Quantum Decoherence Scaling with Bath Size: Importance of Dynamics, Connectivity, and Randomness

1.8 MB·English

by  Fengping Jin

Quantum Decoherence Scaling withBath Size:
Importance ofDynamics, Connectivity, andRandomness
Fengping Jin
Institute for Advanced Simulation, Ju¨lich Supercomputing Centre,
Research Centre Ju¨lich, D-52425 Ju¨lich, Germany
Kristel Michielsen
Institute for Advanced Simulation, Ju¨lich Supercomputing Centre,
Research Centre Ju¨lich, D-52425 Ju¨lich, Germany and
RWTH Aachen University, D-52056 Aachen, Germany
Mark Novotny
3
DepartmentofPhysicsandAstronomy,MississippiStateUniversity,MississippiState,MS39762-5167, USAand
1
HPC2 CenterforComputationalSciences,MississippiStateUniversity,MississippiState,MS39762-5167, USA
0
2
Seiji Miyashita
n
Department of Physics, Graduate School of Science,
a
J University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan and
CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan
1

] Shengjun Yuan
h Institute for Molecules and Materials,
p Radboud University of Nijmegen,
-
NL-6525AJ Nijmegen, The Netherlands
t
n
a
Hans De Raedt
u
Department of Applied Physics, Zernike Institute for Advanced Materials,
q
[ University of Groningen, Nijenborgh 4, NL-9747AG Groningen, The Netherlands
(Dated:December11,2013)

1
v The decoherence of a quantum system S coupled to a quantum environment E is considered. For states
7 chosenuniformlyatrandomfromtheunithypersphereintheHilbertspaceoftheclosedsystemS+Ewederive
7 ascalingrelationshipforthesumoftheoff-diagonalelementsofthereduceddensitymatrixofSasafunction
0 of the size DE of the Hilbert space of E. This sum decreases as 1/√DE as long as DE 1. This scaling
≫
0 predictionistestedbyperforminglarge-scalesimulationswhichsolvethetime-dependentSchro¨dingerequation
. foraringofspin-1/2particles,fourofthembelongingtoSandtheotherstoE.Providedthatthetimeevolution
1
drivesthewholesystem fromtheinitialstatetowardastatewhich hassimilarproperties asstatesbelonging
0
totheclassofquantumstatesforwhichwederivedthescalingrelationship,thescalingpredictionholds. For
3
systemswhichdonotexhibitthisfeature,itisshownthatincreasingthecomplexity(intermsofconnections)
1
oftheenvironmentorintroducingasmallamountofrandomnessintheinteractionsintheenvironmentsuffices
:
v toobservethepredictedscalingbehavior.
i
X
PACSnumbers:03.65.Yz,75.10.Jm,75.10.Nr,05.45.Pq
r
a
I. INTRODUCTION quantumsystemreachesthermalequilibriumandhowthiscan
bederivedfromdynamicallaws.
On the one hand there exists a variety of studies explor-
Decoherence of a quantum system S interacting with a
ingthemicrocanonicalthermalizationinanisolatedquantum
quantum environment E is of importance for two reasons.
system [3–6]. On the other hand there exist various studies
First, decoherence of S is the primary requirement for S to
investigatingtheprocessofcanonicalthermalizationofasys-
relaxtoastatedescribedbyacanonicalensembleatacertain
tem coupled to a (much)larger system [3, 7–13] and of two
temperature[1]. Second,decoherenceisarguablythelargest
finiteidenticalquantumsystemspreparedatdifferenttemper-
impedimentforpractical,realizablequantumcomputers[2].
atures[14,15].
The large interest in technological areas like spintron- In previous work [16, 17], we numerically demonstrated
ics, quantum computing and quantum information process- thataquantumsysteminteractingwithanenvironmentathigh
ing have stimulated the theoretical research of quantum dy- temperaturerelaxesto a state describedby the canonicalen-
namics in open and closed interactingsystems. Besides this semble. In thispaperwe focusoninvestigatingthe dynamic
moreapplicationdriveninteresttherepersiststhefundamental propertiesof the decoherence of a quantum system S, being
andstillunansweredquestionunderwhichconditionsafinite a subsystem of the whole system S+E. We do this both
2
witha theoreticalpredictionandbysimulatingthedynamics indetailthespin-1/2Hamiltonianswehavesimulatedtopro-
ofarelativelylargesystemS+E ofspin-1/2particlesusing videacasestudyforthisscaling.
a time-dependentSchro¨dinger equation (TDSE) solver [18].
Inparticular,weinvestigatethescalingofthedegreeofdeco-
herence of S with the size of E, keeping the size of S fixed. A. Timeevolution
BasedonsimilarargumentsasgiveninRef.[19],wefindthat
the degree of decoherence of S decreases as 1/√DE, where ApurestateofthewholesystemS+E evolvesintimeac-
DE isthedimensionoftheHilbertspaceoftheenvironmentif cordingto(inunitsofh¯ =1)
the state of thewhole system ischosenuniformlyat random
fromtheunithypersphereintheHilbertspace. Inthispaper, Y (t) =e−itH Y (0) =(cid:229)DS (cid:229)DE c(i,p,t) i,p , (2)
we denote states chosen uniformly at random from the unit | i | i | i
i=1p=1
hypersphereintheHilbertspaceofthewholesystemby“X”
where the set of states i,p denotesa complete set of or-
andoftheenvironmentby“Y”. {| i}
thonormalstatesinsomechosenbasis,andD andD arethe
Wealsoaddressthequestionunderwhatcircumstancesthe S E
dimensionsof theHilbertspacesof thesystem andthe envi-
wholesystemevolvestoastatewhichhasthesamedegreeof
ronment, respectively. We assume that D and D are both
decoherence as a state “X”. In particular we study the case S E
finite.
in which the initial state of S+E is a direct product of the
The spin Hamiltonian H models a system with N spin-
state ofSandastate“Y”ofE. Iftheinitialstateofthe S
whole|↑s↓y↑s↓teimS+Eisslightlydifferentfromagivenstate“X”, 1/2particlesandanenvironmentwithNE spin-1/2particles.
the dynamicsmay drivethe wholesystem into a state which Thus,DS=2NS andDE =2NE. ThewholesystemS+E con-
tainsN=N +N spin-1/2particlesandthedimensionofits
is very different from the given state “X”, but which is of a S E
Hilbert space is D=D D . In our simulations we use the
similartype.Weinvestigatethroughoursimulationswhenthe S E
spin-up – spin-downbasis. Numerically, the real-time prop-
dynamicsplaysanimportantroleinthedecoherenceinthatit
agation by e itH is carried out by means of the Chebyshev
candriveS+Etoastate“X”byintroducingsmallworldbond −
polynomialalgorithm[22–25],therebysolvingtheTDSEfor
connectionsinE and/orbetweenSandE andbyintroducing
the whole system starting from the initial state Y (0) . This
randomnessintheinteractionstrengthsoftheenvironment. | i
algorithm yields results that are very accurate (close to ma-
The paperis organizedas follows. In Section II our theo-
chineprecision),independentofthetimestepused[18].
reticalresultsforthescalingofthedecoherenceofSarepre-
sented, together with details of the one-dimensional ring of
spin-1/2particleswhichwesimulatetobetterunderstandthe
B. Computationalaspects
scalingprediction. SectionsIII-Vcontainresultsfortheone-
dimensionalringsunderstudy.Inparticularwelookattheef-
Computer memory and CPU time severely limit the sizes
fectofaddingadditionalbonds(SmallWorldBonds,SWBs)
of the quantumsystems that can be simulated. The required
betweenthesystemandenvironmentspinsand/orbetweenen-
CPU timeismainlydeterminedbythenumberofoperations
vironmentspins only (Section IV) and of randomnessin the
tobeperformedonthespin-1/2particles.TheCPUtimedoes
interaction strengths of the Hamiltonian of the environment
notputahardlimitonthesimulation. However,thememory
(Section V). Section VI contains our conclusions and a dis-
of the computer does severely limit which system sizes can
cussionofourresults.
be calculated. The state Y of a N-spin-1/2 system is rep-
resented by a complex-va|luied vector of length D=2N. In
viewofthepotentiallylargenumberofarithmeticoperations,
II. THEORY,MODEL,ANDMETHODS
it is advisable to use 13 - 15 digit floating-point arithmetic
(correspondingto8bytesforarealnumber). Thus,torepre-
Thetimeevolutionofaclosedquantumsystemisgoverned
sentastateofthequantumsystemofN spin-1/2particleson
bythetime-dependentSchro¨dingerequation(TDSE)[20,21].
a conventionaldigitalcomputer,we needa least 2N+4 bytes.
If theinitialdensitymatrix ofan isolatedquantumsystem is
Hence, the amountof memory that is required to simulate a
non-diagonal then, according to the time evolution dictated
quantumsystemwithNspin-1/2particlesincreasesexponen-
by the TDSE, it remains non-diagonal. Therefore, in order
tiallywith N. Forexample,forN =24(N =36)we needat
to decohere the system S, it is necessary to have the system
least 256 MB (1 TB) of memory to store a single arbitrary
S interact with an environmentE, also called a heat bath or
state Y . Inpracticeweneedthreevectors,memoryforcom-
spinbathiftheenvironmentiscomposedofspins. Thus,the | i
municationbuffers,localvariablesandthecodeitself.
HamiltonianofthewholesystemS+E takestheform
Theelementaryoperationsperformedbythecomputational
kernelareoftheform Y U Y whereU isasparseuni-
H=H +H +H , (1)
S E SE | i← | i
tarymatrix with a verycomplicatedstructure(relativeto the
where H and H are the system and environmentHamilto- computationalbasis). Inherenttotheproblemathandisthat
S E
nian respectively, and H describes the interaction between eachoperationU affectsallelementsofthestatevector Y in
SE
| i
thesystemandenvironment.Inwhatfollows,wefirstdescribe anontrivialmanner.Thistranslatesintoacomplicatedscheme
thegeneraltheorythatleadstothescalingofthedecoherence foraccessingmemory,whichinturnrequiresa sophisticated
of the system S with the size of E and S. We then describe MPIcommunicationscheme[26].
3
C. Reduceddensitymatrix matrix r in the representationthat diagonalizesHS. Clearly,
s (t)isaglobalmeasureforthesizeoftheoff-diagonalterms
ofthereduceddensitymatrixintherepresentationthatdiago-
The state of the quantumsystem S is described by the re- e
duceddensitymatrix nalizesHS. Ifs (t)=0thesystemisinastateoffulldecoher-
ence(relativetotherepresentationthatdiagonalizesH ).
S
rˆ(t) Tr r (t), (3)
E
≡
wherer (t)isthedensitymatrixofthewholesystemS+E at D. Scalingpropertyofs
timet andTr denotesthetraceoverthedegreesoffreedom
E
of the environment. In terms of the expansion coefficients Wecanproveascalingpropertyofs byassumingthatthe
c(i,p,t), thematrixelement(i,j) ofthereduceddensityma- final state of the whole system is a state “X”, a state that is
trixreads pickeduniformlyatrandomfromtheunithypersphereinthe
Hilbertspace. Thewavefunctionofthewholesystemreads,
rˆij(t)=TrEp(cid:229)D=E1q(cid:229)D=E1c∗(i,q,t)c(j,p,t)|j,pihi,q| Y =(cid:229)DS (cid:229)DE C E(S) E(E) , (6)
| i i,p i p
(cid:229)DE i=1p=1 (cid:12) E(cid:12) E
= c∗(i,p,t)c(j,p,t). (4) (cid:12) (cid:12)
(cid:12) (cid:12)
p=1 where E(S) E(E) isthesetofeigenvectorsofH
i p S
Wecharacterizethedegreeofdecoherenceofthesystemby (HE), ann(cid:12)d theEorea(cid:16)lna(cid:12)nd imEaog(cid:17)inary parts of Ci,p are real ran-
(cid:12) (cid:12)
dom var(cid:12)iables. The(cid:12)derivation of the scaling behavior fol-
lows Ref. [19]. In particular Eqs. (A8), (A12) and (A23)
s (t)= D(cid:229)S−1 (cid:229)DS r (t) 2, (5) of Ref. [19] are used. We introduce the following short-
ij
v hand notation for the sum over the off-diagonal elements,
u i=1 j=i+1
ut (cid:12)(cid:12)e (cid:12)(cid:12) (cid:229) Di=Sjk ij=(cid:229) Di=S1(cid:229) Dj=S1(1−d ij)k ij foranyk ij,whered ij isthe
wherer (t)isthematrixelement(i,j)ofthereduceddensity Kr6oneckerdeltafunction.Theexpectationvalueisgivenby
ij
e
2
E 2s 2 =E (cid:229)DS (cid:229)DE C C =(cid:229)DS (cid:229)DE E C C C C
(cid:0) (cid:1)=(cid:229)DSi6=(cid:229)Dj(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Ep=1 i∗,1p−j,dp(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)p,p′ E i6=Cji∗,ppC=1j,,pp′C=i1,p′C(cid:16)∗j,pi∗′,p+j,dpp,ip,p′E′ ∗j,Cp′i∗,(cid:17)pCj,pCi,p′C∗j,p′
i=j p=1,p=1
=i(cid:229)D6=6 Sjp(cid:229)D=E1E′(cid:16)(cid:12)C(cid:16)i,(cid:0)p(cid:12)2(cid:12)Cj,p(cid:12)2(cid:1)(cid:17)=(cid:16)i(cid:229)D6=Sjp(cid:229)D=E1DSDE(D1(cid:17)SDE+1) =(cid:16) DDSDSE−+11 = D1E(cid:17)−(cid:17)+D1DS1S , (7)
(cid:12) (cid:12) (cid:12) (cid:12)
where E() denotesthe expectationvaluewith respectto the ForfixedD >1, itfollowsfromEq.(7) thattheenviron-
S
·
probability distribution of the random variablesC . Equa- mentdoesnothavetobeverylargeforEq.(8)tohold,which
i,p
tion (7) does not require any condition on the Hamiltonian is in agreement with Ref. [27]. Nevertheless, the existence
Eq.(1). Forexample,ifH iscomposedoftwo ormoreen- ofanenvironmentiscrucial. Ifthereisnoenvironment,then
E
vironmentsthatdonotcoupletoeachother,butonlyinteract thes approachestoaconstant(seeAppendixA),evenifthe
with the system, in Eq. (7) D is the productof the sizes of wholesystemisinitiallyinastate“X”.
E
theHilbertspacesofalltheenvironments.Inaddition,Eq.(7)
doesnotimposeanyrequirementonthegeometry.
FromEq. (7) it followsthatforanyfixed valueof D >1
S E. Modelandmethod
andD 1,s scalesas
E
≫
FortestingthepredictedscalingofEq.(8)wesimulatesys-
1 1 D 1 1
s E(2s 2)= S− . (8) tems of spin-1/2 particles. For studying the time evolution
≈ √2q √2rDSDE+1 ∼ √2DE of the whole system S+E, we consider a general quantum
spin-1/2modeldefinedbytheHamiltonianofEq.(1)where
Therefore,ifthesizeofthesystemSisfixed(whichisthecase
consideredinthispaper),s decreasesas1/√D forlargeD .
HlaergneceN,Efo.raspin-1/2systems shoulddecreaseEas2−NE/2foEr HS=−Ni(cid:229)S=−11j=(cid:229)NiS+1a =(cid:229) x.y,zJia,jSia Saj, (9)
4
We will see that these two cases show very different scal-
ing properties of the decoherence depending on the initial
state.Wealsoinvestigatetheeffectsofrandomlyaddingsmall
world bonds (SWBs) between spins in the system and envi-
ronmentandbetweenspinsintheenvironment(seeFig.1).
The initial state of the whole system S+E is prepared in
twodifferentways,namely:
“X”: We generate Gaussian random num-
•
bers a(j,p),b(j,p) and set c(j,p,t = 0) =
FIG. 1. (Color online) An example of a spin system used in the { }
(a(j,p) + ib(j,p))/ (cid:229) (a2(j,p)+b2(j,p)).
simulations. TheNS=4systemspin-1/2particlesarecoloredlight j,p
gray(cyan),andtheNE=18environmentspin-1/2particlesarecol- Clearly this procedure gqenerates a point on the
oreddarkgray(red). Thethinblacksegmentsshowtheconnections hypersphereintheD-dimensionalHilbertspace. Alter-
foraone-dimensional ring,whicharetheonlybonds(interactions) natively,wegeneratepointsin thehypercubebyusing
presentincaseIandII(seetext).Thethick(greenandwhite)bonds uniform random numbers in the interval [ 1,1]. Our
show SWBsinHSE. Thisparticular example shows a spin system general conclusions do not depend on th−e procedure
with K =2, where K denotes the maximum number of subsystem
used(resultsnotshown).
spinsthatareconnectedviaSWBswithoneenvironmentspin(thick
white lines, see also Section IV). The medium thick (blue) bonds
showSWBsinHE. UDUDY: The initial state of the whole system is a
•
product state of the system and environment. In this
paper (N =4), we confine the discussion to the state
S
H = N(cid:229)E−1 (cid:229)NE (cid:229) W a Ia Ia , (10) UDUDY,whichmeansthatthefirst,second,third,and
E − i=1 j=i+1a =x,y,z i,j i j fourthspinareintheup,down,up,anddownstatere-
spectively,andthestateoftheremainingspinsisa“Y”
H = (cid:229)NS (cid:229)NE (cid:229) D a Sa Ia . (11) stateinthe(D/24)-dimensionalHilbertspace. The“Y”
SE −i=1j=1a =x,y,z i,j i j stateoftheenvironmentispreparedinthesamewayas
the“X”stateofthewholesystem.
Here,SandIdenotethespin-1/2operatorsofthespinsofthe
system and the environment,respectively(we use unitssuch
a a
that h¯ and k are one). The spin componentsS and I are
B i j
related to the Pauli spin matrices, for example Six is a direct III. SCALINGANALYSISOFs
productofidentitymatricesandthePaulispinmatrix 1s x=
2
0 1
1 inpositioniofthedirectproductwith1 i N . All simulations are carried out for a system S consisting
2(cid:18)1 0 (cid:19) ≤ ≤ S of four spins (NS = 4) coupled to an environment E with
Forthegeometryofthewholesystem,wefocusontheone-
the number of spins N ranging from 2 to 30. The interac-
E
dimensionalringconsistingofasystemwithN =4spin-1/2 a
S tion strengths J with 1 i N 1 are always fixed to
particlesandanenvironmentwithNE spin-1/2particles, see J= 0.15. Fori,ci+as1eIallno≤n-ze≤roW Sa−andD a arerandomly
Fig. 1. Past simulations have shown that a high connectiv- − i,j i,j
generatedfromtherange[ 0.2,0.2]. ForcaseIIallnon-zero
ity spin-glass type of environment is extremely efficient to W a and D a are equal to−J = 0.15 (isotropic Heisenberg
decohere a system [16, 28–30], so we may expect that the i,j i,j −
model).
one-dimensionalring is one of the most difficult geometries
toobtaindecoherenceinshorttimes.
We assume that the spin-spin interaction strengths of the
a
system S are isotropic, J = J and that only the nearest-
i,j
neighborinteractionstrengthsW a andD a arenon-zero.Note A. Verificationofscaling:casesIandIIwith“X”
i,j i,j
thatforaringthereareonlytwobondswithstrengthD a con-
i,j
nectingSandE. Wedistinguishtwocases: We corroboratethe scalingpropertyofEq.(8)bynumeri-
callysimulatingthequantumspinsystem(seeEq.(9)through
CaseI:Thenon-zerovaluesofW a andD a aregener- (11)). Ifwechoosetheinitialstateofthewholesystemtobe
• i,j i,j
ated uniformlyat randomfrom the range [ W ,W ] and an“X”state,thenduringthetimeevolutionthewholesystem
[ D ,D ],respectively. − will remain in the state “X”. Hence, the condition to derive
−
Eq. (8) are fulfilled. Fig. 2 demonstrates that the numerical
Case II: All non-zero values of the model parameters results for both cases I and II agree with Eq. (8). In par-
• areidentical,W a =JandD a =J. Thiscorrespondsto ticular the insets in Fig. 2 show that for both cases I and II,
i,j i,j
a uniformisotropicHeisenbergmodelwith interaction ln(2s ) N /2,andthats scalesas1/√D evenifN =2
E E E
≈−
strengthJ. andN =4(N <N ).
S E S
5
100 100
10-1 10-1
10-1 10-3 10-1 10-3
10-5 10-5
2 10 20 30 2 10 20 30
10-2 NE 10-2 NE
(t) (t)
s s
10-3 10-3
10-4 10-4
0 200 400 600 800 1000 1200 1400 0 1000 2000 3000 4000 5000 6000 7000 8000
t t
100
10-1 FIG.3. Simulationresultsfor s (t)(seeEq.(5))for caseIfordif-
10-1 10-3 tfheerewnthsoilzeessyNste=mNisEU+D4UoDfYth(esewehteoxlet).syCsutermve.sfTrhoemitnoiptiatlosbtaottetoomf
10-5 correspond tosystemsizesranging fromN=6toN=34insteps
10-2 2 10N 20 30 of2. Theinsetshowsthetime-averagedvaluesofs (t)(pluses)asa
(t) E functionofthesizeNEoftheenvironment.Thedataobeythescaling
s propertyofEq.(8)(solidline).
10-3
10-4
100
10-1
0 200 400 600 800 1000 1200 1400
10-2
t
10-1 10-3
FIG. 2. Simulation results for s (t) (see Eq. (5)) for case I (top) 12 18 24 30
and case II (bottom) for different sizes N =NE+4 of the whole (t) NE
system.Theinitialstateofthewholesystemis“X”(seetext).Curves s 10-2
fromtoptobottomcorrespondtosystemsizesrangingfromN=6
toN=34 instepsof 2. Theinsetsshow thetime-averagedvalues
coofnsfi(rtm)in(pgluthseest)heaosreatifcuanlcptrioednicotfiotnheofsEizqe.(N8E)(osfoltihdeliennev).ironment, 10-3
0 200 400 600 800 1000 1200 1400
t
B. Differentinitialconditions
FIG.4. SameasFig.3forcaseIIinsteadofcaseI.Curvesfromtop
tobottomcorrespondtosystemsizesrangingfromN=16toN=34
Weinvestigatetheeffectsofthedynamicsbypreparingthe
instepsof2.Thesolidlineintheinsetisaguidetotheeyes.
initialstateofthewholesystemsuchthatitisslightlydiffer-
ent from “X”. The initial state of the whole system is set to
UDUDY. IncontrasttoFig.2,wewillseethatthetwocasesI
andIIbehavedifferently.
2. CaseIIandUDUDY
We consider the case in which the whole system is de-
1. CaseIandUDUDY scribed by the isotropic Heisenbergmodel(Ja =W a =
i,i+1 i,i+1
D a =J). In Fig. 4 we present simulation results for dif-
i,i+1
In Fig. 3, we present the simulation results for case I, the ferentsystemsizesN=N +4rangingfrom16to34. From
E
couplings in the Hamiltonians H and H are chosen uni- Fig.4, itisseenthatthebehaviorforcaseIIistotallydiffer-
E SE
formlyatrandom. ThesizeN=N +4ofthewholesystem ent from that of case I (see Fig. 3). In particular, s (t) does
E
rangesfrom 6 to 34. An average overthe long-timestation- not scale with the dimension of the environment. From the
ary steady-state values of s (t) still obeys the scaling prop- present numerical results, we cannot make any conclusions
erty of Eq. (8), showing that s decreases as 1/√D , where aboutthelimitforlargeN . Howeverifs (t)approacheszero
E E
DE =2NE. IfNE ¥ ,s 0. Thissuggeststhatinthether- as NE ¥ (see the fifth column of Table I) it does so very
→ → →
modynamicallimitthesystemSdecoherescompletely. slowly.
6
100 TABLEI.Thetimeaverageofs (t)inthestationaryregimeshown
inFigs.2,3and4.
prediction caseI caseII
10-1 NE ofEq.(8) UDUDY “X” UDUDY “X”
s(t) 42 31..379078××1100−−11 31..471466××1100−−11 31..377257××1100−−11 31..373141××1100−−11
10-2 68 84..525749××1100−−22 84..853948××1100−−22 84..523862××1100−−22 84..429625××1100−−22
10 2.139×10−2 2.286×10−2 2.153×10−2 2.121×10−2
12 1.070×10−2 1.149×10−2 1.071×10−2 1.254×10−2 1.061×10−2
10-3 0 1000 2000 3000 4000 5000 6000 1146 52..364794××1100−−33 52..789656××1100−−33 52..365778××1100−−33 63..795967××1100−−33 52..364663××1100−−33
t 18 1.337×10−3 1.430×10−3 1.349×10−3 2.694×10−3 1.343×10−3
20 6.686×10−4 7.065×10−4 6.736×10−4 2.204×10−3 6.641×10−4
g2Fr2IeGae.nn5dd.Na(sCSho=eldo4rl.inoRen:elidtnhese)oilSnidiimtilaiunlleas:ttiatotheneirisens“iutXilat”ls(sfstoearetestei(sxtU)t).fDoUrcDaYse(IsefoerteNxt=); 22222468 3184....363147573289××××11110000−−−−4455 3194....570546052651××××11110000−−−−4455 3184....363157652481××××11110000−−−−4455 1111....975402989291××××11110000−−−−3333 3184....362145877836××××11110000−−−−4455
30 2.089×10−5 2.338×10−5 2.107×10−5 1.379×10−3 2.104×10−5
10-1
10-2
10-3
canunderstandthe verydifferentbehaviorof cases IandII,
Ds 10-4 seeFigs.3and4,byconsideringthestationarystatesthatare
obtained.Figure5showsthatthefinalvaluesofs (t)forcaseI
10-5 are very close for both initial states “X” andUDUDY. This
suggests that the final stationary state in case I has proper-
ties similar to those of a state “X”, and hence case I obeys
10-6
0 5 10 15 20 25 30 thescalingpropertyofEq.(8)toagoodapproximation. The
N time-averagedvaluesof s (t)in Figs. 2, 3 and4, denotedby
E
s , are listed in Table I. From Table I, we see thatthe values
of s for case II with an initial state UDUDY are very dif-
FIG. 6. (Color online) Difference D s between the time-averaged
ferent from those with an initial state “X”, and do not show
values of s (t) for the initial stateUDUDY and “X” of the whole
the scaling property of Eq. (8). Thus, the numerical results
system(seeTableI)asafunctionofthesizeoftheenvironmentN .
E suggestthatthe initialstate andthe randomnessof the inter-
Pluses: caseI;circles: caseII.Thedottedlineisalinear fittothe
action strengths play a very important role in the dynamical
data (pluses) for theUDUDY initial state, excluding thefirst three
datapoints,resultinginD s =0.049/√DE. evolutionofthedecoherenceofasystemcoupledtoanenvi-
ronment.Inparticular,forcaseII,startingfromastate“X”the
time-averagedvaluesofs (t) scale as s 1/√D , butsuch
E
≈
C. Computationaleffort scalingisnotobservedforstartingfromastateUDUDY.
Inthispaper,thelargestnumberofspinsthatwesimulated FromTableI,itisseenthatthevaluesofs forcaseIwith
is N =34. Using the Chebyshev polynomial algorithm and theinitialstateUDUDY arealwaysslightlylargerthanthose
a large time step (t 10p ), the N =34 simulation for the withtheinitialstate“X”. Therefore,itisinterestingtoexam-
≈
bottomcurvesinFig.2(uptoatimet 600)tookabout0.3 ine the differenceD s between the values of s for the initial
≈
million core hours on 16384 BG/P (IBM Blue Gene P) pro- statesUDUDY and “X”. Figure 6 shows that D s for case I
cessors,using1024GBofmemory.Similarly,ittookabout4 (redpluses)alsoscalesas1/√D (dottedline),exceptforthe
E
millioncorehourstocompletetheN=34curveinFig.3(up firstthreedatapoints,whichisprobablyduetolargefluctua-
toatimet 8000). tionsin thecalculationsforthesesmallsystem sizes. There-
≈
fore, the dynamics of case I will drive the system to a state
“X”onlywhentheenvironmentapproachesinfinity. Figure6
D. Summary:initialstatedependence alsoshowsthatD s forcaseII(circles)isalmostconstantfor
system sizes N rangingfrom 16 to 34. Hence, it is unlikely
Foraninitialstate“X”ofthewholesystemthescalingofs , thatcaseIIwiththeinitialstateUDUDY willdecohere,even
asgivenbyEq.(8),worksextremelywellforbothcaseIand if thesimulationscouldbe performedformuchlongertimes
caseII,asseeninFig.2. WhentheinitialstateisUDUDY,we andforlargersystemsizes.
7
IV. CONNECTIVITY:RINGWITHSMALLWORLD nothaveadramaticeffectons (t)andhasverylittleeffectat
BONDS earlytimes(seeFig.8a).
Just as for case I, it is seen that adding a few SWBs ex-
We investigate the effects of adding small world bonds clusively to HSE for a spin configuration with K =1 signif-
(SWBs)totheHamiltoniansH or/andH forbothcaseIand icantly decreases the time to approach the steady state, and
SE E
caseII(seeFig.1). ToanalyzetheadditionofSWBstoHSE that the SWBs in HSE also lead to a decrease in s (t) for a
we distinguishbetweenspinsystemswith K<2andK 2, fixed time even at early times (see Fig. 8b). For spin con-
where K denotes the maximum number of subsystem s≥pins figurationswith K =2 case I and case II seem to havesimi-
thatareconnectedviaSWBswithoneenvironmentspin. This lar decoherencepropertiesif SWBs are addedexclusivelyto
distinctionismotivatedbythedistinctdecoherencecharacter- HSE,asseenbycomparingFig.7candFig.8c. However,con-
istics forsystems with K <2 and K 2 for case I (see next nectinginadditioneachpairofnon-neighboringenvironment
subsection). An exampleofa spin co≥nfiguartionwith K =2 spinsbyisotropicSWBsdrivesthecurvesveryfarawayfrom
isshowninFig.1. Inparticular,weareinterestedinwhether thevalueofs (t)foraninitialstate“X”(seeFig.8d).
systemswithSWBswillexhibitthesamescaling,andwhether
theywilldecoherefromaninitialstatefasterthaneitherofthe
cases studied thusfar. The additionof manySWBs changes C. Summary:SWBs
thegraphfroma one-dimensionalringtoa graphwith equal
bondlengthsthatcanonlybeembeddedinhighdimensions. AddingSWBstoH or/andtoH changestherateofde-
SE E
Theinitialstates arealwaysUDUDY. Furthermore,inorder coherenceasseenbytheapproachtotheasymptoticvaluefor
nottochangetoomanyparameterssimultaneouslywestartall s (t). In case II,addingisotropicSWBs to H or H effec-
SE E
simulationsfromthesamestate“Y”oftheenvironment.Fur- tivelyalterssomespin-spincorrelationsleadingtoadecrease
thermore,afterchoosingtherandomlocation(andcouplings in the steady-state value of s (t). However, this decrease is
W a andD a forcaseI)ofthefirstSWBwepreservethisbond notsufficienttoreachthesteady-statevalueofs (t)thatcom-
when adding additional SWBs. We will see that case I and plies with the predictionEq. (8). Addingisotropic SWBs to
caseIIstillbehaveverydifferently. H andconnectinginadditioneachpairofnon-neighboring
SE
environmentspins byisotropic SWBs drivesthe curvesvery
farawayfromthevalueofs (t)foraninitialstate “X”,even
A. CaseIandSWBs muchfurtherawaythanthesteady-statevalueforaringwith-
outSWBs. IncontrasttocaseIsystemswithK<2andK 2
Forinvestigatingtheuniversalityofthefinalvalueofs (t) donotbehavesignificantlydifferent. ≥
we add SWBs (randomcouplingsin the interval[ 0.2,0.2]) Comparing case II with case I for K < 2, we conclude
−
intheHamiltonianH or/andH forcaseI,andperformsim- that without introducing the randomness in the x, y, z com-
SE E
ulations for N =24 with N =4. From Fig. 7a, we see that ponentsofthespin-spincouplings,thedynamicscannotdrive
S
adding more and more SWBs to H speeds up the decoher- thesystemtodecoherenceiftheinitialstateisdifferentfrom
E
ence process and that the final value of s (t) corresponds to astate“X”. Increasingthecomplexityoftheenvironmentby
theonegivenbyEq.(8). Asseenintheinset, addingSWBs addingisotropicSWBsbetweenallnon-neighboringenviron-
to H has no noticeable effect on the early time behavior of mentspinsdoesnothelpinthisrespect,evenonthecontrary.
E
s (t). However,forcaseIandconfigurationswithK 2,increasing
≥
Adding SWBs exclusively to H speeds up the deco- thecomplexityoftheenvironmentbyaddingSWBsbetween
SE
herence process even further and even at early times clear allpairsofnon-neighboringenvironmentspinsallowsthedy-
changes in s (t) can be observed (see Figs. 7b, c). For spin namicstodrivethesystemtodecoherence.
configurations with K =1, s (t) reaches the value given by For both case I and case II, adding SWBs in HSE and HE
Eq.(8)forsufficientlylongtimes,ascanbeseenfromFig.7b. separatelyspeedsup the decoherencein thatit evolvesmore
However,forconfigurationswithK=2(seeFig.7c)orK>2 quickly to a stationary state. The asymptotic value for s (t)
(results not shown) s (t) does not obey the scaling property isapproachedmuchfasterwhenaddingSWBstoHSE instead
Eq. (8). Restoring this scaling property seems to require of HE, and the SWBs in HSE also affects (t) at early times.
an environment that is much more complex than the one- Thus a random SWB coupling to the system via HSE is the
dimensionaloneasindicatedbyFig.7dinwhichwepresent mosteffectivewaytodecreasethetimefordecoherence.
simulation results for the case that SWBs between all non-
neighboringenvironmentspinshavebeenadded.
V. RANDOMNESSINTHEENVIRONMENT
B. CaseIIandSWBs SectionIIIAshowsthatfortheinitialstate“X”thescaling
predictedby Eq. (8) is confirmedboth for case I and case II
For case II, isotropic SWBs are added to H or/and H . (see Fig. 2). However,sectionIIIB showsthatstarting from
SE E
From Fig. 8, it is clear that even for long times none of the theinitialstateUDUDY thisscalingisapproachedas1/√D
E
curvesapproachthe dotted horizontalline, the valueof s (t) for case I (see Figs. 3 and 6) but not for case II (see Figs. 4
foraninitialstate“X”. AddingSWBsexclusivelytoH does and6).SectionIVshowsthataddingSWBsincaseIIdoesnot
E
8
100 100
(a) 100 (b) 100
10-1
10-1 10-1
10-2
(t) 10-1 (t) 10-3
s 10-2 0 10 20 30 40 50 s 10-2 0 10 20 30 40 50
10-3 10-3
0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000
t t
100 100
(c) 100 (d) 100
10-1 10-1
10-1 10-1
10-2 10-2
(t) 10-3 (t) 10-3
s 10-2 0 10 20 30 40 50 s 10-2 0 10 20 30 40 50
10-3 10-3
0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000
t t
FIG.7. (Coloronline)Simulationresultsofs (t)forcaseIwithN=24andNS=4withSWBsadded. TheinitialstateisUDUDY. The
dottedhorizontallinerepresentsthevalueofEq.(8).Redsolidline:ringwithoutSWBs.(a)RandomlyaddedSWBsbetweennon-neighboring
environment spins. Greenlong-dashed line: oneSWB;orangedottedline: twoSWBs; purpleshort-dashed line: four SWBs; bluedotted-
dashedline: eightSWBs. (b)RandomlyaddedSWBsbetweenthesystemandenvironmentspinssuchthatK=1. Greenlong-dashedline:
oneSWB;orangedottedline: twoSWBs;purpleshort-dashedline: fourSWBs;bluedotted-dashedline: eightSWBs. (c)Randomlyadded
SWBsbetweenthesystemandenvironmentspinssuchthatK=2.Greenlong-dashedline:twoSWBs;orangedottedline:fourSWBs;purple
short-dashed line: sixSWBs; bluedotted-dashed line: eightSWBs. (d)Sameas(c)except thateachpairofnon-neighboring environment
spinsisconnectedbyaSWB.Insets:timeevolutionforshorttimes.
significantlychangethelong-timebehaviorofs (t)approach- inset of Fig. 9. The inset shows that even one randombond
ingthepredictedvalueofEq.(8). Thereforethenaturalques- sufficestorecovertheasymptoticvalueEq.(8). Howeverthe
tion to ask is how much randomness is required for s (t) to timescaletoreachtheasymptoticvalueofs canbecomeex-
obeythescalingrelationEq.(8). Toanswerthisquestion,we tremelylong. Weleavethequestionofhowfasttheapproach
startfromtheisotropicHeisenbergring(caseII)andreplace tothepredictedvalueofs isforfuturestudy.
the interactionstrengthsof a fewrandomlychosenbondsby
randomW a (seeEq.(10)). Forunderstandingthebehaviorofs (t)incaseIIwithran-
i,j domness,weinvestigatetheindividualcomponentsofthere-
Figure 9 presents the simulation results for s (t) by intro- duced density matrix r for the ring system. We study the
ducing 1, 2, 4, 6 and 8 random bonds in the environment additionofone,two,uptoeightrandomlyreplacedbondsin
Hamiltonian H of Eq. (10). The interaction strengths W a theenvironment.Recallthatoncethepositionforonerandom
E i,j e
of these randomly selected bonds are drawn randomly from bond is chosen, this is also one of the random bonds when
a uniform distribution in [ 0.2,0.2]. Furthermore, the ran- therearetwoormorerandombonds. Similarly,thelocations
−
domlyselectedbondforthecasewith1randombondisalso oftherandompositionsfora largenumberofrandombonds
arandombondforthecasewith2andmorerandomlychosen includethesamepositionsandstrengthsasforasmallernum-
bonds, thereby not changingtoo many parametersat a time. berofrandombonds. Furthermore,thesameinitialstate“Y”
Simulations up to time t =6000 show that introducing 4, 6 of the environment is chosen for all simulations. We stud-
and8randombondsleadsthesystemtorelaxtothepredicted iedtheeffectofvaryingthepositionsoftherandomlychosen
valueofs (seeEq.(8)).Fortimesuptot=6000theeffectof bonds and of different initial states “Y” for the environment
oneortworandombondsisnotapparent.Thereforeforthese for a couple systems and did not find significant changes in
twocasesweperformedextremelylongrunsasshowninthe ourobservations.
9
100 100
(a) 100 (b) 100
10-1 10-1 10-1 10-1
10-2
(t) 10-2 10-2 (t) 10-2 10-3
s 0 10 20 30 40 50 s 0 10 20 30 40 50
10-3 10-3
10-4 10-4
0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400
t t
100 100
(c) 100 (d) 100
10-1 10-1 10-1 10-1
10-2
(t) 10-2 10-3 (t) 10-2 10-2
s 0 10 20 30 40 50 s 0 10 20 30 40 50
10-3 10-3
10-4 10-4
0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400
t t
FIG.8.(Coloronline)Simulationresultsofs (t)forcaseIIwithN=26andNS=4withisotropicSWBsadded.TheinitialstateisUDUDY.
The dotted horizontal linerepresents the value of Eq. (8). Red solid line: ring without SWBs. (a) Randomly added SWBsbetween non-
neighboringenvironmentspins.Greenlong-dashedline:twoSWBs;orangedottedline:fourSWBs;purpleshort-dashedline:sixSWBs;blue
dotted-dashedline:eightSWBs.(b)RandomlychosenSWBsbetweenthesystemandenvironmentspinssuchthatK=1.Greenlong-dashed
line: twoSWBs;orangedottedline: fourSWBs;purpleshort-dashedline: sixSWBs;bluedotted-dashedline: eightSWBs. (c)Randomly
chosen SWBsbetween the system and environment spins such that K =2. Green long-dashed line: two SWBs; orange dotted line: four
SWBs;purpleshort-dashedline: sixSWBs;bluedotted-dashedline: eightSWBs. (d)Sameas(c)exceptthateachpairofnon-neighboring
environmentspinsisconnectedbyaSWB.Insets:timeevolutionforshorttimes.
Figure 10 presents the results of the time evolution of the VI. CONCLUSIONSANDDISCUSSION
absolute value r of the individual components of the re-
ij
| |
duced density matrix. For completeness we show both the ThemaintheoreticalresultofthecurrentpaperisEq.(8)for
diagonalcomponeentsandtheoff-diagonalcomponents. Fig- thedecoherenceofaquantumsystemScoupledtoaquantum
ure 10 shows that most of the 120 off-diagonalcomponents environmentE. Forstudyingdecoherencewe examines (t),
quicklyrelaxtoasmallvalue(114blacklinesinFig.10(b)- which is the square root of the sum of all the off-diagonal
(e)). The slowest decaying r ij are plotted in red. There elements of the reduced density matrix r for S in the basis
are six such components. In|the|steady state all |r ij| oscil- thatdiagonalizestheHamiltonianHSofthesystemS.Wefind
late but have nearly the sameetime-averagedvalue, in agree- (seealsoEq.(8))that e
mentwiththemean-field-typeargumentgiveninAppendixB.
e
Thus, only a few r ij are responsiblefor the lack of scaling s 1 1 1 , (12)
of s in case II wh|en|starting from the initial stateUDUDY, ≈ √2DE (cid:18) −2DS(cid:19)
andalsoforthelongtimesrequiredtoapproachthepredicted
e wherethereduceddensitymatrixr forSisaD D matrix
valueofEq. (8) ofs (t)in the case thatthere areoneor two S× S
whilethedensitymatrixofthewholesystemS+E isaD D
randombonds. ×
matrix with D=D D . Thus D does not have to be very
S E Ee
large in order for the predicted scaling to hold, in particular
the scaling requiresD 1 D 1. In additionthe scaling
E ≫ ≫ −S
requiresthatS+E isdrivenfromaninitialwavefunctionto-
ward a steady state whichis well describedby a state which
wecalled“X”.
10
100
10-2 10-1
(a) (b)
10-1 10-3 ~r||ij1100--53
10-7
(t) 10-2 10-4
s 10000 50000
10-1
10-3 (c) (d)
~|ij10-3
r|
10-5
10-4
0 1000 2000 3000 4000 5000 6000 10-7
t
FIG. 9. (Color online) Simulation results of s (t) obtained by se- 10-1
lectivelyreplacingisotropicspin-spininteractionsbyrandombonds. (e) (f)
The size of the system and whole system are NS=4 and N =26, ~|ij10-3
respectively. TheinitialstateisUDUDY. Redsolidline: 1random r| 10-5
bond; greenlong-dashed line: 2randombonds; purpledottedline:
4 random bonds; orange short-dashed line: 6 random bonds; blue 10-7
0 2000 4000 6000 0 2000 4000 6000
dotted-dashed line: 8 random bonds. Inset: simulation results for t t
oneandtworandombondsforlongtimes.
FIG.10.(Coloronline)Timeevolutionofthecomponentsr ijofthe
reduceddensitymatrixofthesystemwithN=26andNS=4. The
initialstateisUDUDY. (a)CaseII. StartingfromcaseII, 1(b), 2
Wehaveperformedlarge-scalereal-timesimulationsofthe e
time-dependentSchro¨dingerequationforNS spinsinthesys- (licn)e,s4:(ddi)a,g6on(ea)lacnodm8po(nf)enratsndormii ;broenddslianrees:inatrllod6ucsleodwilnyHdEec.aByilnuge
temandNE spinsintheenvironment.Wehavesimulatedspin- componentsfor r ij foroner|and|ombond;blacklines:allother114
1/2systemswithN=NS+NE uptoN=34,allwithNS=4. off-diagonalcom|pon|ents r ij .e
Startingfromastate“X”forS+E thesimulationsagreevery | |
e
wellwiththescalingpredictionEq.(12),asshowninFig.2.
e
In AppendixC we demonstratethatin this case notonlythe
off-diagonalelementsofr obeyascalingrelationbutalsoits berequiredtoapproachthestate“Z”isduetoonlyafewoff-
diagonalelementsobeyascalingrelation,althoughadifferent diagonalelements of r , as seen in Fig. 10. We find that the
one. approachto the state “Z” canbe spedup byaddingrandom-
e
Thereforeaslongasthedynamicsdrivestheinitialstateto nesstoE (Figs.9ande10).
astate“Z”whichhassimilarpropertiesas“X”thescalingre- What do our results say about the approach to the quan-
lationEq.(12)shouldhold.Thenextstepistoexamineunder tumcanonicalensemble?Thecanonicalensembleisgivenby
whatconditionsourtestquantummodelisdriventothestate the diagonalelements of the reduceddensity matrix r if the
“Z”, andstudy the time scale neededto relaxfrom an initial off-diagonalelements(asmeasuredbys (t))canbeneglected
statetothestate“Z”. Fortheone-dimensionalquantumspin- [1,17]. AslongasE hasafiniteHilbertspaceDE ourescaling
1/2ring we find that homogeneouscouplingsdo notlead to resultscan be usedto arguethatin a strictsense, thesystem
an evolution to the state “Z” (Fig. 4), and hence the scaling willnotbeinthecanonicalstateunlessDE ¥ . However,if
→
as 1/√D is not observed. This conclusion is not modified the canonical distribution is to be a good approximation for
E
if some randomly chosen homogeneous small world bonds some temperatures T up to some chosen maximum energy
areadded(Fig.8). Also systemswithrandomcouplingsand Ehold>0,thenthisrequiresthatexp( Ehold/kBT) s which
− ≫
randomsmallworldbondsbetweensystemandenvironment givesforourspin-1/2systemkBT 2Ehold/[NEln(2)]. For
≫
spinssuchthatthemaximumnumberofsystemspinsthatin- thisargumenttoholdinthecanonicaldistributiontheenergies
teractwithoneenvironmentspinistwoorlargerdonotevolve aretakentobepositivevaluesabovethegroundstateenergy.
toastate“Z”(Fig.7c). Inthiscase,theenvironmentrequires Thislackofthermalizationatlowtemperaturesforsmallsys-
amorecomplexconnectivitythanthesimpleone-dimensional temsissupportedbysimulationsinRef.[17].
oneinordertoobservethescalingas1/√D (Fig.7d).There- What do our results say about trying to prolong the time
E
fore,althoughwefindthatsomerandomnessintheinteraction todecoherenceinordertobuildpracticalquantumencryption
strengthsin E or between S and E the dynamicsis very im- or quantum computational devices? The important thing is
portanttodrivethewholesystemtowardthestate“Z”,asseen to ensure that the system is not driven toward the state “Z”,
inFigs.3,5,7a,b,and9itisnotalwayssufficient. Moreover oratleastthatittakesa verylongtimetoapproachthestate
itmaytakealongtimetoevolvetowardthestate“Z”ifthere “Z”. This can be achieved by changing the Hamiltonian of
isonlyalittlerandomness(Fig.9)oriftheenvironmentE is the system, H =H +H +H , such that it has very small
S E SE
large(the N =34results of Fig. 3). The long time thatmay randomness particularly in the coupling between the system