Table Of ContentQuantum nondemolitionmeasurements ofa flux qubitcoupled toa noisy detector
Wei Jiang,1 Yang Yu,1,2,∗ and Lianfu Wei2,†
1NationalLaboratoryofSolidStateMicrostructuresandDepartmentofPhysics,NanjingUniversity,Nanjing210093,China
2Lab. of Quantum Opt-electronics, Southwest Jiaotong University, Chengdu 610031, China
We theoretically study the measurement-induced dephasing caused by back action noise in quantum non-
demolitionmeasurementsofasuperconducting fluxqubitwhichiscoupledtoasuperconductingquantumin-
terference device (SQUID). Our analytical results indicate that information on qubit flows from qubit to de-
tector,whilequantumfluctuationswhichmaycausedephasing ofthequbitalsoinjecttoqubit. Furthermore,
the measurement probability is frequency dependent in a short time scale and has a close relationship with
themeasurement-induced dephasing. Whenthedetuning between driven andbare resonator equals coupling
strength,wewillaccessthestateofqubitmoreeasily. Inotherwords,weobtainthemaximummeasurement
rate. Finally,weanalyzedmixedeffectcausedbycouplingbetweennon-diagonal termandexternalvariable.
1
Wefoundthattheinitialinformationofqubitisdestroyedduetoquantumtunnelingbetweenthequbitstates.
1
0
2 PACSnumbers:03.65.Ta,03.67.Lx,42.50.Lc,85.25.Dq
n
a I. INTRODUCTION that the easier we gain information about one variable, the
J
faster we lose information about its conjugate variable. It
1
means that the dc-SQUID can be used as a quantum-limited
The concept of quantum nondemolition (QND) measure-
mesoscopic detector.17 We also study the deviation of QND
] ment which can be used for detection of weak force acting
h measurementduetothecouplingbetweennon-diagonalterm
onsystemsuchasgravitationalwavehadbeenintroducedto
p ofqubitandvariableofdetector. Thisarticleisorganizedas
allowrepeatabledetection.1ThebasicrequirementofaQND
- follows. In Sec. II the basic models were introducedfirstly,
t measurementisthataseriesofmeasurementsofthevariable
n fromwhichweuse masterequationtodiscussthedephasing
a of system should give predictable results.2 It means that the
caused by quantum noise and its correlation with measure-
u observable Aˆ must commute with Hamiltonian Hˆ that de-
mentprobabilitydistribution. In Sec. III, we discuss the ef-
q scribes interacting system and detector, i.e., [Aˆ,Hˆ] = 0. A
[ fect of non-diagnoseterm on QND measurement. We show
large number of works associated with QND measurements thatthemixeffectcausedbycouplingbetweennon-diagnose
1 have been done in quantum optic,3–5 cavity QED6,7 and cir-
termanddetectorcandemolishtheinformationofqubit.
v cuitQED.8
4
Recently, QND measurements have experimentally been
5
3 achieved on superconducting flux qubit system by investi- II. PROBABILITYDISTRIBUTIONAND
0 gatingthecorrelationbetweentheresultsoftwo consecutive MEASUREMENT-INDUCEDDEPHASING
. measurements.9 However,ingeneral, themeasuringappara-
1
0 tusisamesoscopicsystem(e.g.,dc-SQUID)whichiscoupled TheHamiltonianofsystemisgivenby16
1 toenvironment. Therefore,noisecanaffectdiscriminationof
1 signalwhichreflectstheinformationofqubitviacouplingbe- Hˆ(t)=Hˆ +Hˆ +Hˆ +Hˆ (t), (1)
s meter int drive
: tween meter and qubit (see Fig. 1(b)). There exist a larger
v
Xi nthuimnkbearboofutwtohrekqsuinanmtuemsosmcoecphicansyicssteomfdwehteecretioonnepirsofcoerscse,1d1,t1o2 washaertewHoˆsleivsetlhseyHstaemmiltonianofqubitwhichcanbeconsidered
r andaboutthefundamentalquantumlimitswhichconstrainthe
a performance of the detector.13,15 In practice, noise plays an ǫ ∆
Hˆ = σˆ + σˆ , (2)
importantroleinquantummeasurement:quantumnoisefrom s 2 z 2 x
themeteractsbackonthesystem,suchasqubit,atthesame
time,theinformationaboutthevariableconjugatetothemea- whereσˆz andσˆx are Paulimatrices, ǫ and∆are energydif-
ferenceandtunnelsplittingofqubit. TheHamiltonianofde-
suredvariableisdestroyed.Thisphenomenonisomnipresent,
tectorwhichactsasanoscillatoris
becauseinformationaboutsystemiscarriedawayintothesur-
roundingduetoindirectenvironmentalcouplingviacoupling
todetector.14 Foraweakmeasurement,thedetectionmaybe Hˆmeter =~ωosaˆ†aˆ. (3)
quantumlimitedthatthesignal-to-noiseofmeasurement,de-
The Hamiltonian that describes coupling between qubit and
finedastheratiooftheamplitudeoftheoscillationlineinthe
oscillatoris
outputspectrumtobackgroundnoise,isnomorethan4.15
In this article, we study the correlation between measure- Hˆ =~gσˆ aˆ†aˆ, (4)
int z
ment probability and measurement-induced dephasing. Our
modelwhichisusedtoanalyzetheinteractionbetweenqubit fromwhichweknowthatthevariablethatwewanttomeasure
anddetectoris basedonthe theoryofQND measurementof isσˆ . giscouplingstrength.Inotherwords,wecanconclude
z
flux qubitdynamicsin a shorttime.16 Theconclusionshows thestate ofqubitfromthestatesofdetectoraccordingto the
2
FIG.1: (a)SchematicoffluxqubitcoupledtoaSQUIDwhichused
forqubit readout asadetector. (b)Schematicoftheprocessofin-
formation detraction from qubit and noise injection from one-port
detector.
FIG.2: (coloronline)(a)Probabilitydistributionconsideringzero
point fluctuation for getting the measurement result for 0 and the
valueofσˆz.Weassign 0 toσz =1and 1 toσz = 1.The initialstate|0ih0|, plottedasafunctionofdetuningδω. (b)Proba-
| i | i −
Hamiltonianofexternaldrivingofharmonicoscillatoris bilitydistributionconsidering theback actionnoiseinsteadof zero
pointfluctuationwiththesameparameter. Thedampingrateofres-
Hˆdrive(t)=f(t)(aˆ+aˆ†), (5) onatorκ=0.1GHz,thecouplestrengthbetweenqubitandoscillator
where f(t) = 2fcos(ω t) is the driving force with driving g=0.3GHzandthebackactionnoisespectradensitySII =2/κ.
d
frequencyω . ForanidealQNDmeasurement,itisnecessary
d
that the condition [σˆz,Hˆ] = 0 must be satisfied. However, φ =arctan(δωi) π. (10)
forsuperconductingfluxqubits, duetotheoff-diagonalterm i
κ/2 − 2
∆ = 0, the measurement is not perfect QND. In case that
∆ 6 ǫ is satisfied, σˆ still would be treated as a conserved If we choose a time t 1/ǫ, then the signal difference
qua≪ntityonthetimesczaledeterminedby1/ǫandoneexpects becomes16 ∼
smalldeviationsfromanidealQNDcase16. Afterneglecting
δα(t) √2tA, (11)
the off-diagonalterm and movingin the rotating frame with ≈
frequencyωd,weobtainthenewHamiltonian where A = f(e2iφ0 e2iφ1)/√2. We obtain the outcome
−
ǫ probabilitydistribution
Hˆ = σˆ +~(δω+gσˆ )aˆ†aˆ+f(aˆ+aˆ†) (6)
z z
2 1 δx(t)
withδω = ωos ωd. ThenwecangetaQNDmeasurement P(I,t)= 2[1+Ierf(√2S t)hσˆzi0]. (12)
− II
from the form of the Hamiltonian. In order to find the rela-
tionshipbetween measurement-induceddephasingand prob- IfwemeasurearotatedquadratureXˆ =(ae−iφ+a†eiφ)/√2
φ
ability distribution, we analyze the impact of noise on QND and choose the phase φ = argA. The qubit difference be-
measurementinpartAandmakeamorequantitativeanalysis comesδx(t)= At,andtheprobabilityforthemeasurement
| |
inpartB. is
1 At
P(I,t)= [1+I σˆz 0erf( | | )]. (13)
2 h i √2S t
A. THEIMPACTOFNOISEONQNDMEASUREMENT II
In Fig. 2 we plot the probability of measuring the 0 state
BasedonthemodelmentionedbyRef. [16],wesubstitute undertheconditionthatweconsiderthecontributionofzero
aGaussianwhitenoiseforthezeropointfluctuationbecause pointfluctuation(Fig.2 (a)). We also plotthe probabilityof
ofthefactthatnoisecontributesalargepartintheactualmea- measuringthe0stateundertheconditionthatweconsiderthe
surement. The distribution of qubit signal I conditioned on contributionofbackactionnoiseinsteadofthezeropointfluc-
thequbitstateσ = 1is tuation duringthe measurementprocess(Fig. 2 (b)). Due to
z
± theinfluenceofbackactionnoise,wecanfindthatthestrong
1 (x x )
p(I σˆ )= exp[− − i ], (7) measurementregion,comparedwiththeonethatonlyconsid-
z
| √2πSIIt 2SIIt ered the zero point fluctuation, is reduced. It means that we
wherex (t)=√2Re[α (t)],i= ,S isnoisespectraden- needmoretimetoreachthesamemeasurementstrengthasthe
i i II
± situationthatweonlyconsidercontributionofthezeropoint
sity. Afterturningontheinteractionwithtimet,thecoherent
statequbit-dependentamplitudeisfoundtobe16 fluctuation.
α (t)=A eiφi[1 e−iδωit−κt/2] (8)
i i
−
B. CORRELATIONBETWEENMEASUREMENT
withqubit-dependentdetuningδω =δω gσ andthequbit-
i − z PROBABILITYANDDEPHASINGRATE
dependentamplitudesandphasegivenby
Ai = f (9) We choose the initial state of system to be ρ(0) = ρ0 ⊗
(δω )2+κ2/4 0 0, i.e., at time t = 0 the harmonicoscillator in the vac-
i | ih |
p
3
uumstate. Then,we turnonqubitandoscillator interaction, phasewhichistheconjugatevariablewithamplitude. Atthe
afteraperiodoftime,qubitandoscillatorentanglewitheach same time, the measurement-induceddephasingratewill get
other. At the same time, we perform a strong measurement the maximum value. In Fig. 3, we plot the probability dis-
of the flux quadrature xˆ = aˆ +aˆ†. As a result, the pointer tribution that measuring 0 state conditioned on initial state
state (the oscillator state) is projected to the state x x. If ispreparedon 0 0 andmeasurement-induceddephasingas
| ih | | ih |
themeasurementisstrongenoughthatthestatesofoscillator a function of detuning. When matching one of qubit fre-
x x whichcorrespondtothequbitstates 0 and 1 re- quency, both measurement-induceddephasingrate and mea-
± ±
| ih | | i | i
spectivelyareorthogonal,thecoherenceofqubitisdestroyed surement probability get a maximum value. In the region
bythemeasurement. Todescribethisphenomenon,wemust where erf(At) 1, it corresponds to a strong projective
| | ≈
findouttheequationofmotionthatcandescribesthesystem measurementcase. Atthesametime,thetwocoherentstates
stateevolution.ThemasterequationusedincircuitQEDsys- ofoscillatorarewellseparatedinphasespaceduetodecayof
temisvalidhere.10Thequbitdensitymatrixobeys: coherence term at a measurement-inducedephaing rate Γ .
m
Ingeneralcase,attimet=0,thesystemisinaproductstate
ρ˙ =Lρ=−~i[Hˆ,ρ]+κD[aˆ]ρ+γ1D[σˆ−]ρ+ γ22D[σˆz]ρ, 1
(14) ψ(t=0) = (0 + 1 ) α , (19)
where [Lˆ] is damping superoperator defined by [Lˆ] = | i 2 | i | i ⊗| i
D D
(2LˆρLˆ† Lˆ†Lˆρ ρLˆ†Lˆ)/2. In above expression, κ is de- where α istheinitialstateofoscillator. Attimet, thestate
cayrateo−fresona−tor,γ1 isqubitrelaxationrateandγ2 isde- ofsyste|micanbewrittenas
phasing rate. To figure out the effect induced by measure-
1
mentduetocouplingofqubittotheoscillator,wedonotcon- ψ(t) = (0 α+ + 1 α− ). (20)
| i 2 | i| i | i| i
siderthecouplingbetweenqubitandenvironment.Therefore,
we neglectrelaxation dueto coupledto environmentand set Then, if we know oscillator state is α , the qubit state
i
γ1 = 0, (i.e., for an ideal QND measurement no quantum can be determined exactly. From Eq. (18) we find that the
tunnelingoccursbetweenqubitstates0 and 1 duringmea- measurement-induced dephasing depends on the overlap of
| i | i
surement).Eq.(14)canbesolvedundertheconditionthatthe bothtwooscillatorstateα 18. Incasethattheoscillatorstates
±
densitymatrixwithanexpansioninpositiveP representation. α− and α+ areorthogonal,themeasurementcouldbecon-
| i | i
InRef.[10]thesolutionofthemasterequationwasfoundto sideredasastrongprojectivemeasurement,i.e.,intheregion
be erf(At) = 1. For the maximum dephasing value, the os-
| |
t cillatorstates decayto a steady state significantly,whichex-
a10(t)=a10(0)exp[−i(ǫ−iγ2)t−i2gZ α+(t)α∗−(t)dt′] plainsthefeatureofFig.3thatbothmeasurement-inducedde-
0 phasingrateandmeasurementprobabilitygetahighvaluefor
(15)
matchingthequbitfrequency.Inthissituation,thequbitstate
anda01(t)=a∗10(t). Theamplitudeofcoherencestateα±(t) is encoded in the amplitude of coherence rather than phase.
aredeterminedby
Itmeansthattheeasierwegaininformationaboutamplitude,
α+(t)=αs++exp[ (κ/2+ig+iδω)t](α+(0) αs+) (16) thefasterweloseinformationaboutphase.Themeasurement-
− −
induceddephasingratecanalsobeexpressedas
withαs+ =−if/(iκ/2+ig+δω),and α−(t)α+(t) =e−Γmt. (21)
|h | i|
α (t)=αs +exp[ (κ/2 ig+iδω)t](α (0) αs) (17)
− − − − − − − Inaddition,asshowninFig. 3,withthevalueofdamping
rate κ increasing, both structures of probability distribution
with αs = if/(k/2 ig +iδω), where αs is the steady and measurement-induced dephasing rate become flat with
− − − ±
coherence state value of the oscillator, α+ and α− are am- synchronoustendency,whichconsistentwiththeexpectation.
plitudes of coherentstate α+ and α− , which dependson Increasingofdrivingforcef onlychangetheoverallscaleof
| i | i
qubitstates,a01isthecoherenceterm. (e) (h)ratherthanthestructure,whichexplainsthetrendthat
∼
We can find that γ2 is the intrinsic dephasing rate from withthevalueofdrivingforceincreasing,thestrongmeasure-
Eq. (15) andΓ is the measurement-induceddephasingrate mentregionin (a) (h)becomeswiderand themeasurement
m ∼
whichisdeterminedbyαs foraprojectmeasurementandex- probabilityincreasessignificantly,especiallyformatchingthe
±
pressedasfollowing10 qubitfrequency.
Γ =2gIm(αsαs)= (n++n−)κg2 , (18)
m + − κ2/4+g2+δω2 III. DEVIATIONOFAPERFECTQNDMEASUREMENT
DUETOMEASUREMENT-INDUCEDTUNNELING
where n = αs 2 = f2/[κ2/4 + (δω g)2] is aver-
± | ±| ±
age number of photons in the resonator. We find that for
Taking into account the impact of non-diagonal term of
δω = g2+κ2/4 we will get a large probabilityif mea-
± qubit,weobtainthesystemHamiltonian
suretheapmplitudeofthesignal.Inotherwords,whenmatch-
ing one of the frequencies of qubit (i.e., δω = g), the in- ǫ ∆
formation of qubit is encoded in the amplitude±rather than Hˆs = σˆz + σˆx. (22)
2 2
4
FIG. 3: (color online) (a)∼(d) Probability of measuring 0 state for the initial state |0ih0| with damping rate of oscillator κ = 0.1 GHz,
κ = 0.2 GHz, κ = 0.3GHz, κ = 0.4GHz, respectively, and qubit coupling strength g = 0.3GHz, t = 0.1ns, noise spectra density
SII = 2/κ. (e)∼(h) measurement-induced dephasing rate Γm as a function of detuning δω between resonator and drive frequency with
drivingforcef =1GHz.Theparametersof(e)∼(h)aresamewiththosein(a)∼(d),respectively.
Then we can diagonalize the qubit Hamiltonian Hˆ . In new
s
eigenvectorspace,thesystemHamiltoniancanbewrittenas
1
ρ˙I(t)= [HˆI (t),ρI(t)]. (27)
E i~ int
Hˆ = σˆ , (23)
s z
2 We choose the initial state as the a product of qubit and
meanwhile,theinteractHamiltoniancanbewrittenas detector,i.e.,ρ(0) = ρ0(0) ρD(0). Then,tracingEq. (27)
⊗
overthedetectordegreesandexpandingitsrighthandto the
Hˆ =~gσˆ aˆ†aˆ, (24) secondorder,weobtainthetimeevolutionofreduceddensity
int n
matrixρ0. Eq. (27)becomes
where E = √ǫ2+∆2/2 is energy splitting of qubit and n
is a vector that represents the direction of qubitbasis before
1 t
diagonalizedrelativetotheenergybasis ρ˙I(t)= dτ [σˆI(t),σˆI(t τ)ρI(t τ)] nˆI(τ)nˆ
0 − ~2 Z { n n − − h iD
0
σˆn =cosησˆz +sinησˆx, (25) −[σˆnI(t),ρI(t−τ)σˆnI(t−τ)]hnˆnˆI(τ)iD}.
(28)
with η = arctan(∆/ǫ). The eigenstates are denotedby the
superpositionof|0iand|1i After taking matrix elements between eigenstates of Hˆs,
η η wefinallyobtaintheequationofmotionthatcandescribethe
=cos 0 +sin 1
reduceddensitymatrixevolution
|↑i 2| i 2| i
η η
= sin 0 +cos 1 . (26) t
|↓i − 2| i 2| i ρ˙Ikk′(t)=−Z dτ Mkk′ll′ρIll′(t), (29)
From Eq. (24) we find that fluctuation induced by exter- 0 Xl,l′
nal noise in particle number nˆ = aˆ†aˆ can causes transition
inwhichweintroducedthematrix
betweenqubitstates. Therefore,qubitactsasaspectrumana-
lyzer. WeseparatetheinteractionHamiltonianintotwoparts,
oneis~gcosησˆ aˆ†aˆandtheotheris~gsinησˆ aˆ†aˆ. Thefirst Mkk′ll′ =
z x
partdoesnotaffecttherepeatabilityofQNDmeasurementas g2{hnˆI(τ)nˆi[(σˆnI(t)σˆnI(t−τ))klδl′k′−(σˆnI(t−τ))kl(σˆnI(t))l′k′]
d“sispciunsflseipd”intrapnrseivtiioonusdsueecttioonthse. cTohueplsiencgobnedtwpaeretnwqiullbictaoufsfe- +hnˆnˆI(τ)i[(σˆnI(t−τ)σˆnI(t))l′k′δkl−(σˆnI(t))kl(σˆnI(t−τ))l′k′]}(3.0)
diagonalterm and externalvariable. Accordingto the equa-
tionofmotion,thetimeevolutionofdensitymatrixisobtained Tocalculatetheenergyrelaxationrateanddephasingrate,
ininteractionpictureby we compute the transition rate between qubit eigenstates by
5
computingevolutionofthediagonaltermsofreduceddensity where θ0 = arctan(2g/κ) 2g/κ is phase shift during
≈
matrix ρ11 and ρ00, and the decay of the off-diagonal term measurement derived from Eq. (10), which is identical with
ρ01 withtheinitialstatebeingaσˆx eigenstate. Thetransition Ref. [10] and Ref. [19] with a coefficient of expression dif-
rates between the two qubit eigenstates can be derived from ferencethat is caused by the differenceof the lower limit of
Eq. (29) integral.ItisquiteacontrasttoEq.(18)thathasdoublepeaks
formatchingthefrequenciesofqubit. Withtheamplitudeof
off-diagonal term of density matrix increasing, the pure de-
Γ↓ =g2sin2ηSnn(ω01) phasingratebecomeweakandrelaxationwillbedominant.
Γ↑ =g2sin2ηSnn( ω01), (31) FormEq.(33),wefindthatthedecayofcoherenceiscaused
−
by intrinsic dephasing and aggravated by measurement-
where Γ↑ and Γ↓ are rates at which qubit is excited from induceddephasing. Inaddition,thedeviationfromtheinitial
ground state to excited state and decay from excited state valueof σ becomebiggerasthetimegoesbyduetotran-
z
to ground state, respectively. ω01 = E/~, Snn(ω) = sition bethweein the qubit states. Therefore, the state of qubit
tdτeiωτ nˆ(τ)nˆ(0) . If the noise source is in the thermal is acquired; meanwhile, the repeatability of measurement is
0 h i
eRquilibriumattemperatureT, thetransitionratemustsatisfy destroyed.ItmeansthatitshouldnotbeconsideredasaQND
thebalanceconditionΓ↑/Γ↓ = exp( β~ω01)witheffective measurement. Based on this effect, it can be used for “back
−
temperature β = 1/k T . It means that the qubit energy actioncooling”inmanysystem,suchasacantilevercoupled
B eff
canbeabsorbedoremittedbydetectorandtheratiobetween to a optical cavity,20 a noisy qubit coupled to a mechanical
positive and negative noise spectral density depends on the resonator,(author?)21nanomechanicalresonatorcoupledtoa
effective temperature. We can also obtain the expression of drivensuperconductingresonantor22andcoolingamicromir-
dephasingrateΓ rorbyradiationpressureinsideaopticalcavity.23
φ
1
Γφ = g2sin2η[Snn(ω01)+Snn( ω01)]
2 −
+g2cos2ηS (ω =0) (32)
nn IV. CONCLUSION
Γ +Γ
↑ ↓
= +γ ,
φ
2
We have analyzed the QND measurement of a flux qubit
where γ is qubitpure dephasingrate. For the case ∆ = 0, coupled to a noisy detector. The analytical results reveal
φ
themeasurementisaperfectQNDmeasurementwhichcannot that both of the measurement probability and measurement-
causestransitionbetweenqubitstates,inwhichthedephasing induceddephasinghaveasimilartrendthattheirvalueshave
rateis peaks for matching one of the qubit frequencies and reduce
attheresonancepoint,atwhichdrivingfrequencyequalsthe
∞
Γ =γ =g2 dτ nˆI(τ)nˆ . (33) bare harmonic oscillator frequency. With the resonator de-
φ φ Z−∞ h iD cay rate increasing,both ofthe curvesof dephasingrate and
measurement probability become flat. Due to the coupling
To calculate dephasing rate caused by fluctuation of particle
between the non-diagonal term of qubit and variable of de-
number of oscillator during measurement, we should calcu-
tector,thenoiseofdetectorperturbstheinitialstateofqubit.
late correlator δnˆ(τ)δnˆ , where δnˆ is quantum fluctuations
h i As a result, the transition between the qubit states destroys
around the mean of photons in the resonator. For a driving
the initial informationof qubit. Therefore, there will be en-
resonatorwithfrequencyω ,thefollowingequationisvalid19
d ergy exchange between qubit and detector according to the
value of effective temperature of noise, which can be used
aˆ(t)=e−iωdt[α+dˆ(t)], (34)
for“back action” cooling. Conversely,the behaviorof qubit
which acts as a spectrum analyzer can also be used for ana-
whereαisaclassicpartcorrespondingtothecoherencestate
lyzingthepropertyofnoise.
α anddˆ(t)isaquantumpartwhichcanannihilatethecoher-
| i
encestatetothevacuumstate. Thenthecorrelatorbecomes
δnˆ(τ)δnˆ(0) = dˆ(τ)dˆ(0) =neiδωt−iκ2t, (35)
h i h i
V. ACKNOWLEDGEMENT
wherenistheaveragephotonumberintheresonator. Under
the condition that δω = 0 and weak coupling g κ, the
measurementinduceddephasingrateis ≪ This work was partially supported by the State Key Pro-
gram for Basic Research of China (2006CB921801) and
Γ =κnθ2, (36) NSFC(10725415).
m 0
∗ Electronicaddress: [email protected] † Electronicaddress:[email protected]
6
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