Non-Markovian dynamics in a spin star system: Exact solution and approximation techniques

0.29 MB·English

by  Heinz-Peter Breuer

Non-Markovian dynami
s in a spin star system: Exa
t solution and approximation
te
hniques
1 1 1,2
Heinz-Peter Breuer, Daniel Burgarth, and Fran
es
o Petru

ione
1
Physikalis
hes Institut, Universität Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
2
S
hool of Pure and Applied Physi
s, University of KwaZulu-Natal, 4041 Durban, South Afri
a
N 1
The redu
ed dynami
s of a
entral spin
oupled to a bath of spin-2 parti
les arranged in a
spin star
on(cid:28)guration is investigated. The exa
t time evolutioNn o→f th∞e redu
ed density operator
is derived, and an analyti
al solution is obtained in the limit of an in(cid:28)nite number of
bathspins,wherethemodelshows
ompleterelaxationandpartialde
oheren
e. Itisdemonstrated
that the dynami
s of the
entral spin
annot be treated within the Born-Markov approximation.
The Nakajima-Zwanzig and the time-
onvolutionless proje
tion operator te
hnique are applied to
4 the spin star system. The performan
e of the
orresponding perturbation expansions of the non-
0 Markovian equations of motionis examined througha
omparison with the exa
tsolution.
0
2 PACSnumbers: 03.65.Yz,05.30.-d,75.10.Jm,73.21.La

n
a
J I. INTRODUCTION Se
. IIC we analyze the limit of an in(cid:28)nite number of

0 bath spins, dis
uss the behavior of the von Neumann
1 entropy of the
entral spin, and demonstrate that the
Solidstatespinnanodevi
esareknownasverypromis-
1,2,3,4 model exhibits
omplete relaxation and partial de
oher-
ing
andidates for quantum
omputation and also
2 5 en
e. Several non-Markovian approximation te
hniques
v for quantum
ommuni
ation . They provide a s
alable
aredis
ussedinSe
.III. Thedynami
equationsfoundin
1 system that
an easily be integrated into standard sili-
se
ondorderin the
ouplingareintrodu
edinSe
.IIIA,
5
on te
hnology. A drawba
k of su
h systems
ompared
where it is also demonstrated that the prominent Born-
0 to other proposals for quantum
omputing, su
h as ion
1 traps6 and
avity QED7, are the many degrees of free- Markov approximation is not appli
able to the spin star
0 model. Employing a te
hnique whi
h enables the
al-
domof the surroundingmaterial
ausing dissipation and
4 8
ulation of the
orrelation fun
tions of the spin bath,
de
oheren
e . The (cid:28)rst step in over
oming these disad-
0 we derive in Se
. IIIB the perturbation expansions
or-
/ vantages is to be able to model them.
h responding to the Nakajima-Zwanzig and to the time-
An important
ontribution to quantum noise in solid
p
onvolutionless proje
tion operator te
hnique and
om-
state systems arisesfrom the nu
lear spins, and re
ently
- pare these approximations with the analyti
al solution
t mu
h work has been devoted to the modeling of spin
n forthe dynami
s of the
entral spin. Finally, the
on
lu-
9,10,11,12,13,14,15,16,17
a bath systems (for a review, see Refs. sions are drawn in Se
. IV.
u 18,19). The intera
tion of a
entral spin with a bath of
q environmentalspinsoftenleadstostrongnon-Markovian
v: behavior. The usual derivations of Markovian quantum
20
i masterequationsknown,e.g.,fromatomi
physi
s and II. EXACT DYNAMICS
X 21
quantumopti
s thereforefail formanyspin bath mod-
r els and a detailed investigation of methods is required
a A. The model
whi
h are
apable of going beyond the Markovian ap-
proximation.
29
In this paper, we examine the redu
ed dynami
s of a We
Nons+id1er a (cid:16)spin star(cid:17)1
on(cid:28)guration whi
h
on-
simple spin star system. The advantage of this model sistsof lo
alizedspin-2 parti
les. Oneofthespins
is that, while showing several interesting features su
h islo
atedat the
entie=ro1f,t2h,e..s.t,aNr, while theotherspins,
as partial de
oheren
e and strong non-Markovian be- labeled by an index , surroundthe
entral
havior, it is exa
tly solvable due to its high symme- spin at equal distan
es on a sphere. In the language of
8
try. The model therefore represents an appropriate ex- open quantum systeσms we regard the
entral spin with
ample for a general dis
ussion of the performan
e of Pauli spin operator asan open system living in atwo-
S
various non-Markovian methods. We study here the dimensional Hilbertspa
e H anσd(it)he surroundingspins
Nakajima-Zwanzig22,23,24 and the time-
onvolutionless des
ribed by the spin operators as a sNpin bath with
proje
tion operator te
hnique25,26,27,28 and derive and Hilbert spa
e HB whi
h is given by an -fold tensor
analyzetheperturbationexpansionsofthe
orresponding produ
t of two dimeσnsional spa
es. σ(i)
non-Markovianmaster equations. The
entral spin intera
ts with the bath spins
30
The paperisorganizedasfollows. InSe
. II, we intro- via a Heisenberg XX intera
tion represented through
du
e the model investigated,a spin starmodel involving the Hamiltonian
aHeisenbergXX
oupling(Se
.IIA), anddeterminethe
αH =2α(σ J +σ J ),
+ − − +
exa
t time evolution of the
entral spin (Se
. IIB). In (1)
2
S
where where tr denotes the partial tra
e over the Hilbert
J N σ(i), rsp(ta)
e HvS(t)of the
entral spin. We note tha1t the length
± ≡ ± (2) ρ ≡(t)| | of the Blo
h ve
tor is equal to if and only
Xi=1 if S des
ribSes a pure state, and that the von Neu-
mann entropy of the
entral spin
an be expressed as
r(t)
and a fun
tion of the length of the Blo
h ve
tor:
1
σ±(i) ≡ 2 σ1(i)±iσ2(i) (3) S ≡ trS{−ρSlnρS} (11)
(cid:16) (cid:17) 1 1
i = ln2 (1 r)ln(1 r)+ (1+r)ln(1+r).
represents the raising and lowering operators of the th − 2 − − 2
bath spin. The Heisenberg XX
oupling has been found
t=0
tobeane(cid:27)e
tiveHamiltonianfortheintera
tionofsome Theinitialstateoftheredu
edsystemat istaken
31
quantum dot systems . Equation (1) des
ribes a very to be an arbitrary (possibly mixed) state
simple time independent intera
tion with equal
oupling
α 1+v3(0) v (0)
stitornensgatrhounfdorthaellzb-aatxhiss.pTinhs.e IotpiesraintovrarJian≡t u12ndeNi=r1roσt(ai)- ρS(0)= v+2(0) 1−−v23(0) !, (12)
representsthe total spin an~gu=la1r momentum ofPthe bath
(unitsare
hosensu
hthat ). The
entralspinthus while the spin bath is assumed to be in an unpolarized

ouples to the
olle
tive bath angular momentum. in(cid:28)nite temperature state:
HBW
eoninstisrtoidnug
oefaJstnatOesN|jb,amsi,sνiin.mTthheesbeastthatJHe2silabreertdes(cid:28)pnae
de ρB(0)=2−NIB. (13)
3
as eigenstates of (eigenvalue ) and of [eigenvalue
j(j+1) ν IB B
℄. The index labels the di(cid:27)erent eigenstates in Here, denotes the unit matrix in H , and we have
j,m (j,m) v±
the eigenspa
e M belongingj to aNgiven pajir m ojf dv1e,(cid:28)2ned the aslinear
ombinationsof the
omponents
quantum numbers. As usual, ≤ 2 and − ≤ ≤ . of the Blo
h ve
tor:
The dimension of Mj,m is given by the expression29,32 v1 iv2
v = ± .
±
N N 2 (14)
n(j,N)= .
N/2 j − N/2 j 1 (4)
(cid:18) − (cid:19) (cid:18) − − (cid:19)
Theinitialstateofthetotalsystemisgivenbyanun
or-
ρ (0) ρ (0).
S B
Wefurtherintrodu
etheusualsuperoperatornotation related produ
t state ⊗
for the Liouville operator
ρ(t) i[H,ρ(t)],
L ≡− (5) B. Redu
ed System Dynami
s
ρ(t)
where denotesthedensitymatrixofthetotalsystem
in the Hilbert spa
e HS ⊗HB. The formal solution of Inthisse
tion,wewillρdSe(rti)vetheexa
tdynami
softhe
redu
ed density matrix for the model given above.
the von Neumann equation
One possibility of obtaining the evolution of the
entral
d
ρ(t)=α ρ(t) spin is to substitute Eq. (7) into Eq. (8) and to expand
dt L (6)
theexponentialwithrespe
ttothe
oupling. Thisyields
ρ (t) exp(α t)ρ (0) ρ (0)

an then be written as S B S B
≡ tr { L ⊗ }
ρ(t)=exp(α t)ρ(0). ∞ (αt)k
L (7) = kρ (0) 2−NI .
B S B
k! tr L ⊗ (15)
k=0
Our main goal is to derive the dynami
s of the redu
ed X (cid:8) (cid:9)
density matrix
It is easy to verify that we have
ρ (t) ρ(t) ,
S B
≡tr { } (8) H2n =4n[σ σ (J J )n+σ σ (J J )n]
+ − − + − + + −
(16)
B
wheretr denotesthepartialtra
etakenovertheHilbert
B
spa
e H of the spin bath. The redu
ed density matrix and
is
ompletely determined in terms of the Blo
h ve
tor H2n+1 =2 4n[σ σ (J J )n+σ σ (J J )n].
+ − + − − + − +
v (t) · (17)
1
v(t)= v (t) σρ (t)
2 S S
 ≡tr { } (9) We note that su
h simple expressionsare obtained sin
e
v (t) σ J
3 3 3
a term is missing in the intera
tion Hamiltonian.
 
We substitute the last two equations into
through the relationship
n
n
ρS(t)= 12(cid:18)v1(1t)++v3i(vt2)(t) v1(1t−)−v3i(vt2)(t) (cid:19), (10) Lnρ=inkX=0(−1)k(cid:18)k(cid:19)HkρHn−k (18)
3
to get the formulas (29). We note that the dynami
s
an be expressed
om-
f (t)
12
trB L2k−1ρS(0)⊗2−NIB =0 (19) fp3le(tte).lyTthhirsoufag
htoisnl
yontwneo
rteeadl-tvoaltuheedrfoutna
ttioionnasl symmeatnrdy
(cid:8) (cid:9) of the system.
and Theredu
edsystemdynami
shasbeenobtainedabove
exp(α t)
B 2kρS(0) 2−NIB =( 16)kQkv3(0)σ3 with the help of an expaαnsion of L with respe
t
tr L ⊗ − 2 (20) tothe
oupling
onstant . Itshouldbe
learthat,alter-
(cid:8) k (cid:9) natively, the behavior of the
entral spin may be found
+ (−4)k l=0(cid:18)22kl (cid:19)Rlk−l!(v−(0)σ++v+(0)σ−), fdoirret
htley
formompotshiteessoylsutteimon. oTfhtihsesSol
uhtriöodnin
gaenr eeaqsuilaytiobne
X

onstru
tedbymakinguseofthefa
tthatthesubspa
es
k =1,2,... + j,m,ν j,m+1,ν
whi
h holdfor all . Here, wehaveintrodu
ed spannedbythestates| i⊗| iand|−i⊗| i
the bath
orrelationfun
tions areinvariantundeσrthetimeevolution,where|±ide1notes
3
1 the eigenstate of belonging to the eigenvalue ± .
Q (J J )k ,
k ≡ 2NtrB + − (21) Sometimes it is useful to express the redu
ed dynam-
1 n o i
s in terms of superoperators instead of the Blo
h ve
-
Rlk−l ≡ 2NtrB (J+J−)k−l(J−J+)l . (22) tor. Tothisend,weintrodu
esuperoperatorsS± andS3,
n o Awhi
harede(cid:28)nedbytheira
tiononanarbitraryoperator
:
We will
ome ba
k to these
orrelation fun
tions when 1
A σ Aσ σ σ ,A ,
we dis
uss approximation te
hniques in Se
. III. S± ≡ ± ∓− 2{ ∓ ± } (30)
Using the formulas (19) and (20) in Eq. (15) we
an A σ Aσ A.
3 3 3
express the
omponents of the Blo
h ve
tor as follows, S ≡ − (31)
v±(t) = f12(t)v±(0), With these de(cid:28)nitions we may write the redu
eddensity
(23)
v (t) = f (t)v (0), matrix as follows,
3 3 3 (24) IS
ρ (t)=
S
2 (32)
where we have introdu
ed the fun
tions 1 1
+ f (t) f (t) f (t)( + ) ρ (0),
f12(t) 2 2 3 − 12 S3− 3 S+ S− S
≡ (25) (cid:20)(cid:18) (cid:19) (cid:21)
tr cos 2 J J αt cos 2 J J αt 2−NI , I
B + − − + B S S
⊗ where denotes the unit matrix in H . Due to the
n h p i h p i o non-unitary behavior of the redu
ed system, the super-
and operator on the right-hand side is not invertible for all
f (t) tr cos 4 J J αt 2−NI . times. This point will be
ome important later on when
3 ≡ B + − ⊗ B (26) we dis
uss approximation strategies.
n h p i o
CaJl
ulatinJg2thetra
esoverthespinbathintheeigenbasis
3 C. The Limit of an in(cid:28)nite number of bath spins
of and using
J J j,m,ν =(j m)(j m+1) j,m,ν ,
∓ ±
| i ∓ ± | i (27) The expli
it solution
onstru
ted in the previous se
-
N
tiontakesonarelativelysimpleforminthelimit →∞
we (cid:28)nd of an in(cid:28)nite number of bath spins. It is demonstrated
N
f (t) in the appendix that for large the bath
orrelation
12
≡
fun
tions approa
hthe asymptoti
expression
cos[2h(j,m)αt]cos[2h(j, m)αt]
j,mn(j,N) 2N − , (28) Qk ≈Rlk−l ≈ 2kk!Nk. (33)
X
Nk
and Consequently, in Eq. (15) a termαo2fkthe order always
o

urs together with a fa
tor of . A non-trivial (cid:28)nite
cos[4h(j,m)αt] N
f (t) n(j,N) , limit → ∞ therefore exists if we res
ale the
oupling
3 ≡ 2N (29)
j,m
onstant as follows,
X α
α .
h(j,m) → √N (34)
where we have introdu
ed the quantity ≡
(j+m)(j m+1)
− . Using this approximation in Eq. (20) one obtains
pThus we have determined the exa
t dynami
s of the 2kρ (0) 2−NI
ρ (t) B S B
S tr L ⊗
redu
ed system: The density matrix of the
entral k
(cid:8)( 8N) k! (cid:9)
spinisgiventhroughthe
omponentsoftheBlo
hve
tor − (v (0)σ +v (0)σ +v (0)σ ).
3 3 − + + −
whi
h are provided by the relations (23), (24) and (28), ≈ 2 (35)
4
If we insert this into Eq. (15) we get an in(cid:28)nite series
whi
h yields the following expressions for the fun
tions
f (t) f (t)
12 3
and ,
f (t)=1+g(t), f (t)=1+2g(t),
12 3
(36)
where
π
g(t) αtexp( 2α2t2) (√2αt).
≡− − 2er(cid:28) (37)
r
(x)
Note that er(cid:28) is the imaginary error fun
tion. It is a
real-valued fun
tion de(cid:28)ned by
(ix)
(x) erf ,
er(cid:28) ≡ i (38)
whi
h leads to the Taylor series
2 ∞ x2k+1
(x)= .
er(cid:28) √π k!(2k+1) (39)
k=0
X
Figures 1 and 2 show that this approximation obtained
in thelimitofanin(cid:28)nite numberofbathspinsisalready
N 200
reasonablefor ≈ .
N →∞
Figure2: ComparisonofthelimitN =20 [seeNEq=s.2(0306.)and
(37)℄ with the exva
t fun
tions for and The
3
(cid:28)gure shows the -
omponentof the Blo
h ve
tor.
is des
ribed by the stationary density matrix
1 v−(0)
lim lim ρ (t)= 2 2 .
t→∞N→∞ S v+(0) 1 ! (40)
2 2
The 3-
omponent of the Blo
h ve
tor relaxes to zero,
while the o(cid:27)-diagonal elements of the density matrix
show partial de
oheren
e, i.e. they assume half of their
originalvalues. This behavioris alsore(cid:29)e
tedin the von
Neumann entropy of the redu
ed system. Its dynam-
r(0)
i
s depends on the initial entropy parametrized by
v (0)
3
and on , whi
h is obvious be
ause the entropy is a
s
alar quantity and the system is invariant under rota-
z
tions around the -axis. Figure 3shows the entropyas a
fun
tion of time for di(cid:27)erent initial
onditions.
III. APPROXIMATION TECHNIQUES
N →∞ In this se
tion we will apply di(cid:27)erent approximation
Figure1: ComparisonofthelimitN =20 [seeNEq=s.2(0306.)and te
hniques to the spin star model introdu
ed and dis-
(37)℄ with the evx±a
t fun
tions for and The
ussed in the previous se
tion. Due to the simpli
ity of
plot shows the -
omponentof the Blo
h ve
tor.
thismodelwe
annotonlyintegrateexa
tlytheredu
ed
system dynami
s, but also
onstru
t expli
itly the vari-
(x)
By
ontrast to erf , the imaginary error fun
tion is ousmasterequationsforthedensitymatrixofthe
entral
g(t)
tnot bounded. Hgo(wt)ever, 1 isbounded, and in the limit spin and analyzeand
omparetheir perturbationexpan-
→∞wehave →−2. Thus,inthislimitthesystem sions. In the following dis
ussion we will sti
k to the
5
Q
1
It is important to noti
e that , as well as all other
bath
orrelationfun
tions areindependent of time. This
is to be
ontrasted to those situation in whi
h the bath

orrelation fun
tions de
ay rapidly and whi
h therefore
allow the derivation of a Markovian master equation.
Thetime-independen
eofthe
orrelationfun
tionsisthe
main reason for the non-Markovian behavior of the spin
bath model.
The integro-di(cid:27)erential equation (43)
an easily be
solved by a Lapla
e transformation with the solution
v (t)
± = cos 2√Nαt ,
v±(0) (45)
(cid:16) (cid:17)
v (t)
S(t) v3(0) = cos 2√2Nαt . (46)
Figure 3: Von Neumann entropy of the redNu
e→d s∞ys- 3 (cid:16) (cid:17)
Stemmaxfo≡r ldni(cid:27)2erent initial
onditions in the limit .
is the maximal entropy for a qubit, represent- In many physi
al appli
ations the integration of the
ing a
ompletelymixedstate. integro-di(cid:27)erential equation is mu
h more
ompli
ated
and one tries to approximate the dynami
s through a
master equation whi
h is lo
al in time. To this end, the
v (s) v (s)
Blo
h ve
tornotation. Ea
h of the master equationsob- terms ± and 3 under the integral in Eq. (43) are
v (t) v (t)
tained
aneasilybetransformedintoanequationinvolv- repla
edby ± and 3 , respe
tively. We thus arrive
ing Lindblad superoperators [see Eqs. (30), (31)℄ using at the time-lo
al master equation
the translation rules d
ρ (t)
1 dt S
v σ = ρ ,
3 3 2S3−S+−S− S (41) t
(cid:26) (cid:27) = 4Nα2 ds(v (t)σ +v (t)σ +v (t)σ )
1 − + − − + 3 3
v+σ−+v−σ+ = 3ρS. Z0
−2S (42) = 4Ntα2(v (t)σ +v (t)σ +v (t)σ ),
+ − − + 3 3
− (47)
whi
hissometimesreferredtoasRed(cid:28)eldequation. Also
A. Se
ond order approximations this master equation is easily solved to give the expres-
sions
v (t)
The se
ond order approximation of the master equa- ± = exp( 2Nα2t2),
tionfortheredu
edsystemisusuallyobtainedwithinthe v±(0) − (48)
Bornapproximation8. Itisequivalenttothese
ondorder v3(t) = exp( 4Nα2t2).
of the Nakajima-Zwanzig proje
tion operator te
hnique, v (0) − (49)
3
whi
h will be dis
ussed systemati
ally in Se
. IIIB. In
our model the Born approximation leads to the master TheRed(cid:28)eldequationisequivalenttothese
ondorderof
equation the time-
onvolutionless proje
tion operator te
hnique,
ρ˙S(t) whi
h will also be dis
ussed in detail in Se
. IIIB.
t In order to obtain, (cid:28)nally, a Markovian master equa-
= ds B [H,[H,ρS(s) ρB(0)]] tion, i.e. atime-lo
alequation involvingatime indepen-
− tr { ⊗ }
Z0 dentgenerator,onepushestheupperlimitoftheintegral
t t
= 8α2Q ds(v (s)σ +v (s)σ +v (s)σ ), inEq.(47)toin(cid:28)nity,thatisonestudiesthelimit →∞
1 + − − + 3 3
− (43) of the master equation. This limit leads to the Born-
Z0
Markov approximation of the redu
ed dynami
s. In the
where the bath
orrelationfun
tion is found to be presentmodel,however,itisnotpossibletoperformthis
1 approximationbe
ausetheintegranddoesnotvanishfor
Q = J J t
1 2NtrB{ + −} large . Thus, the Born-Markov limit does not exist for
the spin bath model investigated here and the des
rip-
1
= σ(i)σ(j) tionofrelaxationand de
oheren
epro
essesrequiresthe
2NtrB + −  usage of non-Markovianmethods.
Xi,j 
1
=  σ(i)σ(i)
2NtrB( + − ) B. Higher order approximations
i
X
N
= . A systemati
approa
h to obtain approximate non-
2 (44) Markovian master equation in any desired order is pro-
6
vided by the proje
tion operator te
hniques. We de(cid:28)ne and
a proje
tion superoperatorP through the relation 2 = 2 ,
L pc L
ρ= B ρ ρB (cid:10) 2(cid:11) = (cid:10) 2(cid:11),
P tr { }⊗ (50) L oc L
(cid:10) 4(cid:11) = (cid:10) 4(cid:11) 2 2,
ρB ρB(0) L pc L − L
with the referen
e state ≡ and introdu
e the
notation (cid:10) 4(cid:11) = (cid:10) 4(cid:11) (cid:10)3 (cid:11)2 2,
L oc L − L
(cid:10) 6(cid:11) = (cid:10) 6(cid:11) 2(cid:10) 2(cid:11) 4 +3 2 3,
hXi≡PXP (51) L pc L − L L L
(cid:10) 6(cid:11) = (cid:10) 6(cid:11) 15(cid:10) (cid:11)2(cid:10) (cid:11)4 +3(cid:10)0 (cid:11)2 3,
n L oc L − L L L
foranysuperoperatorX. Notethatthe(cid:16)moments(cid:17) hX i
are operators in the total Hilbert spa
e HS ⊗HB of the (cid:10) (cid:11) ··· (cid:10) (cid:11) (cid:10) (cid:11)(cid:10) (cid:11) (cid:10) (cid:11)

ombined system.
Inthetimeindependent
asetheordered
umulantsare
There are two main proje
tion operator methods:
just the ordinary
umulants know from
lassi
al statis-
The Nakajima-Zwanzig (NZ) te
hnique and the time-
ti
s. To
al
ulate thQeses fun
Rtkio−nls one
an againuse Eq.

onvolutionless (TCL) te
hnique. In our model, the ini- (20). The fun
tions k and l arerealpolynomialsin
N k
tial
onditions fa
torize. The NZ and the TCL method
of order . A method of determining these polynomi-
therefore lead to relatively simple, homogeneous equa-
als is sket
hed in the appendix.
tions of motion. The NZ master equation is an integro-
If we express the resulting master equations in terms
v (t) v (t)
di(cid:27)erential equation forthe redu
ed density matrix with ± 3
(t,τ) of and , we get for the TCL te
hnique
a memory N , whi
h takes the form
∞ s t2n−1
t : v˙±(t) = α2n 2n v±(t),
ρ˙ (t) ρ = dτ (t,τ)ρ (τ) ρ , TCL (2n 1)!! (57)
S B S B n=1 −
⊗ N ⊗ (52) X
Z0 ∞ 2q t2n−1
v˙ (t) = α2n 2n v (t),
3 3
(2n 1)!! (58)
while the TCL master equation is a time-lo
al equation n=1 −
(t) X
of motion with a time-dependent generator K , whi
h
reads and for the NZ method
∞
ρ˙S(t)⊗ρB =K(t)ρS(t)⊗ρB. (53) NZ: v˙±(t) = α2ns˜2nIn(t,τ)!v±(τ), (59)
n=1
X
∞
BoththeNZandtheTCLmasterequation
anof
ourαse v˙ (t) = α2n2q˜ (t,τ) v (τ).
3 2n n 3
be expanded with respe
t to the
oupling strength . I ! (60)
n=1
Sin
e the intera
tion Hamiltonian is time independent, X
this expansion yields s2n s˜2n q2n q˜2n
The quantities , , and represent real poly-
N n
t ∞ nomials in of the order . For example, we have
dτ (t,τ)ρ (τ)= αn (t,τ) n ρ (τ)
Z0 N S n=1 In hL ipc S (54) q2 = −4N,
X q = 32N2,
4
−
q = 1024N +1536N2 1536N3,
for the NZ master equation, and 6 − −
∞ tn−1 ···
(t)= αn n q˜2 = 4N,
K nX=1 (n−1)!hL ioc (55) q˜4 = −32N2,
q˜ = 1024N +1536N2 1280N3,
6
− −
for the TCL master equation, where we have introdu
ed
···
the integral operator s = 4N,
2
−
(t,τ) tdt t1dt tn−3dt tn−2dτ s = 48N +16N2,
In ≡ 0 1 0 2··· 0 n−2 0 (56) 4 −
s = 1024N 384N2+384N3,
R R R nR 6 − −
for the ease of notion. The symbol hL ipc denotes the
n ···
partialn
umulants and hL ioc the ordered
umulants of s˜2 = −4N,
order . Their de(cid:28)nitions
an be found in Refs. 25,26, s˜ = 48N +48N2,
4
−
27,28. In our model we have s˜ = 1024N +2112N2 1216N3,
6
− −
2n+1 = 2n+1 =0 .
L pc L oc ···
(cid:10) (cid:11) (cid:10) (cid:11)
7
(2n)
The th-order approximation of the master equa-
2n 2n
tions (denoted by TCL and NZ , respe
tively)is ob-
tained by trun
ating the sums in Eqs. (57), (58) and in
n
Eqs. (59), (60) after the th term. In the TCL
ase,
the resulting ordinary di(cid:27)erential equations
an be inte-
grated very easily. The equation of motion of the NZ
method
an be solved with the help of a Lapla
e trans-
formation. However,itmaybeveryinvolvedto
arryout
the inverse transformation for higher orders. For exam-
ple, the solution of the twelfth order of the NZ equation
as obtained by standard
omputer algebratools is (cid:28)lling
some hundred pages, whereas the solution of the TCL
equation
an be written in a single line.
The solutions of the master equations in se
ond and
fourth order are plotted in Figs. 4 and 5, together with
the exa
t solutions. We observe that both methods lead
to a good approximation of the short-time behavior of
the
omponentsofthe Blo
hve
tor. Wefurtherseethat
the TCL te
hnique is not only easier to solve, but also
provides a better approximation of the dynami
s within
a given order.
Figure5: ComparisonoftheTCLandtheNZte
hniquewith
theexa
tsolution. Theplotshowsthese
ondandthefovu3r(tth)
order approximations as well as the exa
t solution of
100
[see Eqs. (24)and (29)℄ for a bath of spins.

orresponding solution given by Eqs. (48) and (49)
on-
tains in(cid:28)nitely many orders. Of
ourse, the expansion of
theexa
tsolution
oin
ideswiththeexpansionoftheap-
2n 2n
proximations obtained with TCL or NZ within the
(2n)
th order. The errorofαT4CL2 orNZ2, for example,is
therefore a term of order , as is illustrated in Fig. 6.
Figure4: ComparisonoftheTCLandtheNZte
hniquewith
the exa
t solution. The plαot shows the approximatiovn±s(tto)
se
ond and fourth order in and the exa
t solution of
100
[see Eqs. (23) and (28)℄ for a bath of spins.
Figure 6ǫ:(t)E≡rrorv±o(ft)T−CLv2±apparnodx(tN)Z2: The plot shows thve±d(et)-
Sin
e the TCLand the NZ method lead to expansions viation v±apopfrotxh(te)exa
t solutαiton
oftheequationsofmotionandnotoftheirsolutions,the fromtheappro(cid:12)(cid:12)ximate solution (cid:12)(cid:12) for small .
solutions of the trun
ated equations may
ontain terms
ofarbitraryorderin the
ouplingstrength. Forexample, Con
erningthelong-timebehavior,boththeTCLand
even though TCL2 is a se
ond order approximation, the the NZ method may lead to very bad approximations.
8
v (t)
±
For example, in the fourth order approximationof
(see Fig.4) the TCLaswell asthe NZ solutionleavethe
Blo
hsphere,i.e. fortimeslargerthansome
riti
altime
these solutions do not represent true density matri
es
anymore.
Ifwelookathigherorders,theNZmethodisseentobe
better than the TCL method as far as the 3-
omponent
of the Blo
h ve
tor is
on
erned. An example is shown
in Fig. 7, where we plot the tenth orderapproximations.
v (t)
3
We observe that the solution of the TCL equation
(58) is always greater than zero. This fa
t is obviously

onne
ted to the stru
ture of this equation, whi
h takes
v˙ (t) = (t)v (t) (t)
3 3 3 3
the form K with a real fun
tion K .
3 v
3
If the -
omponent of the Blo
h ve
tor vanishes at
t = t
0
the time , then a time-lo
al equation of motion of
v˙ (t )
3 0
this form
an only be ful(cid:28)lled if is also zero. In
our
ase, however, the exa
t solution passes zero with a
8
non-vanishing time derivative. It is a well-known fa
t
that the perturbation expansion of the TCL generator
only exists, in general, for short and intermediate times
and/or
oupling strengths. This is re(cid:29)e
ted in the fa
t
thatthesuperoperatorontheright-handsideofEq.(32)

annotbeinvertedforalltimes,i.e. itisnotalwayspossi-
v (0) v (t)
3 3
bletoexpress in termsof . Asimilarsituation
also o

urs in open systems intera
ting with a bosoni
reservoir, e. g., in the damped Jaynes Cummings model
whi
hdes
ribesthe intera
tion ofaqubitwith abosoni
Figure 7: The TCL and the NZ approximation of the
om-
100
reservoir at zero temperature. The NZ te
hnique does ponents of the Blo
h ve
tor in tenth order for a bath of
not lead to su
h problems. However, sin
e the
ompo- spins.
v (t)
±
nents do not vanish, the
orresponding high-order
TCL approximation is still more a

urate than the NZ
approximation, as may be seen from Fig. 7. is mu
h simpler than that of the non-lo
al equations of
the NZ te
hnique.
Itshouldbekeptinmind,however,thattheexpansion
IV. CONCLUSION basedontheTCLmethod
onverges,ingeneral,onlyfor
shortandintermediateintera
tiontimes. Forlargetimes
Withthehelpofasimpleanalyti
allysolvablemodelof the perturbation expansion may break down, whi
h has
aspin starsystem,wehavedis
ussedtheperforman
eof been illustratedin ourmodel to be
onne
ted to zerosof
proje
tion operator te
hniques for the dynami
s of open the
omponents of the Blo
h ve
tor. It turns out that
systemsandtheresultingperturbationexpansionsofthe theNZequationofmotionyieldsabetterapproximation
equationsofmotion. Themodel
onsistsofa
entralspin of the exa
t dynami
s in this regime.
surrounded by a bath of spins intera
ting with the
en- In view of the heuristi
approa
h to the Born and to
XX
tral spin through a Heisenberg
oupling, and shows the Red(cid:28)eld equation (see Se
. IIIA) it is sometimes

omplete relaxation and partial de
oheren
e in the limit
onje
tured that a non-lo
al equation of motion should
of an in(cid:28)nite number of bath spins. Due to its high be generally better than a time-lo
al one. The results
symmetry the model allows a dire
t
omparison of the of Se
. IIIB show that this
onje
ture is, in general,
Nakajima-Zwanzig (NZ) and of the time-
onvolutionless not true. The fa
t that in a given order the time-lo
al
(TCL) proje
tionoperator methods with the exa
t solu- TCL equation is at least as good (and mu
h simpler to
tion in analyti
al terms. deal with), if not better than the non-lo
al NZ equation
While the Born-Markov limit of the equation of mo- has also been observed in other spe
i(cid:28)
system-reservoir
8
tion does not exist in the model, the dynami
s of the models , and has been
on(cid:28)rmed by general mathemat-
33

entral spin exhibits a pronoun
ed non-Markovian be- i
al arguments . However,
are must be taken when
havior. It has been demonstrated that both the NZ and applying a
ertain proje
tion operator method to a spe-
the TCL te
hniques provide good approximations of the
i(cid:28)
model: The quality of the
orresponding perturba-
short-time dynami
s. In pra
ti
al appli
ations the TCL tion expansion of the equation of motion may strongly
method is usually to be preferred sin
e it leads to time- depend on the spe
i(cid:28)
properties of the model, e. g., the
lo
al equations of motion in any desired order with a intera
tion Hamiltonian, the intera
tion time, the envi-
mu
h easier mathemati
al stru
ture, whose integration ronmental state and the spe
tral density.
9
For example, in our parti
ular model the TCL expan- Appendix A: BATH CORRELATION FUNCTIONS
sion to fourth order turns out to be more a

urate than
thefourthorderNZexpansion. However,thereisnorea-
In this appendix we outline how to
al
ulate the bath
son why TCL should be generally better then NZ. To

orrelation fun
tions

larify further this point we
onsider the Taylorseriesof 1
k
the 3-
omponent of the Blo
h ve
tor: Qk ≡ 2NtrB (J+J−) , (A1)
v3(t)=a0+a2(αt)2+a4(αt)4+O (αt)6 . (61) Rk−l 1 n(J J )k−ol(J J )l .
(cid:0) (cid:1) l ≡ 2NtrB + − − + (A2)
The
orresponding expansion obtained from TCL2 is n o
J
given by 3
a2 TJ2he tra
e
an be
omputed in the eigenbasis of j and
v3(t)=a0+a2(αt)2+ 2 (αt)4+ (αt)6 , m (see Se
. IIB) yielding a sum of polynomialsin and
2a0 O (62) . However, it turns out that it is easier to use the
(cid:0) (cid:1)
omputational basis of the spin bath
onsisting of the
while NZ2 gives the expansion states
a2
v (t)=a +a (αt)2+ 2 (αt)4+ (αt)6 . s1 s2 sN ,
3 0 2 6a O (63) | i⊗| i⊗···⊗| i (A3)
0
(cid:0) (cid:1) s 0 1
i
where the take on the values or and
In our model the exa
t
oe(cid:30)
ients of the expansion (61)
are found to be 16 σ3(i)|sii=(−1)si|sii. (A4)
a =1, a = 4N, a = N2.
0 2 4
− 3 (64)
With the help of these states the problem is redu
ed to
a2 a
ombinatorial one. Sin
e
Of
ourse, the se
ond order
oe(cid:30)
ient is the same in
N
raellpreoxdpua
nesi
oonrsr,e
wthlyiltehienfoguerntehraolrdneerit
hoeer(cid:30)T
CieLnt2an4o.rHNoZw2- J± = σ±(i) (A5)
ever,inourmodelitturnsoutthattheTCL2approxima- Xi=1
tionismorea

uratebe
ausethefourthorder
oe(cid:30)
ient
we have
2aa22 = 126N2 (65) (J+J−)k = σ+(i1)σ−(i2)σ+(i3)σ−(i4)···σ+(i2k−1)σ−(i2k),
0 i1,X...,i2k
(A6)
found from the solution of the TCL equation is
loser
a
4 where the summation istaken overall possible
ombina-
to the
orre
t fourth order
oe(cid:30)
ient than the
orre- i ,i ,...,i
1 2 2k
tions of the indi
es . Under the tra
e over
sponding
oe(cid:30)
ient
a2 16 thebathwe
ansorttheseindi
es,withoutinter
hanging
2 = N2 theoperatorsbelongingto thesameindex, and
al
ulate
6a 6 (66)
0 the partial tra
es over the various bath spins separately.
Let us denote the partial tra
e over the Hilbert spa
e
i
of the NZ equation (see Fig. 6). Thus we see that it i
a of the th bath spin by tr . For example, we have for
4 k =2
depends
ru
ially on the value of whether TCL2 or
:
NZ2 is better.
σ(1)σ(3)σ(4)σ(1)
Choosing an appropriate intera
tion Hamiltonian and trB + − + −
initial state, one
an easily
onstru
t examples where n o
NZ2 is better thaancoTs(CαLt)2. For example, if v3(t) was a = tr1 σ+(1)σ−(1) tr3 σ−(3) tr4 σ+(4) 2N−3
0

osine fun
tion , then NZ2 would avlr(eta)dy give = 0, n o n o n o
the exa
t solution.a Oexnpt(heαo2tt2h)er hand, if 3 was a (A7)
0
Gaussian fun
tion − , then TCL2 would re- σ(i) = 0 2N−3
produ
etheexa
tsolutionbe
ausethehigher
umulants sin
e tri ± . Note that the fa
tor appears
of a Gaussian fun
tion vanish. (Nn 3o) i I =2
due to − fa
tors of tr { } . These fa
tors arise
I
The features dis
ussed above should be taken into a
-
from the partial tra
es of the unit matri
es in the spin

ount in appli
ations of proje
tion operator methods to
spa
es, whi
h we did not write expli
itly. As a further
k=4
spe
i(cid:28)
open systems. In the general
ase in whi
h an
example, we have for :
analyti
al solution is not known a
areful analyti
al or
numeri
alinvestigationofthehigherordersoftherespe
- trB σ+(1)σ−(3)σ+(1)σ−(1)σ+(3)σ−(1)
toifvteheexTpaCnLsioorntshiestNhZusminetdhisopde,ntshaebilne(cid:29)tuoejnu
degoeftihneitqiaula
loitry- =ntr1 σ+(1)σ+(1)σ−(1)σ−(1) tr3o σ−(3)σ+(3) 2N−2
relations,ortoestimatethetimes
aleoverwhi
hone
an = 0, n o n o
trust the approximationobtained within a given order. (A8)
10
σ(i)σ(i) = 0
be
ause of ± ± . An example of a non-vanishing It should be
lear that the above method of determin-
term is given by ing the
orrelation fun
tions is easily translated into a
Q
k
σ(1)σ(2)σ(2)σ(1) nRukm−leri
al
ode from whi
h one obtains the and the
trB + − + − l in any desired order.
n o
= σ(1)σ(1) σ(2)σ(2) 2N−2
tr1 + − tr2 − + N k N
= 2N−n2, o n o (A9) noFmoirals Q≥k a,nthdeRtelkr−ml iosfrleeapdkreinsegnotredderbyintheosfutmhempaonldys-
with a maximal number of di(cid:27)erent indi
es, be
ause
tr σ(i)σ(i) =1
where we have used that i ∓ ± . these terms have the largest
ombinatorial weight. Af-
n o ter sorting the spin operators, these terms will have the
Inviewofthese
onsiderationswearenowleftwiththe
following form,

ombinatorial problem of determining all nonzero sum-
k l
mandsforthegivenvaluesof and . Asanexample,let
Q
2
us
al
ulate expli
itly the
orrelationfun
tion . From
its de(cid:28)nition we have
1 σ(i1)σ(i1)σ(i2)σ(i2) σ(ik)σ(ik).
Q = J J J J + − + − ··· + − (A12)
2 2NtrB{ + − + −}
1
= σ(i1)σ(i2)σ(i3)σ(i4) .
2N trB + − − + (A10)
i1,iX2,i3,i4 n o
N
There are k di(cid:27)erent ways of assigning the indi
es
The nonzero summands in this expression have the fol- i ,i ,...,i
1 2 k
(cid:0)to(cid:1)this term. Fora(cid:28)xedsetofindi
es, there
lowing stru
ture: k! k!
are · di(cid:27)erenttermsinthesum(A6)whi
hleadtothe
σ+(i)σ−(i)σ+(i)σ−(i) → N possibilities, sortedexpression(Aσ+12),
orrespondingtoaσp−ermutation
σ(i)σ(i)σ(j)σ(j) N(N 1) of the labels of all operatorsand of all2N−okperators.
+ − + − → − possibilities, The tra
e of the expression (A12) yields . Hen
e,
σ+(i)σ−(j)σ+(j)σ−(i) → N(N −1) possibilities, the term of leading order of the polynomial Qk is found
to be
i = j
where 6 in the se
ond and the third line. Colle
ting
these results we (cid:28)nd
1
Q2 = 2N N ·2N−1+2·N(N −1)·2N−2 2−N N k! k!2N−k Nkk!.
N2(cid:0) (cid:1) k · ≈ 2k (A13)
= . (cid:18) (cid:19)
2 (A11)
ARk−silmilar pro
edure must be
arried out to
al
ulate
l . We state some results:
Rk−l N
1 3 3 Asimilarproofholdsfor l . Thus,wehavefor →∞
Q = N N2+ N3, k
3 2 − 4 4 and (cid:28)xed:
3
Q = 2N +5N2 4N3+ N4,
4
− − 2
···
R11 = −12N + 21N2, Qk ≈Rlk−l ≈ 2kk!Nk, (A14)
1 5 3
R1 = N N2+ N3,
2 2 − 4 4
5 23 19 3
R1 = N + N2 N3+ N4,
3 −2 4 − 4 2
.
··· whi
h has been used in Se
. IIC.
1 2
M.A.NielsenandI.L.Chuang,QuantumComputationand D.LossandD.P.DiVin
enzo,Phys.Rev.A57,120(1998).
3
QuantumInformation (CambridgeUniversityPress,Cam- B.E. Kane, Nature 393, 133 (1998).
4
bridge, 2000). D.D. Aws
halom, N. Samarth, and D. Loss, Semi
ondu
-