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Quantum Illumination at the Microwave Wavelengths

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Quantum Illumination at the Microwave Wavelengths Quantum Illumination at the Microwave Wavelengths 1 2 3 4 5 6, Shabir Barzanjeh, Saikat Guha, Christian Weedbrook, David Vitali, Jeffrey H. Shapiro, and Stefano Pirandola ∗ 1Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany 2Quantum Information Processing Group, Raytheon BBN Technologies, Cambridge, Massachusetts 02138, USA 3QKD Corp., 60 St. George St., Toronto, M5S 3G4, Canada 4School of Science and Technology, University of Camerino, Camerino, Macerata 62032, Italy 5Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 6Department of Computer Science & York Centre for Quantum Technologies, University of York, York YO10 5GH, United Kingdom Quantum illumination is a quantum-optical sensing technique in which an entangled source is exploited to improve the detection of a low-reflectivity object that is immersed in a bright thermal background. Here we describe and analyze a system for applying this technique at microwave frequencies, a more appropriate spectral region for target detection than the optical, due to the naturally-occurring bright thermal background in the microwave regime. We use an electro-opto- mechanical converter to entangle microwave signal and optical idler fields, with the former being sent to probe the target region and the latter being retained at the source. The microwave radiation collected from the target region is then phase conjugated and upconverted into an optical field that is combined with the retained idler in a joint-detection quantum measurement. The error probability of this microwave quantum-illumination system, or quantum radar, is shown to be superior to that of any classical microwave radar of equal transmitted energy. PACS numbers: 03.67.-a, 03.65.-w, 42.50.-p, 07.07.Df Introduction.{ Entanglement is the foundation of many problem in a natural way by purposefully injecting am- quantum information protocols [1{3], but it is easily de- plified spontaneous emission noise to thwart eavesdrop- stroyed by environmental noise that, in almost all cases, ping. By contrast, similar noise injection in QI target- kills any benefit such nonclassical correlations would oth- detection experiments has to be considered artificial, be- erwise have provided. Quantum illumination (QI) [4, 5], cause better target-detection performance would be ob- however, is a notable exception: it thrives in environ- tained without it. The appropriate wavelengths for QI- ments so noisy that they are entanglement breaking. enabled target detection thus lie in the microwave region, The original goal of QI was to detect the presence in which almost all radar systems operate and in which of a low-reflectivity object, immersed in a bright ther- there is naturally-occurring background radiation con- mal background, by interrogating the target region with taining many photons per mode on average. In general, one optical beam while retaining its entangled counter- the development of quantum information techniques for part for subsequent joint measurement with the light microwave frequencies is quite challenging [16{18]. returned from that target region. Although the ther- In this Letter, we introduce a novel QI target-detection mal noise destroys the entanglement, theory showed system that operates in the microwave regime. Its that the QI system will significantly outperform a clas- transmitter uses an electro-opto-mechanical (EOM) con- sical (coherent-state) system of the same transmitted verter [19{23] in which a mechanical resonator entan- energy [5{7]. Later, a QI protocol was proposed for gles signal and idler fields emitted from microwave and secure communication [8] whose experimental realiza- optical cavities [19, 20]. Its receiver employs another tion [9] showed that entanglement's benefit could indeed EOM device|operating as a phase-conjugator and a survive an entanglement-breaking channel. Because of wavelength converter|whose optical output is used in this feature, QI is perhaps one of the most surprising arXiv:1410.4008v3 [quant-ph] 16 Feb 2015 a joint measurement with the retained idler. We show protocols for quantum sensing. Together with quantum that our system dramatically outperforms a conventional reading [10{13], it represents a practical example of quan- (coherent-state) microwave radar of the same trans- tum channel discrimination [3], in which entanglement is mitted energy, achieving an orders-of-magnitude lower beneficial for a technologically-driven information task. detection-error probability. Moreover, our system can be So far, QI has only been demonstrated at optical wave- realized with state-of-the-art technology, and is suited lengths [9, 14, 15], for which naturally-occurring back- to such potential applications as standoff sensing of low- ground radiation contains far less than one photon per reflectivity objects, and environmental scanning of elec- mode on average, even though QI's performance advan- trical circuits. Thanks to its enhanced sensitivity, our requires tage the presence of a bright background. The system could also lead to low-flux non-invasive techniques QI communication protocol from [8, 9] deals with this for protein spectroscopy and biomedical imaging. Electro-opto-mechanical converter.{ As depicted in Fig. 1(a), the EOM converter couples a microwave-cavity ∗ [email protected] mode (annihilation operatora ^w, frequency !w, damp- 2 T ing rate κw) to an optical-cavity mode (operatora ^o, r = γM n¯ γM kBTEOM=~!M , where kB is Boltzmann's b ≈ frequency !o, damping rate κo) through a mechanical constant, TEOM is the EOM converter's absolute temper- T !M =(kB TEOM) 1 resonator (operator ^b, frequency ! , damping rate γ ) ature,n ¯ = [e~ 1]− , and the approxima- M M b − [20{22]. In the frame rotating at the frequencies of the tion presumes kBTEOM ~!M , as will be the case for microwave and optical driving fields, the interaction be- the parameter values assumed later. The second term tween the cavities' photons and the resonator's phonons in Eq. (1) is a beam-splitter interaction between the me- is governed by the Hamiltonian [24] chanical resonator and the microwave cavity mode that transfers the entanglement to the propagating microwave ^ ^ ^ ^ ^ ^ H = ~!M byb + ~ ∆0;j + gj(b + by) a^jya^j + Hdri: field d^w, as long as the microwave-mechanical rate satis- j=w;o 2 X h i fies Gw/κw > r [25, 26]. Microwave-optical entanglement.{ The output propa- Here, gj is the coupling rate between the mechanical res- gating modes can be expressed in terms of the intracavity onator and cavity j, which is driven at frequency !j ∆0;j − quantum noise operators,c ^j;in, and the quantum Brow- by the coherent-driving Hamiltonian H^dri [24]. nian noise operator, ^bint, via [24] d^ = A c^ Bc^y C ^b ; (2) w w w;in − o;in − w int d^ = Bc^y + A c^ C ^by ; (3) o w;in o o;in − o int where Aj, B, and Cj depend on the cooperativity terms 2 Γj = Gj /κjγM as given in [24]. Thec ^w;in; c^o;in and ^bint modes in Eqs. (2) and (3) are in independent ther- T T mal states whose average photon numbers,n ¯w,n ¯o , and T n¯b , are given by temperature-TEOM Planck laws at their respective frequencies. It follows that the propagating modes, d^w and d^o, are in a zero-mean, jointly-Gaussian state completely characterized by the second moments, 2 T 2 T 2 T FIG. 1. (a) Schematic of the electro-opto-mechanical (EOM) n¯ d^y d^ = A n¯ + B (¯n + 1) + C n¯ ; w ≡ h w wi j wj w j j o j wj b converter in which driven microwave and optical cavities are 2 T 2 T 2 T n¯ d^yd^ = B (¯n + 1) + A n¯ + C (¯n + 1); coupled by a mechanical resonator. (b) Microwave-optical o ≡ h o oi j j w j oj o j oj b QI using EOM converters. The transmitter's EOM converter ^ ^ T T T dwdo = AwB(¯nw + 1) BAon¯o + CwCo(¯nb + 1): entangles microwave and optical fields. The receiver's EOM h i − converter transforms the returning microwave field to the op- The propagating microwave and optical fields will be tical domain while performing a phase-conjugate operation. entangled if and only if the metric d^wd^o =pn¯wn¯o is greater than 1 [24]. As we can seeE from ≡ jh Fig. 2,ij there is The electro-opto-mechanical coupling rates gj are typ- a wide region where > 1 in the plane of the coopera- ically small, so that we can linearize the Hamiltonian by E tivity parameters, Γw and Γo, varied by varying the mi- expanding the cavity modes around their steady-state crowave and optical powers driving their respective cavi- field amplitudesc ^j =a ^j Nj, where the Nj 1 ties, and assuming experimentally-achievable system pa- are the mean numbers of cavity− photons induced by the p rameters [25, 27]. The threshold condition = 1 almost pumps [19, 25]. In the interaction picture with respect coincides with the boundary between the stableE and un- to the free Hamiltonian, we may then write [24] stable parameter regions, as given by the Routh-Hurwitz ^ ^ ^ ^ ^ criterion [33]. H = ~Go(^cob + byc^y) + ~Gw(^cwby + bc^y ); (1) o w The quality of our microwave-optical source can also where Gj = gj Nj is the multi-photon coupling rate. be evaluated using measures of quantum correlations, as This expression assumes that the effective cavity detun- typical in quantum information. Since the QI's advan- p ings satisfy ∆w = ∆o = !M and that resonator is in its tage is computed at fixed mean number of microwave − fast-oscillation regime, so that the red sideband drives the photonsn ¯w irradiated through the target, the most pow- microwave cavity while the blue sideband drives the opti- erful quantum resources are expected to be those maxi- cal cavity and we can neglect terms oscillating at 2!M .
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