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Terahertz quantum illumination using free-electron lasers

Linfeng Zhang, Leshi Zhao, Haitan Xu, and Zheng Li

Phys. Rev. Applied 21, 064021 – Published 10 June 2024

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Abstract

We propose a terahertz quantum illumination (QI) scheme utilizing emissions from free-electron lasers (FELs) based on joint photon number measurement and quantum state discrimination using maximum a posteriori estimation. We analyze the performance of our QI scheme and quantify it with error probabilities. We numerically demonstrate the quantum advantage of our QI scheme over classical illumination. Our work paves the way to using FELs as entangled photon sources for quantum illumination in the terahertz regime.

Received 10 November 2023
Revised 2 January 2024
Accepted 16 May 2024

DOI:https://doi.org/10.1103/PhysRevApplied.21.064021

Physics Subject Headings (PhySH)

Electron beams & optics Lasers Quantum correlations, foundations & formalism Quantum entanglement Quantum optics

Free-electron lasers

Terahertz techniques

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Linfeng Zhang¹, Leshi Zhao¹, Haitan Xu ^2,3,4,*, and Zheng Li^1,5,6,†

¹State Key Laboratory for Mesoscopic Physics and Collaborative Innovation Center of Quantum Matter, School of Physics, Peking University, Beijing 100871, China
²School of Materials Science and Intelligent Engineering, Nanjing University, Suzhou, Jiangsu 215163, China
³Shishan Laboratory, Nanjing University, Suzhou, Jiangsu 215163, China
⁴School of Physical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
⁵Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
⁶Peking University Yangtze Delta Institute of Optoelectronics, Nantong, Jiangsu 226010, China

^*Coressponding author: [email protected]
^†Coressponding author: [email protected]

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Vol. 21, Iss. 6 — June 2024

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Figure 1
Schematic of terahertz quantum illumination by entangled emission from a FEL. (a) Illustration of entangled emission from a FEL in the laboratory frame. In such a frame, premicrobunched electrons are injected into an undulator and generate terahertz emission via wiggling motion. The resulting emission has nonlinear components dominated by double emission. (b) Schematic of a terahertz quantum illumination system. After filtering, the two-mode emission from the FEL is directed to the QI system. Photons in one beam act as the probe to illuminate the target region, while photons in the other beam are sent to the detector as ancilla. By a joint measurement we can determine whether an object is present in the target region.
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Figure 2
Entangled emission from the FEL. (a) Illustration of the angular configuration parameters in (b),(c). $Z_{1, 2}$ and $A_{1, 2}$ are the zenith angles and the azimuth angles of the three-momentum $k_{1}, k_{2}$ of the emitted photons, which are defined with respect to the point $S$ at the center of undulator for far-field measurement. (b) Normalized double angular distribution of the differential scattering rate $\frac{d \dot{W}}{d k_{1}^{0} d Ω_{1} d Ω_{2}}$ in the laboratory frame by numerical simulation, where $k_{1}^{0} = ω_{1}$ is the energy of photon 1 and $Ω_{1}$ and $Ω_{2}$ are the solid angles of the emitted photon pair. (c) Double angular distribution of the concurrence $C$ by numerical simulation. In (b),(c), the polar angle and the radii refer to $A_{2} \in [0, 2 π]$ and $γ \tan (Z_{2}) \in [0, 1]$ for photon 2, where $γ$ is the Lorentz factor of the incident electrons. The black and purple dots represent different angular configurations of photon 1 for each plot. We choose $ω_{1} ≃ ω_{fd} / 3$ and $ω_{2} ≃ 5 ω_{fd} / 3$ . (d) Reduced density matrix ${\hat{ρ}}_{f}$ for the chosen angular configuration of the photon pair with $γ \tan (Z_{1}) = 0.36$ , $γ \tan (Z_{2}) = 0.55$ , $A_{1} = 90^{\circ} = A_{2}$ , where the concurrence $C = 0.99$ indicates nearly optimal entanglement, and the scattering rate is relatively high. In (b),(c), this angular configuration is marked with the purple point representing photon 1 and the yellow point representing photon 2. The density matrix corresponds to a Bell state approximately, which is useful for QI.
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Figure 3
Discrimination procedure of the QI scheme based on direct joint photon number measurement. (a) Joint photon measurement process. For each illumination pulse, we use THz photodectors to record the photons from the return and idler modes. The filled and open rectangles refer to the cases that at least one photon is observed and that no photon is observed, respectively. With the outcomes from both photodectors for each illumination pulse, we can obtain the total count numbers $M_{O}$ , $M_{R}$ , $M_{I}$ , and $M_{C}$ for different outcome combinations of the joint measurement, and $M = M_{O} + M_{R} + M_{I} + M_{C}$ is the total number of illumination pulses. Here the subscript $O$ denotes the outcome combination that no photons are received by either photodector, subscript $C$ denotes the outcome combination that photons from the return and idler modes are received by both photodetectors, subscript $R$ denotes the outcome combination that only photons in the return mode are received by the corresponding photodetector, and subscript $I$ denotes the outcome combination that only photons from the idler mode are received by the corresponding photodetector. (b) Determination of the presence of the object based on the principle of MAPE with the count numbers $M_{O}$ , $M_{R}$ , $M_{I}$ , $M_{C}$ . Specifically, for a certain prior distribution, we can calculate the conditional or posterior probabilities $P_{po} (0) = P (0 | ⟨ M_{O}, M_{R}, M_{I}, M_{C} ⟩)$ and $P_{po} (1) = P (1 | ⟨ M_{O}, M_{R}, M_{I}, M_{C} ⟩)$ via the Bayesian theorem. If $P_{po} (0) < P_{po} (1)$ , we decide that there is an object in the target region, otherwise the object is absent.
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Figure 4
Error probability of the QI scheme compared with that of the classical illumination scheme. (a) Schematic of the detectors used in the QI scheme as shown in Fig. 1, based on the quantum capacitance detector [13]. A shielded cryogenic chamber is used to suppress the thermal noise for the THz photodetector. (b) Numerical results of the error probabilities of different illumination schemes. The error probability of the QI scheme with direct joint photon number measurement and MAPE is significantly lower than that of the classical illumination scheme. The parameters used in the numerical calculations are ${\bar{N}}_{bR} = 200, {\bar{N}}_{bI} = 0.01, N_{S} = 0.018, N_{I} = 0.059, N_{C} = 0.005, κ = 0.01, ζ = 0.5$ .
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Physical Review Applied

Terahertz quantum illumination using free-electron lasers

Linfeng Zhang, Leshi Zhao, Haitan Xu, and Zheng Li

Phys. Rev. Applied 21, 064021 – Published 10 June 2024

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