Chaos can act as a decoherence suppressor

0.15 MB·English

by  Jing Zhang

Chaos canact asa decoherence suppressor
JingZhang,1,2,3,∗ Yu-xiLiu,4,2 Wei-MinZhang,5 Lian-AoWu,6 Re-BingWu,1,2 andTzyh-JongTarn7,2,3
1Department of Automation, Tsinghua University, Beijing 100084, P. R. China
2Center for Quantum Information Science and Technology,
TsinghuaNationalLaboratoryforInformationScienceandTechnology, Beijing100084, P.R.China
3Department of Physics and National Center for Theoretical Sciences,
National Cheng Kung University, Tainan 70101, Taiwan
4Institute of Microelectronics, Tsinghua University, Beijing 100084, P. R. China
5Department of Physics and Center for Quantum Information Science,
National Cheng Kung University, Tainan 70101, Taiwan
6Department of Theoretical Physics and History of Science,
TheBasqueCountryUniversity(EHU/UPV)andIKERBASQUE-BasqueFoundationforScience,48011,Bilbao,Spain
1
7DepartmentofElectricalandSystemsEngineering, WashingtonUniversity, St.Louis,MO63130, USA
1
(Dated:January18,2011)
0
2
Wepropose a strategyto suppress decoherence of a solid-statequbit coupled tonon-Markovian noises by

n attachingthequbittoachaoticsetupwiththebroadpowerdistributioninparticularinthehigh-frequencydo-
a main.Differentfromtheexistingdecoherencecontrolmethodssuchastheusualdynamicsdecouplingcontrol,
J high-frequencycomponentsofourcontrolaregeneratedbythechaoticsetupdrivenbyalow-frequencyfield,

7 andthegeneration of complexoptimized control pulsesisnot necessary. Weapply theschemetosupercon-
1 ducting quantum circuits and find that various noises in a wide frequency domain, including low-frequency
1/f,high-frequencyOhmic,sub-Ohmic,andsuper-Ohmicnoises,canbeefficientlysuppressedbycouplingthe

] qubitstoaDuffingoscillatorasthechaoticsetup. Significantly,thedecoherencetimeofthequbitisprolonged
h
approximately100timesinmagnitude.
p
-
t PACSnumbers:05.45.Gg,03.67.Pp,05.45.Mt
n
a
u
Introduction.— Solid state quantum information process- encetimeofthequbitbycouplingittoachaoticsetup[8].Al-
q
ing[1]developsveryrapidlyinrecentyears.Oneofthebasic thoughitiswidelybelievedthatthechaoticdynamicsinduces
[
featuresthatmakesquantuminformationuniqueisthequan- inherent decoherence [9–13], e.g., the quantum Loschmidt

1
tum parallelism resulted from quantum coherence and en- echo [11], we find surprisingly that the frequency shift of
v
tanglement. However, the inevitableinteraction between the the qubitinducedby the chaotic setup, which has notdrawn
4
9 qubit and its environmentleads to qubit-environmententan- enoughattentionin the literature,can helpto suppressdeco-
1 glementthatdeterioratesquantumcoherenceofthequbit. In herenceofthequbit.
3 solidstatesystems,thedecoherenceprocessismainlycaused
. Themainmeritsofthismethodare: (1)thehighfrequency
1 by the non-Markoviannoises induced, e.g., by the two-level
components,whichcontributeto thesuppressionofthe non-
0 fluctuatorsin the substrate and the charge and flux noises in
1 thecircuits[2–4]. Markoviannoises,canbegeneratedbythechaoticsetupeven
1 drivenby a low-frequencyfield; and (2) generatingcomplex
v: There have been numbers of proposals for suppressing optimalcontrolpulsesisnotnecessary.
non-Markoviannoises in solid-state systems. Most of them
i
X suppress noises in a narrow frequency domain, e.g., low- Decoherence suppression by chaotic signals.— Consider
r frequency noises [2, 3]. Among the proposed decoher- thefollowingsystem-environmentmodel[4]
a
ence suppression strategies, the dynamical decoupling con-
trol (DDC) [5] is relatively successful in suppressing non-
Markovian noises in a broad frequency domain and has re-
Hˆ =(ω +δ )Sˆ + ω σˆ(i)+ g σˆ(i)Sˆ +h.c. ,
centlybeendemonstratedinthesolid-statesystemexperimen- q q z i z i + −
Xi Xi (cid:16) (cid:17)
tally [6]. The main idea of the DDC is to utilize high fre- (1)
quencypulsestoflipstatesofthequbitrapidly,averagingout where ω (ω ), Sˆ (σˆ(i)), and Sˆ (σˆ(i)) are the angular fre-
thequbit-environmentcoupling. Thehigherthefrequencyof q i z z ± ±
quency,thez-axisPaulioperator,andtheladderoperatorsof
the control pulse is, the better the decoherence suppression
thequbit(thei-thtwo-levelfluctuatorintheenvironment);g
i
effects are. Efforts have been made to optimize the control
is the coupling constant between the qubit and the i-th two-
pulses[7]intheDDC,however,therequirementsofgenerat-
levelfluctuator;andδ (t) δ isanangularfrequencyshift
q q
ing extremely high frequencycontrol pulses or complex op- ≡
inducedbychaoticsignals. Iftheinitialstateofthesystemis
timized pulses make it difficult to be realized in solid-state
separable, ρˆ(0) = ρˆ ρˆ , we can reduce the dynamical
S0 B0
quantuminformationsystem,inparticularinsuperconducting ⊗
equationofthetotalsystembytracingoutthedegreesoffree-
circuits.
domofthebath.Theinfluenceofthechaoticsignalδ (t)falls
q
In this letter, we proposea methodto extendthe decoher- intotwoaspects: (1)δ (t)affectstheangularfrequencyshift
q
2
ofthequbitinducedbythebath Hamiltonian
∆ωq =Z ωc2 dωJ(2ω)ImZ tei(ωq−ω)(t−t′)+iRtt′δqdt′′dt′; HˆDuf =ωoaˆ†aˆ− λ4 aˆ+aˆ† 4−I(t)√12 aˆ+aˆ† (2)
ωc1 0 (cid:0) (cid:1) (cid:0) (cid:1)
and(2),moreimportantly,itmodifiesthebath-induceddeco- anddampingrateγ, whereaˆ andaˆ† arethe annihilationand
herencerateofthequbitwhichcanbewrittenas: creationoperatorsofthenonlinearDuffingoscillator;ω /2π
o
isthefrequencyofthefundamentalmode;λisthenonlinear
Γq =2 ωc2 dωJ(ω)Re tei(ωq−ω)(t−t′)+iRtt′δqdt′′dt′, constant; andI(t) = I0cos(ωdt) denotesthe classicaldriv-
Z Z ingfieldwithamplitudeI andfrequencyω /2π. Weemploy
ωc1 0 0 d
the interaction between the qubit and the Duffing oscillator,
whereJ(ω)= igi2δ(ω−ωi)isthespectraldensityofthe HˆI = gqoSˆzaˆ†aˆ (gqo - couplingstrength), whichcan be ob-
bath. Here, sincPethefrequenciesofthefluctuatorsdistribute tained, e.g., by the Jaynes-Cummingsmodel underthe large
inafinitedomain,∆ω andΓ arerestrictedtobeintegrated detuningregime[15].
q q
inthefinitefrequencydomain[ωc1,ωc2]. Wedemonstrateour Bytracingoutthedegreesoffreedomoftheoscillatorini-
resultsusingthezero-temperaturebath. tially in a coherentstate α , we find that the interactionbe-
Wenowcometoshowhowthedampingratecanbereduced tween the qubit and the D| uiffing oscillator introduces an ad-
bythefrequencyshiftδq(t),usingthefunctionδq(t)asalin- ditionalfactorforthenon-diagonalentriesofthestate ofthe
ear combination of sinusoidal signals with small amplitudes qubit[9]:
andhighfrequencies,i.e.,δ (t)= A cos(ω t+φ ).
q α dα dα α
Here ωdα should satisfy the conditPions: ωdα ≫ |ωc2 −ωq|, f01(t)= αeit(HDuf+gqoaˆ†a)e−it(HDuf−gqoaˆ†a) α . (3)
ω ω ,andA /ω 1. UsingtheFourier-Besselse- h | | i
| q − c1| dα dα ≪
ries identity [14]: eixsiny = nJn(x)einy with Jn(x) as Therearetwoaspectsofthefactorf01(t)=eΣq(t)+iΘq(t).
the n-thBessel functionofthePfirst kindandthe approxima- The phase shift Θ (t) tδ (t′)dt′, induced by f (t), is
tionJ (x) 1 x2/4forx 1,wehave q ≈ 0 q 01
0 ≈ − ≪ related to δq(t) in Eq. (1R), which can be used to suppress
Z0tei2ω−(t−t′)+iRtt′δq(t′′)dt′′dt′ ≈Feiω−t(cid:18)sinωω−−t(cid:19), tMehceh(otd)e[c1=o0h,e|1fr1e0]n1,c(lete)oa|2dfsth=teoqeau2dΣbdiqitt(.ito)H,noai.wled.e,evtcehore,hteqhrueeanancmteupmelfitfuLedcotesscsohqfmuatihrdeet
qubit.Suchdecoherenceeffectshavebeenwellstudiedinthe
where the correction factor F = exp π ∞ Sδq(ω)dω ;
Sδq(ω) = αA2dαδ(ωdα−ω)/2π is(cid:16)−theRpωocwderωs2pectru(cid:17)m lciotehrearteunrecefsourprpergeuslsaironansdtracthegaoytiicsvdaylnidamwihcesn[t9h–e1d3e]c.oOheurrendcee-
density of thPe signal δq(t); ω− = (ωq−ω)/2; and ωcd is suppression induced by δq(t) is predominantin comparison
the lower bound of the frequencyof δq(t) such that ωcd withtheoppositedecoherenceaccelerationprocess.
≫
|ωc2 − ωq|, |ωq − ωc1|. Here, we omit the higher-order Wenowcometoshownumericalresults,usingsystempa-
Bessel function terms because 1/(i(ωq ω)+nωdα) rameters:
− ≪
1/(i(ω ω)) and J (A /ω ) J (A /ω ) under
q n dα dα 0 dα dα
− ≪
the conditions. The analysisshowsthatδq(t) inducesa cor- (ωo,gqo,γ,λ)=(ωq,0.03ωq,0.05ωq,0.25ωq). (4)
rectionfactorF fortheenvironment-inducedfrequency-shift
∆ωq anddampingrateΓq,i.e., Thebathhasa1/f noisespectrum(see,e.g.,Ref.[2,3])with
J(ω) = A/ω and A/ω = 0.1. The evolution of the co-
∆ωq = F Z ωc2 dωJ(ω)[12−(ωcos(ωωq)−ω)t] =F∆ωq0, herenceCxy = hSˆxi2+qhSˆyi2 ofthequbitandthespectrum
ωc1 q− analysisoftheangularfrequencyshiftδq(t)arepresentedin
ωc2 2J(ω)sin(ωq ω)t Fig.1. AsshowninFig.1(b),(c),ifwetunetheamplitudeI0
Γ (t) = F dω − =FΓ ,
q Zωc1 ωq−ω q0 3o0f,ththeesisnigunsoaildsaδld(rti)vienxghfiibeiltdpIer(ito)dsicucahndthcahtaIo0t/icωqbe=ha5vioanrsd.
q
where ∆ω and Γ are the frequency-shift and damping As shown in Fig. 1(a), in the periodic regime, the decoher-
q0 q0
rate when δ (t) = 0. The correction factor F may become enceofthequbitisalmostunaffectedbytheDuffingoscilla-
q
extremely small when δ (t) is a chaotic signal which has tor. Thetrajectoryintheperiodiccase(greencurvewithplus
q
a broadband frequency spectrum in particular in the high- signs)coincideswiththatofnaturaldecoherence(blacktrian-
frequencydomain,suchthatthedecayrateandfrequencyshift glecurve),asinFig.1(a). Inthechaoticregime,thedecoher-
canbesuppressedbyachaoticsignal. enceofthequbitisefficientlysloweddown(seethebluesolid
Generation of chaotic signals and suppressing 1/f curve in Fig. 1(a) representing the trajectory in the chaotic
noises.— To show the validity of our method, as an exam- case). Thisdemonstratesthat,withtheincreaseofthedistri-
ple,we showhowtosuppressthe1/f noisesofaqubitwith bution of the spectral energy in the high-frequencydomain,
freeHamiltonianHˆ =ω Sˆ bycouplingittoadrivenDuff- thedecoherenceeffectsaresuppressedasexplainedinthelast
q q z
ing oscillatorwhich is used to generatechaoticsignals, with section.
3
1 60 quantum devices, e.g., the superconductingqubit system, as
Cxy00..68 CIdheaaol tcica scease (a) ωS() [dB]δq24000 Periodic spectrum (b) srekaedtcohuetdciirncuFitig[.162]., bIuttiswsoirmksilainr taoqtuhietewdiidffeelyrenutsepdarqaumbei-t
ter regime. In this superconducting circuit, a single Cooper
0.4 −20
Periodic case −40 pair box (SCB) is coupled to a dc-SQUID consisting of two
0.2 JosephsonjunctionswithcapacitancesC˜ andJosephsonen-
−60 J
00 20 40 60 80 t/τ 100 −800 5 10 15 20 25 30ωτ35 ergiesE˜J and a paralleledcurrentsource. TheSCB is com-
60 0.6 posed of two Josephson junctions with capacitancesCJ and
ωS() [dB]δq23450000 Chaotic spectrum (c) Γ/ω0q.4 PreeΓgrqiimom/dωeicq CrΓehqga0i/omωtqiec (d) JFJooigssee.pp2hhassnoodnntejhunenecrrgetiaioednsoEiustJrc.eirpTclhuaecitedidinffbReyreefan.cd[1ec6-bS]eQitswUtehIeDant-tthhteehecrfic-rbchuiaaiostteiindc
10 0.2 λ/ωq setup.
0
−10 0
−20
−300 5 10 15 20 25 30ωτ35 −0.2 10 20 30 40 50I0/ωq60 E
J
FIG. 1: (color online) Decoherence suppression by the auxiliary CJ E~J E~J
chShˆayoit2icofsetthuep.sta(tae)othfetheevoqluubtiito,nwohfertehethceohreedreanscteerCiskxycu=rvehSaˆnxdi2th+e EJ φφφφC~J φφφφe C~J Ie
black triangle curve represent the ideal trajectory without any de- Vg CJ
coherenceandthetrajectoryundernaturaldecoherenceandwithout
corrections; andthegreen curve withplussigns and thebluesolid
curve denote the trajectories with I0/ωq = 5 and 30. With these
parameters,thedynamicsoftheDuffingoscillatorexhibitsperiodic
FIG. 2: (color online) Schematic diagram of the decoherence sup-
andchaoticbehaviors. τ = 2π/ωq isanormalizedtimescale. (b) pression superconducting circuit in which a SCB is coupled to a
and (c) are the energy spectra of δq(t) with I0/ωq = 5 (the peri- current-biaseddc-SQUID.
odiccase)and30(thechaoticcase). TheenergyspectrumS (ω)
δq
isinunitofdecibel(dB).(d)thenormalizeddecoherenceratesΓ/ωq
versusthenormalizeddrivingstrengthI0/ωq. TheHamiltonianofthecircuitshowninFig.2canbewrit-
tenas:
φˆ
We compare in Fig. 1(d) the average natural decoherence Hˆ = EC(nˆ ng)2 2EJcos cosθˆ+E˜Cnˆ˜2
− − 2
rateΓ andmodifieddecoherencerateΓ ofthequbitver-
q0 qm φ
sus different strengths I0 of the driving field. Figure 1(d) −2E˜Jcos 2e cosφˆ−φ0Ieφˆ, (5)
showsthat,whenthestrengthI ofthedrivingfieldincreases,
0
the decoherence process is efficiently slowed down. It is where EC = 2e2/(Cg+2CJ) is the charging energy of
interesting to note that there seems to exist a phase transi- SCB with Cg as the gate capacitance; ng = CgVg/2e is
−
tion around I /ω = 20, i.e., a sudden change of the mod- thereducedchargenumber,inunitofthe Cooperpairs, with
0 q
ified decoherence rate Γqm (see the black solid curve with Vg as the gate voltage; nˆ is the number of Cooper pairs on
plus signs). It is noticable that, around this point, the dy- the island electrode of SCB with θˆ as its conjugate opera-
namics of the Duffing oscillator enters the chaotic regime tor; E˜ = e2/C˜ is the chargingenergy of the dc-SQUID;
C J
which is indicatedbya positiveLyapunovexponent(see the nˆ˜isthechargeoperatorofthedc-SQUIDandφˆtheconjugate
greendash-dottedcurveinFig.1(d)). Themodifieddecoher- operator; and φ and I are the external flux threading the
e e
ence rate Γqm changesdramaticallyin the parameterregime loopofthedc-SQUIDandtheexternalbiascurrentofthedc-
I0/ωq [20,35]whichisthesoft-chaosregimeoftheDuff- SQUID.Here,weconsiderazeroexternalfluxthreadingthe
∈
ing oscillator. When the dynamics of the Duffing oscillator loopofthecoupledSCB-dcSQUIDsystem. Inthiscase,the
entersthehard-chaosregimeatI0/ωq 35,themodifiedde- phase dropacrossthe SCB is equalto the phase dropacross
≈
coherencerateΓqm isstabilizedatavaluemuchsmallerthan the dc-SQUID φ˜. We further introduce the ac gate voltage
thenaturaldecoherencerateΓq0. Thesimulationresultsshow Vg =Vg0cos(ωgt)withamplitudeVg0andangularfrequency
thatthe decoherenceofthequbitisefficientlysuppressedby ω . WiththeconditionthatC V E /2e ω = E ω ,
g g g0 C q J g
≪ −
ourproposal,evenifthereexistsanadditionaldecoherencein- the SCB works near the optimal point, and we only need
troducedby the auxiliarychaotic setup. Furthercalculations to worry about the relaxation of the SCB. By expandingthe
show that the modified decoherence rate Γqm in the chaotic Hamiltonianof the SCB in the Hilbert space of its two low-
regimeisroughly100timessmallerthantheunmodifiedde- est states and leaving the lowest nonlinear terms of φˆ in
coherencerateΓq0,meaningthatthedecoherencetimeofthe the rotating frame, we can obtain the effective Hamiltonian
qubitcanbeprolonged100times. Hˆ = Hˆ + Hˆ + Hˆ discussed in the foregoing sec-
eff q Duf I
Experimentalfeasibilityinsuperconductingcircuits.—Our tion. This dc-SQUID, acting as the auxiliary Duffing oscil-
general study can be demonstrated using the solid state lator, can be used to suppress low frequency 1/f noises of
4
Typeofnoises Frequencydomain Γ¯q0 Γ¯qm T1 =T2 60904034,60836001,60635040.T.J.Tarnwouldalsoliketo
1/fnoise [10kHz,1MHz] 0.58MHz 5.4kHz 187µs acknowledge partial support from the U. S. Army Research
Ohmic [2ωq/3,3ωq/2] 0.35MHz 5.4kHz 187µs Office underGrantW911NF-04-1-0386. L. A. Wu hasbeen
Sub-Ohmic [2ωq/3,3ωq/2] 0.35MHz 5.4kHz 187µs supportedbytheIkerbasqueFoundationStart-up,theBasque
Super-Ohmic [2ωq/3,3ωq/2] 0.36MHz 5.4kHz 187µs
Government(grantIT472-10)andtheSpanishMEC(Project
TABLEI:Decoherencesuppressionagainstvariousnoisesforexper- No. FIS2009-12773-C02-02).
imentallyaccessibleparameters:EJ/2π =5GHz,ωg/2π =4.999
GHz,E˜C/2π=0.188MHz,andE˜Jcosφ2e/2π =12.032MHz.
thequbit. Usingtheexperimentallyaccessibleparametersas ∗ Electronicaddress:[email protected]
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J. Zhangwouldlike tothankDr. H. T.TanandDr. M.H.
[16] D.Vionetal.,Science296,886(2002);I.Siddiqietal.,Phys.
Wu for helpful discussions. We acknowledge financial Rev.Lett.93,207002(2004);Phys.Rev.B73,054510(2006).
support from the National Natural Science Foundation of
China under Grant Nos. 60704017, 10975080, 61025022,