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Optomechanical Microwave Quantum Illumination in Weak Coupling Regime ∗ Wen-Juan Yang1,2, Xiang-Bin Wang1,2,3 1State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People’s Republic of China 2 CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 3 Jinan Institute of Quantum Technology, Jinan, Shandong 250101, China We propose to realize microwave quantum illumination in weak coupling regime based on multi- modeoptomechanicalsystems. Inourproposal themultimodetogetherwith afrequency-mismatch process could reduce mechanical thermal noise. Therefore, we achieve a significant reduction of error probability than conventional detector in weak coupling regime. Moreover, we optimize the 7 signal-to-noise ratio for limited bandwidth by tuningthedelay time of entangled wave-packets. 1 0 2 I. INTRODUCTION tonsincavitiescaninducebistabilitywhichmaygiverise n to experimentaldifficulties[27]. Therefore,it is meaning- a J Quantumentanglementisakeyingredientinquantum ful to study the generation of entanglement in weak in- 1 information processing[1–3]. In practice, entanglement teraction regime. Also, the existing study is limited to 2 canbeeasilydestroyedbyenvironmentnoise[4,5]. Quan- thesituationthattheoutputsignalsaremonochromatic. tum illumination(QI) can benefit from entanglement in A study for the more practical case of finite bandwidth ] target detection even it is under these entanglement- is needed. h destroying noise[3, 6–11]. In this paper, we propose a novel realization of quan- p - The aim of quantum illumination is to detect low- tumilluminationinweakcouplingregimebasedonmulti- nt reflective target which is embedded in a bright back- modeoptomechanicalsystems. Thetransmitterisanop- a ground thermal bath. Half of a pair of entangled optomechanicalsystemconsistingofatwo-modemicrowave u tical beams is sent out to interrogate the target region. cavitycouplingwithatwo-modeopticalcavityviaame- q Thenthereturnedandtheretainedsignalbeamareused chanicalresonator. Inthissystem,theoutputmicrowave [ to decide the presence or absence of the object. Even signal and the optical signal of the two cavities are en- 1 though the fragile entanglement is easily destroyed by tangled. Thereceiverisasimilaroptomechanicalsystem v the bright thermal noise, quantum-illumination proto- which converts the reflected microwave signal into opti- 5 col still has remarkable advantage over classical probe cal signal. The retained optical mode of the transmit- 9 protocol[7, 9–11]. Several experiments have realized the ter and the optical mode of the receiver are then sent 9 practical protocol in quantum sensing[12–14]. Recently, to the photon-detectors to make a joint measurement. 5 Weedbrooketal.[11]showthatquantumdiscordexhibits We use a two-mode and off-resonant(with frequency de- 0 . the role of preserving the benefits of quantum illumina- tuning δ) process to minimize the mechanical thermal 1 tion while entanglement is broken. noise. Consequently, our method can achieve a signifi- 0 It is advantageousto operate the frequency of the sig- cant reduction of error probability than classical system 7 1 nal which interrogates the object in microwave region[9, ofthesametransmittedenergy. Ourmethodworksinthe : 10, 12–14]. But so far there is no efficient way to detect weak many-photoncoupling regime where rotating-wave v single photon in microwave region while in the visible approximation works well. Moreover, we optimize the i X light frequency region, ultrasensitive detection of single- delay time ofthe microwavesignal’s filter function. This r photon have been achieved. Such that the detection of makes it possible for our method to work with a finite a microwave signals via the detection of their entangled bandwidthofsignals. Inthiscase,wecanstillachievean optical signals is a more efficient way[15–18]. Optome- improvementofsignal-to-noiseratioby48%thanthatof chanicalsystemscouldbeagoodcandidatetorealizethe classicaldetection given that signals-bandwidth σ which entanglement of microwave and optical fields by using equals cavities-bandwidth κ . mechanical motion[19–28]. However, the existing pro- The paper is organized as follows. In Sec. II we de- posal is supposed to work with rotating wave approx- scribethesystemandderivethequantumLangevinequa- imation(RWA) which requests weak many-photon cou- tionsfortheHamiltonianofourproposedsystem. InSec. plings of optomechanical systems[10]. In addition, it is III we calculate the error probability of the proposed QI extremely difficult to achieve strong coupling in optical system, give an analytical expression for the delay time regime for a system to generate entanglement between and analyze the limited bandwidth situation. Finally, optical and microwave modes. Moreover, plenty of pho- Sec. IV summarizes the main conclusions of this work. ∗EmailAddress: [email protected] 2 quency mismatch fulfilling ω δ = ω J ω , M M M − ≪ − ≪ under the resolved-sidebandapproximationand rotated- wave approximation, the Hamiltonian can be linearized as[30, 31] G G Hˆ =δˆb†ˆb 1(aˆ†ˆb+aˆ ˆb†) 2(aˆ ˆb+aˆ†ˆb†), (2) 1 − 2 + + − 2 − − where G /2 = g α , G /2 = g α , α = −iΩ±e−iφ± 1 + + 2 − − − ± κ/2 and κ is the cavity decay rate. Coupling terms G and 1 G can be chosen to be real and positive by adjusting 2 the laser phases φ and φ respectively. The Hamilto- + − nianshowsthattheemittedphotonsfrommodesaˆ and + aˆ could be entangled via intermediate phonon mode ˆb. − Taking cavity dissipation κ and mechanical dissipation γ into consideration, the Langevin equations of Eq. (2) are[32] FIG.1: (a)Optomechanicalsystemconsistingofatwo-mode microwave cavity coupling with a two-mode optical cavity via a mechanical resonator. (b) Realization of microwave G κ quantumillumination inhybridoptomechanical system. The aˆ˙+ = i 1ˆb aˆ+ √κaˆ+,in, 2 − 2 − transmitter entangles microwave signal and optical signal. G κ The receiver converts the reflected microwave signal into op- aˆ˙ = i 2ˆb† aˆ √κaˆ , (3) − − −,in tical signal and performs a phase-conjugate operation. 2 − 2 − ˆb˙ = iG1aˆ +iG2aˆ† (iδ+ γ)ˆb √γˆb . 2 + 2 −− 2 − in II. MULTI-MODE OPTOMECHANICAL SYSTEM The equations can be solved after Fourier transforma- tions where Oˆ[ω] = ∞ Oˆ(t)eiωtdt(with Oˆ = aˆ ,aˆ ,ˆb). −∞ + − As illustrated in Fig. 1 (a), we consider an optome- Applying the inputR-output relations Oˆ = √ΓOˆ + chanical system including a single-mode mechanical res- out Oˆ (with Γ=κ,γ)[32], we obtain onatorcoupledtobothatwo-modemicrowavecavityand in atwo-modeopticalcavity. Thetwomicrowavemodesare qatuefrnecqieusenωcies,ωω+,p,reωs+peacntidvetlwy.oTophteicmalemchoadneiscaalrmeaotdfereis- aˆ+[ω]=A+aˆ+,in[ω]−Baˆ†−,in[−ω]+C+ˆbin[ω], (4) at frequenc−y,pωb.−Cavity modes aˆ+,p and aˆ−,p are reso- aˆ−[−ω]=B∗aˆ†+,in[ω]+A−aˆ−,in[−ω]+C−ˆb†in[ω]. nantly driven by lasers. The Hamiltonian of the system is[18, 29–31] The correlation functions of input mode satisfy Hˆ = (ω aˆ† aˆ +ω aˆ†aˆ )+ω ˆb†ˆb j,p j,p j,p j j j M jX=± haˆ†+,in(t)aˆ+,in(t′)i= nT+δ(t−t′), jg (aˆ† +aˆ†)(aˆ +aˆ )(ˆb+ˆb†) aˆ† (t)aˆ (t′) = nTδ(t t′), (5) −X j j,p j j,p j h −,in −,in i − − j=± ˆb† (t)ˆb (t′) = nTδ(t t′), + Ω (aˆ eiωj,pt+iφj +H.c.), (1) h in in i b − j j,p X j=± where nT, nT and nT are thermal photon numbers of + − b where ω =ω J, Ω and φ are the intensities and the two cavity modes and phonon thermal number of ± ±,p j j ± phases of the driving lasers respectively. Given the fre- mechanical mode respectively. We obtain G2γκ n [ω,ω′]= aˆ† [ω]aˆ [ω′] =2π[4nT 1 + h +,out +,out i b G2 G2+[γ+2i(δ ω)](κ 2iω)2 | 1− 2 − − | (G G κ)2 +4 1 2 ]δ(ω ω′), (6a) [G2 G2+(γ+2i(δ ω))(κ 2iω)](κ 2iω)2 − | 1− 2 − − − | 3 G2γκ n [ ω, ω′]= aˆ† [ ω]aˆ [ ω′] =2π[4(1+nT) 2 − − − h −,out − −,out − i b G2 G2+[γ+2i(δ ω)](κ 2iω)2 | 1− 2 − − | (G G κ)2 +4 1 2 ]δ(ω ω′), (6b) [G2 G2+(γ+2i(δ ω))(κ 2iω)](κ 2iω)2 − | 1− 2 − − − | x[ω, ω′]= aˆ [ω]aˆ [ ω′] =2πδ(ω ω′)[ 2G G κ +,out −,out 1 2 − h − i − − G2(κ 2iω)+(κ+2iω)(G2 (γ+2i(δ ω))(κ 2iω))+2(1+nT)γ(κ2+4ω2) 1 − 2− − − b ]. (6c) (G2 G2+(γ+2i(δ ω))(κ 2iω))(G2 G2+(γ 2i(δ ω))(κ+2iω))(κ2+4ω2) 1− 2 − − 1− 2 − − Here we assume that nT =nT =0. In practice, in order + − to minimize the error probability of output signals, an increase of repeat rate is needed. Therefore, we need to consider output signals within certain frequency bandwidth. After projecting output signals onto wave-packet modes, aˆ = ∞ f∗(t)aˆ (t)dt. Here f (t) are fil- ±,f −∞ ± ±,out ± ter functions whRich satisfy ∞ f (t)2dt=1. −∞| ± | We use logarithmic negaRtivity to measure the entan- glementoftwocavitymodes. Thelogarithmicnegativity is given by[33] E =max(0, log 2η−), (7) N − 2 where η− is the smallest partial transposed symplectic eigenvalue of matrix V. The components of the covari- FIG. 2: (a) The spectrum of photon number n+[ω] around ance matrix have the form V = 1 ξ ξ + ξ ξ where ω = 0 and around ω = δ where G/κ = 1.0, γ/κ = 0.001, ij 2h i j j ii T ξ = [xˆ ,pˆ ,xˆ ,pˆ ]T with xˆ = (aˆ +aˆ† )/√2 and δ/κ = 1.5, nb = 61.945(ω+/2π = 10GHz, Tb = 30mK), pˆ± =(+aˆ±,f+−aˆ−†±,f−)/(i√2). T±hus we±h,afve ±,f nET+N.=(0c), nTT−he=sp0e.c(tbru)mThoefslopgeacrtirtuhmmiocf-nloeggaartiitvhitmyico-vneergoautitvpiutyt photon numberEN/n+. V + 1 0 V V 11 2 13 14   0 V + 1 V V V= 11 2 14 − 13 , (8) in Fig. 2. As shownin Fig. 2(a), output photonnumber  V13 V14 V33+ 21 0  n+[ω] has two peaks. One is at ω ≈ 0 with width ∼ κ   and the other is at ω δ with width Γ. When δ ap-  V V 0 V + 1  ≈ ∼  14 − 13 33 2  proaches 0, the two peaks emerge together. In this case, n [ω =0] 1 and the maximum entanglement appears where + ≫ at ω = 0, and the ratio(E /n ) of the entanglement to N + ∞ V = aˆ† aˆ = f [ω]2n [ω]dω, the photon number goes to the minimum. This is the 11 h +,f +,fi Z | + | + reason why the optimum ratio E /n in the previous −∞ N + ∞ literature Ref. [10] appears at G1 G2 but not at the V33 = haˆ†−,faˆ−,fi=Z−∞|f−[ω]|2n−[ω]dω, oInptoiumrucmaseenwtahnegrleemδ =ent0,peovinentwthhoeurge≫hpathraemraettieorEG1/≈nG2is. ∞ 6 N + V = Re( aˆ aˆ )=Re( f∗[ω]f∗[ ω]x[ω]dω), the secondminimum valleyataroundω =0,itstillhave 13 h +,f −,fi Z + − − a considerable value. −∞ ∞ V = Im( aˆ aˆ )=Im( f∗[ω]f∗[ ω]x[ω]dω), 14 h +,f −,fi Z + − − −∞ III. QUANTUM ILLUMINATION with 1 ∞ The input-output relation for the receiver is n [ω]= n [ω,ω′]dω′, + 2π Z + −∞ aˆ [ ω]=B∗aˆ† [ω]+A aˆ′ [ ω]+C ˆb′†[ω], (9) 1 ∞ −η − +R,in − −,in − − in n [ω]= n [ω,ω′]dω′, − 2π Z−∞ − where aˆ+R,in is the signal reflected from interrogated x[ω]= 1 ∞ x[ω, ω′]dω′. target region. We have aˆ+R,in = aˆB when there is 2π Z − no low reflective object in the target region(hypothesis −∞ H0) and aˆ+R,in = √ηaˆ+ + √1 ηaˆB when there con- − Wedrawthespectrumofn [ω],E [ω]andE [ω]/n [ω] tainslowreflectiveobjectinthetargetregion(hypothesis + N N + 4 H )[9]. Here aˆ is the background thermal mode in the (iid) signal-idler-mode pairs. Thus the error probability 1 B interrogated region. The mean photon number is n or of receiver is given by[9] B n /(1 η) providing the absence or presence of low re- B − flective object. We measure the photon numbers of mixed modes erfc( SNRM/8) QI cˆη,± =(aˆ−η±â−)/√2. The object’s absenceorpresence PQMI = q 2 , (10) is determined by the photon number difference which is Nˆ = M (Nˆ(k) Nˆ(k)) where Nˆ(k) = cˆ(k)†cˆ(k) and c k=1 c,+ − c,− c,± η,± η,± M = tPmWm is the independent, identically distributed where the signal-to-noise ratio satisfies 4M[( Nˆ Nˆ ) ( Nˆ Nˆ )]2 SNRM = h c,+iH1 −h c,−iH1 − h c,+iH0 −h c,−iH0 , (11) QI ( ( ∆Nˆ ∆Nˆ )2+ ( ∆Nˆ ∆Nˆ )2)2 q h c,+− c,−iH0 q h c,+− c,−iH1 where Nˆ Nˆ =0, (12a) h c,+iH0 −h c,−iH0 Nˆ Nˆ =2√ηRe[ Baˆ aˆ ], (12b) h c,+iH1 −h c,−iH1 h +,f −,fi ( aˆ† aˆ aˆ† aˆ )2 ( ∆Nˆ ∆Nˆ )2 = Nˆ ( Nˆ +1)+ Nˆ ( Nˆ +1) h −η,f −η,fiH0 −h −,f −,fi , h c,+− c,−iH0 h c,+iH0 h c,+iH0 h c,−iH0 h c,−iH0 − 2 (12c) ( aˆ† aˆ aˆ† aˆ )2 ( ∆Nˆ ∆Nˆ )2 = Nˆ ( Nˆ +1)+ Nˆ ( Nˆ +1) h −η,f −η,fiH1 −h −,f −,fi h c,+− c,−iH1 h c,+iH1 h c,+iH1 h c,−iH1 h c,−iH1 − 2 2(√ηIm[ Baˆ+,faˆ−,f ])2. (12d) − h i In Fig. 3 (a), we plot F SNRM/SNRM with co- [e−iωtd x′[ω]ei(ϕ(0)+ϕ′(0)ω+...)],(13) ≡ QI coh | | forpeeqruaetnivceypωar=am0.etHeresreC1SN=RGκMγ21 ainsdthCe2s=ignGκaγ22l-tino-nreosiosenarnat- wthheerpehaxs′e[ωo]f x=′[ω[A].+T|Bhe|2el+imCin+aCti−onB(onfTbin+teg1r)a]ndan’sdphϕasies coh tioofclassicalmicrowavetransmitterwhichisequivalent dependence on ω would improve the value of signal-to- to 4ηMV /(2n +1)[10]. From the figure, we can see noise ratio. If bandwidth σ is small, then t = ϕ′(0) = 11 B d tlahragtetFhecmouaxldimbuemobFtaainpepdeawrsitahtwCe1a≈k mC2anayn-dphaotroelnatoivpe- dϕd(ωω)|ω=0 could be the optimal delay time which reads tomechanical coupling where G /2,G /2 κ. Photon 1 2 γ number n+[0] is shown in fig. 3(1b) wit2h C1≪and C2. topt = γ[1+4(δ)2 +4κ γ Let us now discuss the situation where the output sig- nT + 1 +C nals have a bandwidth σ, i.e., filter functions f+[ω] and + b 2 ], (14) f [ω] have a bandwidth σ. The filter functions have the (nT + 1 +C)2+(δ)2 − b 2 γ afonrdmff+[ω[ω]]==0efioωrtdω/√σ(, f−[,ω]σ=) 1o/r√(σσ,for).ωH∈er[e−fσ2,[ωσ2]] where we assume that C = C = C and nT = nT = 0. ± ∈ −∞ −2 2 ∞ + 1 2 + − has a time-delay term eiωtd as discussed in Ref. [30]. To InFig. 4(a),weshowSNRQI withbandwidthσfordelay find optimal topt at which the signal-to-noise ratio de- time td = 0 and td = topt. It can be easily seen that creases slowly with the bandwidth, we revisit Eq. (12b). the signal-to-noise ratio indeed decreases slowly with σ We have[30] increasesfortd =toptcomparedtothatfortd =0. Figure of meritF with σ is shownin Fig. 4 (b). The advantage σ σ Baˆ aˆ = 1 2 e−iωtdx′[ω]= 1 2 of QI over classical system decreases rapidly for td = 0 h +,f −,fi σ Z−σ σ Z−σ while QI always has advantage over classical ones even 2 2 5 ter performance in error probability than classical sys- 0 −5 P g10−10 o l −15 PQMI PM FIG. 3: (a) QI advantage over classical system. F versus coh C1 and C2. Parameter settings are γ/κ = 0.001, δ/κ = 1.5, −204 5 6 7 nTb =61.945, nT+ =0, nT− =0, η =0.07 and nB =610(room log10M temperature T =273K). (b) Photon number n+[ω =0] versus cooperative parameters C1 and C2. For C1,C2 ≫ 1 and FIG. 5: Error probability PQMI and PcMoh versus log10M for κ≫γ the stability condition could be C1 ≥C2[28]. σ/κ=1. See Fig. 4 for theother parameter values. tem. a x 10-5 5 4 IV. CONCLUSION QI3 R N S 2 Insummary,wehaveproposedaschemetoachieveop- t=t 1 d opt t=0 tomechanicalQIinweakcouplingregime. Thefrequency d b 00 0.5 σ1/κ 1.5 2 mismatch δ minimizes the mechanicalthermal noise and increasestheratioE /n . Asaresult,weachieveasub- 1.5 N + stantial reduction of error probability than classical sys- 1.2 tematC =C comparedtothatofRef.[10]atC C . 0.9 1 2 1 ≫ 2 F Consequently, our quantum illumination proposal works 0.6 in weak coupling regime. Furthermore, we optimize the t=t 0.3 td=0opt delay time of microwave signal’s wave packets. Finally, d 00 0.5 1 1.5 2 we find a 48% improvement of signal-to-noise ratio for σ/κ σ =κ. FIG. 4: (a) Signal-to-noise ratio SNRQI versus bandwidth σ/κ. Parameter settings are C1 = 500, C2 = 500, γ/κ = T T T ACKNOWLEDGMENTS 0.001, n+ = 0, n− = 0, nb = 61.945, nB = 610, δ/κ = 1.5, η=0.07. (b) F versusbandwidth σ/κ. We acknowledge the financial support in part by the 10000-Plan of Shandong province (Taishan Scholars), for relative large σ for td =topt. The performance of QI NSFCgrantNo. 11174177and60725416,OpenResearch is about 48% better than classical system at σ =κ. Fund Program of the State Key Laboratory of Low- Fig. 5 plots error probability PM and PM versus Dimensional Quantum Physics Grant No. KF201513; QI coh log M for σ = κ. Microwave(optical) photon number and the key R&D Plan Project of Shandong Province, 10 n¯ 0.09(n¯ 0.09). QI has orders of magnitude bet- grant No. 2015GGX101035. + − ≈ ≈ [1] M.A.Nielsen andI.L.Chuang,QuantumComputation [9] S. Guha and B. I. Erkmen, Phys. 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