sub-exponential spin-boson decoherence in a finite bath

Sub-exponential spin-boson decoherence in a finite bath

V. Wonga,1, M. Gruebelea,b,*

a Department of Chemistry, University of Illinois, Urbana, IL 61801, USAb Department of Physics, The Center for Biophysics and Computational Biology, University of Illinois, 600 S. Mathews Ave. Box 5–6,

Urbana, IL 61801, USA

Received 17 October 2001

Abstract

We investigate the decoherence of a two-level system coupled to harmonic baths of 4–21 degrees of freedom, to baths

with internal anharmonic couplings, and to baths with an additional ‘solvent shell’ (modes coupled to other bath

modes, but not to the system). The discrete spectral densities are chosen to mimic the highly fluctuating spectral

densities computed for real systems such as proteins. System decoherence is computed by exact quantum dynamics.

With realistic parameter choices (finite temperature, reasonably large couplings), sub-exponential decoherence of the

two-level system is observed. Empirically, the time-dependence of decoherence can be fitted by power laws with small

exponents. Intrabath anharmonic couplings are more effective at smoothing the spectral density and restoring expo-

nential dynamics, than additional bath modes or solvent shells. We conclude that at high temperature, the most im-

portant physical basis for exponential decays is anharmonicity of those few bath modes interacting most strongly with

the system, not a large number of oscillators interacting with the system. We relate the current numerical simulations to

models of anharmonically coupled oscillators, which also predict power law dynamics. The potential utility of power

law decays in quantum computation and condensed phase coherent control are also discussed.

� 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The spin-boson Hamiltonian (SBH) has been

used as a model for a variety of processes in the

presence of dissipation [1,2]: spectroscopy of

nonadiabatically coupled electronic surfaces [3],

electron transfer [4–7], and decoherence of qubitsat low temperature [8], to name but a few. The

dynamics have been solved exactly by path integral

techniques [9]. Asymptotically nonexponential

dynamics or ‘strange kinetics’ [10] occur in this

model under a variety of conditions: perturbation

approaches predict power law tails at 0 K under

certain conditions [11]; oscillations can occur for

parameter choices relevant to electron transfer[12,13]; at higher temperatures and fairly large

coupling strengths, path integral calculations show

that power law tails can persist after initially ex-

ponential relaxation [14].

Baths with discrete spectral densities JðxÞ, fromthe Jaynes–Cummings model on up, have also

Chemical Physics 284 (2002) 29–44

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*Corresponding author. Fax: +1-217-244-3186.

E-mail address: [email protected] (M. Gruebele).1 Present address: Northwestern University.

0301-0104/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0301-0104 (02 )00534-7

mail to: [email protected]

received some attention [15,16]. The dynamics of a

TLS coupled to a discrete bath is relevant in many

physical situations: disorder leads to localized bath

modes, only a small subset of which can directly

interact with a localized system. Although most

model studies have used smooth spectral densities,such as the Ohmic type with a high frequency

cutoff, recent molecular dynamics simulations

show that the spectral density is indeed very often

highly structured over the frequency range of in-

terest. Examples include electron transfer and ex-

citon dynamics in proteins [5,17], a solid state

qubit whose decoherence is controlled by sparse

impurities [18], or conformational transitions be-tween two states of a molecule embedded in a

glassy environment (our guiding example).

Structure in JðxÞ arises from a shell-like dis-

position of couplings from the environment to the

two-level system. As illustrated in Fig. 1, not all

bath modes have equal access to the system be-

cause they are partially localized by disorder and

spatially separated from the system. The assign-ment of system–bath interactions to an inner shell,

next-to-inner shell, etc., depends of course on the

time scales of interest. On sufficiently long time

scales (which may be quite fast in liquids), diffu-

sion processes can mix shells that are valid on

shorter time scales. For this reason approxima-

tions such as instantaneous normal mode analysis

[19] eventually break down. Nevertheless, the

fluctuations of JðxÞ of real baths make it inter-esting to consider few-mode quantum models with

one or more shells of interactions. This type of

hierarchical structure has already been of great usein studying vibrational dephasing and relaxation

phenomena [20–23].

In this paper, we consider the decoherence DðtÞand population decay dynamics P ðtÞ of the TLSusing the SBH and SABH (spin-anharmonically

coupled boson Hamiltonian). The baths are dis-

crete with one or two shells, and with highly fluc-

tuating spectral densities mimicking real systems.The bath size N ¼ 4–21 is certainly not in thePoincar�ee limit, but our exact quantum dynamics

simulations reveal interesting deviations from ex-

ponential behavior before SðtÞ and P ðtÞ settle intosmall-scale fluctuations about a small average va-

lue r (which happens long before large scale Po-incar�ee recurrences occur). The calculations arecarried out at temperatures comparable to the TLSself-splitting as well as for closer to microcanonical

initial conditions of the bath. We find that sub-

exponential decays occur under a wide range of

conditions when the bath is discrete. We also study

those properties of the bath which tend to restore

single exponential decay behavior of DðtÞ and P ðtÞ.

Fig. 1. Left: a schematic rendering of a hierarchical coupling structure, with the system a central box, and modes in each shell

symbolized by black dots. Only first shell modes couple strongly to the system; second shell modes couple to shell 1, but only weakly to

the system (dotted lines). Right: the panel labeled single shell shows the first shell spectral density seen by the TLS. The panel labeled

multiple shells demonstrates how addition of a second shell of modes ‘broadens’ the first shell modes: additional lines appear in the

spectral density due to mixing between first and second shell modes (see Eq. (12)).

30 V. Wong, M. Gruebele / Chemical Physics 284 (2002) 29–44

In particular, strong intrabath anharmonic cou-

plings neglected by the SBH restore exponential

dynamics, while multiple bath shells or additional

bath degrees of freedom are much less effective at

smoothing the spectral density JðxÞ.

2. Hamiltonian and computational approach

We write the SBH in the prediagonalized formfor a degenerate two-level system with coupling D:

H ¼ hcD2

rz þXNi¼1

hcmi ayi ai

�þ 12

�

þ hc2

rx

XNi¼1

Viðayi þ aiÞ: ð1Þ

The three terms are Hsys;Hbath, and Hint in the bi-linear coupling approximation. (In the undiago-

nalized form often used in the literature, rx and rz

are interchanged [11].) Because we are mainly in-

terested in the decoherence properties under reso-

nant tunneling conditions (degenerate states when

D ¼ 0), no rx term is added to Hsys. Eq. (1) thuscorresponds to a symmetric double well where D isthe TLS tunneling splitting, mi is a bath frequency,and Vi a system–bath coupling.In Eq. (1), only the ratios Vi=D and mi=D are

independent quantities. We arbitrarily set D ¼ 100and use propagators with phase factors of the type

�0:188imt. As a result, D; mi and Vi can be inter-preted as having units of cm�1, and t as having

units of ps, commonly used in frequency- or time-

resolved spectroscopy. However in the following,

we will treat D; mi and Vi and t as unitless scalablequantities. For numerical evaluation, bath fre-quencies were chosen at random in the interval

mi=D � ð0:06; 10Þ; in most calculations the systemfrequency was thus embedded in the manifold of

bath frequencies. Couplings ranged from

Vi=D ¼ 0:008 (weak coupling to a low frequency

mode) to Vi=D ¼ 0:4 (strong coupling to a highfrequency mode). With D ¼ 100 cm�1 and t in ps,

the energy and time scales correspond to a low-barrier proton tunneling mode interacting with

low frequency skeletal molecular vibrations. If the

energies are scaled down by 100 and the times are

scaled up correspondingly, the numbers would be

closer to conformational tunneling of a large-mass

mode, or to moderately fast electron transfer be-

tween equipotential states in a protein. When

scaled down by 104, the energy scale roughly cor-

responds to a nuclear spin in a moderate external

field in a disordered bath of impurities with com-parable frequencies.

The couplings Vi and frequencies mi in Eq. (1)can be related to the spectral density JðxÞ by

Jðxi ¼ 2pcmiÞ ¼0:6

dmi

Viq0

� �2: ð2Þ

Here, q0 is the TLS displacement coordinate (dis-tance from the minimum of the double well tobarrier). The factor dmi ¼ ðmiþ1 � mi�1Þ=2 correctsfor the discrete density of bath modes and replaces

the d-function in the usual continuum representa-

tion of the spectral density ðJðxÞ ¼ pc2i dðx � xiÞ=ð2mixiÞÞ. If q0 is taken in �AA, and dmi and Vi aretaken in cm�1, then J in Eq. (2) is in amu ps�2,

where amu are the common atomic mass units in

which mproton � 1.The range of couplings and the random bath

frequency spacings discussed above lead to spec-

tral densities of the type illustrated in Fig. 2. Some

of our JðxÞ could be roughly fitted by an Ohmicdensity with cutoff, such as the one introduced by

Caldeira and Leggett [1],

Fig. 2. Spectral density functions for the protein exciton system

discussed in [17] (solid line), an ohmic spectral density with an

exponential cutoff (dashed line), and examples of two of the

spectral densities (see Eq. (2)), corresponding to results shown

in Fig. 4 and Fig. 10.

V. Wong, M. Gruebele / Chemical Physics 284 (2002) 29–44 31

JðxÞ ¼ gxe�x=xc ; ð3Þwhich we also study as one example in Section 3.

Usually JðxÞ more closely resembles the stronglyfluctuating spectral densities determined by mo-

lecular dynamics simulations for the region

xJxc (Fig. 2) [17]. One can interpret the ViðxiÞ ofa single shell calculation as corresponding to the

peaks in a continuous spectral density function. In

a multi-shell calculation, these ‘peaks’ are broad-

ened by coupling the first bath shell to the next

higher bath shell, better mimicking a continuous

spectral density distribution (Fig. 1). To examine

this, we also consider a Hamiltonian closely re-

lated to Eq. (1), but with

Hbath ¼XNi¼1

hc~mmi ayi ai

�þ 12

�

þ hcVX6n¼3

YNi¼1

0 jnii ða

yi þ aiÞni : ð4Þ

The additional term is a factorized anharmonic

potential which was introduced previously tostudy intramolecular vibrational dephasing [24–

27]. V sets the overall energy scale, n is the order of

the coupling (cubic to sixth-order in our present

calculations), and ji is a unitless Born–Oppenhei-

mer scaling parameter for bath mode i in the range

0.05 to 0.2. (We use the scaling of R ¼ V 1=nj givenby Eq. (5) in [28], which are typical for a strongly

coupled molecular bath.) The 0 after the productoperator in Eq. (4) indicates that the ni are con-strained by

Pni ¼ n, and that only off-diagonal

matrix elements are to be calculated. In a typical

two-shell calculation, all modes i are coupled toone another by the third term in (4), but only the

‘first shell’ modes i ¼ 1 to N 0 < N are coupled to

the system, while the modes i > N 0 would serve to

dephase the first shell modes and only indirectlyaffect the system dynamics (Fig. 1).

The main dynamical quantities reported in this

work are the decoherence

DðtÞ ¼ jhþjqTLSðtÞj�ij ð5Þ

and the relative population decay

P ðtÞ ¼ h�jqTLSðtÞj�i � h�jqTLSð1Þj�ih�jqTLSð0Þj�i � h�jqTLSð1Þj�i ; ð6Þ

where jþi and j�i are the ground and excitedenergy eigenstates of the uncoupled TLS, and qTLSis the reduced density matrix for the TLS obtained

by tracing over the bath degrees of freedom in the

total density matrix.At t ¼ 0, the total density matrix has the fac-

tored form qTLS � qbath. We start with the TLS inthe pure state

jLi ¼ 1ffiffiffi2

p ðjþi þ j�iÞ or

qTLSð0Þ ¼1

2

1 1

1 1

� �: ð7Þ

In the absence of an environment, DðtÞ would re-main constant. The main question at issue in this

paper is the qualitative form of the coarse-grained

DðtÞ and PðtÞ decays when the system–bath cou-pling is turned on. Both quantities also exhibit

small beat structure superimposed on the coarse-

grained decay behavior. The beating is due to thefinite and discrete nature of the environment and

will not be analyzed in detail.

All computations involve propagation of a total

wavefunction for the system plus bath. The total

density matrix remains pure, although the TLS

reduced density matrix decoheres (that is, the

system becomes entangled with the bath degrees of

freedom). Previous numerical studies of closed,pure quantum baths have revealed that even a

bath with as few as six degrees of freedom evolves

to a state whose reduced properties appear to be

those of a thermally equilibrated bath, as long as

the bath has couplings which can exchange pop-

ulation among its degrees of freedoms [22,23]. In

this work, the couplings are either provided via the

system, or directly among the bath degrees offreedom, depending on whether Eq. (1) or (4) de-

scribes the bath Hamiltonian.

The initial bath wavefunction was constructed

as follows. From harmonic oscillator basis func-

tions jnji for degrees of freedom j ¼ 1 to N , a fullproduct basis set fjiig ¼ f

QNj¼1 jnjig was gener-

ated. A general bath state is represented by jWi ¼Pjcijeiui jii. For the initial state, the phases /I were

chosen from a uniform distribution. The square-

amplitudes jcij2 were chosen from one of two

Gaussian distributions: (i) with a specified average

total energy NkT and standard deviationffiffiffiffiN

pkT


and (ii) with the same average but a much nar-

rower width (but still sufficient to include many

bath basis states, so the bath dephases). An upper

energy cutoff was introduced at three standard

deviations of the Gaussian, and a conservative

upper quantum number cutoff was chosen. (i)Corresponds to a ‘temperature’ bath, and (ii) to a

‘quasi-microcanonical’ bath. The effective tem-

perature T 0 differs from the final computed tem-

perature T by a small amount because the

couplings and system energy slightly modulate the

total energy compared to the energies of zero-

order bath states.

To ‘prepare’ the baths, bath wavefunctionswere propagated using the bath Hamiltonian for a

time long enough to ensure that the survival

probability jhWbathðtÞjWbathð0Þij2 reached its longtime average value. Only then was the system–bath

interaction switched on and the decoherence of the

system studied. The initial ‘bath only’ propagation

is trivial for the uncoupled harmonic oscillator

bath, but when couplings among bath oscillatorsare included (as in Eq. (4)), population dynamics

can occur. The baths generally dephased with a

variety of behaviors ranging from Gaussians with

small tails (uncoupled baths) to exponential (cou-

pled baths). Fig. 3(a) illustrates three typical bath

survival probabilities for each type of calculation

described in more detail in Section 3: ‘quasi-mi-

crocanonical’ initial conditions, initial conditionswith a computed ‘temperature’ T, and anhar-

monically coupled (which also has an well-defined

temperature). The important thing to note here is

that no subexponential dynamics occurs in these

baths until jhWbathðtÞjWbathð0Þij2 < 10�3–10�4,whereas the subexponential reduced TLS dynam-

ics described in the next section occur in the region

0.1–10�2.Fig. 3(b) illustrates that a computed tempera-

ture T can indeed describe the bath when only

reduced properties of the bath (e.g. reduced den-

sity matrices for 1 or a few degrees of freedom of

the bath) are examined. It shows the relative mode

populations for three modes of a seven mode bath

discussed in Section 3 as a function of occupation

number on a log-linear plot. A temperature ofkT ¼ 3:2D (460 K if energies are interpreted in

cm�1) provides a very good fit to the populations.

It is clear that in a steady-state, any sub-ensemble

of a large bath is effectively measured or decohered

by the remainder of the bath, such that the reduced

density matrix of the sub-ensemble can have

thermal equilibrium properties: the bath as a

whole may be in a pure state, but any realisticallymeasurable property is thermal. Fig. 3(b) merely

illustrates that this is possible even for very small

baths if not too high-order a coherence is mea-

sured.

Numerical propagation of the wavefunction was

performed with the Shifted Update Rotation

(SUR) algorithm [29]. This propagator is a com-

putationally more efficient and more memory-sav-ing version of the symplectic leapfrog algorithm

well known in classical dynamics simulations,

Fig. 3. (a) Bath survival decays evolved under Hbath only; (b) populations of three bath modes in the harmonic N ¼ 7 ‘temperature’case ðE ¼ NkT ; dE ¼

ffiffiffiffiN

pkT Þ; the straight lines are fits to a temperature of kT ¼ 3:2D.


and first applied to quantum dynamics by Gray

et al. [30]. We also tested Visscher’s S3 algorithm

[31], but found SUR to be more efficient in this

case.

Because of the finite size of the bath, DðtÞ andP ðtÞ cannot decay to zero before the Poincar�ee re-currence time. Rather, they fluctuate about a small

average r [23,32]. Functional forms used to fit DðtÞwere exponential

DðtÞ ¼ rD þ ð0:5� rDÞ expð�t=sDÞ; ð8Þpower law

DðtÞ ¼ rD þ ð0:5� rDÞð1þ t=sDdDÞ�dD ð9Þand stretched exponential

DðtÞ ¼ rD þ ð0:5� rDÞ expð�ðt=sDÞbDÞ; ð10Þwhere rD is the long time average value of DðtÞ.Whenever the exponential is sufficient, only an

exponential fit is reported. This corresponds to

d ! 1 in Eq. (9) and b ! 1 in Eq. (10). A variant

of Eq. (9) has been used previously to provide anunbiased determination of exponential vs. power

law behavior in dynamics of random matrix sys-

tems [33] and in simulations of vibrational dy-

namics [22,34]. Fits to P ðtÞ were of the same formas Eqs. (8)–(10) after replacing 0.5 by 1. The fits

were made to logarithmically smoothed decays

(time window proportional to time), whose r val-ues were taken to be the average value after thedecay had reached a steady-state baseline.

3. Reduced TLS dynamics with finite-dimensional

baths

The time evolution of the decoherence DðtÞ andpopulation P ðtÞ for a 7 degree of freedom bath isshown in Fig. 4 (see Table 1 for parameters and

Fig. 2 for JðxÞ). Although exponentials provide asuperior description of the short time behavior of

P DðtÞ and P ðtÞ, the long time behavior of DðtÞ andP ðtÞ is better represented by power laws. The slowdecay of DðtÞ or P ðtÞ between the initial defla-tionary period and the asymptotic leveling is ge-

neric, and occurs in the great majority of allsimulations with random JðxÞ. Fig. 4b shows thatthe population decay is closely linked to the de-

coherence in functional form and time scales, and

this is also typical of our results. Here, we are in-

terested mainly in the average properties of the

decays, not in the small superimposed beat pat-

terns. Further plots of DðtÞ and P ðtÞ are shownlogarithmically smoothed. As can be seen in Fig. 4,the logarithmic smoothing window was kept small

enough not to distort the average decay behavior.

Fig. 5 shows how DðtÞ for the system summa-

rized in Table 1 is affected solely by variation of

the system–bath coupling strength (by applying

the same multiplicative factor to all Vi in Eq. (1)).At low coupling strengths, increasing coupling

strength speeds decay of DðtÞ. A turnover point isreached at higher coupling strengths, beyond

which DðtÞ decays more slowly with increased

Fig. 4. (a) DðtÞ and (b) P ðtÞ for the 7 degree of freedom bath in Table 1. Smoothed decays were obtained from decays computed by fullquantum dynamics by averaging over the neighboring 7 data points whose spacing grows logarithmically with time. Both DðtÞ and P ðtÞare best fitted by a power law function (Eq. (8)). Exponential functions (Eq. (7)) which fit the early time dynamics well decay to rmuchfaster than DðtÞ and PðtÞ.


coupling. Values of rD are not greatly affected atthe turnover point and beyond, indicating that the

available state space is fully accessed. P ðtÞ (notshown) also exhibits turnover behavior. Turnover

behavior of various types has been discussed in the

literature on quantum dynamics in the presence of

dissipation [1,16], and generally follows the trends

of classical friction descriptions [35]. Although our

TLS dynamics are not strictly described by a single

rate, they are qualitatively no exception to a

Kramers-like turnover. It should also be noted

that at short enough times (just after the ballistic

regime), we observe that the speed of decay for

both DðtÞ and P ðtÞ simply increases monotonicallywith the coupling strength. ‘Friction’ effects in ourquantum dynamics simulations are not evident

until well after the ballistic phase is over.

Fig. 5 also shows that although power laws no

longer best describe DðtÞ beyond the turnover

point, multiscale decoherence in the form of

stretched exponential decays is preserved far be-

yond the turnover coupling strength. One may

speculate that this is caused by the discrete natureof the spectral density not being altered by merely

increasing system–bath coupling strength.

We now turn to the effect that various bath

properties have on the TLS DðtÞ. Trends for thefollowing bath properties are discussed: Ohmic vs.

random spectral densities; increased spectral den-

sity near xbath ¼ xsys; ‘quasi-microcanonical’ ini-

tial conditions vs. baths having a nominaltemperature; tuning the low-frequency cutoff of

the bath above the system frequency, or the high-

frequency cutoff below the system frequency; the

number of degrees of freedom in the bath; the in-

trabath coupling strength (0 for Eq. (1)); and fi-

nally, the addition of a second bath shell which

does not interact with the TLS.

The decoherence DðtÞ for a discretized Ohmic,rather than random, spectral density is shown in

Fig. 6 (see Table 2). Note that the bath frequencies

and average coupling strength are identical to the

N ¼ 7 calculation presented in Figs. 3 and 4 andTable 1 for a random JðxÞ. At a similar averagecoupling strength, it appears that DðtÞ is still betterdescribed by a stretched exponential than by an

exponential. It appears that a ‘smooth’ distribu-tion of Vi does not make up for the discrete dis-tribution of xi.

Next, we examine the dependence of the decays

on the density of near-resonant bath modes. This

is done to see if providing a ‘width’ to peaks in the

bath spectral density has any effect on the decay

exponent (see Fig. 1). Fig. 7a shows a comparison

of DðtÞ for cases where the TLS is coupled to anincreasing number of near-resonant bath modes.

(In calculations reported elsewhere in this paper,

Table 1

First and second column: frequencies and coupling constants

for the bath in Fig. 4, which illustrates decoherence and pop-

ulation decay

xi=D Vi=D xi=D Vi=D

0.118 0.050 0.179 0.050

0.249 0.063 0.188 0.050

0.536 0.079 0.633 0.074

1.000 0.095 0.974 0.084

1.140 0.099 1.857 0.100

2.108 0.119 1.994 0.104

3.411 0.137 2.390 0.108

3.132 0.118

5.038 0.136

Third and fourth column: frequencies and coupling con-

stants for the bath in Fig. 5, which illustrates the turnover effect;

the fourth column corresponds to Vrms=D ¼ 0:12 in Fig. 5.

Fig. 5. Smoothed decays and fits for DðtÞ as a function system–bath coupling strength. Vrms is reported in units of D. For thesystem of Table 2, turnover in DðtÞ occurs between Vrms ¼ 0:24Dand Vrms ¼ 0:59D and a stretched exponential (Eq. (9)) eventu-ally becomes the best fitting function at very large values of Vrms.


there is at most one mode of frequency xi=D � 1in the bath; here, the coupling strength of the

resonant modes was lowered to approximately

preserve the initial 1=e decay time constant of DðtÞ,while the remaining coupling strengths were leftunaltered.) The observed decay exponent increases

only marginally when the number of resonant

modes is increased threefold. Related results onhow the width of features in JðxÞ induced by asecond bath shell affect the TLS dynamics are

discussed further below.

Fig. 8 displays DðtÞ for an initially ‘quasi-mi-crocanonical’ environment (as defined in Section

2) and for the same bath initially at an effective

temperature kT � Ebath=N . The decays are wellfitted by power laws with the same lifetime and

Fig. 6. DðtÞ in the case of a discrete Ohmic spectral density forthree values of coupling strength. Note that except for the Vi ,the bath of Table 1 is identical to the bath in this calculation

(bath frequencies, initial populations, and phases are identical).

The spectral density has been made Ohmic by scaling the system

bath couplings of the calculation shown in Fig. 4 by the ap-

propriate factor.

Table 2


for the slowest decay in Fig. 6 (Vrms=D ¼ 0:086)

xi=D Vi=D xi=D Vi=D

0.118 0.0041 0.249 0.050

0.249 0.0095 0.536 0.063

0.536 0.0250 0.993 0.038 (3)

1.000 0.0275 1.000 0.076–

0.038 (1)

1.140 0.0540 1.007 0.038 (2)

2.108 0.1505 1.140 0.079

3.411 0.2195 2.108 0.095

3.411 0.110

The faster decays have coupling constants scaled by factors

of 2 and 4. Third and fourth column: frequencies and coupling

constants for DðtÞ in Fig. 7. The number in parenthesis indicatesin which order the additional resonant modes were introduced

in Fig. 7. The first was rescaled to a smaller coupling in cal-

culations (2) and (3) to approximately preserve the lifetime.

Fig. 8. Comparison of DðtÞ for a bath intially in a ‘temperature’state versus a quasi-microcanonical bath. The similarity be-

tween the two cases is striking given that the internal dynamics

of the bath (before interaction with the TLS) are markedly

different from one another, as illustrated in Fig. 3.

Fig. 7. DðtÞ as a function of the number of resonant environ-mental modes. Coupling strengths between TLS and resonant

modes were reduced to approximately conserve the decay time

of DðtÞ. Under these conditions, the power law form is pre-

served.


exponent, the only noticeable difference being in

rD. It is remarkable that different internal bath

dynamics can lead to strikingly similar reduced

dynamics for the TLS. In the case of the quasi-

microcanonical bath, the bath survival probability

is Gaussian with recurrences to < 10�4. For theeffective temperature bath, the bath survival de-

cays as a Gaussian until it reaches approximately

10�3 after which the decay has a slow tail. The fact

that the Gaussian regime of the microcanonical

bath is several times longer than in the tempera-

ture bath case, whereas the DðtÞ are very similar,suggests a certain degree of insensitivity of the

reduced dynamics to the dephasing properties ofthe bath (see, however, below for the case of strong

intrabath couplings).

Fig. 9 shows the effect of scaling the bath fre-

quencies for a 7 degree of freedom bath. This was

achieved by multiplying all bath frequencies and

their corresponding system–bath couplings by the

factors indicated in Fig. 9 (equivalent to scaling Dalone, barring a shift in overall time scale). At thetwo extremes in Fig. 9, D is either much greater

than the highest bath oscillator frequency, or

much lower than the lowest bath frequency. For

the intermediate cases, the system is still resonant

with one of the bath oscillators. Tuning the split-

ting far out of the range of the bath frequencies

slows down DðtÞ, and brings about single-expo-nential character (although the slowest decays in

Fig. 9 are too incomplete to judge whether discrete

off-resonant baths induce strictly single-exponen-tial dynamics in the TLS).

The effect of simply increasing the number of

bath modes N coupled to the system over a given

frequency range may be seen in Fig. 10, which

compares baths of 4, 7 and 21 degrees of freedom

under otherwise similar conditions of average

coupling, temperature, etc. (see Table 3). Of

course, a completely fair comparison with spectraldensities JðxÞ of same shape and exactly equalaverage couplings is not possible with discrete

baths. However, the comparison of the three ran-

domly generated baths shows that the power law

behavior is gradually suppressed as the number of

degrees of freedom in the bath is increased: the

exponent d increases from 1.3 to 2.9 as N increasesfrom 4 to 21. Nonetheless, it is remarkable thatsub-exponential dynamics still far from exponen-

tial can be obtained with a rather large number of

bath modes at kT � D. The N ¼ 21 case is close tothe first shell of a polyatomic solvent interacting

with a solute undergoing conformational switch-

ing on a 300 fs time scale at 300 K.

Fig. 9. DðtÞ as a function of bath frequency scaling. System–bath couplings were scaled by the same factor used to scale bath

frequencies. At the two extremes, the TLS splitting is either

lower than the lowest bath frequency, or higher than the highest

bath frequency. In each of the intermediate cases, D is resonantwith one environmental xi.

Fig. 10. The effect on DðtÞ of varying the number of environ-ment modes interacting with the TLS. The power law form of

DðtÞ is preserved in these cases, although the decay exponentincreases with the number of interacting modes.


Fig. 11 shows the effect of introducing cou-plings among bath degrees of freedom (see Eq. (4)

and Table 4). The decays speed up at small values

of the intra-bath coupling, but the power law form

of DðtÞ is preserved. A turnover value of the cou-pling strength is reached, after which the decay

slows and becomes single-exponential. Strong in-

trabath couplings, which allow the bath to relax in

the absence of the system, are the most effectivemechanism we found for eliminating sub-expo-

nential dynamics in favor of exponential decays.

Note that the highest coupling strengths in Fig. 11

are fairly large and that significant sub-exponential

TLS decoherence can still occur for realistic values

of the intrabath coupling (Vi several % of D).Finally, Fig. 12 illustrates the effect of adding a

second shell of bath modes, which are not directlycoupled to the TLS via the coupling term of Eq.

(1), but are coupled to all other bath modes as

shown in Eq. (4). This second shell allows the re-

duced density matrix of the first bath shell to

decohere, but does not directly affect the system

dynamics. The computed TLS DðtÞ decays speed

up slightly with each additional ‘second shell’

mode, but without losing the power law form en-

tirely. The source of this behavior is essentially the

same as that which accounts for the trend seen inFig. 11 and is discussed in Section 4. Other com-

putations (not shown) have verified that artificially

large couplings between even only a single second

shell mode and the first shell bath modes can also

Table 3


for 4 mode (top) and 7 mode (bottom) baths in Fig. 10

xi=D Vi=D xi=D Vi=D

0.138 0.0500 0.185 0.100

0.342 0.0656 0.343 0.103

1.001 0.0906 0.383 0.103

1.348 0.0991 0.497 0.105

0.576 0.106

0.675 0.107

0.762 0.107

0.775 0.107

1.004 0.109

1.071 0.109

1.149 0.110

1.262 0.110

7 mode 1.437 0.111

1.545 0.111

0.118 0.050 1.761 0.112

0.249 0.063 2.076 0.113

0.536 0.079 2.124 0.113

1.000 0.095 2.414 0.114

1.140 0.099 2.424 0.114

2.108 0.119 2.522 0.114

3.411 0.137 2.594 0.114

Third and fourth column: frequencies and coupling con-

stants for 21 mode bath in Fig. 10.

Fig. 11. Smoothed decays and fits for DðtÞas a function of in-tra-bath coupling strength. Vrms is reported in units of D. As inthe case of Fig. 4, a turnover effect can be seen. However, here

DðtÞ eventually becomes exponential as Vrms is increased: cou-plings within the bath are most effective at restoring exponential

decoherence.

Table 4

First and second columns: frequencies and system–bath cou-

pling constants for all calculations in Fig. 11

xi=D Vi=D xi=D Vi=D

0.082 0.050 0.138 0.0500

0.189 0.064 0.342 0.0656

0.583 0.090 0.861 0.0000 (1)

1.007 0.106 1.001 0.0906

1.925 0.129 1.257 0.0000 (2)

1.965 0.130 1.348 0.0991

2.755 0.144 1.971 0.0000 (3)

3.389 0.153

4.065 0.161

Anharmonic couplings within the bath were varied as in-

dicated in Fig. 11. Third and fourth columns: frequencies and

system–bath couplings for calculations in Fig. 12. Numbers in

parentheses indicate order of inclusion into the second shell for

successively faster calculations.


restore exponential dynamics. However, addition

of a second shell, as well as adding more modes of

xi � D (Fig. 7) is far less effective than increased

intrabath couplings in the first shell (Fig. 11) atrestoring single exponential dynamics.

4. Discussion

The discrete SBH behaves in analogy to the

continuous SBH in some respects (decoherence

turnover), but shows subexponential dynamicseven at high temperatures. We begin with a dis-

cussion of rate turnovers and conditions for non-

exponential decays, and conclude by discussing

how these findings are relevant to applications

such as condensed phase control and quantum

computation. Although the results presented here

are numerical only, we believe that the same pro-

cess is acting as in previous analytical studies ofboson–boson decoherence: [21] anisotropic quan-

tum diffusion caused by non-smooth coupling

structure [22,24,33].

Whatever the exact functional form of the

decay may be, turnover effects similar to those

observed here have been extensively discussed in

the literature [36], and in this regard the discrete

SBH shows no anomalous behavior. In the context

of classical activated rate processes, rates are

proportional to friction for small friction values

(energy diffusion regime) and inversely propor-tional to friction for large values (spatial diffusion

regime) [35,36]. The discrete SBH in Section 3 is a

suitable quantum-mechanical model for a rate

process connecting an isoenergetic product and

reactant. In the context of a microscopic quantum

model, it is useful to reiterate that the classical

connection between friction and the time depen-

dent external force expressed by the fluctuation-dissipation theorem is already inherent in the

coupling terms Vi of Eq. (1). Turnover in the ratesof tunneling from a metastable state has also been

couched in the terminology of quantum measure-

ment theory: [1] Ever more frequent position

measurements of the TLS via the last term in Eq.

(1) suppresses tunneling of the system initially in

either the symmetric or antisymmetric combina-tion of jþi and j�i. A variety of turnovers in

electron transfer rates also have been simulated

[16]. In that work, the model consisted of a TLS

coupled to a single harmonic oscillator. Further

environmental relaxation was added into the

master equation in the form of complex rate con-

stants. In particular, turnover effects in tunneling

splitting and oscillator relaxation were observed.In the present treatment, the former effect only

occurs through adjusting the strength of the final

term of Eq. (1) (see Fig. 5), while the latter can be

incorporated through the last term in Eq. (4) (see

Fig. 11).

The decoherence turnover for a discrete bath

strongly coupled to the system may be understood

in analogy to the standard treatment of a discretestate embedded in a continuum [37]. In that

treatment it arises from the interplay of energy

shifts and broadening caused by increasing the Viin Eq. (1). Let fjiig be a dense discrete prediago-nalized manifold coupled to a state j0i at E0 bycouplings Vi ¼ Vai. If we place a window in an

energy range E � DE, we can define the overlap

DN0ðE; V Þ ¼X

En�ðE�DE;EþDEÞjh0jnij2; ð11Þ

Fig. 12. The effect of additional modes in a second coupling

shell, not directly coupled to the TLS; the decay exponent

characterizing DðtÞ gradually increases at the intra-bath cou-pling strengths used here, but exponential decoherence is not

reestablished. The bath Hamiltonian used here was Eq. (4).


where jni are eigenstates of the full Hamiltonian inthe basis fj0i; fiig, and En are the eigenenergies.

As V is increased for a given E 6¼ E0;DN0 goesthrough a maximum as shown in Fig. 13. Initially

the increased coupling mixes zero-order states nearE more efficiently with j0i to yield an enhancedoverlap of the resulting eigenstates with j0i. Athigher coupling j0i is diluted over many eigen-states, and the overlap decreases. For a coarse-

grained Lorentzian distribution � C=½C2þðE � E0Þ2� of the overlaps in Eq. (11), it followsthat at small coupling strengths DN0 is propor-tional to C � V 2, whereas at large couplingstrengths, it is proportional to C�1 � V �2. The

exact V where turnover occurs of course will de-

pend on how the population is distributed over

states as a function of E.We now turn to the conditions for nonexpo-

nential decays in the discrete SBH at high tem-

perature. Increased system–bath couplings lead to

a slightly more exponential form of the decays inthe turnover regime. However, we were able to

observe only stretched exponentials, not single

exponentials, even for very high system–bath

coupling strengths Vi ¼ 1:19D, Fig. 5. The reasonfor this small effect on the functional form of the

decays is that increased system–bath couplings do

not affect the discreteness of the bath. Although

stronger system–bath couplings enhance the abil-

ity of the bath to decohere the system, they do notqualitatively modify the spectral density the way

anharmonicity does at high temperatures. Con-

versely, one may expect the effect of system–bath

couplings to dominate over anharmonicity at very

low temperatures. Related to this is the fact that a

discrete Ohmic, just like the discrete random,

spectral density still produces power law decays

(Fig. 6). Although the Ohmic spectral density issmoother than the random one in a certain sense,

the actual JðxÞ is still discrete with the same finitenumber of bath oscillators.

The spectral density can also be filled in by

adding a second shell of bath modes not coupled

to the system, but which decoheres the first shell of

bath modes via bilinear or higher order couplings

(Fig. 12). Unlike first-shell anharmonicity, onewould expect this to fill in JðxÞ even as T ! 0.

This can be seen easily as follows: Let Hbath ¼Hshell1 þ Hshell2 þ Hshell12 in Eq. (4), where shells 1and 2 are distinguished only by the fact that Vi ¼ 0for the shell 2 modes, and Hshell12 is the part of Eq.(4) which couples those modes with Vi 6¼ 0 withthose that have Vi ¼ 0. Then prediagonalize thisbath Hamiltonian and apply the same unitarytransformation to Hint (which couples only the firstshell to the system) to obtain

Hint ¼Xs0sb0b

Vs0sb0bjs0ihsj � jb0ihbj

¼Xs0sb0b

Vs0sb0bjs0ihsj �

Xb0b

jb0ihb0jb0ihbjbihbj

¼Xs0sb0b

Vs0sb0b

Xb0b

hb0jb0ihbjbi !

js0ihsj � jb0ihbj;

ð12Þ

where jbi are eigenstates of the uncoupled firstshell, and jbi are full bath eigenstates. All modesappear in the interaction Hamiltonian [12], as

opposed to just the first shell modes directly cou-

pled to the TLS; presumably the second shell

Fig. 13. Explanation of the turnover effect for a discrete

quantum system. As the coupling is increased, broadening of

the lineshape initially increases its overlap with an off-resonant

energy window. Eventually the component of the lineshape

within the window begins to decrease as broadening continues

to increase.


modes still make a smaller contribution to JðxÞbecause they are only indirectly coupled to the

system. Nonetheless, the spectral density function

fills in, as in the case discussed in the previous

paragraph and shown in Fig. 1.One should keep in mind that the results pre-

sented in Fig. 12 deal with the case in which second

shell modes are as strongly coupled to first shell

modes as they are among themselves. The only

difference is that the second shell is not directly

coupled to the system. This is a realistic situation,

whether the sphere in Fig. 1 is thought of in state

space or in real space. In real disordered systems(e.g. glasses) modes are localized. Only a small

number of environmental oscillators can interact

with the system. Other degrees of freedom only act

by decohering the modes coupled to the system,

but have no significant coupling matrix elements

with the system itself. The effect of the second shell

on system coherence in Fig. 12 is already rather

weak. Additional outer shells might be expected tohave even smaller effective couplings to the system

because the connecting coupling chains are even

longer. This raises the possibility, as in molecular

vibrational dephasing, that even a large number of

shells will not destroy power law decoherence of

the system, as long as the shells do not interdiffuse

on the time scale of the power law decays. A strict

assignment of inner and outer shells is of coursenot possible over a time scale comparable or

longer to the diffusion time scale for the molecular

(or other) carriers of bath modes. Nonetheless,

given the large deviations in the 0.1–0.01 range

from single exponential decays discussed in Sec-

tion 3, it may be possible to observe such behavior

even in liquid crystals or liquids.

By far the most efficient way of restoring anexponential decay of DðtÞ was to introduce cou-plings amongst the environmental degrees of

freedom (Fig. 11). Diagonal and off-diagonal an-

harmonicity in the bath effectively introduces new

frequencies into the spectral density as the tem-

perature is increased. Each bath mode has an array

of frequencies mn associated with it such that

mn�1!n > mn!nþ1, where n is the occupation number.At low temperatures, the population in levels with

occupation numbers n > 0 is small and anharmo-nicity merely shifts the m0!1 mode frequency. At

temperatures large enough for multiple levels of

each oscillator to be occupied, the spectral density

is enhanced at frequencies below the nominal os-

cillator frequencies m0!1. As a result, the spectraldensity becomes smoother and loses the fluctua-

tions required for sub-exponential decays. Wetherefore expect the effectiveness of anharmonicity

in destroying multi-scale dynamics to be most

important at high temperatures. Many important

physical and biological systems usually described

by the SBH operate precisely at those high tem-

peratures.

A caveat raised by this picture is that popula-

tions and coherences among the bath degrees offreedom are linked by anharmonicity in a partic-

ular way, which cannot be exactly preserved when

anharmonicity is mimicked by additional har-

monic degrees of freedom with lower frequencies.

It would be interesting to study to what extent the

dynamics are affected by mapping the anharmonic

environment into a harmonic environment. Cer-

tainly a different harmonic spectral density wouldbe required at each temperature, whereas anhar-

monicity introduces a temperature dependence

into JðxÞ naturally.The computations in Section 3 indicate that

sub-exponential decays occur when JðxÞ is notsmoothed too much by anharmonicity, and a well-

defined first shell of bath modes exists. The crite-

rion for observing sub-exponential decays thenbecomes simply the following: if JðxÞ is fitted to asum of resonant lineshapes (e.g. Lorentzians), then

the widths of the most prominent of these features

must be sufficiently less than their spacings. This

may be the case for the spectral density of a pro-

tein pertinent to exciton decay in Fig. 2 [17], and

similar spectral densities have been computed for

electron transfer problems [5]. Based on the resultsdiscussed here, it is very likely that the peaks of

JðxÞ can be represented by a small number of first-shell modes localized near the tunneling system of

interest, and that their width can be characterized

by anharmonicity and/or couplings to higher shells

not interacting with the system directly. If experi-

mental decays at higher temperatures are found to

be exponential in the D or P < 0:05 region, weconclude that anharmonicity in those bath modes

localized near the system, and not a continuous


distribution of harmonic bath modes, provides the

best physical description for the decoherence pro-

cess.

We now discuss some implications that power

law decays of the discrete SBH may have for

quantum computation and for condensed phasecoherent control of molecules or other mesoscale

systems such as quantum dots. The general im-

plication is that the qualitatively longer time scale

of the decoherence allows more time for the en-

tanglement process necessary for quantum com-

putation, and a longer time scale over which

molecular coherences can be controlled.

Quantum computation trades the (sub)expo-nentially long time required by certain classical

computations for a polynomial time performance,

by making use of the exponential dependence of

total wavefunction complexity on the number of

degrees of freedom. This is achieved at the expense

of fitting the computation within a time window

shorter than the decoherence time of a register.

This decoherence is usually predicted to occurexponentially in time and in the size of the register.

For example, the effect of non-dissipative interac-

tions between a qubit coupled to an oscillator bath

with a continuous spectrum has been studied for

the purpose of estimating timescales over which a

collection of such qubits might function in a

quantum register [38,39]. Aside from the sobering

conclusion that decoherence is exponentially rapidin the number of qubits in the register, both re-

ports identified three time regimes with distinct

decoherence characteristics. At times shorter than

x�1c , where xc is the high frequency cutoff of the

bath, the time dependence is Gaussian. For

x�1c < t < T�1, where T is the environment tem-

perature, coherence decays as t�1. For times longerthan thermal time T�1 decoherence is exponential.The author of [38] therefore concluded that the

thermal time imposes an insurmountable restric-

tion on the time period over which a quantum

computation is feasible. These results apply gen-

erally to the case of continuous spectral densities

of the type xs with a high-frequency cutoff.

The results obtained in this work suggest that in

baths with fluctuating spectral density, subexpo-nential decoherence of qubits occurs beyond the

thermal time scale. This could result in a signifi-

cant extension of the time over which a quantum

computation can be executed. In particular, if de-

coherence is only a polynomial in time, the time

window over which a calculation could be per-

formed could be dramatically longer. The im-

provement is most significant when there is acorrelation between the noise sources acting on the

qubits. If each qubit in a register were coupled to

an independent bath, the decay exponent could

still be quite high: a power law decay raised to a

high power becomes virtually indistinguishable

from an exponential function. On the other hand,

the possibility has been raised that under certain

conditions (e.g. qubit separations shorter thanenvironmental coherence lengths), independent

decoherence may not be a valid assumption [38].

Such correlations also reduce the effective dimen-

sionality of the bath. To exploit slow decoherence

practically, our results suggest that the qubits

should be embedded in a low-symmetry glassy

environment which satisfies the locality conditions

necessary to produce a shell structure in the bath.Such an environment would produce the highly

structured spectral densities required for slow as-

ymptotic decoherence at low temperature. Finally,

it is worth noting that subexponential decay of

coherences has already been observed in NMR

spectra [40], inferred from spectra [33], and inter-

preted as a manifestation of the multiple timescale

nature of quantum diffusion of a wavefunction instate space [21,22].

The other obvious application of our results is

in condensed phase coherent control. It has been

demonstrated that molecular coherence of isolated

molecules can be controlled by interaction with

lasers to direct the outcome of chemical reactions

[41,42]. Our earlier work has shown that power

law decays of vibrational coherence in gas phasemolecules can be used to control their wavepacket

dynamics for extended times [23,43]. It is not clear

at present whether these approaches can be gen-

erally extended to the condensed phase. The re-

sults provided here offer some hope that this is

possible. In clusters, and on a sufficiently short

time scale also in liquid crystals or in solution,

molecules are surrounded by small and well-de-fined solvent shells. The low symmetry of these

materials results in mode localization, which at


very short times can even be described by ‘‘in-

stantaneous normal modes’’ [44]. Only a limited

number of solvent modes can effectively dephase

the molecule; the 21 mode system disussed in

Section 3 is quite realistic in this context.

If the molecular phase randomization caused bythe accessible solvent modes is in the form of a

power law, laser-induced coherence could be

maintained for 1 or 2 orders of magnitude longer

time than suggested by exponential dephasing

models. As discussed in Section 3, the coupling of

the first solvent shell to further solvent shells does

not substantially alter the dephasing dynamics.

This would open up the possibility of control, atleast up to the time scale sdiff of diffusion processeswhich switch solvent shells (picoseconds in liquids,

much longer in clusters or glasses).

The decoherence decays discussed in Section 3

all show an initial exponential decay down to

D � 0:1–0.01. This initial exponential phase offersthe most serious challenge for condensed phase

control. However, modern pulse shaping equip-ment can produce pulses with 100 to 1000 phase

and amplitude channels. It has been shown for

vibrational dephasing processes that the inverse of

the survival probability jhtj0ij2 for an initial stateis a direct measure of the number of parameters

required in the control field, as long as the

coupling Hamiltonian has the form of Eq. (4) or

the form of a closely related local random matrixmodel [33]. The very general criterion for the

validity of this type of coupling Hamiltonian is

the correlation among coupling constants intro-

duced by mode localization [33]. If this is satisfied

by solvent shells at t < sdiff , then even the ex-ponential phase can be controlled unless 1=Dincreases above the number of control channels

available.In summary, we find that before the coherence

of a two-level system coupled to a discrete oscil-

lator bath decays to its long-time average, the

dynamics undergo a subexponential phase. Mod-

ifications of the Hamiltonian which lead to

smoother spectral densities (most importantly bath

anharmonicity) tend to bring the dynamics back

towards single exponential decoherence. We thusattribute the subexponential dynamics to highly

structured spectral densities. Computed spectral

densities for realistic model systems indeed show

such structure, which can be described by discrete

modes with an associated width. The location and

width of such modes can be mimicked using an

SBH-like model with two shells of modes. The first

shell determines mode frequencies, and the secondshell determines mode widths. Slow dephasing

dynamics might be exploited to extend the time

scale for quantum computation or molecular co-

herent control in condensed media. Glassy baths

are the ones best described by this model because

they offer the localized mode structure necessary

for a structured spectral density, and the ‘‘shell’’

approximation is valid even at long time, while itfails in liquids. Finally, in baths of localized

modes, anharmonic couplings are more important

than a large number of oscillators in restoring

exponential decoherence.

Acknowledgements

This research was funded by a grant from the

National Science Foundation (CHE 9986670), and

by a University of Illinois University Scholarship.

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sub-exponential spin-boson decoherence in a finite bath

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