Decoherence of Rabi oscillations of electronic spin states in a double quantum dot

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by  Alessandro Romito

Decoherence of Rabi oscillations of electronic spin states in a double quantum dot
Alessandro Romito and Yuval Gefen
Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
(Dated: January 19, 2009)
9 We study the role of charge fluctuations in the decoherence of Rabi oscillations between spin
0 states|↑↓i,|↓↑ioftwoelectronsinadoubledotstructure. Weconsidertheeffectsoffluctuationsin
0 energy and in the quantum state of the system, both in the classical and quantum limit. The role
2 of state fluctuations is shown to be of leading order at sufficiently high temperature, applicable to
actualexperiments. Atlowtemperaturethelowfrequencyenergyfluctuationsaretheonlydominant
n
contribution.
a
J
PACSnumbers:

9
1
I. INTRODUCTION II. THE MODEL

]
l
al The system (Cf. Petta et al.3) is schematically pre-
Thekeystoneofquantuminformationprocessingisthe
h sented in Fig. 1. It consists of a gate confined semi-
coherentdynamics ofthe quantumlogicalbits (qubits)1.
-
s Although such coherent behavior is well established in
e atomic systems, it canbe maintainedonly for veryshort V (a) Energy (b)
m time scales, of the order of few nanoseconds, in charge VR VT VL |Se" |T−" |T0"
t. based solid state based systems. To overcome this prob- E0 !ωs |T+"ǫ
ma lem one may employ the spin degree of fredom of the ǫB ǫX J(ǫ) ǫA
electrons residing in a quantum dot as a qubit2. In fact, BnL BnR |Sg"
- asaconsequenceoftheconfinedgeometry,thecoherence
d
timeofthespinmaybeextendedtobeoftheorderoftens
n
o of microseconds3, primarily restricted by the coupling of
c the nuclear spin environment via the hyperfine interac- ǫ(t) fast adiabatic
[ tion4. Theseresultsmotivatedtheexperimentalprogress ǫA passage (c)
2 incontrollingelectronicspininGaAsgatedquantumdots ǫǫX
v systems3,5. ǫB
4 In a recent experiment3 the use of spin states of two slow 0 τ t
3 adiabatic passage |↑↓$ ρ(τ)
electronsina double dotasthe holderofquantuminfor-
3
3 mation has been investigated. In that configuration the
0 systemisgovernedby(i)the hyperfineinteractionwhich Figure 1: (Color online). (a) The schematics of a double
7 tends to mix singlet and triplet states and (ii) the ex- dot with a nearby QPC; BNL, BNR are the respective nu-
0 changeinteractionwhichtends topreservethe totalspin clear magnetic fields. (b) Energy levels of lowest singlet and
t/ of the electron pair. The interplay between the two ef- triplet states vs. the detuning parameter ǫ (c.f. Ref. 3). (c)
a fectshasbeenstudiedtheoretically6 andanalyzedexper- The time variation of ǫ needed to drive Rabi oscillations be-
m imentally7. In particular Ref. 3 reports Rabi oscillations tween |↑↓i and |↓↑i states (refer to panel (b)). The system
- between spin states driven (electrostatically) by tuning is initially in its ground state |Sg(ǫA)i. ǫ is then varied adi-
nd the exchange energy. Such oscillations (faster than the ianbaittsicalollwyes(t~ǫ˙S≪z =J(00)s2t/aEte0,)etitoheǫr=|↑↓ǫiB,orke|e↑p↓iin.gDthuerinsygsttehmis
o typical spin decoherence time) are mainly hindered by variation the point ǫ = ǫX is crossed at a time faster than
v:c chaInrgtehflisupcatupaetriownes8a,n9,a1l0y.zethe decoherenceeffects inthe ~w/it(hgµ~Bǫ˙m≪axJ{(B0)N2/LE,B0)N;Rap})prtooacahvionigdǫa=traǫBns,itiniosntetaod,|Tth+eiv(abruit-
Xi Rabi oscillations due to charge fluctuations. We con- ation of ǫ is slowed down (~E0ǫ˙ ≪ hT0|HN|Sgi2 ≪ J(0)2)
toguaranteeadiabaticity with respecttothenuclearinterac-
sider both the effects of exchange energy fluctuations
r tion. FollowingthepulsethatinducesRabioscillationsinthe
a and fluctuations of the singlet hybridized state which
interval[0,τ],theadiabaticvariationofǫatt>τ isreversed.
is affected by charge fluctuations as well. In particu- This allows to identify the state of the system at t = τ by
lar we calculate the time dependence of the Rabi os- mapping the states |↑↓i and |↓↑i at ǫ = ǫB to |Sg(ǫA)i and
cillations in the presence of gate voltage and tunneling |T0iat ǫ=ǫA respectively,wherethelattercan bemeasured
amplitude fluctuations, both in the classical (high tem- employing theQPC.
perature, Eq. (6)) and the quantum case (Eq. (13)). We
describe the crossover of the decoherence rate between conducting double quantum dot. Tunnel barriers con-
lowandhightemperatureregimes,whichcanberelevant necteachdottotheadjacentreservoirsallowingdot-lead
in the actual experiments. Classical energy fluctuations tunneling of electrons. The gate voltages, V , V , and
T L
have been analyzed in Ref. 9. V , control the tunnel between the dots, and the dots’
R
2
chargeconfiguration(n ,n ),respectively. Itis possible (S (ǫ= 1) T )/√2 = (for +) or . or
L R g 0
| − i±| i |↑↓i |↑↓i |↑↓i
to measure such a charge configurationusing a quantum ( ): thespininthetwodotsareoppositelyoriented.
|↑↓i −
point contact (QPC) located near one the dots. The di- Hereafter we consider state.
|↑↓i
mensionlessdetuningparameter,ǫ VL VRcontrolsthe In the experiment described in Ref. 3 the detuning
∝ −
differencenL nR. InRef.3thesystemwasoperatedbe- parameter is varied in time to induce Rabi oscillations
−
tween (1,1) and (0,2). In the (0,2) charge configuration between and . The time dependence of the pa-
|↑↓i |↑↓i
(ǫ = ǫA), the antisymmetric nature of the electron wave rameter ǫ used to drive the oscillations is depicted in
function enforces the ground state of the system to be a Fig. 1. The system is prepared in the state (or
|↑↓i
singlet. The excitation energy to the lowest triplet state equivalently ) at ǫ = ǫ (cf. Fig. 1). Subsequently,
B
|↓↑i
is experimentallyestimatedto be &400µeV, largerthan a gate voltage pulse at t = 0 modifies ǫ to a point
the charging energy of the single (1,1) state, E0. In the where T0 HN Sg J(ǫ), thus inducing oscillations be-
h | | i≪
(1,1)configuration(ǫ=ǫB)thesingletandtripletstates tween and over a time interval τ. The follow-
|↑↓i |↓↑i
areinsteadpracticallydegenerate. Thelowenergystates ing manipulation of ǫ (cf. Fig. 1) allows to relate the
of the system are two singlets, SL , SR , correspond- measured conductance of the QPC with the probabil-
| i | i
ing to (1,1) and (0,2) respectively, and the three triplet ity of finding the system in right after the pulse,
|↑↓i
rsteaspteesctfiovrelychwaritghe tchoenfisgpuinratcioomnp(o1n,e1n)t, |0T,01i,,|T1+ini,t|hTe−iˆz, P(τ)=|h↑↓|exp(−iHˆ0τ)|↑↓i|2.
−
direction perpendicular to the dots’ plane. An exter-
nal magnetic field B = Bzˆ is applied to split the states
III. CLASSICAL NOISE
T , T bythe Zeemanenergy∆ =gµ B 2.5µeV.
+ − z B
| i | i ∼
We neglect the role of these states (see below) and we
write the Hamiltonian of the system in the S = 0 sub- .
z
space as The Rabi oscillations are obtained by tuning the en-
ergy difference J(ǫ) between S and T , which results
g 0
Hˆ = E ǫ(T T + S S S S ) in the different charge of the|triiplet a|ndihybridized sin-
0 0 0 0 L L R R
| ih | | ih |−| ih |
glet. Rabi oscillations will therefore be extremely sensi-
+E (λ S S +h.c.). (1)
0 s L R
| ih | tive to an environment coupled to charge as opposed to
We introduced a tunneling amplitude between the two the nuclear spin environment. Decoherence effects will
singlet states, λ , which is the only possible tunneling originate both from fluctuations of the (exchange) en-
s
matrix element assuming conservation of total spin. It ergy J(ǫ), analyzed in Ref. 9, and fluctuations of the
canbe chosenrealand positive. This leads to hybridiza- hybridizedsingletstate Sg(ǫ) . We analyzethe Rabios-
| i
tion of the ground and excited states in the singlet sub- cillations taking into account fluctuations of VL, VR and
space, VT. The respectivegatesarecontrolledindependently of
each other, hence it is natural to assume that their fluc-
S (ǫ) = sinθ S +cosθ S , tuations are independent. In principle it is possible to
g R L
| i − | i | i
determineacorrelationmatrixforthefluctuationsofthe
S (ǫ) = cosθ S +sinθ S , (2)
e R L
| i | i | i parameters in the Hamiltonian, ǫ and λ , by consider-
s
thence ing a specific potential form for the double dot. Instead
we assumethat V affects only the tunneling matrix ele-
~ω T
Hˆ0 = s(Se(ǫ) Se(ǫ) Se(ǫ) Se(ǫ))+E0ǫ T0 T0 , ments, λs, which is reasonablefor weak tunneling. Then
2 | ih |−| ih | | ih | ǫ is affected only by the fluctuations of V V . The
L R
−
Hamiltonian is
with ~ω = 2E ǫ2+λ2, tanθ = ( ǫ2+λ2 + ǫ)/λ .
s 0 s s s
The energylevelspas a function ofǫ arepplotted in Fig. 1. Hˆ = Hˆ0( ǫ ǫ+ξc(t), λs λs+ξλ(t) )
Transitionsbetweensingletandtripletstatesaremade { → → }
possible due to the hyperfine interaction of the electrons = Hˆ0+E0(Vˆcξc(t)+Vˆλξλ(t)), (3)
in the dot with nuclear spins, which can be written in
terms of the effective nuclear magnetic field in each of where ξi(t) are Gaussian distributed with ξi(t) = 0,
tchael ddyonta,mHiNcal=scgaµleBo(Bf tNhLe·nSucLle+arBeNnvRir·oSnRm)e.ntTihseotfytphie- Vhˆξi(=t)ξTj(t′)iT=+2~S2Γi/ES02δ(t−St′)δiS,j fo,rVˆi,j=∈hS{c,isS} an+d
c 0 0 L L R R λ L R
| ih | | ih |−| ih | | ih |
order of tens of microseconds, and therefore it acts as if S S . Theassumptionofwhitenoiserendersthedy-
R L
| ih |
it is a frozen external field over the duration of the ex- namics Markovian, leading to an exact master equation
periment. With B ,B 5mT B, the hyperfine for the density matrix of the double dot,
NL NR ∼ ≪
interaction is effective only at ǫ ǫ , where it can mix
X
∼
t|Shgeiloawnedr|eTn+erig,yanstdataersouSndaǫn∼d ǫTB :≪H1 w=hgerµe i(tBmixes ∂tρ=−i/~[Hˆ0,ρ]− Γj(Vˆj2ρ−2VˆjρVˆj+ρVˆJ2). (4)
B ) z T S +h.c|.,gwihile t|he0ienerNgy diffeBrencNeLbe−- j∈X{c,λ}
NR · | 0ih g|
tween them is J(ǫ) = E (ǫ+ ǫ2+λ2) T H S . This differential equation is solved with the initial
0 s ≪ h 0| N| gi
The ground state of the system at ǫ = ǫ is therefore condition ρ = by neglecting terms of order
B
p |↑↓ih↑↓|
3
(J(ǫ)/(~ω )=sin2θ). Under this assumption the den- Here the operators Vˆ acting on the system are the
s c(λ)
O
sity matrix at any time t>0 can be written as same as in Eq. (3) while the classical fluctuators are re-
placedby the operatorsA = a (b †+b )
1 c(λ) i c(λ)i c(λ)i c(λ)i
ρ(t) = 2[|T0ihT0|+Y(t)|SgihSg|+(1−Y(t))|SeihSe| of bosonic baths, Hbathc(λ) = iP~ωibc(λ)†ibc(λ)i. The ef-
fect of the reservoirs on the dynamics of the electrons in
+(X(t) S T +h.c.)] , (5)
g 0 P
| ih | thedoubledotisentirelycharacterizedbytheirsymmet-
in terms of the two functions Y(t) and X(t) describ- ric and antisymmetric spectral functions
ing the evolution of the state populations and co-
herency respectively. The explicit expressions, X(t) =
e(iJ(ǫ)/~−γ2)t, Y(t) = (1 + e−γ1t)/2, allow to deter- Sc±(s)(ω)=1/(2π) Rdteiωth[Ac(λ)(t),Ac(λ)(0)]±i. (10)
mine the surviving probability P(t) = ρ(t) = Z
h↑↓| |↑↓i
1/4[1+Y(t)+2 e X(t) ],
ℜ { } Wetakebothbathstobeatequilibriumatthesametem-
P(τ)= 1[3+e−γ1τ +4cos(J(ǫ)τ/~)e−γ2τ] , (6) perature k T = 1/β. Their bosonic nature guarantees
8 B
γ =4(Γ sin2(2θ)+Γ cos2(2θ)), that S+(ω) = coth(β~ω/2)S−(ω), where S−(ω) is tem-
γ1 =4Γ csin2(θ)+Γ .λ (7) perature independent, Sc−(λ)(ω) = iac(λ)2i[δ(ω+ωi)−
2 c λ δ(ω ω )].
- classical- − i P
We assumethatthe bathisweaklycoupledtothe sys-
tem, and we determine the evolution of the density ma-
The measured probability consists of damped oscil-
trix to second order in a /ω . Following the Bloch-
lations around a mean that approaches an asymptotic c(λ)i s
value. Thedecayofthe oscillationsisrelatedonly tothe Redfield approximation11, we introduce a (short) bath
decoherence of ρ(t), while the relaxation of the popula- correlation time, τ¯, characterizing the typical time scale
tions, encoded in Y(t), determines the slow time varia- at which any correlation of the system and the reservoir
tion of the mean. The dependence of the decay rates, disappears. The time evolution of the reduced density
γ , γ , on ǫ is quite different from what we would have matrix of the system, ρ, coarse grained at time scales
1 2
obtained by simply accounting for fluctuation of J(ǫ), in ∆t τ¯,isMarkovian. Itisdeterminedbythefirstorder
≫
which case linear differential equation
γ =0,γ =4Γ sin4θ+Γ sin2(2θ). (8)
1 2 c λ ∂ ρ = iω ρ ρ e−i(ωa,b−ωc,d), (11)
- classical energy fluctuations only - t a,b − a,b a,b− Ra,b,c,d c,d
c,d
X
This means that the effects of fluctuations of the state
Sg cannot be neglected with respect to (exchange) en- written in the basis of eigenstates of Hˆ where ~ω =
| i 0 a,b
ergyfluctuations. Wealsonotethattheasymptoticvalue
aHˆ a bHˆ b and istheBloch-Redfieldten-
of the probability is P(τ) = 3/8, corresponding to a h | 0| i−h | 0| i Ra,b,c,d
sor:
steady state density matrix with equally populated sin-
glet states (S , S ). This is a signature of the high
g e
temperature|(kiT| i~ω )limit,andisrelatedtotheas-
sumption of claBssic≫al gatse voltage fluctuations. By con- Ra,b,c,d = " δb,dha|Vj|nihn|Vj|cigj(ωc,n)
trast, at low temperature we expect that only the lower jX=c,λ Xn
singlet level is populated. In the experiment oscillations aVj c dVj b gj(ωc,a) aVj c dVj b gj(ωb,d)
−h | | ih | | i −h | | ih | | i
with J(ǫ) rangingfrom tenths to few µeV have been ob-
cseornvdeidtioant kkBBTT ∼∼1~0ωµseV≫3. JW(ǫe)thisertehfaotreoebxtpaiencetdtheaxtpethrie- +Xn δa,chd|Vj|nihn|Vj|bigj(ωn,d)# , (12)
mentally, in which case quantum state fluctuations are
important and we explore their dependence on the tem- withg(ω)=1/4[S+(ω)+S−(ω)] i Pdx/(2π)(S+(x)+
perature. Sj−(x))/(x − ω). jAt timjes t −& R~R/J(ǫ) ≫ ~/ωjs, ne-
glecting terms of order (J/ω ), the sum in Eq. (11)
s
involves only terms suchOthat ~(ω ω ) 0. Ex-
IV. QUANTUM NOISE a,b c,d
− ≪
plicitly the only relevant entries of the Bloch-Redfield
tensor are: = , =
. R,Sg,Sg,Sg,Sg −R=Se,S∗e,Sg,Sg RS.e,SIte,Sfoe,lSloews
In order to extend our analysis beyond the high tem- −RSg,Sg,Se,Se RSg,TL,Sg,TL RTL,Sg,TLSg
that state population and coherency evolve indepen-
perature limit we need to consider the quantum nature
dently of each other and therefore the density matrix
ofthegatevoltagefluctuations. Wethereforemodifyour
has the same form presented in Eq. (5), with different
classical model (Eq. (3)):
functions X(t) and Y(t).
Hˆ =Hˆ +Vˆ A +Vˆ A +H +H . (9) Oncetheexpressionofρ(t)isknown,itcanbeusedto
0 c c λ λ bathc bathλ
4
calculate the survival probability, (a) 4 1 2 3 4 3(b) 1 2 3 4
Q Q
3 3
1 ~βω 2 2
P (τ)= 3+e−γ1τ +tanh s (1 e−γ1τ) 2 2
qm 1 1
8 2 − 1 1
(cid:18) (cid:19)
(cid:2)+4cos((J(ǫ¯)/~+∆ )τ)e−γ2τ , (13) 1 2 3 40 1 2 3 40
J J(µsec) J(µsec)
γ1 =2π[sin2(2θ)Sc+(ωs)+cos2(2θ)Sλ+(ωs)](cid:3)/~2, (14) Q(c3) 1 2 3 4
γ2 =π sin2(2θ)(Sc+(ωs)−Sc−(ωs))+cos2(2θ) 1 2 30 mK 2
((cid:2)Sλ+(ωs)−Sλ−(ωs))+(cos(2θ)−1)2Sc+(0) Pq0m(.τ)8 1715 mK 135 mK 1
+sin2(2θ)Sλ+(0) /(2~2), (15) 0.6 290 mK 1 2 J3(µsec)40
andPd~ω2/∆(ωJ =+[siωn2)(2(θsi)n2R((cid:3)2Pθ)d(ωS/+ω(ω(S)λ−(+ω) −S−S(c−ω()ω))) ++ 0.4 hilgohw T T
R s R c c 0.2
cos2(2θ)(S+(ω) + S−(ω)))]/2. ∆ is a shift in the 1 2 3 4 5 6
R λ λ J τ(nsec)
frequency of the Rabi oscillations that can be neglected
compared with J(ǫ)/~, consistent with our second order
perturbation expansion. The Bloch-Redfield approxima- Figure 2: (color online) Surviving probability versus time τ
tion employed implies that Eq. (13) is valid in the limit (Eq.(13))atJ(ǫ)=3.9µeV(blacksolidline),J(ǫ)=2.4µeV
γ ω ,1/τ¯. We note that, unlike γ , γ depends (orange dashed line) and J(ǫ) = 2µeV (cyan dotted line).
1(2) s 2 1
≪
only on the symmetric (classical) correlators, S+ (ω). For all the curves J(0) = 5 µeV, T = 135 meV as estimated
γ consists of contributions from the bath corrce(λla)tion from the experiment3, and αc = αλ = 7×10−3 obtained by
2 fitting with the experimental curves. The high and low tem-
function at frequency ω (which describes the relaxation
s perature asymptotic values are depicted (dotted gray lines).
process between the two singlet eigenstates and the Insets: Q = J/(2π~γ2) ∝ (numberof visible oscillations) as
corresponding contribution to the dephasing), and from a function of J(ǫ) for threepossible scenarios: (a) αc =0.02,
the zero frequency correlation function (corresponding αλ =0; (b) αc =αλ =7×10−3; (c) αc =0, αλ =7×10−3.
to the contribution of pure dephasing)12. Differentcurvescorrespond todifferenttemperaturesaccord-
In the high temperature limit, β~ω 0, S− (ω ) ing to theirlabels in theinset (c); in all plots J(0)=5µeV.
s → c(λ) s
is negligibly small as compared with S+ (ω ), we thus
c(λ) s
expect a classical result. If we furthermore assume a transitions are exponentially suppressed by e−βωs. Note
Ohmic bath, i.e. Sc−(λ)(ω)= αc(λ)~2ω, such that at high howeverthatexperimentallytheregimeβ~ωs .1canbe
temperatureSc+(λ)(ω)∼Sc+(λ)(0)for0<ω <ωs,Eq.(13) reached, hence the effect of fluctuations of the state |Sgi
reduces to Eq. (6) with can be of interest. In particular this fluctuations affect
thesteadystatevalueofthesurvivalprobabilitywhichis
γ1 =2π/~(Sc+(0)sin2(2θ)+Sλ+(0)cos2(2θ)), (16) (acffu.nFctigio.n2)ofaβnωdsc,aPnqmb(eτd→ire∞ctl)y=ob1s/e8r[v3e+dtiannhth(βe~eωxsp/e2r)i-]
γ =π/(2~)(4S+(0)sin2(θ)S+(0)).
2 c λ ments.
- quantal high T - A comparison with the experimental results of Ref. 3
is obtained assuming Ohmic baths with spectral den-
We in fact recover the classical result for white sities S− (ω) = α ~2ω. These properly describe
noise fluctuations (Eq. (7)) by identifying Γ = c(λ) c(λ)
c(λ) chargefluctuationsduetotheexternalcircuitcontrolling
π/(2~2)S+ (ω ). At low temperature, β~ω 1,
c(λ) s s ≫ the gate voltages. An analysis of the possible scenarios
Sc+(λ)(ωs) ≃ Sc−(λ)(ωs) and the quantum nature of the αc Rαλ,showsthattheexperimentalfactthatthenum-
bath becomes important. The rate γ disappears from ber of visible oscillations as function of J(ǫ) is constant
1
the expression for P(t) and we also note the finite fre- is correctly reproducedfor αc .αλ (cf. insets in Fig.2).
quency contribution to γ2 vanishes. Only the zero fre- The time dependence of Pqm(τ) is depicted in Fig. 2 for
quency component of the spectral density of the bath different values of J(ǫ). We obtain a fit with the experi-
(which is responsible for pure dephasing) survives, mental data for αc =αλ =7 10−3, consistent with the
×
strength of electromagnetic environment in other solid
γ =π/(2~2)(4S+(0)sin4θ+S+(0)sin2(2θ)). (17) state systems13.
2 c λ
- quantal low T -
In this limit the dependence of γ on θ can be explained V. EXTENDED MODEL
2
entirelyintermsofclassicalfluctuationsoftheoscillation
frequency, J(ǫ) (cf. Eq. (8)). Indeed, the effects of fluc- The previous analysis is now extended to include the
tuations of the state S do involve transitions between lowest energy triplet state in the charge configuration
g
the latter state and t|heisinglet excited state, yet these (0,2), T′ . This might be necessary if the energy of
| 0i
5
hen|Tn.ec0ew′.ir)giH+ysacEtmoo0mi(lttδphoae−nriaatλbrnsilpeǫr)leew|atTid0t′sshit,haTttHhˆe0′e0|′,inC=wohtuhHelˆero0em(+0δb,2Eies)n0et(crhλogetny|fiTeEgx0uic0irh.taTat0Tt′i|ioohn+ne. smpkmBroeTadsleilf≪nyicne~Peωoq−mftβ,(a~flτωu)“t.c/st2wu.ReaeNettimoopntaesorkiitnnahtbat”lht(y,θeeien=vneetnrϕhgi)ays1t4Jc,l(aoǫwws)eh,(ticleefem.ve∂FpnǫeiJgri.an1=tu(tcrh)0ee),,
Owing to electron tunneling, λ , the two triplet states, fluctuations of the tunneling rates are important due to
t
T and T′ , do hybridize (cf. Fig. 1(c))14. The en- the difference between electron tunneling in the triplet
|erg0iyspect|ru0mi ofsingletandtripletstatesinthesubspace and singlet states.
S = 0 is depicted in Fig. 1(c). The energies of the hy-
z
bridized triplet states, T (ǫ) = sinϕ T′ +cosϕ T
and T (ǫ) = cosϕ T′|+gsiniϕ T− , are|E0iδ ~ω /|20=i
E δ | eE i (ǫ δ)2|+0iλ2] respe|c0tiively, w0ith∓tantϕ = VI. CONCLUSIONS
0 ∓ 0 − t
( (ǫ δ)2+λ2 +ǫ δ)/λ . The Hamiltonian for this
− p t − t
modelincludes fourparameters,ǫ,δ,λ ,λ whichdepend We have presentedhere a simple model describing the
s t
p
on three fluctuating gate voltages only, V ,V ,V . In effectofchargefluctuations onRabioscillationsbetween
L R T
principle it is possible to determine a correlation ma- spinstates and ofelectronsinadoubledot. We
|↑↓i |↓↑i
trix for the fluctuations of the parameters in the Hamil- haveaccountedfordecoherenceeffectsduetobothenergy
tonian by considering a specific potential form for the and quantum state fluctuations, by including the quan-
double dot. Instead we assume that V affects only tumeffectsofafluctuatingenvironmentwithintheBorn-
T
the tunneling matrix elements, which is reasonable for Markov approximation —Eqs.(13–15). We have shown
weak tunneling. The fluctuations of λ λ + ξ (t) that not only in the high temperature limit does the
t t t
→
and λ λ + ξ (t) will then depend on the same result reproduce that of classical fluctuations (compare
s s s
→
bath and will therefore be correlated, ξ (t) = fξ (t) Eq.(16) to Eq. (7)), but also the low temperature result
t s
with f = (∂λ /∂V )(∂V /∂λ ). At the same time the has a classical interpretation in terms of energy fluctu-
t T T s
gate voltage difference V V will affects only ǫ. The ations only (not state fluctuations, compare Eq. (18) to
L R
density matrix now evolv−es as (cf. Eq. (5)) ρ′(t) = Eq. (8)). In fact the role of the state fluctuations is sig-
ρ(t)+1/2W(t) T T +1/2(1 W(t)) T T ,andthe nificantatatemperaturethatexceedsthe singletexcita-
g g e e
| ih | − | ih |
probability of finding the system in the at time τ is tion energy, a regime which is accessible experimentally.
P′ (τ) = P (τ) 1/8(1 tanh(β~ω /|↑2↓)i)(1 e−γ3τ), Notethatathightemperaturethe“classicallimit”refers
qm qm − − t −
with γ = 2π/~2[sin2(2ϕ)S+(ω ) + f2cos2(2ϕ)S+(ω )] toclassicalenvironmentinducedfluctuations. The latter
3 c t λ t
andγ replacedbyγ′ =γ /4(1 tanh(β~ω ))+γ /4(1 can still cause fluctuations in the quantum state of the
tanh(β2~ω )) + π/(22~2)[(1cos2(2−θ) cos2(s2ϕ))S+3(0) +− system. Atlowtemperaturestatefluctuationsarefrozen
(sin2(2θ) t fsin2(2ϕ))S+(0)]. The−physical mecchanism out.
− λ
that induces decoherence in P(τ) is the same described WearegratefultoAmirYacobyandSandraFolettifor
in the previous paragraphs. Similarly to the decoher- useful discussions. We acknowledge the support of U.S.-
ence in the singlet subspace, fluctuations in T do in- IsraelBSF,theISFoftheIsraelAcademyofSciencesand
g
| i
volve now the exited triplet state T , an effect that is DFG project SPP 1285.
e
| i
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14 M. Stopa and C. M. Marcus (2006), cond-mat/0604008. tify a “sweet point” where ∂ǫJ = 0 at which fluctuations
Such a model can better fit the experimentally observed in ǫare less effective.
behavior of J(ǫ) around ǫ∼0. Furthermore one can iden-