Optimal quantum parameter estimation in a pulsed quantum optomechanical system

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by  Qiang Zheng

Optimalquantum parameter estimationinapulsed quantum optomechanical system
Qiang Zheng 1,2, Yao Yao 1,3, and Yong Li 1,4
1 Quantum Physics and Quantum Information Division,
Beijing Computational Science Research Center, Beijing 100084, China
2 School ofMathematicsandComputerScience, GuizhouNormalUniversity, Guiyang, 550001, China
3 MicrosystemsandTerahertzResearchCenter,ChinaAcademyofEngineeringPhysics,ChengduSichuan610200,China
4 Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Wepropose that apulsedquantum optomechanical systemcanbeappliedfor theproblem ofquantum pa-
rameterestimation,whichtargetstoyieldhigherprecisionofparameterestimationutilizingquantumresource
than that using classical methods. Mainly concentrating on the quantum Fisher information with respect to
6 themechanicalfrequency,wefindthatthecorrespondingprecisionofparameterestimationonthemechanical
1 frequencycanbeenhancedbyapplyingapplicableopticalresonantpulseddrivingonthecavityoftheoptome-
0 chanicalsystem. Furtherinvestigationshowsthatthemechanicalsqueezing resultingfromtheopticalpulsed
2 drivingisthequantumresourceusedinoptimalquantumestimationonthefrequency.

n
a PACSnumbers:42.50.Wk;06.20.Dk;42.50.Pq;03.65.Yz;07.10.Cm
J

8
I. INTRODUCTION (CW)laser.

] Different from the above case of CW laser driving, the
h
Quantum metrology [1] is an active research field in re- so-called pulsed quantum optomechanics [42], is also real-
p
- cent years. According to the quantum Crame´r-Rao inequal- ized by driving the optical cavity with (very) short optical
nt ity, the quantum Fisher information (QFI) plays a key role pulses. Originally, this strategy has been proposed in the
a in this subject [2–6], which bounds the minimal variance of systemsofqubits[43],andlatelyextendedtoatomicensem-
u the unbiased estimator. The QFI gives the quantum limit to bles[44]andlevitatedmicrospherestrappedinanopticalcav-
q the accuracy of the estimated parameter with any positive- ity[45]. ComparedwiththeCW-laser-drivingcase,thebene-
[ operator-valued-measure measurement. One of the central fitofthepulsedschemeisthatitdoesnotneedtheexistence
1 ideasofquantummetrologyistobeattheshot-noiselimitand of a stable steady-state for the optomechanicalsystem. The
v approachtheHeisenberglimitbyvirtueofquantumresource, pulsedinteractionhasalsodisplayeditssuperiorityinprepa-
1 suchasquantumentanglementorsqueezing.Therehavebeen ration and reconstruction of quantum state of the mechani-
0 manystudiesonprecisionof parameterestimationwith sub- calresonator[46,47],enhancingtheoptomechaicalentangle-
8 shot-noiselimitindifferentphysicalsystems,suchastheop- ment[48,49]andEPRsteering[50],andcoolingthemechan-
1
tical interferometers [7–9], Bose-Einstein condensates [10], icalmode[51,52].
0
. atomicinterferometers[11],andsolid-statesystems(e.g.,the Inspired by the experimental progress in pulsed quantum
1
nitrogen-vacancycentres)[12,13]. Tothebestofourknowl- optomechanical systems [42, 49, 51], it is a natural idea to
0
edge,onlyafewpapers[14–16]havedevotedtoinvestigating investigatethe highprecisionofparameterestimationbyap-
6
1 the quantum metrology in the newly-developed novel quan- plying applicable optical pulsed driving on the cavity of the
: tumoptomechanicaldevice. optomechanicalsystem. Hereweinvestigateaspecialpulsed
v
Withthe rapidadvanceoftechnology,quantumcavityop- optomechanicalsystem,wherethecouplingbetweentheme-
i
X tomechanics [17–19], in which the mechanical resonator is chanicalmodeandcavityfieldisquadraticaltothemechanical
r coupled to the optical field by radiation pressure or pho- motionandthecavityfieldisresonantlydrivenwithexternal
a tothermal force, has excited a burst of interest [20] due to optical pulses. We mainly focus on the QFI with respect to
the followingtwo reasons: On one hand, the cavity optome- the mechanical frequency, which is equivalent to estimating
chanical system provides a new platform to investigate the onthemassofthemechanicalresonatorandcouldbeusedfor
fundamental questions on the quantum behavior of macro- massprecisiondetection. With theCrame´r-Raoinequality,a
scopic system [21] and even the quantum-to-classicaltransi- largerQFIimpliesthatthemechanicalfrequencycanbeesti-
tion[22,23];Ontheotherhand,itbringsanovelquantumde- matedwitha higherprecision. We showthattheQFIcanbe
viceforapplicationsinultra-highprecisionmeasurement[24– greatlyenhancedwhentheperiodofthedrivingpulsematches
28], gravitation-wave detection [29], quantum information thatofthemechanicalmotion.Wealsoshowthatthemechan-
processing [30] and quantumillumination [31]. Many inter- icalsqueezingresultingfromtheresonantdrivingpulsesisthe
esting researches in cavity optomechanicalsystems, such as quantumresourcestrengtheningtheQFI.
optomechanicallyinducedtransparency[32,33],ground-state Thispaperisorganizedasfollows.Thepulsedquantumop-
coolingofthemechanicalresonator[34–38],optomechanical tomechanicalmodelisdiscussedinSec.II.Weinvestigatein
entanglement [39, 40], optimal state estimation [41], have Sec.IIItheQFIofthepulsedquantumoptomechanicalsystem
been reported. These studies mainly rely on the enhanced with respect to the mechanical frequency. Then, we display
coupling strength between the phonic and photic fields by that the quantum squeezing is the resource used in optimal
strongly pumping the optical cavity with a continuous wave quantum estimation. Finally, a summary is given in the last
2
section.ThebasicpropertiesoftheQFI,especiallytheQFIof
asingle-modeGaussianstate,arereviewedinAppendixA.
II. THEPULSEDQUANTUMOPTOMECHANICS
Optomechanicalsystems have been implementedin many
physical systems, such as suspended mirrors in the Fabry-
Pe´rot resonators [53], toroidal whispering gallery mode res-
onators [54], trapped levitating nanoparticles [55], ultra-
cold atomic clouds in cavities [56]. Here we focus on a
membrane-in-the-middle cavity optomechanical setup [57],
which has been used for quantum nondemolition measure-
mentofthephononnumberstate[58],coolingofmechanical
resonator [59] or investigationof Landau-Zener-Stu¨ckelberg FIG. 1: (Color online) Schematic diagram of the membrane-in-
dynamics [60]. The linear and quadratical optomechanical middle cavity optomechanical setup considered in this paper. The
couplings between the cavity mode and the mechanical res- couplingbetweenthecavityfield(withthedecayrateκ)andtheme-
onatorcanexistinthismembrane-in-the-middleoptomechan- chanicalresonator(withtheresonancefrequencyωmandthedamp-
ical system. Very recently, the optomechanical quadratical ingrateγm)isquadraticaltothemechanicalmotion,andthedriving
fieldiscomposedbyaseriesofperiodicpulses.Thedurationofone
couplingisalsoachievedinacrystaloptomechanicalsystem
[61],exceptforthemembrane-in-middlesetup. pulseisτp,andthetwoconsecutivepulseshasthetimeintervalτ.
The membrane-in-the-middle quadratical coupling setup
underconsiderationisshowninFig.1andthecorresponding
foraninitialGaussianstateofthemechanicalresonator(such
Hamiltonianisexpressedas[62]
as the thermal equilibrium state with mean phonon number
n ).HerethesuperscriptTrepresentsthetransposition.Here
H = h¯ωm(pˆ2+qˆ2)+¯hω aˆ aˆ+¯hg aˆ aˆqˆ2 th
2 c † 2 † (1)
+i¯h[E0(t)e−iωdtaˆ† h.c.]. 0 2ω 0
− m
U = ω γ ω , (5)
Hereωmisthefrequencyofthemechanicalresonator,pˆandqˆ A  −0m −2ωm 2mγ 
arethe dimensionlessmomentumandpositionoperatorssat- − m − m
isfying the relationship [qˆ,pˆ] = i, aˆ is the annihilation op-  e 
erator of the cavity mode with resonance frequency ω and and−→N =(0,0,(2nth+1)γm)T.e
c ThesolutionofEq.(4)isformallyexpressedas
decay rate κ, and g is the quadratic optomechanical cou-
2
tphleinogpsttirceanlgitnhp.uFtinpaolwlye,rE. W0(et)f=urthe2rPas0s(ut)mκe/(th¯haωtct)hweicthavPit0y(tis) −→v(t)=eUAt−→v(0)+ 0teUA(t−t′)−→Ndt′ (6)
drivenresonantlywithωd =ωc.p =MA(t)−→v(0)+R −→vinh.
For the mechanical resonator, by linearizing the optome-
chanical coupling, the corresponding quantum Heisenberg- HIebreeiMngA3(t)3=uenUitaArtyamndat−→rvixi.nh = U−A1[I3−MA(t)]−→N with
Langevinequationisobtainedas 3 ×
Now, we study the case that the driving field is the peri-
dqˆ=ω pˆ, odic Gaussian pulses with the duration τp and period τ, i.e.
dt m (2) P(t) = P exp[ (t nτ)2/τ2]. Herewekeepthecon-
ddtpˆ=−ωm(t)qˆ−γmpˆ+ξ, dition1/τ0p <nc/2L(−Lis−thecavityplength),whichmeansthe
P
optical driving pulses will not excite the near cavity modes
where ω (t) = ω + A(t) with A(t) = 2g n (t) and
m m e 2 a except the desired one and thus the cavity field can be al-
n (t)= aˆ aˆ . Hereγ isthemechanicaldampingrate,and
a † m
h i ways considered as a single mode one. For remaining the
ξ denotesfor the Browiannoise with nullmean andcorrela-
e quadratic coupling during the pulsed driving, the membrane
tionfunctionsatisfying ξ(t)ξ(t ) = 2n γ δ(t t) in the
′ th m ′
h i − shouldbe lockedata cavitynode. Accordinglythe effective
high-temperaturelimitk T ¯hω . Here k is the Boltz-
B m B
≫ frequency of the mechanical resonator is periodically mod-
mann constant, T is the temperature of the mechanical res-
ulated in time via the optical driving pulses. An alternative
onator,andnth =[eh¯ωm/(kBT) 1]−1 kBT/(h¯ωm)isthe
− ≈ scheme to achieve periodic modulation of the effective fre-
thermalmeanphononnumber.
quencyofthemechanicalresonatorisgiveninRef.[63]with
FromEqs.(2),thedynamicsofthesecondordermoments
atwo-tonedrive.
ofthemechanicalsystem
Moreover,weassumethatthesystemworksinthefollow-
−→v(t)≡(hqˆ2i,hpˆqˆ+qˆpˆi/2,hpˆ2i)T (3) i1n/gωpa.raTmheetecrornedgiitmioens:(i()i)im1/pτlie≪stκhe,(ciia)v1it/yτpis≪rapκid,l(yiiie)xτcpit≪ed
m
canbefullydescribedbytheequations by one pulse and damps to the vacuum state beforethe next
pulsearrives[46].Andthecondition(ii)meansthattheband-
ddt−→v(t)=UA−→v(t)+−→N (4) width of the pulses is much smaller than that of the cavity,
3
whichguaranteesthepulseenteringintothecavityspectrally. equivalentto the estimation onω . In principle, the param-
m
The last condition (iii) makes sure that the free rotation of eter tobe estimated in thismodelcanbe the onesotherthan
themechanicalresonatorisfrozenintheprocessofthepulse- ω (m). Here we just concentrateon the case of ω via its
m m
mirror interaction. In this case, the intracavity photon num- relatedQFIF F .
≡ ωm
ber can be approximatedas a series of Dirac delta functions The numerical values of the parameters used in this pa-
n (t) δ(t nτ)inthetypicalevolutiontimeofthe per are based on the state-of-the-artexperimentsreported in
a ∝ n=0 −
mechanicalresonator. Asaresult,thewholedynamicsofthe Ref.[65]. We choosethecavitydecayasκ 102GHz, and
P ≃
optomechanicalsystemisdividedintotwosteps:(1)onekick the driving pulses with duration τ = 0.1ns. We also set
p
attimet=nτ,whichcanbedescribedbytheunitaryoperator themechanicalfrequencyω =0.5 106Hzandthedamp-
m
UK =e−iθqˆ2 withθ =g2 ∆tna(t)dtbeingthekickstrength ing rate γm = 102Hz unless otherw×ise stated. By carefully
(∆t means the integral time domain and is of the order of choosingthemechanicalmass,reflectivity,andinitialequilib-
thetypicaltimeforaGausRsianpulse)andthuscorrespondsto riumposition,thekickstrengthθisintherangeof(0.01,10)
the linear transformation q q and p p 2θq, and (2) forthetypicalcouplingstrengthg2.
→ → −
thefree-evolutionlastingtimeτ betweentwoadjacentkicks,
whosecorrespondingevolutionisdeterminedbyEq. (2)with
A(t)=0. B. NumericalresultsoftheQFI
Combined these two evolving processes, the equation of
motioninaτ circleisgivenas[46]
With substituting Eq. (9) into Eq. (A10), the QFI F can
v((n+1)τ)=M (τ)Kv(nτ)+ v (τ), (7) be obtained straightforwardly. Fortunately, the last term in
−→ 0 −→ −→inh Eq. (A10) vanishes since there is no first-order moment of
whereM (τ) M (t) ,and the mechanicalmotion. In what follows, we mainly explore
0 A=0 t=τ
≡ | F by numerical simulations as the analytical solution is too
1 0 0 cumbersome.
K= 2θ 1 0 (8) Withtheextensivenumericalsimulations,we findthatthe
−4θ2 4θ 1  evolutionofF intermsofthepulsenumbernshowstwodis-
−
  tinct behaviors, as shownin Fig. 2(a). In this figure, the pe-
denotingtheeffectofthekickonthesecondordermoments.
riod of pulse τ matches that of the mechanical resonator T
That is, K is the representation of U based on the second 0
K ( 2π/ω )thoughτ = T /k,with k takingtherepresenta-
m 0
ordermoments. MakinguseofEq.(7),thestroboscopicstate ti≡vevalues 1, 1, 2, 4, 5, 10 denotingthepulsenumberin
ofthemechanicalresonatorattimet=nτ isobtainedas {2 }
one period of the mechanical motion. With k 4, the QFI
≤
v(nτ)=(M (τ)K)n v(0) F increasesveryquicklyattheinitialpulsenumbernandar-
−→ 0 −→
+[I (M (τ)K)n](I M (τ)K) 1 v (τ). rivesto a largeconstantvalueinthe largenlimit. However,
3 0 3 0 − −→inh
− − (9) with the increase of k 5, the QFI F shows the behaviors
≥
ofinitially increasingwith n andthengraduallygoingdown
to zero with large n. In this case, the value of the QFI is
III. QFIOFTHEPULSEDOPTOMECHANICS very small compared to that of k 4. When the pulse pe-
rioddoesn’tmatchthemechanical≤period,e.g.,k T0 isan
≡ τ
Afterdetailedpresentationofthepulsedquantumoptome- irrational number, we also find that the value of QFI F be-
comestomuchmoresmaller,comparedtotheperiodsmatch
chanicalmodelintheprevioussection,herewemovetoinves-
case. Moreimportantly,itis shownin Fig. 2(a)thattheQFI
tigatethequantumparameterestimationviatherelatedQFIin
F withk = 4is optimal,andthereasonofthisoptimalwill
thismodel.Wewillalsoshowthatthequantumresourceused
bediscussedinSet.IIIB.
forparameterestimationisthesqueezingproducedbypulsed
driving. ThephysicsofthesebehaviorsoftheQFIF canbeunder-
stood as following. Note that the parameter to be estimated
isthe mechanicalfrequencyω . Iftheperiodofthedriving
m
A. Primarydiscussions pulsesτ matchesthatofthemechanicalperiodT0,theinfor-
mation of ω can be extracted to the greatest extent. As a
m
result, the value of QFI F is large necessarily in the match-
Theparametertobeestimatedinthispaperisthefrequency
ing cases. A constant value of the QFI in the large n limit
ω of the (harmonic) mechanical resonator. Choosing this
m
originsfrom the balance between the pulsed driving and the
parameteris based on the following consideration. With the
mechanicaldamping.
relation ω = k /M (k and M being the spring con-
m m m Theinfluenceofthemechanicaldecayrateγ ontheQFI
stantand the mass, respectively),the QFI with respectto M m
p F is studied in the resonant driving regime, as shown in
isproportionaltothatofwithrespecttoω ,thatis
m Fig.3(a).Thisfiguredisplaysthatwiththeincreaseofγ ,the
m
F =µF , (10) valueofF decreasesconsiderably. Moreover,F displaysthe
M ωm
oscillationbehaviorwhenγ isverysmall.Thereasonofthis
m
where µ = km is the scaling factor. As a result, the esti- oscillationissimple:thecoherentevolutionofthemechanical
4M3
mation on M, just as done in the mass spectrometer[64], is resonator, determinedby ω , is dominatedif the mechanics
m
4
10 10
]
)
2 ]
−Hz 0 −2z)
log[F(10−10 kkkk====1124/2 (a) log[F(H10−05 (a) γγγγmmmm====0111..00102
k=5
k=10
−200 1 2 0 1 2
104 n 105 n
5 0
]
)
2
−
z
H
(
r4 [F −20 θ=10
0
g1 θ=1
k=1 o
k=2 l θ=0.1
k=4 (b) (b)
θ=0
30 2 4 −400 1 2
104 n 105 n
FIG.2:(Coloronline)(a)and(b)correspondstothevariationofthe FIG.3: (Coloronline)TheQFIF asafunctionofthepulsenumber
QFI F and the squeezing degree r defined in Eq. (12) in terms of nwithdifferentdecayrateγm(inunitsofHz)forθ =1.0andk ≡
thepulsenumbern,respectively. Differentlinescorrespondstothe T0/τ = 1(a)andwithdifferent kickstrengthθ forγm = 102Hz
different valuesoftheparameter k = T0/τ (τ beingtheperiodof andτ =10−7s(b).Theotherparameterisnth=100.
thepulsesandT0 =2π/ωm),respectively.Theotherparametersare
nth =100andθ=1.0.
quadraturesexperimentally,whicharerequiredfortheQFIof
hasveryhighquantityfactor ωm. themechanicalresonator.Forthepulsedoptomechanicalsys-
γm tem, there existatleasttwo experimentallyfeasible schemes
Theeffectofthekickstrengthθ ontheQFIF isshownin
toachievethisaim.Themainideaofthefirstone[51]isbased
Fig. 3(b). It is obvious that F also goes down considerably
onthehomodynedetectionofthemixingbetweenthesignal
with thedecreaseofθ. Thiscanbe easily understood: with-
pulses,whichinteractwiththemechanicalresonator,andthe
outexternaldriving,themechanicaldampingwillsuppressits
local oscillator (LO) pulses, as shown in Ref. [51] (wherein
coherenceinthelong-timelimit. Asaresult,itisnaturalthat
Fig. 1(a)). Thesecondschemeisbasedonthe beam-splitter
theQFIF decreases.
interaction,wherethemechanicalquadraturesaretransferred
In order to show the advantage of our pulsed-driving es-
intotheopticalreadoutpulses(latterinjected),asdisplayedin
timation protocol, we study in Fig. 4(a) the relationship be-
Ref. [49] (whereinFig. 2). Thus, the informationofthe me-
tweenthegrowing-uppartoftheQFIF andthepulsenumber
chanicalresonatorcanbegainedwiththeoutputofthereadout
n. WefindthatF nαbynumericallyfitting,withtheindex
pulses.
∝
αdependentontheparameterk,asshowninFig.4(b).More-
over,we alsocheckednumericallythatthisdependenceofα
onkisindependentoftheparametersγ andθ.
m
FromFig.4(b),it’sclearthattheQFIF withrespecttothe
IV. MECHANICALSQUEEZINGASAQUANTUM
pulsenumbernapproachestheHeisenberglimitwithα = 2
RESOURCE
fortherelativelylargek > 10,similartotheresultsobtained
previouslyinthesystemsofthepulseddrivingqubit[66,67].
Fork =2,4,theQFIshowsthebehaviourbeyondtheHeisen- Generally,itiswell-knownthatthesqueezedstate[70],as
berglimit[68]withα=3. TheHeisenberglimitandbeyond anessentialresourceforquantummetrology[71,72],canen-
itdisplaythatthepulseddrivingisanessentialwaytoenhance hance the precision of parameter estimation. Motivated by
theprecisionofparameterestimation. thisfact,westudytherelationshipbetweentheQFIF andthe
Inthefollowing,wediscusshowtoreadoutthemechanical mechanicalsqueezinginthissection.
5
x 10−4
k=2 80
fitting, α=3.0 7
k=100
] 0 fitting, α=2.0 B A 6
)
−2 α=1 5
z
H
( q 0 4
F
[
n−15 3
l
2
(a)
(a) 1
−30 −8−08 0 0 80
0 3 6 p
ln (n)
x 10−4
3 80
7
A
6
B 5
2.4
q 0 4
α
3
2
2
(b) 1
(b)
−80
−80 0 80
0 2 4 6 10 16 20 p
k x 10−5
80
6
B A
FIG.4: (Coloronline)(a)Fittingthegrowing-uppartofF withre- 40 5
specttothepulsenumbern. Theindexαisdeterminedbynumeri-
callyfittingF ∝nα.Boththex-axisandy-axisarescaledbasedon 4
thenaturallogarithm. Theshot-noiselimit(α = 1)isdisplayedas q 0 3
thegreenline. (b)Thedependenceoftheindexαontheparameter
k.TheotherparametersarethesameasthatinFig.2. −40 (c) 2
1
Anysingle-modeGaussianstatecanbeexpressedas[73] −80
−80 −40 0 40 80
p
ρ=Dˆ(α)Sˆ(r,φ)ρ (n)Sˆ (r,φ)Dˆ (α), (11)
th † †
where Dˆ(α) = exp[αaˆ h.c.] is the displacementoperator FIG.5: (Coloronline)Thecontour ofWignerfunctionsoftheme-
ofbosonicmodeaˆ, Sˆ(r,−φ) = exp[r(e 2iφaˆ2 h.c.)]isthe chanical resonator before (after) the n-th kicked pulses are labeled
2 − − asA (B).(a), (b) and (c) corresponds tothepulse number n = 1,
squeezingoperator,andρth(m) = ∞m=0 (m+m1)2m+1|mihm| n=3andn=103,respectively.Itisobviousthatherethemechan-
denotes the thermal state with m the mean particle number. icalsqueezingcanbestrengthenedbythekicks. Theparametersare
ThesqueezingstrengthrandthesquPeezingangleφaredeter- k=4,nth =100,andθ=1.0.
minedby
r = 1arcsinh[1( γΣ )12], (12) squeezing angle φ are obtained according to Eqs. (12) and
2 2 det ϕ (13). Note that α = 0 for the mechanical resonator studied
inthispaper. InFig.2(b),we plotthemechanicalsqueezing
2φ= −arcsin(2Σ√ϕγ,12), if Σϕ,11 <Σϕ,22, (13) sBtryencogmthbrinainsgaFfiugn.c2t(ibo)naonfdtFhieg.p2u(las)e,nituimsbaperpanrefnotrthkat≤bot4h.
( π+arcsin(2Σ√ϕγ,12), if Σϕ,11 >Σϕ,22, themechanicalsqueezingandtheQFIareenhancedwiththe
increaseofk. Asaresult,thissqueezingasaquantummetrol-
withγ =(Σ Σ )2+(2Σ )2. HereΣ (i, j = ogyresourcestrengthenstheQFIF. AlthoughFig.2(b)only
ϕ,22 ϕ,11 ϕ,12 ϕ,ij
1, 2)istheeleme−ntofthecovariantmatrixΣ asdefinedin showsthecasek 4,thesimilarresultshavebeenfoundfor
ϕ
≤
Eq.(A6). theotherkvalues(notdisplayedhere).
Byrearrangingthesecond-ordermomentsgiveninEq.(9) BasedonthecorrelationbetweentheQFIandthemechan-
as the covariant matrix, the squeezing strength r and the ical squeezing, we provide an intuitive way to understand
6
x 10−4 ingkicks,thesqueezingstrengthofthemechanicalresonator
80
progressivelyincreases,asdisplayedinFigs.5(b)and5(c).
7
Moreover,thesqueezingangleisalsograduallyapproaches
B A 6
to φ π/4 under some (e.g., n 102) repetitive kicks
≃ ≈
5 (as well as the free evolution between the kicks). Once
q 0 4 the squeezing angle becomes to φ = π/4, which coincides
with the counterpartangleproducedby the squeezingaction
3 U˜ =exp[ i(aˆ2+h.c.)/2]inthekickoperatorU ,wefind
θ=1 K
(a) 2 numerically t−hat it will remain unchanged for large n limit.
1 Wewouldliketopointoutthatherethematchingbetweenthe
−80 squeezingangle by kick and the free rotationangle plays an
−80 0 80
p essentialrolein strengtheningthe mechanicalsqueezing. As
x 10−4 a consequence, the QFI of the mechanical resonator is also
80 enhanced.
A 7 WealsoshowthemechanicalWignerfunctionforthecase
B 6 k = 5 in Fig. 6, which correspondsto a free rotation angle
5 ϑ=ωmT0/5=2π/5.Thus,afterthefreerotation(withcon-
sideringthemechanicaldamping)representedbytheoperator
q 0 4 M (τ = T /5),themechanicalresonatorcannotevolveinto
0 0
3 the squeezedstate with the squeezingangleφ = π/4by the
2 kickoperatorUKinthelong-timelimit.Thisenablesthekicks
squeeze the mechanicalresonator ineffectively,as display in
(b) 1
Fig.6(c).
−80
−80 0 80
p
80 x 10−4 V. CONCLUSION
7
6 In summary, the quantum pulsed optomechanical system
A
B isproposedtoapplyforthequantummetrologyincontextof
5
quantum parameter estimation by focusing on investigating
q 0 4 theQuantumFisherinformation.Wefindthatthemechanical
3 frequency can be estimated with very high precision if the
mechanicalperiodmatchesto thatof the drivingpulses. We
2
alsodisplaythatthemechanicalsqueezingisthequantumre-
(c) 1 sourceusedinoptimalquantumestimationonthefrequency.
−8−08 0 0 80 In future, it is an interesting subject to utilize coherence of
p
themulti-modecavityoptomechanics[74,75]toenhancethe
accuracyofquantumparameterestimation.
FIG.6: (Coloronline)ThesametoFig.5exceptfortheparameter Acknowledgments. We thank X. G. Wang, X. W. Xu, H.
k = 5,corresponding tothefreerotationangleϑ = 2π/5. Inthis Fu and X. Xiao for their helpful discussions. This work is
case,thekickscannoteffectivelyproducethemechanicalsqueezing.
supported by the National Natural Science Foundation of
China (Grant Nos. 11365006, 11422437, and 11121403,
11565010)andthe973program(GrantsNo. 2012CB922104
the QFI for k = 4 (corresponding to a free rotation time and No. 2014CB921403), Guizhou province science and
τ =T /4)beingoptimalwiththekickstrengthθ =1.Tothis technologyinnovationtalentteam(GrantNo. (2015)4015).
0
aim,itisusefultoinvestigatethedynamicsofthemechanical
Wignerfunction,obtainedaccordingtoEq.(A8).
The kick operator K produces the mechanical squeez-
ing, which remainsinvariantunderfree rotationM (τ) (ne- AppendixA:QFIofasingle-modeGaussianstate
0
glecting the mechanical damping). After the first kick acts
on the mechanical resonator, the initial mechanical thermal Inordertokeepthecompletenessofthispaper,herewere-
state translates into a squeezed state with the squeezing an- viewthemainaspectsoflocalquantumestimationtheory,es-
gle φ(n = 1) π/8, as shown in Fig. 5(a). Here φ is de- peciallyfocusingontheQFIofasingle-modeGaussianstate.
≃
finedastheanglebetweentheq-axisandthedirectionofthe Notethatthemechanicalresonatorstudiedinthispaperstays
squeezed quadrature. Then, the squeezed state is rotated by inasingle-modeGaussianstate.
ϑ = ω T /4 = π/2 along the clockwise direction by the Letϕdenoteasingleparametertobeestimated,andp(ζ ϕ)
m 0
freeevolutionM (τ =T /4).Undertheeffectofthefollow- betheprobabilitydensitywithmeasurementoutcome ζ |for
0 0
{ }
7
acontinuousobservableW conditionedonthefixedparame- canbeusedtoequivalentlyrepresentthecorrespondingquan-
terϕ. Thevalueoftheparameterϕcanbeinferredfromthe tumstateρ,asthereisaone-to-onecorrespondencebetween
estimatorfunctionϕˆ=ϕˆ(ζ , ..., ζ ),basedonthemeasure- them.Gaussianstate,asaspecifickindofcontinuous-variable
1 N
mentresultsζ ,...,ζ ofN replicasofthesystem. Usually, states, has wide applications in actual quantum information
1 N
thisisachievedbythemaximumlikelihoodestimation. With processing[82]. Theycan be reproducedefficiently and un-
thedefinitionoftheclassicalFisherinformation[76] conditionallyintheexperiment.Theunconditionednessisone
advantage of the continuous-variablestate, which is hard to
Hϕ = dζp(ζ|ϕ)[∂∂ϕlnp(ζ|ϕ)]2, (A1) achieveinthequbit-baseddiscrete-variable.
the classical CramRe´r-Rao inequality [77] gives the bound of AstateissaidtobeGaussianincaseitsWignerfunctionis
Gaussian. A Gaussian state can be completelycharacterized
thevarianceVar(ϕˆ)foranunbiasedestimatorϕˆ
bythefirst-ordermomentandthesecond-ordermoment
Var(ϕˆ) 1 , (A2)
≥ Hϕ
X = Xˆ ,
Extending from classical to quantum regime, the condi- i i
h i
tional probability p(ζ ϕ) is determined by positive operator (A6)
svtaaltueedρm,epas(uζrϕe)op=era|tTorr[{EˆEˆζρ}]f.oTroa pdaertaemrmeitneeriztehdequultainmtuamte Σϕ,ij == 21hW(Xˆ(iX~Xˆ)jX+iXXˆjjdX2ˆXi~)i.−hXˆiihXˆji
ϕ ζ ϕ
|
bound to precision posed by quantummechanics, the Fisher
R
information must be maximized over all possible measure-
Here
ments[78].
By introducing the symmetric logarithmic derivative L
ϕ
determinedby X~ˆ (qˆ,pˆ), Tr(ρ ), (A7)
ϕ
≡ h···i≡ ···
∂ρ 1
ϕ
= (ρ L +L ρ ),
∂ϕ 2 ϕ ϕ ϕ ϕ and ϕ is the parameter to be estimated in the quantum state
ρ . The Wigner function is related to the the second-order
ϕ
the so-called quantum Crame´r-Rao inequality gives a bound momentsas(settingthefirst-ordermomentsbeingzeros)
tothevarianceofanyunbiasedestimator[79]:
Var(ϕˆ)≥ H1ϕ ≥ F1ϕ, (A3) W(X~)= 2π√D1etΣϕ exp(−21X~Σϕ−1X~T). (A8)
where
Fϕ =Tr[ρϕL2ϕ] (A4) Basedonthefidelitybetweenarbitrarysingle-modeGaussian
statesρ andρ ,
isthequantumFisherinformation. 1 2
The two boundsfor the precision of parameter estimation
[ap8na0drt]itchhlaeevnHeuebmiesbeeennrbfceoorugnntrldiim,butihttee1d/sNtoo-.cqauUlalensdutuasmlhlyoe,ts-htniemorieasteNiolnimi.sitth1e/t√otNal f(ρ1,ρ2)= √|Σ1+Σ2|2+e(x1p−[−|Σ211∆|)X(1T−(Σ|Σ12+|)Σ−√2)−(11−∆|XΣ]1|)(1−(A|Σ9)2|),
ItisnoteasytogivetheexplicitformulaofQFIforagen- makinguseofEq.(A5),theQFIofthesingle-modeGaussian
eralsystem. Fortunately,the QFI is relatedto the Buresdis- stateisfoundtobe[83,84]
tance[79]through
DB2[ρϕ,ρϕ+dϕ]= 41Fϕdϕ2, (A5) Fϕ = Tr[2((Σ1+−ϕP1Σ2)′ϕ2] +21Pϕ′P24 +∆X~ϕ′TΣ−ϕ1∆X~ϕ′.
ϕ − ϕ
(A10)
wherethedefinitionoftheBuresdistancebetweentwoquan-
Here ∆X~ = X~ X~ is the mean relative displacement,
tumstatesρandσisas[81] P = Σ 1/h2d1en−otes2tihepurityofthestate,and
ϕ ϕ −
| |
D [ρ,σ]=[2(1 Tr ρ1/2σρ1/2)]1/2.
B
−
TheWignerfunctionforanarpbitrarygivenstateρ,defined ∆X~ϕ′ =dhX~ϕ+ǫ−X~ϕi/dǫ|ǫ=0. (A11)
as
WenoticethatthequantumCrame´r-Raoboundoftwo-mode
W(q,p)= 1 ∞ dse ips q sρq+s , Gaussianstates[85]wasinvestigatedpreviously.
−
2π h − | | i
Z−∞
8
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