A Simple Scheme for Quantum Non Demolition of Phonons Number of the Nanoelectromechanics Systems

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by  F. R. de S. Nunes

ASIMPLESCHEMEFORQUANTUMNONDEMOLITIONOFPHONONSNUMBEROFTHE
NANOELECTOMECHANICSSYSTEMS
F. R. de S. Nunes1, J. J. I. de Souza1, D. A. Souza1, R. C. Viana2, e O. P. de Sá Neto1
1 CoordenaçãodeCiênciadaComputação,UniversidadeEstadualdoPiauí,CEP:64202220,Parnaíba,Piauí,Brazil.and
2 Centro Cirúrgico do Hospital Dirceu Arcoverde, Parnaíba, Piauí, Brazil.
(Dated:18deJaneirode2016)
Inthisworkwedescribeaschemetoperformacontinuousovertimequantumnondemolition(QND)mea-
surementofthenumberofphononsofananoelectromechanicalsystem(NEMS).Ourschemealsoallowsusto
describethestatisticsofthenumberofphonons.
PACSnumbers:
6
I. QUANTUMMECHANICSMEASUREMENTPROBLEM system*CITAR*variesdependingontheeigenvectors
1
oftheobservablebeingmeasured;
0
2 In general, the measurement of an observable in a given
2. TheoperatoroftheobservableO mustcommutewith
quantum system disturbs its state, such that the observable S
n H . Thisobservablecannotbechangedduringtheme-
a variance is greater in a future measurement [1]. This is ea- I
asurementprocess;
J sily illustrated by a simple system, a harmonic oscillator of

4 massmandmomentumoperatorpandinathermalstate, as 3. ∂HS (cid:54)= 0. This is the main feature of QND measu-
1 previously considered by references [2]-[3]. It’s possible to ∂OC
S
initially make a precise measurement in the x position, the rement: after the interaction of S with A the conju-
h] canonically conjugate operator to moment p. However, due gateobservableOSC ischangeduncontrollably. Sothat
p to Heisenberg’s uncertainty principle, δp ≥ (cid:126)/(2δx), and p this increase in variance does not affect the observa-
ble being measured, we have to demand that the Ha-
- is disturbed. However, in an evolution following this mea-
nt surement, p induces a variation in x: x˙ = [x,p2/2m]/i(cid:126), miltonianofthesystemdoesnotdependontheconju-
gateobservable. Soamorerestrictivewayistorequire
a resulting in, x(t) = x(0)+p(0)t/m. Therefore, using the
u uncertaintyrelationtocalculatetheuncertaintyinxforfuture [HS,OS]= 0,becausethentheobservablebeingmea-
q measurements (δx(t))2 ≥ (δx(0))2 +((cid:126)/2mδx(0))2t2, we suredisaconstantofmovement.
[
concludethatpositionandmomentumareuncorrelated. The

1 measurementapparatusactedrandomlydisruptingtheobser-
v vablebeingmeasured. III. MODEL
0
5
ThecapacitivecouplingbetweenQuantumBit(Qubit)and
7
II. PROTOCOLTOMEASUREGENERALQND nanoelectromechanical system (NEMS) [6]-[7] is illustrated
3
0 in the figure 1. In quantum bit notation, the Hamiltonian of
. The Quantum non demotion (QND) measurement is cha- theBoxofCooperPairs(??)iswrittenas
1
0 racterizedasonethatcanbeperformedwithoutdisturbingthe
E
16 oobfstherevsaybslteemstaSte.isIinnfaeQrreNdDbymmeaesausruerminegnta,nthoebsoebrsvearbvlaebOleOoSf Hqb =(E1−E0)σz− 2Jσx, (2)
A
: an auxiliary system A, without disturbing the next evolution
v whereσx =|1(cid:105)(cid:104)0|+|0(cid:105)(cid:104)1|,σz =|0(cid:105)(cid:104)0|−|1(cid:105)(cid:104)1|,andeisthe
Xi oSfrOemS.aiAnsftearnaeifigneintestnautemobfeOrof.successivestepsthefinalstate eletroncharge.En =2EC(n−ng)2isthechargingenergyof
r Formally,ifwehavethetoStalHamiltonian: ncooperpairs,withEC =e2/2C(cid:80),C(cid:80) =CN+Ccpb+CJ.
a Also,n =n +n ,wheren =C V /2eisthegate
g N cpb cpb cpb cpb
charge, C is the capacitance and V the potential diffe-
H = H +H +H , (1) cpb cpb
S A I rence of the Cooper pair box. n = C V /2e, is the gate
N N N
charge, C is the capacitance and V is the potential diffe-
with H being the system Hamiltonian, H being the ap- N N
S A
renceofNEMS.E isthecapacitiveenergyofQubitJoseph-
paratus Hamiltonian, and H being the Hamiltonian of the J
I
sonjunction.Therefore,thenecessarychargingenergyforthe
apparatus-system interaction. The QND measurement O
S
transitionofoneCooperpairwillbe:
mustsatisfythefollowingproperties:
1. ∂∂OHSI (cid:54)= 0and[OA,HI] (cid:54)= 0. Thisconditionisbecause En+1−En = 2EC(cid:2)(n+1−ng)2−(n−ng)2(cid:3),
wewanttomeasureO throughO . Thisimpliesthe
S A
interactionHamiltonianshouldbeafunctionofO and forn=0
S
thatO variesaccordingly,tointeractwiththesystem.
A
E −E = 2E (1−2n )
Infact,thisconditionmustbeobservedforanytypeof 1 0 C g
measurement, since it simply requires that the pointer =2E (1−2n −2n )
C N cpb
2
AssumingsmallNEMSoscillationamplitude,wegettheex- whereλ=g/∆,∆=ω−ν ,ν =E /(cid:126)andX =b†σ +
a a J −
pressionCN = CN(0)+(∂∂CxN)x,withxbeingtheNEMS’s bσ+resultsintheefectiveHamiltonian
flexionaxisdeformationposition.Thusthecapacitiveinterac-
tionbetweentheQubitandNEMSmodeis: (cid:20) g2 (cid:21) (cid:126)(cid:20) g2(cid:21)
H ≈ (cid:126) ω+ σ b†b+ ν + σ . (8)
H = (cid:126)gσz(b+b†), (3) eff ∆ z 2 a ∆ z
Q−N
(cid:113) Now,withtheequationsofdynamicsofthedensityoperator,
where, g = 2m(cid:126)ω ×[4nN(0)EC(∂∂CxN)]/((cid:126)CN), and (cid:126) is
thePlanckconstantdividedby2π. −i γ
TheCompleteHamiltonianforthismodelis: ρ˙ = (cid:126) [Heff,ρ]+κD[b]+γD[σ−]+ 2ϕD[σz]
H = −EJσx+(cid:126)ωb†b+(cid:126)gσz(cid:0)b+b†(cid:1). (4) = Lρ (9)
|0(cid:105),|1(cid:105) 2
whereD[α]=(2αρα†−α†αρ−ρα†α)/2.
IV. RESULTS
Withthiswecancalculethecorrelation
(cid:104)σ (t)σ (0)(cid:105) = Tr(cid:2)σ eLt(|+(cid:105)(cid:104)−|)(cid:3), (10)
− + s −
andfinallytheQubitabsorptionspectrum.
1 (cid:90)
S(ω) = dteiωt(cid:104)σ (t)σ (0)(cid:105) . (11)
2π − + s
−
WeusedtheQutip[?]packagetoobtainnumericalresultsfor
thecorrelation(fig. 2.a), spectrum(fig. 2.b), anditsstatistic
distribution(fig. 2.c). Forourpresentcalculationweusedthe
Qubitintheexcitedstate,andtheNEMSinthevacuumstate,
with the number of thermal occupation of its reservoir being
equaltoone.
Figura1:SchematicModel.
However, in this measurement protocol QND that measu-
resthenumberofphonos,cantoconducetheQubitStarkfre-
TheHamiltonianH iswrittenintheCooperpairbasis.
|0(cid:105),|1(cid:105) quence displacement in ν = ν +ng2/∆, followed by the
However,changingtheatomicbasistothenewrepresentation, n a
independentmeasureofQubitstate, oncethatthenumberof
σz →σx,σx →−σz, (5) phonosisnotchangedinthisprocess.
theHamiltoniantermsH become:
|0(cid:105),|1(cid:105)
H = EJσz+(cid:126)ωb†b+(cid:126)gσx(cid:0)b+b†(cid:1) (6) V. DISCUSSION
|−(cid:105),|+(cid:105) 2
withσx =|+(cid:105)(cid:104)−|+|−(cid:105)(cid:104)+|andσz =|−(cid:105)(cid:104)−|−|+(cid:105)(cid:104)+|. Motivatedbyaseteofdiscovery[4]-[5]-[6]-[7]-[8],weex-
ploredanelectromechanicalinteractioninahighlydispersive
Making the rotation wave approximation to the Hamilto-
regimeinpromotingforQNDmeasurementscheme.Wehave
nian(6),wehave
demonstrated that the spectrum of the phonons of NEMS in
H˜ =(cid:126)ωb†b+ EJσz+(cid:126)g(cid:0)σ b†+σ b(cid:1), (7) the Qubit state resolution, thereby have access to each num-
2 − + berofstateandstatisticsofBosen-Einsteisthisressoandor.
where, σ+ = |+(cid:105)(cid:104)−|, σ = σ†, |−(cid:105) is the fundamental
− +
atomicstate,|+(cid:105)istheexcitedatomicstate.
For our case, considering a tightly dispersive regime, VI. ACKNOWLEDGEMENTS
we can to expand the Hamiltonian with a Baker-Campbell-
Hausdorasfollows,
PartofthecalculationswereperformedwiththeQuantum
OpticsToolbox.O.P.deSáNetoisgratefultoLeonardoDan-
(cid:104) (cid:105) λ2 (cid:104)(cid:104) (cid:105) (cid:105)
e−λXH˜eλX = H˜+λ H˜,X + H˜,X ,X +... tasMachadoforhelpfuldiscussions.
2!
3
(a)
(b)
(c)
Figura 2: (a) Excited states correlation in time function, for χ =
g2/∆ >> κ,γ; (b) Qubit absorption spectrum given resolution
numberstatesofNEMSintermalstate;(c)Visualizationofthequan-
tumstates.
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