, i.e. when employing the on-off photon detection, the SMPC receiver achieves the 3 dB advantage in a low-$N_{S}$ regime. When the input signal includes the internal thermal noise, i.e. $\nu>1$, the receiver may achieve the lower quantum advantage as shown in Fig. 7. (b). Note that when $\nu>1$, the signal photon number is lower-bounded to $(\nu-1)/2$, so that the unavailable region exists in the domain.

In Fig. 7. (c) and (d), we show that the SMPC receiver is robust against memory loss. It exhibits the quantum advantage by $\mathcal{F}_{{\rm SMPC},K=1}$ in different memory efficiencies, where the TMSV state achieves the maximally feasible quantum advantage of $2\eta$, implying that the SMPC receiver works well regardless of memory loss.

We presented a testbed of microwave QI with a lossy optical memory. To take quantum advantage under the memory loss, it is necessary to increase the input two-mode squeezing parameter that is proportional to increase the cooperativity of an optical cavity in an EOM trasmitter. It was quantitatively analyzed by maximum error exponents for local measurement. Under the testbed, we proposed an SMPC receiver that can achieve the maximum quantum advantage for local measurement. Our new receiver takes quantum advantage even with an on-off detection, which is also robust against the optical memory loss.

Since the memory loss profoundly affects the quantum advantage, our contribution lies on enhancing the performance of microwave QI with quantum memory in a practical setup. We expect that our idea can be further developed by taking a feedback method that was considered in microwave QI [38].