Double dot chain as a macroscopic quantum bit

0.17 MB·English

by  F. de Pasquale

Double dot chain as a macroscopic quantum bit
Ferdinando de Pasquale,1,2,∗ Gian Luca Giorgi,1,2 and Simone Paganelli2,3
1INFM Center for Statistical Mechanics and Complexity
2Dipartimento di Fisica, Universit`a di Roma La Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy
3Dipartimento di Fisica, Universit`a di Bologna, Via Irnerio 46, I-40126, Bologna, Italy
5
0 We consider an array of N quantum dot pairs interacting via Coulomb interaction between ad-
0 jacent dots and hopping inside each pair. We show that at the first order in the ratio of hopping
2 and interaction amplitudes, the array maps in an effective two level system with energy separation
n becomingexponentiallysmallinthemacroscopic (large N)limit. Decoherenceat zerotemperature
a is studied in the limit of weak coupling with phonons. In this case the macroscopic limit is robust
J with respect to decoherence. Some possible applications in quantum information processing are
6 discussed.
2
PACSnumbers: 03.67.Mn,03.65.Yz,05.50.+q,73.43.Nq

]
l
l
a I. INTRODUCTION step is to show that in the long time limit the array be-
h havesasatwolevelsystemwithenergyseparationwhich
- vanishes for large N. The model is exactly equivalent
s In recent years a lot of attention has been devoted to
e to a one-dimensional antiferromagnetic Ising model in
the existence of quantum superposition states in macro-
m a transverse field. The system is characterized by two
scopic systems. The first suggestion to understand this
. phenomenonisduetoSchr¨odinger[1]whointroducedthe equivalent charge configurations and in the macroscopic
t
a paradoxofthecatinquantumsuperpositionbetweenlive limit the ordering at zero temperature implies a sepa-
m ration of phase space in two regions around each of the
anddeath,stronglystressingthedifferentbehaviorofthe
- quantumworldwithrespecttothehumanexperience. It degenerate configurations. However,for a finite size sys-
d tem, hopping induces oscillations between the two con-
is commonly accepted that quantum behavior vanishes
n
figurations. We associate to this behavior a macroscopic
o as the system size increases. Remarkable exceptions are
quantum bit. The antiferromagnetic Ising model in a
c quantum systems which undergo a phase transition, as
[ superconductors and superfluids. A quantum superpo- transverse field has been studied since the pioneering
sition of mesoscopic states has been observed in SQUID work of Bethe [18, 19, 20], and has received recently a
3
renewed attention as a model for quantum computation
v devices [2]and seems to be a promisingtool for the real-
[21, 22, 23]. We found convenient to introduce a sim-
2 ization of a quantum computer.
1 Quantummacroscopicstatesareexpectedtoberobust pleapproximationwhichmakestransparenthowthe two
4 withrespecttodecoherenceandthusidealcandidatesfor level behavior appears asymptotically.
5 quantuminformationstorage. Moreover,itiswellknown The study of decoherence in such a system is analo-
0 that a quantum system which undergoes a phase transi- goustodecoherenceinaquantumregister[24]. Weshow
4
tion lives in one of a particular set of states, for a time that, atleast inthe weak coupling andzero temperature
0
/ whichbecomesinfinitelylargeinthelimitoflargesystem limit and for a three-dimensional environment, the sys-
at size. In particular, if the ground state is twofold degen- tem exhibits a robustness growing with the size of the
m erate, one can associate these states to a macroscopic array.
quantum bit. The availability of macroscopic quantum Decoherence with respect to phonons of a single two
-
d bits is relevant for quantum information processing, as level system has been studied with various methods (see
n shown, for instance, in Ref. [3], in the case of spin clus- reviews by Leggett et al. [16] and Weiss [17]). We found
o
ters. Hereanewapplicationinteleportationprocessesis however the resolvent method [25, 26], already intro-
c
: also shown. duced in the discussion of electron-phonon interaction
v Decoherence of a single qubit has been extensively problems [27], convenient to obtain results at zero tem-
Xi studied [4, 5, 6, 7, 8, 9, 10]. One of the most relevant perature in the double dot chain.
causesofdecoherenceisthecouplingwithabosonicbath The paper is organized as follows. In section II we
r
a [11, 12, 13, 14, 15, 16, 17] whose effects are relevant also introduce the model of the double dot chain and its in-
at zero temperature. teraction with a phonon bath. The introduction of the
In the present paper we investigate the coherence of resolventmethodtodiscussdecoherencewillbetheargu-
an array of N double quantum dots coupled through mentofsectionIII. InsectionIVweshallapplythesame
Coulomb interaction in order to show that such system method to show that the double dot chain, in the limit
is a suitable candidate as a macroscopic qubit. The first w/U <<1,beingwthehoppingamplitudebetweendots
inside eachpair andU the Coulombinteractionbetween
different pairs, behaves as an effective two level system
withenergyseparationdecreasingexponentiallywithN.
∗Electronicaddress: [email protected] In section V we study decoherence in our system in the
2
U
approximation introduced above. Finally, section VI is H = w σx+ σzσz +1 , (4)
devoted to conclusions. In appendix we review, inside S − l 2 l l+1
l l
the present approximation,decoherence effects for a sin- X X(cid:0) (cid:1)
H = ω a†a , (5)
gle dot pair. B q q q
q
X
H = g n eiqcosθl a† +a , (6)
SB q l q −q
II. THE MODEL
q,l
X (cid:0) (cid:1)
In a previous work [28] we proposed an array of few where a mapping between the spin states and
|↑i |↓i
coupled quantum dot pairs as a channel for teleporta- andthechargestates 1,0 and 0,1 hasbeenperformed
tion. We want here to show the extension of the model and nl = (σlz +1)/2.|Weiindic|ateiwith θ the angle be-
toacaseofN pairsanddiscusstherobustnessofthesys- tween the phonon mode q and the the dot chain direc-
tem with respect to decoherence due to interaction with tion. This notation is useful for describing a generic d-
an external phonon bath at zero temperature. We ex- dimensionalenvironmentcoupledwithaone-dimensional
pect that the extensive character of the interaction will system. The constant gq represents the coupling of the
increasedecoherencewhilethemacroscopicnatureofthe dot charge with the mode q. The explicit mathematical
firsttwoenergystates willenhance the robustnessofthe expressionforgq dependsonthespecificconfigurationof
system. It will be shown that the latter feature prevails. the system and the type of interaction. In Ref. [29] the
Neglectingspineffects,thedoubledotarrayischaracter- explicit form of gq in some remarkable case is given.
ized by the Hamiltonian
N−1 2 N III. RESOLVENT METHOD FOR WEAK
HS =U nl,αnl+1,α−w c†l,1cl,2+h.c. , (1) COUPLING DECOHERENCE
Xl=1 αX=1 Xl=1(cid:16) (cid:17)
where c† creates an electron on the l (th) dot on the α Decoherence at zero temperature is studied using the
l,α
resolvent method. At the initial time t = 0 system and
(th) row of the array and n =c† c .
l,α l,α l,α bath are decoupled: Ξ(t=0) = S 0 where 0 is
To extend the teleportation scheme described in [28], | i | i⊗| i | i
the vacuum phonon state.
we introduce an initial superposition of two spin config-
The time evolution of the state Ξ(t) =
urations of zero potential energy of N pairs: | i
exp( iHt) Ξ(t=0) is studied in terms of the
− | i
S =α Φ +β Ψ , (2) complex Laplace transform defined as
| i | i | i
∞
where |Φi = |↓,↑,↓,↑,....↑i and |Ψi = |↑,↓,↑,↓,....↓i. Ξ(ω) =ilim eiωt−δt Ξ(t) dt. (7)
LetusconsideraninitialsystemofN−1doublequantum | i δ→0Z0 | i
dots with hopping inside any pair and without Coulomb
The resolvent method allows to write
interaction. The ground state of this system is repre-
sented by the tensor product of ( + )/√2 for each 1
|↑i |↓i Ξ(ω) = Ξ(t=0) . (8)
pair. By an adiabatic switching of electrostatic repul-
| i ω H | i
sion between adjacent pairs, the system is driven in its −
new ground state, which, for w/U <<1, is well approx- Using the identity
imated by (Φ + Ψ )/√2. Φ (Ψ )
N−1 N−1 N−1 N−1 1 1 1 1
| i | i | i | i
is the same state than Φ (Ψ ), defined on N 1 = + HI (9)
sites. Thestate S isobta|inied|conisideringanextrad−ou- ω−H ω−H0 ω−H0 ω−H
| i
ble dot (as usual called Alice) in a superposition state andperformingaprojectiononthevacuumphononstate,
α + β and its interaction with the first pair of we define a new system state Φ (ω) = 0Ξ(ω) that
|↑i |↓i | S i h | i
(ΦN−1 + ΨN−1 ). If the interaction is adiabatically obeys to the evolution equation
| i | i
switched on again, then the system is driven in a state
1
close to S . A proper manipulation of system param-
Φ (ω) = Φ (t=0) +
eters per|miits to transfer the information, i.e. α and β, | S i ω HS | S i
−
encodedpreviouslybyAlice,tothelastdoubledot(Bob) 1 1
0 H Ξ(t=0) . (10)
[28]. h |ω H SBω H | i
S
− −
The model described above is suitable to be rep-
Here the bath ground state energy is set to zero and
resented by a spin Hamiltonian through the mapping
σz = (n n ) and σx = (c† c +h.c.). This pic- H0 =HS +HB and HI =HSB.
i l,1 − l,2 l l,1 l,2 In the weak coupling limit only corrections to the
tureisusefultostudythedecoherenceeffectsinducedby
imaginary part of Φ (ω) will be taken into account.
the interaction with a phonon bath. In the spin repre- | S i
We first perform an iteration inside Eq. 10 replacing
sentation the overallHamiltonian becomes
(ω H)−1 with the right hand side of Eq. 9, and then
−
H = H +H +H , (3) introduce a complete set of intermediate phonon states:
S B SB
3
1 1 1
Φ (ω) = Φ (t=0) + 0 H Ξ(t=0)
S S SB
| i ω H | i h |ω H ω H H | i
S S S B
− − − −
1 1 1
+ 0 H H k k Ξ(t=0) . (11)
SB SB
h |ω H ω H H | ih |ω H | i
S S B
k − − − −
X
Inaperturbativeapproach,terms involvingpowersofg where
q
are small and can be neglected, unless self-energy con-
tributesappear. Inthelattercase,anotnegligibleimag- 1
inary part can arise performing the sum over q in the G(HS)= 0 HSB HSB 0 (13)
h | ω H H | i
S B
continuous limit. − −
The contribution involving self-energy in the sum cor-
responds to k = 0, since 0 (ω H)−1 Ξ(t=0) is ex- is the self-energy operator acting on the system sub-
actly Φ (ω) . Then, all ohth|er l−inear an|d quadratiic con- space. The right term of Eq.12 describes the evolu-
S
tribut|ions iniH will be neglected. Hence, Eq. 11 be- tion of the macroscopic state isolated from phonons. As
SB
comes we will show in section IV, in the limit of w/U 1,
≪
the macroscopic dot chain behaves as a two level sys-
1 1 tem oscillating between the H asymptotic eigenstates
1 G(H ) Φ (ω) = Φ (t=0) , S
(cid:18) − ω−HS S (cid:19)| S i ω−HS | S i |±i= 2−1/2(|Φi±|Ψi) with energies E±. So, Eq.12 be-
(12) comes
1 1 1
1 G(H ) Φ (ω) = + +Φ (t=0) + Φ (t=0) . (14)
S S S S
− ω H | i ω E | ih | i ω E |−ih−| i
(cid:18) − S (cid:19) − + − −
Noting that the operator G(H ) maps the subspace is calculted assuming first G++ = 0, obtaining for the
S
spanned by into itself, it is possible to reduce Eq.14 pole ω = E , and then substituting this value inside
+
in terms of t|w±oicoupled equations: G++, which depends on ω. After, the principal value of
G++ will be ignored, and only the imaginary part will
ω E+ G++ +ΦS(ω) G+− ΦS(ω) = matter. We compared, in appendix, the results of our
− − h | i− h−| i
+Φ (t=0) , (15) approximation,withthoseknownforsinglequantumdot
(cid:0) (cid:1) h | S i
pairs.
ω E G−− Φ (ω) G−+ +Φ (ω) =
− S S
− − h−| i− h | i
Φ (t=0) , (16)
(cid:0) (cid:1) h−| S i IV. DOUBLE DOT ARRAY EVOLUTION
where G±± = G .
h±| |±i
To the leading order in the system-bath coupling, we As first step we calculate the evolution of the system
obtain when it is decoupled from the bath. We find convenient
to explicitly solvethe evolutionfromaninitial state cor-
1
h+|ΦS(ω)i = ω−E+−G++ h+|ΦS(t=0)i, (17) r2e.sponding respectively to |Φi or |Ψi introduced in Eq.
1
Φ (ω) = Φ (t=0) .(18) We distinguish in the system Hamiltonian the hop-
h−| S i ω E− G−− h−| S i ping term H = w σx from the potential energy
− − I − l l
The solution in the time domain is obtained assuming H0 = U/2 l σlzσlz+1P+1 . This is the antiferromag-
first the correction introduced by the matrix elements netic version of the well known one-dimensional Ising
(cid:2)P (cid:0) (cid:1)(cid:3)
of G as negligible, and then calculating the latter in in Model in a transverse field [31]. It is worth noting that
ω =E or ω =E . the absence of periodic boundary conditions implies a
+ −
For instance, the integral relaxationmechanism of an initially orderedstate where
a single domain wall propagates between the two end
e−iωt points of the array. This feature makes a difference in
dω
ω E G++ the excitation spectrum which is relevant for an arrayof
ZC − +−
4
finite size. greater than U.
Applying H on Φ(t=0) the system is driven in a In each step of a repeated application of H it is pos-
I | i I
new configuration labeled as Φ (t=0) . The action of sible to go towards new configurations or to come back.
1
| i
H generates a sum of states each of them differentiates Then, for n>0, we write
I
from Φ(t=0) due to one spin flip in a different place
| i
along the array. Here it is important to note that flips H Φ (t=0) = w[Φ (t=0) + Φ (t=0) ].
I n n−1 n+1
| i − | i | i
on the first and the last qubit put the system in a state (19)
with Coulomb energy U, while all intermediate transi- After N steps the system reaches Ψ and after 2N
| i
tions lead to a state with a 2U electrostatic energy. In steps it comes back to the initial configuration. Defin-
thelimitofU largewithrespecttow,weshallneglectall ing Φ = Ψ and Φ = Φ = Φ , and taking into
N 0 2N
| i | i | i | i | i
configurations involving intermediate states with energy account the time evolution we obtain
(ω U) Φ (ω) = Φ (t=0) w[Φ (ω) + Φ (ω) ] U(δ +δ ) Φ (ω) . (20)
n n n−1 n+1 n,0 n,N n
− | i | i− | i | i − | i
The system is solved by means of the discrete Fourier The asymptotic behavior is determined by values of ω
transform defined as close to zero. Then a(ω) << 1 and the denominator of
B (ω) reads as geometric series:
n
2N−1
1
Φ (ω) = Φ (ω) eink,
(cid:12)(cid:12)(cid:12)eΦk (ω)E = √21N 2nXN=−01|Φn(ω)i e−ink. (21) Bn(ω)= 21N ω−UU 2XqN=−01e−inNπqXl∞=0al(ω)cosl Nπ q,
| n i √2N k (27)
Xk=0 (cid:12) E or
(cid:12)(cid:12)e
As a consequence of periodicity conditions, k= 2πn be-
2N 1 U 2N−1 ∞ l l a(ω) l
ing n=0,1,2,...,2N 1. B (ω)=
From Eq. 20 follow−s n 2N ω U m 2
− q=0 l=0m=0(cid:18) (cid:19)(cid:18) (cid:19)
X X X
π
exp i (l 2m n)q . (28)
[ω U +2wcosk] Φ (ω) = Φ (t=0)
k k × N − −
−
U (cid:12) E (cid:12) E h i
Φ(cid:12)(cid:12)0e(ω) +eiN(cid:12)(cid:12)keΦN(ω) . (22) The sum over q gives
−√2N | i | i
(cid:0) (cid:1)
It’s now possible to extract two equations connecting 1 U ∞ l l al(ω)1 e2iπ(l−2m−n)
B (ω)= − .
Φ0 to ΦN : n 2N ω U m 2l 1 eiNπ(l−2m−n)
| i | i − Xl=0mX=0(cid:18) (cid:19) −
(29)
[1+B (ω)] A (ω) B (ω) A (ω)
Φ (ω) = 0 | 0 i− N | N i, The condition for a nonvanishing B (ω) is
| 0 i [1+B0(ω)]2−BN2 (ω) (23) (twl−ee2nm−na)nd=+2N: K, where K is any intneger be-
−∞ ∞
[1+B (ω)] A (ω) B (ω) A (ω)
|ΦN(ω)i= 0(1+|BN0(ω))i2−−BNN2 (ω)| 0 i, Bn(ω)= ω UU ∞ l m!(ll! m)!al2(lω)δ(l−2m−n),2NK.
(24) − l=0m=0 −
X X
where (30)
or, using the Kronecker Delta function
1 2π(2N2N−1)e−ink Φk(t=0)
A (ω) = , (25) U ∞ ∞ l! al(ω)
| n i √2N kX=0 ω−U(cid:12)(cid:12)(cid:12)e+2wcoskE Bn(ω)= ω U l+n+2NK ! l−n−2NK ! 2l .
− l=0K=−∞ 2 2
X X
(31)
(cid:0) (cid:1) (cid:0) (cid:1)
1 U 2N−1 e−inNπq SincethecoefficientsofaNewton’sbinomialformulahave
B (ω)= , (26) tobe realandpositive,inthelimita(ω)<<1weobtain
n 2N ω U 1 a(ω)cos πq
− q=0 − N
X
U
B (ω) (1+M), (32)
with a(ω)=2w/(U ω) and noting that B =B . 0
N −N ≃ ω U
−
−
5
where equation (3.32c)) the eigenvalue of Eq. 39 was derived.
Onthebasisofthisresultthephenomenonofasymptotic
2N−1
1 1 degeneracy was established and shown to be directly re-
M =1 (33)
− 2N 1 2w cosq latedto the appearanceofthe orderedphaseinthe large
Xq=0 − U N limit.
contains powers of w/U and has to be calculated at the
desired order in q, and
V. DOUBLE DOT ARRAY DECOHERENCE IN
N THE LONG TIME LIMIT
1 2w
B (ω) . (34)
N ≃−2N U
(cid:18) (cid:19)
Accordingtothepreviousanalysiswecanlimitourself
Herewenotethatthelastcontributioncannotbeignored to consider only the first two states of the array.
|±i
because it gives rise to the energy separation between The decoherence rate will be however modified by the
Φ and Φ . extensive interaction with the bath.
0 N
| i | i
Furthermore we obtain We have to calculate the matrix elements of G(H )
S
in the subspace of + and taking into account the
1 | i |−i
A (ω) Φ (t=0) (35) particularsystem-bathinteractionH definedinEq. 6.
0 0 SB
| i≃ U | i
Here
and
1 G(HS)= eiqcosθ(l−l′)|gq|2nl′ω H1 ω nl,
|AN(ω)i≃ U |ΦN(t=0)i. (36) qX,l,l′ − S − q
(40)
where the sum over l,l′ runs over the array sites where
As a result, after an inverse Laplace transform, we get,
apart from corrections containing powers of w/U, electrons are present.
We choose the basis elements + and defined re-
| i |−i
Φ (t) =eiMUt[Φ (0) cos∆t i Φ (0) sin∆t] spectively as the sum and the difference of Φ and Ψ .
| 0 i | 0 i − | N i | i | i
(37) We introduce the form factor Λ defined through
qcosθ
and
n eiqcosθl Φ =Λ Φ (41)
Φ (t) =eiMUt[Φ (0) cos∆t i Φ (0) sin∆t], l | i qcosθ| i
| N i | N i − | 0 i (38) Xl
having introduced the energy gap or
∆=2w(2w/U)N−1. (39) nleiqcosθl Ψ =eiqΛqcosθ Ψ (42)
| i | i
l
X
We eventually obtain the long time behavior of a two
level system with energy separation exponentially van- . Explicitly, Λ = 1 ei2qcosθN / 1 ei2qcosθ .
qcosθ
− −
ishing in the large N limit. Actually, in Ref. [19] (see The matrix elements of the self-energy operator are
(cid:0) (cid:1) (cid:0) (cid:1)
cos2 qcosθ sin2 qcosθ
G++ = g 2 Λ 2 2 + 2 , (43)
q qcosθ
q | | | | "ω−E+−ωq ω−E−−ωq#
X
cos2 qcosθ sin2 qcosθ
G−− = g 2 Λ 2 2 + 2 , (44)
q qcosθ
q | | | | "ω−E−−ωq ω−E+−ωq#
X
qcosθ qcosθ 1 1
G−+ = (G+−)∗ =i g 2 Λ 2cos sin . (45)
q qcosθ
| | | | 2 2 ω E ω − ω E ω
q (cid:20) − +− q − −− q(cid:21)
X
We find convenient the introduction of the following generalized densities of states
qcosθ qcosθ
ρ+−(ǫ) = (ρ−+)∗(ǫ)= i g 2 Λ 2cos sin δ(ǫ ω ), (46)
q qcosθ q
− | | | | 2 2 −
q
X
qcosθ
ρ (ǫ) = g 2 Λ 2cos2 δ(ǫ ω ), (47)
1 q qcosθ q
| | | | 2 −
q
X
6
qcosθ
ρ (ǫ) = g 2 Λ 2sin2 δ(ǫ ω ), (48)
2 q qcosθ q
| | | | 2 −
q
X
from which follows
ρ (ǫ) ρ (ǫ)
G++ = dǫ 1 + 2 , (49)
ω E ǫ ω E ǫ
Z (cid:20) − +− − −− (cid:21)
1 1
G+− = i dǫρ+−(ǫ) , (50)
− ω E ǫ − ω E ǫ
Z (cid:20) − +− − −− (cid:21)
ρ (ǫ) ρ (ǫ)
G−− = dǫ 1 + 2 . (51)
ω E ǫ ω E ǫ
Z (cid:20) − −− − +− (cid:21)
The real part of G gives a negligible contribution to the VI. CONCLUSIONS
polelocationifcomparedwithE andE . Thus,assum-
− +
ingadensityofstatedifferentfromzeroonlyforpositive
The existence of a macroscopic degenerate ground
ǫ, as in Ref. [12], the only non vanishing contribution is
state is a general feature of a system exhibiting a phase
γ = ImG−−:
transition. It would be of interest to exploit such degen-
−
erate states as elements of a macroscopic qubit. Here
γ = πρ2(∆), (52) we considered an array of N interacting dot pairs. We
−
showed that, in the long time limit and neglecting cor-
where ∆ = E E is the energy gap of the two level rection of order w/U, the system oscillates between the
− +
−
system and is positive (being + the ground state). twoconfigurationscharacterizedbyzeroelectrostaticen-
| i
Then the solution for +Φ (t) and Φ (t) is ergy. Such a system has a robustness which increases
S S
h | i h−| i with the size as shown calculating the decoherence due
+Φ (t) = eiE+t +Φ (t=0) , (53) to a phonon bath at zero temperature. Decoherence cal-
S S
h | i h | i culation has been performed in the framework of an ap-
Φ (ω) = eiE−t−γt Φ (t=0) . (54)
h−| S i h−| S i plication of the resolvent method. It is however impor-
tant to note that a general conclusion about robustness
As expected, the ground state is not affected by de- of macroscopic qubits should consider nonzero tempera-
coherence, while the excited state relaxes. Damping is tureeffectsandotherdifferentmechanismsfordaphasing
proportional to the density of states calculated at the in semiconductors (such as cotunneling and background
energy gap. The density of states is however quite dif- charge fluctuations).
ferent from that of a single dot pair. Two competitive Assaid,coherentmanipulationof S isusefulinorder
| i
effects appear. The first one is represented by the pres- to realizea supportforquantuminformationtransferal-
ence of the form factor Λqcosθ inside ρ2, which, in the lowinga solidstate teleportation. The basic information
large N limit, increases the dephasing rate by a factor processing steps, i.e. initialization of the system, local
proportional to N2. The second, predominant, effect to gatesandreadoutare respectivelyobtainedby adiabatic
beconsideredistheexpontentialreductionwithN ofthe variation of system parameters, oscillations between Φ
| i
energy separation. and Ψ ) and local charge measurement. Moreover, it is
For instance, in the simple case of g 2 = 1/N and wort|htionotethat,becauseoftheanalogywithspinclus-
q
| |
ω =cq (longitudinal phonons) ters behavior, most of considerations on quantum com-
q
puting developed in Ref. [3] should be valid also for our
sin2qN qcosθ system, with the advantage that instead of measuring
γ(∆) dcosθddq sin2 δ ∆ c2q2 , spin states, we need to detect local charges. Due to the
∝ sin2qcosθ 2 −
Z asymptotic behavior of the effective hopping amplitude,
(cid:0) (5(cid:1)5)
sucha qubitrequiresgatetimes exponentially increasing
where d is the dimension of the bath and c is the speed
with the size of the system and the optimal chain length
of sound. If we compare this quantity with the system
will be determined by the specific application.
oscillation frequency we obtain
γ(∆)
N2∆d/2−1. (56) APPENDIX A: DECOHERENCE RATE IN A
∆ ∝
DOUBLE QUANTUM DOT
This result indicates that, for a phonon bath in three
dimensions, the macroscopic limit involves a growth of We introduce a double quantum dot in contact with a
the robustness with respect to decoherence. bosonic bath with Hamiltonian
7
N N
+ −
R = + + . (A-4)
| i 2∆ | i 2∆ |−i
H = H +H +H ,
S B SB
ε
H = σ +Tσ ,
S z x
2
Eq. 12 has now to be solved using
H = ω a†a ,
B q q q
q
X
1
H = σ g a†+a , (A-1)
SB 2 z q q q
q 1 1
X (cid:0) (cid:1) G(H )= g 2σ σ . (A-5)
S q z z
4 | | ω ω H
discussed in Ref. [12]. Here a one-dimensional bath is q − q− S
X
considered for simplicity. Labeling with L and R the
| i | i
eigenstates of σ with respective eigenvalues +1 and 1,
z
−
the eigenstates of HS are We need to calculate + G(HS) + and G(HS) .
Actually, obtaining Gh++| will be|eniough,hd−u|e the int|r−ini-
1
= [ 2T L +(∆ ε) R ], (A-2) sic robustness of the ground state [30] which implies
|±i N± ± | i ∓ | i thatG−− hastobe zero(this featu|−reiiseasilycheckedin
the presentformalism). To do itfirstwe write + in the
where ∆=√ε2+4T2 and N = (∆ ε)2+4T2 while | i
± L,R basis, then apply σ , come back in the basis
the respective eigenvalues are ε± =p±12∓∆. |inordierto apply(ω ωq zHS)−1, rewritethe n|±ewi state
By inversion we obtain − −
through L,R to apply the second σ operator, and fi-
z
| i
nally re-express the result in terms of + and . The
∆+ε ∆ ε | i |−i
L = N + N − , (A-3) result is
+ −
| i 4T∆ | i− 4T∆ |−i
1 1 ε 2 1 ∆ ε 2
G++ = g 2 + − . (A-6)
4Xq | q| "ω−ωq− ∆2 (cid:16)∆(cid:17) ω−ωq+ ∆2 (cid:18) ∆ (cid:19) #
The sum over q is performed as an integral through the The density matrix in the basis is then
|±i
introduction of the density of states ρ which is assumed
to be different from zero only for positive values of its
argument[12]. Thesecondterminsidethesquarebracket
gives the contribution to the imaginary part, which is ρ++(0)e−2γt ρ−+(0)e−γtei∆t
γ = πT2ρ(∆) /∆2. The evolution is thus ρ(t)= ρ−+(0)e−γtei∆t 1 ρ++(0)e−2γt . (A-9)
−
(cid:18) − (cid:19)
+Φ(cid:2)S(t) = (cid:3)+ΦS(t=0) e−i∆2te−π∆T22ρ(∆)t, (A-7)
h | i h | i
with the same dephasing rate obtained in [12], in the
regime of zero temperature, using markovian assump-
ΦS(t) = ΦS(t=0) ei∆2t. (A-8) tions.
h−| i h−| i
[1] E. Schr¨odinger, Naturwissenschaften 23, 807, 823, 844 [6] D. F.Walls and G. J. Milburn, Phys. Rev. A 31, 2403
(1935). (1985).
[2] Y. Nakamura, Y. Pashkin, and J. S.Tsai, Nature (Lon- [7] M. J Collet, Phys. Rev.A 38, 2233 (1988).
don) 398, 786 (1999). [8] S.Paganelli, F.dePasqualeandS.M.Giampaolo, Phys.
[3] F.Meier, J.Levy,andD.Loss, Phys.Rev.B68,134417 Rev. A 66, 052317 (2002).
(2003). [9] W. H. Zurek,Rev.Mod. Phys. 75 715 (2003).
[4] W. H.Zurek,Phys. Today 44, 36 (1991). [10] D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I. O. Sta-
[5] M. Brune, E. Hagley, J. Dreyer,X.Maˆitre, A. Maali, C. matescu and H. D. Zeh Decoherence and the appearence
Wunderlich,J.M.Raimond,andS.Haroche,Phys.Rev. of a Classical World in Quantum Theory (Springer,
Lett.77, 4887 (1996). Berlin) (1996).
8
[11] A.O.CaldeiraandA.J.Leggett,Phys.Rev.A31,1059 [22] A. Osterloh, L. Amico, G. Falci and R. Fazio, Nature
(1985). (London) 416, 608 (2002).
[12] T. Brandes and T. Vorrath, Phys. Rev. B 66, 075341 [23] T.J.OsborneandM.A.NielsenPhys.Rev.A66,032110
(2002). (2002).
[13] G. M. Palma K. A. Suominen and A. K. Ekert , Proc. [24] J. H. Reina, L. Quiroga and N. F. Johnson, Phys. Rev.
RoyalSoc. London A,452 567 (1996). A 65, 032326 (2002).
[14] C. Anastopoulos and B. L.Hu,Phys.Rev.A 62 033821 [25] S.Fujita,IntroductiontoNon-EquilibriumStatisticalMe-
(2000). chanics (Saunders,Philadelphia, 1968).
[15] V. Privman and D. Mozyrsky, J. Stat. Phys. 91 787 [26] K. D. Schotte,Phys. Rep.46, 93 (1978).
(1998). [27] A. Suzukiand D.Dunn,Phys. Rev.A 25, 2247 (1982).
[16] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. [28] F. de Pasquale, G. Giorgi and S. Paganelli, Phys. Rev.
Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, Lett. 93, 120502 (2004).
1 (1987). [29] L. Fedichkin and A. Fedorov, Phys. Rev. A 69 032311
[17] U.Weiss, Quantum Dissipative Systems, 2nd ed. (World (2004).
Scientific,Singapore, 1999). [30] A.M. Childs,E. FarhiandJ. Preskill, Phys.Rev.A 65,
[18] H.Bethe, Z. Physik 71, 205 (1931). 012322 (2002).
[19] E.Lieb,T.SchultzandD.Mattis,Ann.Phys.(N.Y.)16, [31] S.Sachdev,QuantumPhaseTransitions,CambridgeUni-
407 (1961). versity Press, Cambridge (1999).
[20] P.Pfeuty, Ann.Phys(N.Y.) 57, 79 (1970).
[21] V.MurgandJ.I.Cirac,Phys.Rev.A69,042320(2004).