Open quantum system description of singlet-triplet qubits in quantum dots

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by  L. K. Castelano

Openquantum system descriptionofsinglet-triplet qubits inquantum dots
L. K. Castelano,1,∗ F. F. Fanchini,2,3 and K. Berrada3,4
1DepartamentodeF´ısica, Universidade Federal deSa˜oCarlos, 13565-905, Sa˜oCarlos, SP,Brazil
2Faculdade deCieˆncias, UNESP-Universidade Estadual Paulista, Bauru, SP,17033-360, Brazil
3TheAbdusSalamInternationalCentreforTheoreticalPhysics,StradaCostiera11,Miramare-Trieste, Italy
4PhysicsDepartment,CollegeofScience,AlImamMohammadIbnSaudIslamicUniversity(IMISU),Riyadh,SaudiArabia.
(Dated:January13,2017)
Wedevelopatheoreticalmodeltodescribethedissipativedynamicsofsinglet-tripletqubitsinGaAsquantum
dots.Usingtheconcurrenceexperimentallyobtained[M.D.Shulmanetal.,Science336,202(2012)]asaguide,
wefoundthateachlogicalqubitfluctuatesundertheactionofarandomtelegraphnoise(RTN)thatsimulatesthe
1/fαnoise.WealsostudythedynamicsofconcurrenceasafunctionoftheamplitudeoftheRTN,thecorrelation
7
timeoftheRTN,thepreparationtimeofstates,andthetwo-qubitcoupling.Furthermore,weshowthatthetwo-
1
0 qubit coupling together with the preparation time strongly affect the entanglement dissipative dynamics and
2 bothphysicalquantitiescanbeemployedtoenhancetheentanglementbetweensinglet-tripletqubits.

n
PACSnumbers:73.63.Kv,03.67.a,73.21.La
a
Keywords:
J

1
1 I. INTRODUCTION Thepresentpaperisorganizedasfollows. Inthenextsec-
tion,wepresentthemodelthatdescribesthedynamicsofthe
]
h The development of quantum information processing has openquantumsystem. InSec. III,we introducethe concept
p enabledthediscoveryofnewtechniquesandplatformswhich ofentanglement,measuredbyconcurrence,togetherwithre-
- sults of ourtheoreticalmodel. We also include in Sec. III a
t arepavingthewaytoaccomplishquantumtechnologiesinthe
n nearfuture.1Amongtheseplatforms,spinqubitsinquantum detailedstudyonthemostimportantphysicalparametersthat
a dots(QDs)2 iscertainlyoneofthemoststrikingsystemsbe- rulestheentanglementdynamics. Finally,Sec. IVcontainsa
u
q causeoftheirpotentialscalabilityandminiaturization.3–5Fur- summaryofourresults.
[ thermore,electricalreadoutandcontrolofspinsqubitsinQDs
have been achieved in several different approaches,6 where
1 II. THEORETICALMODEL
spin blockade and charge sensors enable the observation of
v
single/two-spindynamics.7 Indoublequantumdots(DQDs),
9
The main focus of this work is related to the study of the
6 alogicalqubitcanbeencodedbymeansofsinglet-triplet(S-
1 T ) states of two electron spins8–10 and the inter-qubitinter- dissipativedynamicsoftwoS-T0 qubits,wheretheinforma-
0
3 action can be implemented through a capacitive coupling.11 tion is stored in the spin states of two electrons. Such states
0 By controling and coupling S-T qubits, the entanglement canbeexperimentallyachievedbyconfiningtwoelectronsin
0
1. between two S-T0 qubits has been experimentally demon- eachDQDsystem.12 Thelogicalqubitcomposedbythetwo-
0 strated.12 Togetherwiththesuccessofsuchademonstration, levelsystem (S ,T0 ) canbe isolatedbyap-
| i ≡ | ↑i| i ≡ | ↓i
7 theubiquitousnoisehasbeenprobedintheexperimentaldata plyinganexternalmagneticfieldintheplaneofthedevicein
1 sucha waythattheZeemansplittingmakesthe parallelspin
ofref.[12]. ForasingleS-T qubit,thenoisehasbeenchar-
v: acterizedandshowntobecon0sistentwiththepower-law1/fα states T+ and T− energeticallyinaccessible.
| i | i
i noise.13 Also,ithasbeenexperimentallyverifiedthattheex- Toextendsuchatwo-levelsystemtoatwo-qubitsystem,it
X
poentαhasatemperaturedependence;forinstance,α 0.7 isnecessarytocoupletwoS-T0 qubits,wherethetunnelling
ar forT=50mK andα 0forT=100mK.13 Sucha tem≈pera- betweenthemissuppressedandtheircouplingiselectrostatic
≈ (formoredetails, seeref.[12]). Thus,theeffectiveHamilto-
turedependencecanbeascribedtophonon-induceddecoher-
encemechanism.14The1/fα noisecanbepresentinavariety nianforthetwo-qubitsystemcanbewrittenasfollows:12
of systems15 and particullarly in other QDs systems.16,17 To 1
modelthe1/fαnoise,randomtelegraphnoise(RTN)hasbeen Hˆ2-qubit = 2(cid:16)J1σz(1)⊗I+J2 I⊗σz(2)(cid:17)+
employedindifferenttheoreticalworks.18–21
J
Inthiswork,weemploytheRTNtodescribethedecoher- 412 (cid:16)σz(1)⊗σz(2)−σz(1)⊗I−I⊗σz(2)(cid:17)+
ence caused by the interactionbetweentwo S-T qubitsand
0 1
theirenvironmentinthelowtemperaturelimit,i.e. α6=0.By 2(cid:16)∆Bz,1σx(1)⊗I+∆Bz,2 I⊗σx(2)(cid:17), (1)
using such a model, we are able to quantitatively reproduce
experimental results obtained for S-T0 qubits in two GaAs whereσx,y,z arethePaulispinmatrices,Iistheidentityand
coupledDQDs.12 Throughourdescriptionofthisopenquan- theindex1(2)isrelatedtothefirst(second)qubit(hereafter,
tumsystem,weexploittheroleoftheamplitudeoftheRTN, we use units of ~ = 1). This Hamiltonian is able to imple-
thecorrelationtimeoftheRTN,thepreparationtimeofstates, mentuniversalquantumcontrol,whichisgivenbytwophys-
andthetwo-qubitcouplingintheentanglementdissipativedy- ically distinct local operations, x and z, and by the interac-
namics. tionbetween thequbitsgivenbyσ(1) σ(2). The exchange
z z
⊗
2

splitting, J , between Si and Ti applies rotations in the
qubit i=1,2iaround the|z iaxis, w|h0ilie rotations around the x 1.0
axisaredrivenbyamagneticfieldgradient∆B . Moreover, 0.5
z
∆Bz isresponsibleforthepreparationofeachqubitinasu- 0.0
perposition between |Si and |T0i. The two-qubit coupling, )c-0.5
J12, depends on the energy between levels of the left and (t, -1.0
the right DQD and it can be switched on and off during the RTN 1 1.0
quantum dynamics.12 Due to the Pauli exclusion principle, J
0.5
S and T states have different charge configurations and
0
| i | i
becausebothqubitsareelectrostaticallycoupled,thestateof 0.0
thefirstqubitisconditionedtothestateofthesecondqubit.In -0.5
otherwords,whensimultaneouslyevolving,theyexperiencea
-1.0
dipole-dipolecouplingthatgeneratesanentangledstate. Fol-
0 200 400 600 800 1000 1200
lowing the experimental steps,12 each qubit is initialized in
Time (ns)
the S state, then rotated by π/2 around the x axis when
J =| Ji = 0, ∆B /2π 30MHz, for i=1,2. After this FIG.1: (Coloronline)SimulationoftheRTNasafunctionoftime,
stiage, a1l2arge exchanzg,ie spli≈tting is switched on correspond- consideringtheamplitudeJ0 =1MHzandtwodifferentcorrelation
times:τ =10ns(toppannel)andτ =30ns(bottompannel).
ing to J /2π 280MHz, and J /2π 320MHz. Experi- c c
1 2
≈ ≈
mentally,itwasfoundthatthetwo-qubitcouplingisgivenby
J =J J .12
12 1 2 Furthermore,tocheckthe1/fαnatureoftheRTN,wenumeri-
To include the dissipative dynamics, we consider a phe-
callycalculatethepowerspectrumS(f),whichistheFourier
nomenologicalapproach, where both qubits are subjected to
transformof the autocorrelationfunction(Eq.(6)). Such re-
localRTNfluctuationsonexchangesplittingtermsJ andJ .
1 2 sults are shownin Fig. 2 for a fixed amplitudeJ = 1 MHz
Thus,theRTN-Hamiltoniancanbewrittenas 0
andfordifferentcorrelationtimes:τ =1ns(magentadotted
c
Hˆ = JRTN(t,τ )σ(1) I+JRTN(t,τ )I σ(2). (2) curve),τc = 10ns(blacksolidcurve),andτc = 30ns(blue
RTN 1 c z ⊗ 2 c ⊗ z dashed curve). The solid red curve in Fig. 2 is a plot of the
Forsuchakindofnoise,JRTN(t,τ )jumpsbetweentwoval- function4 10−3/f0.89,whichisusedforcomparisontothe
k c ×
ues J andJ accordingto18 highfrequencybehaviorofthenumericallycalculatedpower
0 0
− spectrum S(f). Such a power-law noise modelis similar to
JRTN(t,τ )=( 1)f(t,τc,k)J , (3) theonededucedforonlyonequbitinDQDs.13
k c − 0 Toperformtheanalysisofourresultsandtocomparetothe
where k = 1,2 and the function f(t,τ ,k) is related to the experimentalwork,12 the quantumcorrelationcalled concur-
c
timeswherethejumpsoccurbythefollowingexpression renceisemployed. Concurrenceisa wellknownmeasureof
entanglement,whichisbroadlyacceptedtoberesponsiblefor
f(t,τ ,k)= Θ(t tk), (4) asetofimportanttasksinquantuminformationtheory,suchas
c X − j quantum teleportation22 and quantum key distribution.23 For
j
whereΘ(x)istheHeavisidefunctionand
j c=1 ns
0.1
tkj =−Xτc log(pkn). (5) c=10 ns
n=1 0.01 c=30 ns
0.89
IthneEcqo.rr(e5l)a,tpioknnatriemuenτifodremtelrymdiinsetrsibthuetefdrerqaunednocmyonfumjubmeprssaanndd S( f )1E-3 c/f
c
isrelatedtotheautocorrelationfunctionasfollows
1E-4
JRTN(t,τ )JRTN(t′,τ ) =exp( 2t t′ /τ ), (6)
h k c k c i − | − | c
1E-5
where representsanaverageoverthefluctuations.
h···i
1E-6
0.01 0.1 1 10 100
III. RESULTS f (MHz)
FIG.2: (Coloronline)Numericalcalculationofthepowerspectrum
TheresultsfortheJ1RTN(t,τc)areshowninFig.1,consider- S(f)consideringtheRTNamplitudeJ0 =1MHzandthefollowing
ingJ0 =1MHzandτc =10ns(black-solidcurveinthetop valuesforthecorrelationtime: τc=1ns(magentadottedcurve),τc
panel)andτc =30ns(red-dashedcurveinthebottompanel). =fu1n0ctinosn(bc/lfac0k.8s9oliisdpcluorttveed),aasnadsτocli=dr3e0dncsu(rbvleufeodracsohmedpacruisrovne).. The
As expected, there are less jumps for a higher value of τ .
c
3
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 J0=0 0.2
=3ns
c
0.0 0.0
-0.2 J0=11.6MHz -0.2 =9ns
c
-0.4 -0.4
=18ns
J0=23.2 MHz c
-0.6 -0.6
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200
Time (ns) Time (ns)
FIG.3: (Coloronline)Time-evolutionoftheDDSEconsideringthe FIG.4: (Coloronline)Time-evolutionoftheDDSEconsideringthe
RTNwithfixedcorrelationtimeτc=9nsandfordifferentnoiseam- RTN with fixed noise amplitude J0 = 11.6 MHz and for different
plitudes: J0 = 0 (dashed blue curve), J0 = 11.6 MHz (solid black valuesofthecorrelationtime: τc =3ns(dashedbluecurve),τc =9
curve),andJ0 =23.2MHz(dottedredcurve). Opencirclesdenote ns(solidblackcurve),andτc=18ns(dottedredcurve).Opencircles
theDDSEsextractedfromexperimentaldata.12 denotetheDDSEsextractedfromexperimentaldata.12
two qubits, there is an analytical solution to concurrence,24 thatoccurduringthepreparationofthestate( + )/√2.
whichisgivenby |↑i |↓i
Suchanegativevalueoccursduringthepreparationtimeasa
result of the interaction of the π/2 rotation around the x di-
C(ρ)=max 0,λ λ λ λ , (7)
{ 1− 2− 3− 4} rection and the RTN, which acts in the z direction. Indeed,
where λ (i = 1,2,3,4) are the eigenvalues of R = to take into accountsuch a negativevalue of DDSE, we just
i
√ρρ˜√ρ listed in descendingorder. ρ˜is the time-reversed needtouseτprep = 25nsandJ0 = 0,ascanbeobservedin
pdensityoperator,whichcanbewrittenas Fig.(3). Furthermore,onecansee6 inFig.(3)thattheDDSE
oscillatesintimeandithasanenvelopefunctionthatdecays
ρ˜=(σ(1) σ(2))ρ∗(σ(1) σ(2)), (8) faster as the value of J is increased. We also perform an
y ⊗ y y ⊗ y 0
analysis of DDSE as a function of time for a fixed value of
where ρ∗ is the conjugate of ρ in the standard basis of two theRTNamplitudeJ = 11.6MHzanddifferentcorrelation
0
qubits. times τ , which is shown in Fig. (4)). The increasing of the
c
The initial state of each DQD is set to = S , then correlationtimehasasimilareffectwhencomparedtothein-
|↑i | i
a π/2 rotation around the x axis is performed during the creasingoftheRTNamplitudeinFig.(3);i.e.,thebiggerthe
preparationtimeτprep,whichputseachqubitinasuperposed correlationtime,thefasterthedecayoftheenvelopefunction
state( + )/√2. Followingtheexperimentaldescription as a function of time. For τ = 9 ns and J = 11.6MHz,
|↑i |↓i c 0
given in ref. [12], we use ∆Bz,1 = ∆Bz,2 = π/(2τprep), thereisagoodmatchingbetweenexperimentalresults(open
J1/2π = 280MHz, and J2/2π = 320MHz. The system circlesinFigs. (3)and(4)andDDSE extractedfromthedy-
dynamicscanbeobtainedbynumericallysolvingtheunitary namicsincludingtheRTN.ResultsshowninFigs.(3)and(4)
trajectoriesdescribedbythesystemHamiltonian(Eq.(1))to- might suggest that it is possible to find different pairs of τ
c
gether with the RTN Hamiltonian (Eq. (2)). We perform an and J thatadjust the experimentaldata, butthis conception
0
average over different unitary trajectories to extract the dy- is misleading because otherpairs of τ and J cannotrepro-
c 0
namicsofthesystemincludingtheRTN.25 ducetheexperimentaldatainwholerangeoftime.
We begin our analysis of the dynamics of the system By means of our description of the decoherence mecha-
throughtheevolutionintimeofthedifferenceofthedescend- nism, we cananalyzethe entanglementdissipativedynamics
ing sorted eigenvalues (DDSE) λ1 λ2 λ3 λ4 of the throughourtheoreticalmodel. Particularly,wefocusontwo
− − −
matrix R = √ρρ˜√ρ, which is equal to the concurrence aspects: the role of the preparation time τ and the two-
p prep
C(ρ) when it assumes positive values. In Fig. (3), we plot qubitcouplingJ . The system is interactingwith the envi-
12
λ1 λ2 λ3 λ4 as function of time, assuming different ronmentduringthepreparationtime,whichaffectstheentan-
− − −
values for the RTN amplitude J0 and for τc = 9 ns. When glementefficiency. Tounderstandtheroleofsuchaphysical
J0 =0,thedynamicsisunitaryandtheconcurrenceoscillates parameter,weanalyseeffectsonthemaximumvalueofcon-
withoutdissipationwithaperiodof280nsafterτprep. Sucha currencecausedbydistinctpreparationtimesτprep. Another
periodiscompletelydefinedbythesystemHamiltonianterm crucial physicalparameter that rules the entanglementis the
J /4σ(1) σ(2). DDSE assume zero values for t τ coupling between each qubit. The two-qubit coupling J ,
12 z z prep 12
⊗ ≤
when J = 0, but the first experimental value of DDSE at that can be increased by controlling the dipole-dipole inter-
0
t = 25 ns is negativeand it must be related to noise effects action, determinesthe time τ for achieving the state with
ent
4
decreased. This behavior is related to the fact that a maxi-
1.0
mally entangled configuration is faster achieved for a larger
J eventhoughtheRTNdisturbstheidealobtainmentofthe
12
0.9 superposedstateduringthepreparationtimeτ .InFig.(5),
prep
onecanseethatforR & 20themaximumvalueofentangle-
)
) 0.8 prep=10 ns mentisapproximatelyconstant.
(
max(C 0.7 pprreepp==2450 nnss ingFluyrethnehramncoerse,thaesmmaaxlliminucmrevaaseluienotfheentvaanlguleemofeJnt1,2assucrapnribse-
observedbythesteepjumpinmax(C(ρ))forasmallvariation
0.6
ofRinFig.(5). Tounderstandfurtherhowthesecharacteris-
tictimesaffectthemaximallyentangledstate,inFig.(6),we
0.5 plotthemaximumvalueofconcurrenceasafunctionofτ
prep
forR = 1, R = 2, R = 5, R = 10, andR = 50. One can
0 10 20 30 40 50
notice in Fig. (6) that the maximum concurrence monotoni-
R
cally decreasesas a functionofτ andthatthe maximum
prep
FIG. 5: Numerical solution for the maximum value of the concur- concurrencerapidlyincreaseswiththeincreasingofR.
renceasafunctionofRandforτ = 10ns,τ = 25ns,and
prep prep
τ = 40 ns, considering the description of thenoise that better For example, by doubling the experimental value of the
prep
fitstheexperimentaldata,i.e.τc =9nsandJ0 =11.6MHz. two-qubit coupling (R = 2), the maximum value of entan-
glement has a growth of 40% for 5 τ 50. These
prep
≤ ≤
resultsaddressthewaysuchphysicalparameterscanbetuned
highestvalue ofentanglement(concurrence). Thistime τent in order to substantially enhance the entanglement between
can be extracted from the term J /4σ(1) σ(2) of Eq. (1) S-T qubits.
12 z z 0
⊗
and is given by J = π/τ . In the experimental results,
12 ent
τexp = 140 ns and the maximum obtained value for entan-
ent
glementisaround0.44.12Inotherwords,bothτ andτ
prep ent
aretwofundamentalcharacteristictimesthatareintrinsically
related to the success of achieving an maximally entangled
state.
IV. CONCLUSION
To illustrate the role of the preparation time and the de-
pendencyonthetwo-qubitcouplingJ in theentanglement
12
dynamics,weplotinFig.(5) themaximumvalueofconcur-
renceasafunctionofR=J /Jexp,whereJexpisthevalue Insummary,weproposedamodelbasedontheRTN,which
12 12 12 mimicsthe1/fαnoise,todescribethedissipativedynamicsof
extractedfromtheexperimentaldata,fordifferentpreparation
twoS-T qubitsintwoDQDs. Byemployingsuchamodel,
timesτ . Asexpected,suchresultsshowanenhancement 0
prep we were able to determine a suitable description of the ex-
oftheentanglementwhenJ isincreasedandwhenτ is
12 prep perimentaldatashowninref.[12]. Moreover,westudiedthe
roleofthepreparationtimeandthetwo-qubitcouplinginthe
dissipativedynamics.Weshowedthatthetwo-qubitcoupling
1.0 playsacrucialroleintheentanglementevolutionandasmall
R=1 increaseinJ canleadtoaconsiderableamplificationofthe
12
0.9 R=2 entanglement. Such results can be used as a reference for
R=5
R=10 further studies in quantum systems where the 1/fα noise is
)) 0.8 R=50 present.
(
C
x( 0.7
a
m
0.6
0.5 V. ACKNOWLEDGEMENTS
10 20 30 40 50
We thank M. Shulman and F. Brito for helpful discus-
(ns)
prep sions. LKC and FFF are grateful to the Brazilian Agencies
FIG. 6: Numerical solution for the maximum value of the concur- FAPESP (grants12/13052-6and 15/05581-7),CNPq (grants
renceasafunction of τprep for R=1, R=2, R=5, R=10, and R=50, 304841/2015-3and474592/2013-8),andCAPESforfinancial
considering the description of the noise that better fits the experi- support. KB and FFF would like to thank the International
mentaldata,i.e.τc =9nsandJ0 =11.6MHz. CentreforTheoreticalPhysics(ICTP)forfinancialsupport.
5
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