Table Of ContentAn efficient scheme for numerical simulations of the spin-bath decoherence
V. V. Dobrovitski1 and H. A. De Raedt2
1Ames Laboratory, Iowa State University, Ames IA 50011, USA
2Department of Applied Physics/Computational Physics,
Materials Science Centre, University of Groningen,
Nijenborgh 4, NL-9747 AG Groningen, The Netherlands
(Dated: February 1, 2008)
We demonstrate that the Chebyshev expansion method is a very efficient numerical tool for
studying spin-bath decoherence of quantum systems. We consider two typical problems arising
in studying decoherence of quantum systems consisting of few coupled spins: (i) determining the
pointer states of the system, and (ii) determining the temporal decay of quantum oscillations. As
our results demonstrate, for determining the pointer states, theChebyshev-based schemeis at least
3 a factor of 8 faster than existing algorithms based on the Suzuki-Trotter decomposition. For the
0 problemsofsecondtype,theChebyshev-basedapproachhasbeen3–4timesfasterthantheSuzuki-
0 Trotter-based schemes. This conclusion holds qualitatively for a wide spectrum of systems, with
2
different spin baths and different Hamiltonians.
n
a PACSnumbers: 75.10.Jm,02.60.Cb,75.10.Nr,03.65.Yz
J
3
I. INTRODUCTION properties of the central system, we can consider the re-
2
duced density matrix of the central system, i.e. the ma-
1 trix ρ (t) = Tr Ψ(t) Ψ(t), where Tr means trac-
Recently,agreatdealofattentionhasbeendevotedto S B B
| ih |
v ing over the environmental degrees of freedom. Initially,
thestudyofquantumcomputation[1,2]. Formanyphys-
0 ρ (0) = ψ ψ , the system is in pure state, and its
icalsystems,basicquantumoperationsneededforimple- S 0 0
3 density m|atriihx is| a projector, i.e. ρ2(0) = ρ (0). At
1 mentationofquantumgateshavebeendemonstrated. To S S
t > 0, this property is lost, and the system appears in a
1 be practical,aquantumcomputershouldcontainalarge
0 number of qubits (some estimates give up to 106 qubits mixed state. It has been shown that, even for relatively
3 [3]), and be able to perform many hundreds of quantum small integrable and non-integrable systems, the mixing
0 is sufficient for the time-averaged, quantum dynamical
gate operations. However, these requirements are not
/ propertiesofthesubsystemtoagreewiththeirstatistical
h easytosatisfyinexperiments. Arealtwo-statequantum
p system is different from the ideal qubit. The system in- mechanics values [11]. Diagonalizing the density matrix
- teracts with its environment, and this leads to a loss of ρS, we can find the (instantaneous) states of the system
t
n phase relations between different states of the quantum qi(t) and (instantaneous) occupation numbers of these
| i
a computer (decoherence) [4, 5, 6, 7], causing rapid accu- states wi(t). It is generally assumed (and is true for all
u cases we know) that in “regular” situations, the states
mulationoferrors. Detailedtheoreticalunderstandingof
q
q (t) quickly relax to some limiting states p , called
the decoherence process is needed to prevent this. i i
: | i | i
v “pointer states”. This process (decoherence) is, in most
More generally, decoherence is an interesting many-
i cases, much faster than the relaxation of the occupation
X body quantum phenomenon which is fundamental for
numbersw (t)to theirlimitingvalues(whichcorrespond
many areas of quantum mechanics, quantum measure- i
r
to thermal equilibrium of the system with the bath).
a menttheory,etc [4, 5]. It alsoplaysanimportantrolein
solidstatesystems,andmightsuppressquantumtunnel- The theoretical description of decoherence, i.e. a de-
ing of defects in crystals [7], spin tunneling in magnetic scription of the evolution of the central system from its
molecules and nanoparticles [8, 9], or destroy Kondo ef- initialpure state ψ0 to the final mixedstate, andfinding
fect in a dissipationless manner [10]. I.e., decoherence the final pointer states pi , is a very difficult problem
| i
in many physical systems can have experimentally de- of quantum many-body theory. Some simple models can
tectable(andsometimesconsiderable)consequences,and be solved analytically, for some more complex models
extensivetheoreticalstudiesofdecoherenceareneededto different approximations can be employed, such as the
understand behavior of these systems. Markovapproximationfor the bath, which assumes that
the memory effects in the bath dynamics are negligible.
Formally speaking, decoherence is a dynamical devel-
A special case of environment consisting of uncoupled
opmentofquantumcorrelations(entanglement)between
oscillators,so-called“bosonbath”,isalsoratherwellun-
the central system and its environment. Let us assume
derstoodtheoretically. But,althoughthemodelofboson
that initially the central system is in the state ψ and
0
| i bath is applicable for description of a large number of
the environment is in the state χ , so that the state
0
| i possible types of environments (phonons, photons, con-
of the compound system (central system plus bath) is
duction electrons, etc.) [7], it is not universal.
Ψ(t=0) = ψ χ . Inthecourseofdynamicalevo-
0 0
| i | i⊗| i
lution, the direct product structure of the state Ψ(t) A particularly important case where the boson bath
| i
is no longer conserved. If we need to study only the description is inapplicable is the decoherence caused by
2
anenviromentmadeofspins,e.g.nuclearspins,orimpu- of the system made of many spins 1/2. Such systems
rityspins(socalled“spinbath”environment). Similarly, arealsoofprimaryinterestforstudyingmanysolidstate
decoherencecausedbysomeothertypesofquantumtwo- problems, since an electron is a particle with the spin
levelsystems canbe described in terms of the spin bath. 1/2, and its orbital degrees of freedom are often irrele-
Analytical studies of the spin-bath decoherence are dif- vant. Thus, a system made of several coupled spins 1/2
ficult, and the spin-bath decoherence of many-body sys- is a good model for investigating a large class of impor-
tems is practically unexplored yet. In this situation, nu- tant problems both in quantum computing, and in solid
merical modeling of spin-bath decoherence becomes an state theory. The approach described below can be eas-
invaluable research tool. ily extended to arbitrary spin values, but discussion of
The most directapproachto study spin-bath decoher- simulations with arbitrary spins is beyond the scope of
ence is to compute the dynamical evolution of the whole this paper.
compoundsystembydirectlysolvingthetime-dependent We consider the following class of models. There is a
Schr¨odinger equation of the model system. Even for a central system made of M coupled spins Sm (Sm =1/2,
modest amount of spins, say 20, such calculations re- m=1...M). ThespinsSm interactwithabathconsist-
quireconsiderablecomputationalresources,inparticular ing of N environmentalspins In (In =1/2,n=1...N).
because to study decoherence we have to follow the dy- The Hamiltonian governingbehavior of the whole “com-
namical evolution of the system over substantial periods pound” system(centralspins Sm plus the bathspinsIn)
of time. Therefore it is worthwhile to explore ways to is
significantly improve the efficiency of these simulations.
= + = + + , (1)
In this paper, we apply the Chebyshev’s expansion H H0 V HS HB V
method to simulate models for the spin-bath decoher-
where and are the “bare” Hamiltonians of the
S B
ence. This method has been widely applied before H H
central system and the bath, correspondingly, and is
[12, 13, 14, 15, 16, 17, 18] to study dynamics of large V
the system-bath interaction. Below, we present simula-
quantumsystems,but, toourknowledge,hasneverbeen
tion results for the following general form the Hamilto-
used for simulations of systems made of large number
nians:
of coupled quantum spins. We show that for realistic
problems and typical values of parameters this method HS = Jmαm′SmαSmα′ + HmαSmα,
is a very efficient tool, giving significant increase in the hmX,m′iα=Xx,y,z Xm α=Xx,y,z
simulations speed, sometimes up to a factor of eight, in
comparisonwiththealgorithms[19,20]basedonSuzuki- HB = Γαnn′InαInα′ + HnαInα,
Trotter decompositions [21]. We illustrate this point by hnX,n′iα=Xx,y,z Xn α=Xx,y,z
test examples that we have encountered in our previ- = Aα SαIα. (2)
ous studies of the dynamics of the spin-bath decoher- V X X mn m n
hm,niα=x,y,z
ence. We also briefly discuss two other approaches, the
shortiterativeLanczos(SIL)method[14,22,23]andthe We assume that the Hamiltonian does not explicitly
H
multi-configurationaltime-dependentHartree(MCTDH) depend on time, i.e. all exchange interaction constants
[24, 25, 26] method, which are known to demonstrate J, Γ, and A, and all external magnetic fields H are con-
very good performance in many problems of quantum stant in time. Although this makes impossible to model
chemistry. the time-dependent quantum-gate operation, the inves-
The remainder of the paper is organized as follows. tigation of the fundamental properties of spin-bath de-
In Section II, we describe the model and the approaches coherence is not seriously affected by this requirement.
used for the decoherence simulations. In Section III, we The dynamics of the model (1) is already too complex
describe the specific details of application of the Cheby- to be studied analytically, and for general , when no a
H
shev’s expansion method to the spin-bath decoherence priori knowledge is available, the only option is to solve
simulations. In Section IV, we present the results of our the time-dependent Schr¨odinger equation of the whole
testsimulations. AbriefsummaryisgivenintheSection compound system numerically. I.e., we choose some ba-
V. sis states for the Hilbert space of the compound system
(thesimplestchoiceisthedirectproductofthestates
and for each spin S , I ). We represent an ini|ti↑ail
m n
| ↓i
stateofthecompoundsystemΨ asavectorinthisbasis
II. SIMULATIONS OF THE SPIN-BATH 0
set, and the Hamiltonian is represented as a matrix,
DECOHERENCE: THE MODEL AND
H
NUMERICAL APPROACHES so that the Schr¨odinger equation
idΨ(t)/dt= Ψ(t) (3)
Wefocusondecoherenceinquantumsystemsofseveral H
coupledspins. Thistypeofquantumsystemsisofpartic- is a system of first-order ordinary differential equations
ularinterestforquantumcomputations,sinceaqubitcan with the initial condition Ψ(t=0)=Ψ .
0
be represented as a quantum spin 1/2, and qubit-based The length ofthe vectorΨ is 2M+N; for typicalvalues
quantum computation is, in fact, a controlled dynamics M = 2 and N = 20, an exact solution of about 2 106
·
3
differential equations becomes a serious task. Moreover, by using an approximate form of the wave function of
the interaction between the central spins is often much the total system (central system plus bath). In partic-
bigger than the coupling with environment or coupling ular, the multi-configurational time-dependent Hartree
between the bath spins, so that the system (3) is often (MCTDH) method [24, 25, 26] is known to be very effi-
stiff. Simple methods, e.g. predictor-corrector schemes, cient, e.g. for modeling of boson-bath decoherence. The
perform rather poorly in this case, and very small inte- MCTDHapproachusesanapproximaterepresentationof
gration steps are needed to obtain a reliable solution. thewavefunction,basedontheassumptionthatthewave
AlgorithmsbasedontheSuzuki-Trotterdecomposition function of the total system can be written as a super-
[19, 20] cansolve (3) for sufficiently long times (essential positionofarelativelysmallnumberof”configurations”,
to determine the pointer states of the central system). i.e. products of time-varying single-spin wavefunctions.
They can handle Hamiltonians with explicit dependence MCTDH is a method of choice when the dimension-
on time, are unconditionally stable, exactly preserve the ality of a single-particle Hilbert space is large, and the
unitarity of quantum evolution, and the time step can multi-particle quantum correlations are associated with
be made more than an order of magnitude bigger than a superposition of a small number of products of single-
in the typical predictor-corrector method. Moreover, as particle wavefunctions. The problems considered in our
our experience shows, for the scheme based on Suzuki- paperpresentanoppositesituation. Thebathconsistsof
Trotterdecomposition,alargepartofthetotalnumerical many spins 1/2, i.e. we have only 2 orbitals per particle
error is accumulated in the total phase of the wavefunc- (spin), and the single-particle evolution is very simple,
tion Ψ(t) , and does not affect any measurable physical while the complex many-particle quantum correlations
| i
quantities (observables). However, for reasonably large are responsible for most of the physical effects (i.e., the
systems,thisschemeisstillslow,andsimulationsofdeco- number of important single-spin-wavefunctionsproducts
herencelastedforupto200CPUhoursonaSGI3800su- isverylarge). Itisprobablethatmanyproblemsofspin-
percomputer. The problem of long simulation times be- bath decoherence can be efficiently treated by MCTDH,
comesespeciallyprominentifweneedtofindthepointer butcorrespondingstudyrequiresaseparateextensivere-
states, or if the dynamics of the central system is much search effort, which is beyond the framework of our pa-
fasterthanthedecoherencerate. We foundthatinthese per.
cicumstances, the method based on Chebyshev’s expan-
sion becomes a very efficient tool to study problems of
decoherence. III. CHEBYSHEV’S METHOD FOR SPIN-BATH
Along with the Chebyshev’s expansion method, the DECOHERENCE
shortiterativeLanczos(SIL)approach[14,22,23],which
is also based on the power-series expansion of the evolu- For a time-independent Hamiltonian, the solution of
tion operator, was found to be efficient for many sim- Eq. (3) can be formally written as
ilar problems of quantum chemistry. We have tested
this method, but our results are negative. Low-order Ψ(t)=exp( it )Ψ =U(t)Ψ (4)
0 0
− H
SIL method (with small number of Lanczos iterations
per step, usually, less than 25) gives an unacceptable er- where U(t) = exp( it ) is the evolution operator. An
− H
ror, even for very short time steps. On the other hand, effective way [12, 13, 14, 15, 16, 17, 18] of calculation
high-orderSIL method (with morethan 25Lanczositer- of the exponent of a large matrix is to expand it in
H
ations per step) is noticeably slower the approach based a series of the Chebyshev polynomials of the operator
on Chebyshev’s expansion. . Below, we describe the specific details of application
H
We believe that low performance of SIL method orig- of the Chebyshev method to the spin-bath decoherence
inates from the fact that for a small number of Lanc- simulations.
zos iterations (i.e., for low-order SIL), only a very lim- The Chebyshev’s polynomials Tk(x) = cos(karccosx)
ited part of the spectrum is described correctly. For a are defined for x [ 1,1]. Thus, the Hamiltonian
∈ − H
typical problem where SIL is known to be very effective firstshouldberescaledbythefactorE0 (therangeofthe
(e.g., a wavepacket propagation), most of relevant basis valuesofthesystem’senergy)andshiftedbyEc (median
states have energy close to the energy of a wavepacket. value of the systems’ energy):
Onlytheserelevantstatesshouldbeaccuratelydescribed,
1
while accurate description of the whole energy spectrum E = (E +E ), E =E E
c max min 0 max min
is excessive. In contrast, in a typical spin-bath decoher- 2 −
ence problem, a large number of bath states with very Emin = min = min Φ Φ ,
hHi hΦ|Φi=1h |H| i
different energies are involved in the decoherence pro-
E = max = max Φ Φ . (5)
cess. Correspondingly, a large part of spectrum should max hHi hΦ|Φi=1h |H| i
be takeninto account,and the high-orderSIL integrator
shouldbeemployed,reducingtheperformanceoftheSIL In this way, the rescaled operator = 2( E )/E
c 0
method. is also bounded by 1 and 1: 1 G H1,−i.e. 1
− − ≤ hGi ≤ − ≤
Wealsonotethatsignificantspeed-upcanbe achieved Φ Φ 1foranystatevector Φ suchthat ΦΦ =1.
h |G| i≤ | i h | i
4
Forspinsystems,the Hamiltonianis boundedboth from
35
aboveandfrombelow,andthe operator canbe found.
G n
In the specific case considered in this paper, when o 30
the Hamiltonian is defined by Eq. (2), we take E = si
H 0 n 25
2max(Emin , Emax ). For this choice, E0/2 a
| | | | − ≤ hHi ≤ p
E /2. Correspondingly, we can take E = 0; this choice
0 c x 20
is legitimate, and, although might be not optimal for e
some problems, still results in very good performance of of 15
Chebyshev’s method (see below). Since max = r e = 10 -5
is the norm of the Hamiltonian, the valuehHofiE0kcHank de 10
be estimated using the Cauchy’s inequality: E0/2 ≤ or 5 e = 10 -6
+ + . Similarly, ,
kHSk kHBk kVk K
0
kHSk ≤ hmX,m′iα=Xx,y,z|Jmαm′|·kSmαk·kSmα′k (6) 0 5 t = 1E0 t /215 20
+ Hα Sα 1
| m|·k mk
X X
m α=x,y,z
FIG. 1: Dependence of the order of the Chebyshev’s expan-
1 1
= hmX,m′iα=Xx,y,z 4|Jmαm′|+Xm α=Xx,y,z 2|Hmα|, rsieosnpoKndsontoththeevmaliuneimoufmτ v=aluEe1tǫ/2=. 1T0−h5e osoflitdhecierxcpleasnscioorn-
coefficient ck;the open circles corresponds to ǫ=10−6.
and and canbeestimatedinthesamemanner.
B
kH k kVk
As a result, we have an estimate E E , where
0 1
≤ The high precision of this scheme originates from the
1 1 fact that, for k τ, the value of a Bessel function de-
E1 = hmX,m′iα=Xx,y,z 2|Jmαm′|+hnX,n′i2|Γαnn′| creases super-exp≫onentially Jk(τ) ∼ (τ/k)k, so that ter-
mination of the series (8) at k = K leads to an error
1
+ Aα + Hα + Hα , (7) which decreases super-exponentially with K. In prac-
X 2| mn| X| m| X| n| tice, K = 1.5τ already gives precision of 10−7 or bet-
hm,ni m n
ter in most cases. Due to the same reason, this scheme
and the operator can be defined as =2 /E , which is asymptotically more efficient than any time-marching
1
satisfies the inequGality 1 1.G H scheme. For given sufficiently small error ǫ, the num-
The Chebyshev’s exp−ans≤ionhGoif≤the evolution operator ber of operations Nop needed for finding the wavefunc-
U(t) (see Eq. 4) now looks like tion at time T, i.e. Ψ(T), grows linearly with T for
the Chebyshev-based scheme. For a marching scheme
∞ of order r with the time step ∆t, the numerical error
U(t)=exp( iτ )= ckTk( ) (8) is ǫ (∆t)rT, so that for given ǫ and T, the number
− G kX=0 G of op∼erations needed is Nop = T/∆t T1+1/r, grow-
∼
ing super-linearly with increasing T. For very long-time
where τ = E t/2. The expansion coefficients c can be
1 k simulations, and when very high precision is necessary,
calculated using the orthogonal property of the polyno-
the Chebyshev method is more efficient than any time-
mials T (x):
k marching scheme known to us. However, in practice,
a precision better than 0.5%–1% is very rarely needed.
a 1 T (x)exp( ixτ)
c = k k − dx=a ( i)kJ (τ), (9) Similarly, very long-time simulations are rarely of inter-
k π Z−1 √1 x2 k − k est: in most cases, the simulations are interesting only
−
until the dynamics of the system exhibits some non-
whereJ (τ)istheBesselfunctionofk-thorder,anda =
k k trivial behavior. Therefore, in spite of its asymptotic
2 for k = 0 and a = 1 for k 1. The successive terms
k ≥ efficiency, the Chebyshev method is not always the best
in the Chebyshev’s series can be efficiently determined
choiceforrealresearch,anditsefficiencyshouldbestud-
using the recursion
ied in every separate case.
T ( )=2 T ( )+T ( ) (10)
k+1 k k−1
G G G G
with the conditions T ( ) = 1, T ( ) = . Thus, to IV. SIMULATION RESULTS
0 1
G G G
find the vector Ψ(t), we just need to sum successively
the terms of the series (8), using Eq. (10) for calculation WeassesstheusefulnessoftheChebyshev’smethodfor
ofthe subsequentterms,until wereachsomepre-defined a wide spectrum of decoherence problems, by consideri-
value K of k, which is determined by the required preci- ong two central problems of decoherence, description of
sion. damping of quantum oscillations in a system, and deter-
5
mination of the pointer states. In fact, there is no strict
0.50
boundary: studyingbothproblems,wetrackevolutionof
thesystemcheckingitsstateatregularintervalsoflength
T, but in studying the oscillations decay the interval T 0.25
is muchsmallerthanthe characteristicdecoherencetime )
t
Tdec,whileinstudyingthepointerstates,T islargerthan z (1 0.00
T . S
dec
In spite of the asymptotic advantages of the
Chebyshev-based scheme, it is not a priori clear if it is -0.25
efficientforrealisticproblems,whenthe requirednumer-
ical error δ is modest (say, δ = 10−2–10−3). Also, if we
-0.50
track the dynamics of the decoherence process, we make
many steps of modest length T, and the overhead asso- 0 5 10 15
ciated with the use of the Chebyshev’s expansion might Time
be significant, see Fig. 1.
To study this issue, we have performed several types
FIG.2: Timedependenceoftheoscillationsoftheexpectation
of numerical tests. The timing information reported in value of Sz(t) in the two-spin system decohered by a spin
1
this paper has been obtainedfrom calculations on a SGI
bath.
Origin 3800 (500 MHz) system, using sequential, single
processor code. The order of Chebyshev’s expansion K
have been defined by the pre-specified precision ǫ. We
spin S = 1 and S = 0, and the state with the total
z
determined the minimum value of K such that c < ǫ
| k| spinS =0. These stateshavedifferentenergies,and, for
for k K, starting from the value K = [1.1τ] ([x] is
the int≥eger part of x), and adjusting it0as needed. Each example,the dynamics of S1z(t) is representedby oscilla-
tions with the frequency J. Due to interaction with the
simulationhasbeen performedthree times: (i) using the
spin bath, these oscillations are damped, see Fig. 2. To
Chebyshev’s method with ǫ = 10−12, the reference run,
study this damping indetail, we takethe Suzuki-Trotter
(ii) using Chebyshev’s method with ǫ=10−5–10−6, and
time step ∆t = 0.035, T = 2∆t, and watch the system
(iii) using the scheme based on Suzuki-Trotter decom-
since t = 0 till t = 800T. If we do not need such a
max
position [19, 20]. Previously we have used the latter to
high resolution, we increase T. In Table I, we present
study spin-bath decoherence [10, 27]. In this paper, we
the CPU time needed to perform the simulations using
havechosentoconsiderthesameproblemsasinourpre-
theSuzuki-TrotterandChebyshev’smethods,alongwith
viousworksonthissubject,inordertoavoidtheimpres-
the resulting error δ (which should not be confused with
sion that the tests have been constructed to favor one
the “nominal” precision of the Chebyshev’s scheme ǫ).
particular method.
The errorδ hasbeenobtainedfromcomparisonwith the
First, we consider the problemof oscillations damping “reference”Chebyshev’srun(ǫ=10−12), andis equalto
inthe centralsystemoftwospins coupledby Heisenberg
themaximumofabsoluteerrorsofthequantities(allnor-
exchange, interacting with the bath. We studied this
malized to unity) 2Sα, 2Sα, 4SαSβ (α,β = x,y,z), and
problemusingtheSuzuki-TrotterschemeinRef.27. The 1 2 1 1
the so-called “quadratic entropy” [28] S(2) = 1 Trρ2.
Hamiltonians describing the bath and the system are: − S
These quantities have been calculated and compared at
regular intervals of length T. Their calculation increases
=JS S , =0, = A (S +S )I ,
HS 1 2 HB V n 1 2 n the number of computations, so that the tests 1, 2, and
X
(11) 3, which are otherwise equivalent for the Suzuki-Trotter
with N = 16 bath spins. The exchange parameter method, require more and more CPU time.
J = 16.0 (antiferromagnetic coupling between the cen-
tral spins), while A are uniformly distributed between
n
0 and 0.5. The initial state of the compound system
−
Ψ0 = ψ0 χ0 is the product of the initial state ψ0 TABLE I: Comparison of the Suzuki-Trotter scheme (abbre-
| i | i⊗| i | i
of the central system, and χ of the bath. In this case, viated as ST) with the Chebyshev’s scheme (abbreviated as
0
| i
ψ = , i.e. the first central spin is in the state Ch) for theproblem of oscillations decay.
0
|Sz(it = 0|)↑=↓i+1/2, and the second spin is in the state
S1z(t = 0) = 1/2. The initial state of the bath χ is Test ∆t T tmax δ ǫ CPU Time
2 − | 0i 1, Ch — 200∆t 8T 1·10−5 10−6 22 min
the linear superposition of all basis states with random
coefficients. Physically, this situation corresponds to the 1, ST 0.035 200∆t 8T 0.44·10−2 — 80 min
case of the temperature θ which is high in comparison 2, Ch — 8∆t 200T 0.3·10−4 10−6 59 min
with the bath energies An, but is much lower than the 2, ST 0.035 8∆t 200T 0.48·10−2 — 89 min
system’s energy J (note that J An in this case). 3, Ch — 2∆t 800T 0.55·10−3 10−6 226 min
≫
The initial state of the central system is a superpo- 3, ST 0.035 2∆t 800T 0.48·10−2 — 156 min
sition of two eigenstates of : the state with the total
H
6
TABLEII:Comparison oftheSuzuki-Trotterscheme(abbre- 0.50
viated as ST) with the Chebyshev’s scheme (abbreviated as
S
Ch)fortheproblemofoscillations decay,employingthetwo-
r 0.25
leap approach with different T1 and T2.
f
o
4T,eCsth ∆—t 15T01∆t ∆T2t 0.4·δ10−4 10ǫ−6 CP61UmTiinme ts 0.00
n
4, ST 0.02 150∆t ∆t 0.2·10−2 — 144 min e
5, Ch — 300∆t ∆t 0.4·10−4 10−6 75 min m -0.25
e
5, ST 0.02 300∆t ∆t 0.3·10−2 — 221 min El
-0.50
0 500 1000 1500 2000
As one can see from Table I, for realistic values of
Time
maximum error δ 0.5 10−2, and even for not very
∼ ·
long runs, the Chebyshev’s scheme can be faster than
the Suzuki-Trottermethodbyafactorofupto four,and FIG. 3: Temporal evolution of different elements of the den-
the efficiency of the Chebyshev’s scheme growsfast with sity matrix ρ: diagonal elements corresponding to the states
| ↑↑i, | ↑↓i, | ↓↑i, and | ↓↓i (the four upper curves), and the
increasing T. However, this straightforward comparison
is too crude, and Table I is only an illustration of basic non-diagonal element ρ1S2 (the lowest curve). Very slow re-
laxation is betterseen for theuppermostcurve(thediagonal
featuresoftheChebyshev’smethod. Tomodelfastoscil-
element corresponding to the state | ↑↑i) which has a small
lationswhichdecayslowly(often, withthe decaytime of
negative slope. Note that the two lines in the middle (the
orderofdecoherencetimeT ),weshouldmakeT signif-
dec second and thethird lines from above, thediagonal elements
icantlysmallerthantheoscillationperiodtosc =2π/J,in corresponding to the states |↑↓i and |↓↑i, correspondingly)
ordertocorrectlydeterminetheamplitudeofoscillations areveryclosetoeachotheratt≥200,asexpectedforanear-
at given time. equilibrium (although not completely relaxed) situation.
Therefore,to trackthe damping ofoscillations,we use
the two-leap approach: first, we make a large time leap
of length T (T t , but T T ), and then we
1 1 ≫ osc 1 ≪ dec In our simulations we used h = 0.1 and Unn′ randomly
make a number n (usually, 15–20) of smaller steps T
2 2 distributed in the interval [ 0.013,0.013]. This Hamil-
such that T t but n T t , resolving in detail −
2 ≪ osc 2 2 ≥ osc tonian is known to result in stochastic behavior [30]; we
one period of oscillations and extracting the amplitude.
havecheckedthelevelstatisticsindependently,andfound
By repeating this two-stage sequence n times, we can
tot that it closely follows the Wigner-Dyson distribution.
reliably track the change of the oscillations amplitude
To determine the pointer states, we need to find the
with time. The test example of this approach have been
elementsofthe reduceddensitymatrixρ (t)inthelong-
takenfrom our recentwork [29]. We have performedthe S
time limit t . We start at t = 0 from the state
same kind of simulations as described above, with N = → ∞
of the compound system which is the product of the
16 bath spins, repeating the two-leap sequence n = 8
tot states of the bath and the central system (as above),
times, each time making one long leap T followed by
1
but the initial state of the central spins now is the sin-
n = 21 short leaps T . The results of these tests are
2 2 glet ψ = (1/√2)[ ]. Because of decoher-
presented in Table II. Again, Chebyshev-based method | 0i | ↑↓i−| ↓↑i
ence, the final state of the central system is mixed, and
can be up to three times faster than the Suzuki-Trotter
ρ = w p p + w p p , where p and p are
algorithm [19, 20]. S 1| 1ih 1| 2| 2ih 2| | 1i | 2i
the pointer states, which are superpositions ofthe states
Finally, we have tested the Chebyshev scheme in the
, , and . As we have found in our
problem of determining the pointer states, employing an | ↑↑i | ↓↓i | ↑↓i | ↓↑i
work [10], the form of this superposition is determined
example from our work [10]. This example is interesting
by the ratio J/b, where b = A2. For J/b 1, the
also because it deals with a physically important case of n n ∼
pointer states are very closePto the singlet S = 0 and
a spin bath possessing chaotic internal dynamics, which
triplet S = 1, S = 0 states, and for J b, the pointer
is relevant for majority of realistic spin baths (such as z ≪
states are close to and . Thus, the quantities
nuclearspinsorimpurity spins baths). The Hamiltonian | ↑↓i | ↓↑i
characterizingthetypeofthepointerstatearethevalues
describing the system is
of the non-diagonalelements of the density matrix ρ in
S
HS =JS1S2, V =XAnS1In, (12) ethleembeanstisρ|1S2↑↑=i, h|↑↓↓↓i|ρ|S↑|↓↓i↑,ianisda| ↓v↑eir.y Isnuiptaabrtleicuqluaarn,ttihtye
to characterize the pointer state. This non-diagonal ele-
i.e.,thebathspinsarecoupledonlywiththe firstcentral
ment is close to zero for J b, and gradually increases
spin, and the bath Hamiltonian is now
≪
in absolute value with increasing J.
HB = hzIkz + Unn′InxInx′. (13) Typical results for temporal evolution of the elements
Xn hnX,n′i of the density matrix ρS are shown in Fig. 3. One can
7
mechanics and quantum measurement theory, quantum
TABLE III: Comparison of the Suzuki-Trotter scheme (ab-
computing,solidstatephysicsetc. Decoherenceisacom-
breviated as ST) with the Chebyshev’s scheme (abbreviated
plex many-body phenomenon, and numerical simulation
as Ch) for the problem of determining thepointer states.
is an important tool for its investigation. In this paper,
Test ∆t T tmax δ ǫ CPU Time wehavestudiedefficiencyofthe numericalschemebased
6, Ch — 100∆t 500T 0.2·10−3 10−6 19 min onthe Chebyshevexpansion. We havepresentedspecific
6, ST 0.14 100∆t 500T 0.7·10−2 — 105 min detailsoftheapplicationofthis methodtothespin-bath
decoherence modeling. To assess the efficiency of the
7, Ch — 10∆t 5000T 0.1·10−2 10−6 52 min
simulation method, we have used model problems which
7, ST 0.14 10∆t 5000T 0.8·10−2 — 117 min
we have encountered in our previous studies of the spin-
8, Ch — 1000∆t 50T 0.3·10−6 10−6 13 min bath decoherence. We compared the Chebyshev-based
8, ST 0.14 1000∆t 50T 0.8·10−2 — 107 min scheme with a fast method based on the Suzuki-Trotter
decomposition. We have found that in many cases, the
formergivesaconsiderableincreaseinthespeedofsimu-
see thatin this situation, we do not need to use the two- lations,sometimesuptoafactorofeight(fortheproblem
leap approach with different T and T . The relaxation of finding the system’s pointer states), while in studying
1 2
(after some initial period) is slow, and no fast oscilla- the decoherence dynamics, the increase in speed is less
tions of considerable amplitude exist at long times, so drastic(afactorof2–3),butstillconsiderable. Thiscon-
that the one-leap approach is sufficient. Thus, the ef- clusion holds for many types of centralsystems and spin
ficiency of the Chebyshev-based scheme is expected to baths, with different Hamiltonians.
be very good. This is indeed the case, as Table III Acknowledgments
demonstrates. The results presented there correspond
to J = 0.1. The Chebyshev-based scheme is faster than
the Suzuki-Trotter method up to a factor of 8.
We havecheckedourconclusionsonmanyothercases,
withthecentralsystemsmadeofuptoM =4spins,and
withthebathsmadeofuptoN =22spins,withdifferent
HamiltoniansanddifferentvaluesoftheHamiltonianpa-
rameters. We foundthatChebyshev-basedmethod gives
asignificantincreaseinthesimulationsspeedforallprob- This work was partially carried out at the Ames Lab-
lems wherethe value ofT canbe made sufficiently large. oratory, which is operated for the U. S. Department of
Energyby IowaState Universityunder ContractNo.W-
7405-82and was supported by the Director of the Office
V. SUMMARY of Science, Office of Basic Energy Research of the U. S.
DepartmentofEnergy. Supportfromthe Dutch “Sticht-
Theoretical studies of the spin-bath decoherence are ingNationaleComputerFaciliteiten(NCF)”isgratefully
important for many areas of physics, including quantum acknowledged.
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