Optomechanical Microwave Quantum Illumination in Weak Coupling Regime

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by  Wen-Juan Yang

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 op- tomechanicalsystemconsistingofatwo-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 band-
width. 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±ˆa−)/√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] ver-
sus 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.
+ −
≈ ≈
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