Overdamping by weakly coupled environments

Overdamping by weakly coupled environments

Massimiliano Esposito

Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium

Fritz Haake

Fachbereich Physik, Universität Duisburg-Essen, 45117 Essen, Germany 共Received 27 May 2005; published 15 December 2005兲

A quantum system weakly interacting with a fast environment usually undergoes a relaxation with complex frequencies whose imaginary parts are damping rates quadratic in the coupling to the environment in accord with Fermi’s “golden rule.” We show for various models 共spin damped by harmonic-oscillator or random- matrix baths, quantum diffusion, and quantum Brownian motion兲 that upon increasing the coupling up to a critical value still small enough to allow for weak-coupling Markovian master equations, a different relaxation regime can occur. In that regime, complex frequencies lose their real parts such that the process becomes overdamped. Our results call into question the standard belief that overdamping is exclusively a strong cou- pling feature.

DOI:10.1103/PhysRevA.72.063808 PACS number共s兲: 42.50.Ct, 05.30.⫺d, 03.65.Yz, 76.20.⫹q

I. INTRODUCTION

The dynamics of an isolated and finite quantum system consists of a reversible superposition of oscillations with 共real兲 Bohr frequencies␻S. In order to understand the irre- versible processes occurring in finite quantum systems, such as relaxation to equilibrium or decoherence, one needs to take into account the interaction between the system and its environment. The weak-interaction limit together with the Markovian approximation already allow a good understand- ing of such irreversible processes and has some universal features. The generator of the evolution of the共reduced兲 den- sity matrix of the system obtained by second-order perturba- tion theory 共often called the Redfieldian兲 is not an anti- Hermitian generator any more. Its eigenvalues⌫+i⍀ acquire a real part ⌫ describing irreversible decay to equilibrium.

The imaginary parts of the eigenvalues are shifted Bohr fre- quencies ⍀=␻S−␦␻. The two shifts ⌫ and ␦␻^{, normally} increase 共quadratically兲 as the strength of the coupling grows.

We show here that Markovian perturbative master equa- tions such as the Redfield equation关1–7兴 allow for more than just describing the well known normal damping just men- tioned. When the coupling strength is increased, it can hap- pen that, at a critical value, a shifted frequency⍀ vanishes and for yet stronger coupling goes imaginary. The pertinent eigenvalues⌫−兩⍀兩 are real and, interestingly, decrease with growing coupling. The resulting relaxation is nonoscillatory, i.e., overdamped. The principle purpose of this paper is to show that contrary to common belief the transition to over- damping is still compatible with perturbative treatment. In brief, overdamping can be a weak-coupling effect.

All models to be studied here have Hamiltonians such as Hˆ = Hˆ_S+ Hˆ

B+ SˆBˆ , 共1兲

where Hˆ

S and Hˆ

B, respectively, generate the free motion of the system and the environment共bath兲 while the interaction involves respective coupling agents Sˆ and Bˆ .

It may be well to emphasize that the so-called rotating- wave approximation关7,8兴, extremely useful as it may be for very weak damping, in particular in quantum optics, is defi- nitely not allowable for strong damping and overdamping.

Indeed, the rotating-wave approximation is based on the as- sumption that the Bohr frequencies of the system are very large compared to the system damping rate such that all “an- tiresonant” terms can be time averaged out when writing the master equation in the interaction picture. But overdamping occurs precisely when the Bohr frequencies of the system become of the order of or smaller than the system damping rate. In a recent study of low-quality resonators 关9兴, the rotating-wave approximation was shown to still be afford- able for overlapping resonances. But the Hamiltonians to be employed in the present paper must retain the antiresonant terms that the rotating-wave approximation would suppress.

A word on physical contexts where overdamping shows up is in order. One is diffusion, a topic to be dealt with below 共Sec. III兲. Another is temporal fluctuations in critical phe- nomena, described by time dependent Ginzburg-Landau equations without inertial terms 关10兴; Ref. 关11兴 describes a derivation of such a Ginzburg-Landau equation from an un- derlying unitary evolution of a “larger” system.

The plan of the paper is as follows: In Sec. II, we solve the Redfield master equation for a two-level system interact- ing with a general environment. When the environment is made of harmonic oscillators共spin-boson model兲, we show in Sec. II B that the transition from normal damping to over- damping occurs at a critical value of the coupling which can be made arbitrarily small and therefore accessible to pertur- bation theory. For environment operators Hˆ

Band Bˆ modeled by random matrices from the so-called Gaussian orthogonal ensemble 共spin-GORM model兲, we show in Sec. II C that weak-coupling overdamping is compatible with the exact dy- namics computed numerically. In Sec. III, we show that the transition from a nondiffusive to a diffusive regime, identi- fied for a particle traveling in a spatially extended system 1050-2947/2005/72共6兲/063808共9兲/$23.00 063808-1 ©2005 The American Physical Society

while interacting with an environment, corresponds in fact to a transition from normal damping to overdamping; that tran- sition will turn out amenable to perturbative analysis. Finally, in Sec. IV we study the transition from normal damping to overdamping for a central harmonic oscillator interacting with a large collection of harmonic oscillators 共quantum Brownian motion兲. We show that overdamping again allows for perturbative treatment, by comparison with the exact re- sults known for this model. Conclusions are drawn in Sec. V.

II. DAMPED SPIN

A. Hamiltonian and Markovian master equation Any two-level system has the Pauli matrices␴^ˆx,␴^ˆy, and

␴^ˆz共together with unity兲 as a complete set of observables. If such a “spin” interacts with a general environment we may choose the Hamiltonian as

Hˆ =q␻0

2 ␴^ˆz+ Hˆ

B+␴^ˆxBˆ . 共2兲

Inasmuch as the interaction␴^ˆxBˆ does not commute with the Hamiltonians for the uncoupled spin and bath, it allows for transitions between the unperturbed energy levels. Denoting the means of the spin observables by

x共t兲 = Tr␳^ˆ共t兲␴^ˆx, y共t兲 = Tr␳^ˆ共t兲␴^ˆy, z共t兲 = Tr␳^ˆ共t兲␴^ˆy, 共3兲 we write the Redfield equation as共see 关5,6兴兲

z˙共t兲 = 2⌫关z共⬁兲 − z共t兲兴, 共4兲 x˙共t兲 = −␻0y共t兲,

y˙共t兲 =共⍀²+⌫²兲

␻0

x共t兲 − 2⌫y共t兲,

with the time dependent damping rate ⌫共t兲 and frequency

⍀共t兲 and the stationary inversion z共⬁兲

⌫共t兲 = 2 q²

冕

0

t

d␶cos共␻0␶兲C共␶兲, 共5兲

⍀共t兲²+⌫共t兲²=␻02+ 4 q²␻0

冕

0

t

d␶^sin共␻0␶兲C共␶兲,

⌫共t兲z共⬁兲 = 2 q²

冕

0

t

d␶sin共␻0␶兲D共␶兲.

Properties of the bath are represented by the functions C共t兲 and D共t兲, respectively, the real and imaginary parts of the equilibrium autocorrelation function␣共t兲=具Bˆ共t兲Bˆ共0兲典 of the bath coupling agent Bˆ 共for a definition and properties see Appendix A兲.

The Markovian approximation consists of taking the up- per bounds of the time integrals in共5兲 to infinity, such that the damping constant and frequency become time indepen- dent, ⌫共⬁兲⬅⌫, ⍀共⬁兲⬅⍀. That approximation is legiti- mate when the spin dynamics characterized by the rates␻0,

⍀, and ⌫ is much slower than the decay of the bath correla- tion function ␣共t兲 and requires that we restrict the further

discussion to times much larger than the bath correlation time. We may then rewrite共5兲 as

⌫ = ␲

q²关␣^˜共␻0兲 +␣^˜共−␻0兲兴, 共6兲

⍀²+⌫²=␻02+ 4

q²␻02

冕

^d␻P_␻^˜␣₀^2共−^␻␻^兲2, z共⬁兲 =␣^˜共−␻0兲 −␣^˜共␻0兲

␣

˜共−␻0兲 +␣^˜共␻0兲,

with ␣^˜共␻兲 the Fourier transform of ␣共t兲. The solutions of Eqs.共4兲 in the Markovian limit read

z共t兲 = z共⬁兲 + 关z共0兲 − z共⬁兲兴e^−2⌫t, 共7兲 x共t兲 =x共0兲⌫ − y共0兲␻0

⍀ sin共⍀t兲e^−⌫t+ x共0兲cos共⍀t兲e^−⌫t

y共t兲 = x共0兲关共⍀²+⌫²兲/␻0兴 − y共0兲⌫

⍀ sin共⍀t兲e^−⌫t

+ y共0兲cos共⍀t兲e^−⌫t.

The reduced density matrix␳⁼¹₂^{+ x}␴^ˆx+ y␴^ˆy+ z␴^ˆzcan thus be written as a superposition of four modes,

␳^ˆ共t兲 =

兺

␰=1 4

c_␰共0兲␳^ˆ␰es␰t. 共8兲

For normal damping, s₁= 0, s₂= −2⌫, s3= −⌫+i⍀, and s₄= −⌫−i⍀. Overdamping occurs when

⍀²⬍ 0, 共9兲

and then the rates of 共8兲 are given by s1= 0, s₂= −2⌫, s₃= −⌫+兩⍀兩, and s4= −⌫−兩⍀兩.

B. The spin-boson model

Taking the bath as a collection of harmonic oscillators 关5,12兴 we have for its free Hamiltonian and coupling agent

Hˆ_B=1 2

兺

n=1 N

共Pˆn2+␻n2Qˆ

n

2兲, Bˆ =

兺

n=1 N

⑀nQˆ_n. 共10兲

We assume a quasicontinuum of bath frequencies␻n, employ a spectral function ␥共␻兲=兺n⑀n

2␦共␻n−␻兲, and adopt Ullers- ma’s choice共see 关13兴 and Appendix A兲,

␥共␻兲 = 2

␲

␬␣²␻²

␣²+␻^{2, 共11兲} where␣ is the decay rate of the autocorrelator of the bath coupling agent and ␬ an overall coupling strength. Thus equipped we can evaluate the rates in共5兲. In the limits of high temperature, i.e.,␤q␻0⬅q␻0/ kBTⰆ1, we get

⌫ =^␤→02 ␬␣²

␤^q2共␣²⁺␻0 2兲 =

共␻0/␣兲→0 2␬

␤^{q2, 共12兲}

⍀²+⌫² =

␤→0␻0

2+ 4 ␬␣␻02

␤^q2共␣²⁺␻02兲 =

共␻0/␣兲→0

␻0

2, 共13兲

⌫z共⬁兲 = − ␬␣²␻0

q共␣²+␻0 2兲 =

共␻0/␣兲→0

−␬␻0

q ; 共14兲

here the limit␻0/␣→0 has been taken to remain consistent with the Markovian approximation; note that in the present section, ␬, has the dimension of an action such that ⌫ is a rate.

The critical value of␬ at which overdamping occurs is now found with the help of Eq.共9兲 by subtracting Eq. 共12兲 to the power of 2 to Eq.共13兲. We find

␬c =

␤→0q²␤␻0

2 . 共15兲

If␬⬎␬c, and if ␬cis small enough to be treated by pertur- bation theory we have a self-consistent theory of overdamp- ing. Clearly, high temperatures are favorable for that theory to apply since the pertinent ␬c is suppressed by the factor

␤q␻0Ⰶ1.

One might fear that our way of obtaining␬cis not com- pletely consistent if solely restricted to the second-order per- turbation theory because⌫²is of the order ␬^{2 while}⍀²+⌫² is of the order␬and does not include the␬²corrections. That fear will be eased by the following argument. If we were to add O共␬²兲 corrections to the right-hand sides of Eqs. 共12兲 and共13兲, the results 共15兲 for the critical coupling would be generalized to a series in powers of the leading terms dis- played in共15兲. 关In fact, Jang et al. Ref. 关14兴 found ⍀²+⌫²

=␻02共1+4␬^{2/ q2}兲; by recalculating␬c, we again find Eq.共15兲 if␤^q␻0Ⰶ1.兴

Another look at the high-temperature rates reveals an in- teresting feature of overdamping. We have from共8兲

s₁= 0, 共16兲

s₂= − 4␬ q²␤^, s₃= − 2␬

q²␤⁺ 2␬

q²␤

冑

^{1 −}

冉

^␬␬^c

冊

^2,

s₄= − 2␬ q²␤⁻

2␬

q²␤

冑

^{1 −}

冉

^␬␬^c

冊

^2.

Most remarkably, the slowest relaxation rate of the spin, 兩Re关s3兴兩, decreases when the coupling to the environment in-

creases. For strong overdamping,␬c/␬Ⰶ1, we even have s₃= −␤␻0

2

4␬ ^+O

冉

^q2␤␬^33␻40

冊

^{. 共17兲}

This is in contrast to the normal-damping case, accessible from the above by replacing

冑

−1→ +i, where the two slow- est rates 兩Re关s2兴兩 and 兩Re关s3兴兩 increase as the coupling be- comes stronger.

C. The spin-GORM model

We retain the overall Hamiltonian共2兲 but modify the en- vironment so as to let the free-bath Hamiltonian Hˆ

B and the coupling agent Bˆ be represented by random matrices from the Gaussian orthogonal ensemble 共GOE兲. The resulting spin-GORM model was studied in Refs.关6,15兴. We use the results of that work; in particular, we adopt a unit of time that makes the Hamiltonian dimensionless and the bath correla- tion time of order q 关see Eq. 共19兲 below兴. Specifically, we write

Hˆ_B= Xˆ

冑

8N, Bˆ =␩ ^Xˆ⬘

冑

8N; 共18兲 here Xˆ and Xˆ⬘^{are random}共N/2兲⫻共N/2兲 GOE matrices with mean zero. Their nondiagonal共resp. diagonal兲 elements have standard deviation␴ND= 1共resp. ␴D=

冑

2兲. The parameter␩ serves as a coupling strength.

To study this model it is convenient to assume that the environment is initially in a microcanonical distribution with the共dimensionless兲 energy⑀. The autocorrelator of the bath coupling agent then reads

␣共⑀,t兲 =

N→⬁

␩^{2J1关t/共2q兲兴}

4t/q e^i⑀t/q 共19兲 and has the Fourier transform

␣

˜共⑀,␻兲 =

N→⬁␩^2q

2␲

冑

¹4−共⑀+ q␻兲². 共20兲 It is worth noting that here we meet Wigner’s semicircle law for the mean level density of the GOE.

The general rates of the Markovian Redfield equation given in Eq.共6兲 can be evaluated and read

⌫共⑀兲 =␩²

2q

冋 ^冑

^{14−共⑀− q␻0兲2+}

^冑

^{14−共⑀+ q␻0兲2}

册

共21兲 and

⍀共⑀兲²+⌫共⑀兲²=␻02+␩²␻02−␩² q␻0

冑

共⑀^{+ q}␻0兲²−¹₄

␲ ^arctan

冉 冑

^共共^⑀⑀^{+ q+ q␻}␻⁰0^{兲 +}兲²−²¹₄¹

冊

^−␩q2␻0

^冑

^{共⑀+ q␲␻0兲2−14arctan}

冉 冑

^共共^⑀⑀^{+ q+ q␻}␻⁰0^{兲 −}兲²−²¹¹₄

冊

+␩² q␻0

冑

共⑀^{− q}␻0兲²−¹₄

␲ ^arctan

冉 ^冑

^共共⑀⑀^{− q− q␻}␻⁰0^{兲 +}兲²−²¹¹₄

冊

^+␩q2␻0

^冑

^{共⑀− q␲␻0兲2−14arctan}

冉 ^冑

^共共⑀⑀^{− q− q␻}␻⁰0^{兲 −}兲²−¹²¹₄

冊

^{. 共22兲}

If ⍀²⬍0, we have overdamping. To discuss that case, we momentarily set⍀²= A − B where A =⍀²+⌫²and is given by 共21兲 and B=⌫² by 共22兲. When ␩ is large we could have overdamping because B⬎A, but then the perturbation theory may fail and our approach will lose self-consistency. How- ever, since all terms in A共but none in B兲 carry explicit fac- tors ␻0 or ␻02, and since the other quantities containing ␻0

关i.e.,

冑

. and arctan共.兲兴 are bounded away from zero in the limit␻0→0, it is always possible to choose ␻0sufficiently small such that A⬍B for small␩. This is illustrated in Fig. 1.

The dependence of the smallest rates on the coupling strength is similar as in the spin-boson model. The rates 兩Re关s3兴兩 and 兩Re关s4兴兩 关see Eqs. 共8兲兴, grow with the coupling constant␩in the normal-damping regime, have a cusp at the transition, and then decay into the regime of overdamping, as illustrated in Fig. 2.

We have numerically solved the exact dynamics in order to verify that the perturbative equation predicts the correct dynamics for normal damping as well as for overdamping.

The agreement is excellent as illustrated in Fig. 3. We can conclude that the spin-GORM model allows for overdamp- ing at weak coupling.

III. DIFFUSION MODEL

We now consider a particle moving on a one dimensional closed loop while interacting with an environment. The per- tinent dynamics has been studied recently in Refs.关6,16,17兴 by using the Redfield equation. A transition from nondiffu- sive to diffusive relaxation has been identified. We shall use the results of this study to show that the transition men- tioned, in fact, is one from normal damping to overdamping.

The Hamiltonian of the loop constituting the subsystem is represented by an N⫻N matrix

Hˆ

S=

冢

^{− A− AE00⯗00 − A− AE0000 − AE……
00 − A− A
0000 − A…E…
00 − A− AE0000 − A− AE⯗0000}

冣

^N⫻N _共23兲

taken in the site basis兩l典, where l=0,1,…,N−1 labels the N sites on the loop. The diagonal elements of Hˆ

Sare the on-site

energies of the particle while the off diagonal elements gen- erate hopping to neighboring sites.

A weak interaction with an environment is described by the Redfield master equation. The correlation time of the environment is assumed much shorter than all characteristic time scales of the loop and therefore the correlation function of the environment can be modeled by

␣ll⬘共␶兲 = 2Q␦共␶兲␦ll⬘. 共24兲 By using the Bloch theorem, the Redfield generator共contain- ing N⁴elements兲 can be simplified in N independent sectors 共with N² elements兲, each corresponding to a given value of the Bloch number q. For our finite loop, periodicity yields q = n2␲/ N, where n = 1 , 2 ,…,N. By diagonalizing a given sector we get N eigenvalues depending on q. The complete spectrum of the Redfield generator then consists of the N² eigenvalues obtained by varying q.

As already mentioned, two relaxation regimes have been identified in this model. In the nondiffusive regime all eigen- values are complex with real parts of similar magnitude, pro- portional to the coupling constant Q,

Re关s兴 ⬇ −2Q

q² +O

冉

N¹

冊

^{. 共25兲}

However, in a given sector 共therefore at a given q兲 when the coupling term is increased beyond the value Q

= 2qA sin共q/2兲, one of the N eigenvalues separates from the other N − 1 ones. This eigenvalue is always real and is called the diffusive one. The diffusive branch is made of the diffu- FIG. 1.␩^{= 0.2,}⑀=0, and q=1. This figure shows that the con-

dition for overdamping can be satisfied at weak coupling if␻0is sufficiently small. This is still true for any generic choice of⑀.

FIG. 2. ␻0= 0.01,⑀=0, and q=1. The upper figure shows that for a fixed and small value of␻0there exists a critical and small value of the coupling␩c⬇0.14 above which overdamping occurs.

The lower figure illustrates the qualitative change of the coupling dependence of the slowest relaxation rates when going from the normal damping regime to the overdamped regime.

sive eigenvalues of the different sectors. These eigenvalues have a smaller magnitude than the real parts of the nondif- fusive eigenvalues. They therefore control the long time re- laxation of the subsystem. The diffusive eigenvalues are given by

s = −2Q q² +2Q

q²

冑

^{1 −}

冉

^{2qAQ sinq2}

冊

^{2. 共26兲}

Lets define Qc⬅2qA sin共␲/ N兲⬇2␲qA / N. For Q⬍Qc no diffusive eigenvalues are present in the spectrum and the relaxation regime is nondiffusive关see Eq. 共25兲兴. As soon as Q⬎Qc, at least two diffusive eigenvalues exist in the spec- trum and the relaxation regime is called the diffusive regime.

The two smallest diffusive eigenvalues controlling the long time scale relaxation are

s = −4␲^2A2

QN² . 共27兲

Notice that the perturbative approach is consistent, because Qccan be made as small as desired by choosing A / N small.

It is already clear at this point that the nondiffusive共resp.

diffusive兲 regime implies normal damping 共resp. overdamp- ing兲. Indeed, as for normal damping 共resp. overdamping兲, the smallest relaxation rates increase共resp. decrease兲 with grow- ing coupling in the nondiffusive 共resp. diffusing兲 regime.

Furthermore, as in the normal-damping共resp. overdamping兲 regime, the small Redfield eigenvalues are complex 共resp.

real兲 in the nondiffusive 共resp. diffusing兲 regime. We can make that association even clearer if we assume the environ- ment made of harmonic oscillators which we model by using Ullersma’s spectral density共see Appendix A兲. In this case, we find that at high temperature, the zero-frequency limit of the Fourier transform of the environment correlation function is given by

␻→0lim␣^˜共␻兲 = lim

␻→0

J共␻兲

␻ ⁼

␬

␲␤^{. 共28兲}

Since our instantaneous-decay assumption共24兲 implies

␣

˜共0兲 =Q

␲^{, 共29兲}

we conclude

Q =␬

␤ ^共30兲

and thus find the diffusive-branch eigenvalues

s = − 2␬ q²␤⁺

2␬

q²␤

冑

^{1 −}

冋

^{4qA sin}

^冉

^q2

^冊

^2␤␬

册

²

= −␤^A2

␬ ^q

2+O

冉

^q2␤␬^33A4q

4

冊

^{. 共31兲}

The similarity between these diffusive eigenvalues and the smallest eigenvalue of the spin-boson model in the over- damping regime关s3 in Eq. 共16兲兴 is obvious as is similarity between the real part of the small nondiffusive eigenvalues 关共25兲 with 共30兲兴 and the real parts of the small eigenvalues of the spin-boson model in the normal damping regime关s3and s₄in共16兲兴.

IV. QUANTUM BROWNIAN MOTION (QBM) A. Hamiltonian

In this section we study overdamping in an exactly solv- able model of the Brownian motion. The model is made of a central harmonic oscillator interacting with an environment which itself is a collection of harmonic oscillators共see, e.g., 关7,13,18,19兴; these references will lead the reader to earlier work兲. The exact solution proves extremely valuable for our FIG. 3. 共Color online兲 Transition from normal damping to overdamping in the spin-GORM model. The full lines represent the exact dynam- ics of the three spin observable x共t兲, y共t兲, and z共t兲 obtained numerically by diagonalizing the full Hamiltonian and the dashed lines represent the dynamics predicted by the Redfield equation 共second order perturbation theory兲. The two re- sults give curves which are so close to each other that the dashed lines are almost invisible. The situation depicted here is the same as in Fig. 2, where␩^{varies and}␻0= 0.01,⑀=0, and q=1. As predicted by Redfield theory, the transition occurs at ␩c⬇0.14. The initial condition is x共0兲=^冑₃⁸, y共0兲=0, and z共0兲=₃¹. For the exact dynamics we have taken N = 3000 and a width of the initial energy shell␦⑀=0.025.

endeavor since it will be seen to yield, in the Markovian limit, precisely the same condition for overdamping as the perturbative treatment.

We write the total QBM Hamiltonian as共see 关20兴兲 Hˆ =1

2共Pˆ²+␻0 2Qˆ²兲 +1

2

兺

n=1

N

冋

^Pˆn2+␻n2

冉

^Qˆn−␻^⑀nn2Qˆ

冊

²

册

共32兲 and thus have the system and bath parts

Hˆ

S=1

2

冋

^Pˆ2+

冉

^␻02+

^兺

^{n=1N ␻⑀n2n2}

冊

^Qˆ2

册

^{, 共33兲}

Hˆ

B=1 2

兺

n=1 N

共Pˆn2+␻n2Qˆ

n

2兲. 共34兲

The coupling agents of the system and bath read Sˆ = Qˆ

0; Bˆ = −

兺

n=1 N

⑀nQˆ

n. 共35兲

The QBM Hamiltonian共32兲 is a sum of squares and thus manifestly positive. A not manifestly positive variant of that Hamiltonian 关13,19兴, discussed in Appendix B, can be mapped onto the QBM Hamiltonian by a renormalization of the bare frequency of the central oscillator. That observation allows us to use the exact results of Ref.关19兴 for our present study of QBM.

B. Exact treatment

The Hamiltonian共32兲 generates the Heisenberg equations of motion

Pˆ˙ 共t兲 = −

冉

^␻02+n=1

^兺

^{N ␻⑀2n2}

冊

^{Qˆ 共t兲 −}

^兺

^{n=1N ⑀nQˆn共t兲,}

Pˆ˙

n共t兲 = −␻n 2Qˆ

n共t兲 −⑀nQˆ 共t兲, Qˆ˙ 共t兲 = Pˆ共t兲,

Qˆ˙

n共t兲 = Pˆn共t兲. 共36兲

The solution of共36兲 can be written as Qˆ

␯共t兲 =

兺

_␯=0^{N 关A˙␮␯共t兲Qˆ␯共0兲 + A␮␯共t兲Pˆ␯共0兲兴, 共37兲}

Pˆ

␯共t兲 = Qˆ˙

␯共t兲;

the indices␮and␯step from 0 to N and Qˆ

0⬅Qˆ, Pˆ0⬅ Pˆ. All A_␮␯共t兲’s can be expressed in terms of the function

g共z兲 = z²−␻0 2−

兺

n=1 N ⑀²

␻n2−

兺

n=1 N ⑀n2

z²−␻n2. 共38兲 The zeros of g共z兲 yield the eigenfrequencies of Eqs. 共36兲.

Assuming the bath frequencies form a quasicontinuum we employ a spectral function ␥共␻兲=兺n⑀n

2␦共␻n−␻兲 to replace the sum in共38兲 by an integral,

g共z兲 = z²−␻02−

冕

0

⬁

d␻␥共␻兲

␻^{2 −}

冕

0

⬁

d␻ ␥共␻兲

z²−␻^{2. 共39兲} We adopt an initial condition with statistical independence of the central oscillator and bath without restriction for the density operator␳共0兲 of the central oscillator,

␳^ˆtot共0兲 =␳^ˆ共0兲e^−␤HˆB ZB

. 共40兲

The time dependent density operator of the central oscillator then obeys the exact master equation

␳^ˆ˙共t兲 = − i

2q关Pˆ²− fpq共t兲Qˆ²,␳^ˆ共t兲兴 + i

qfpp共t兲†Qˆ,关Pˆ,␳^ˆ共t兲兴+‡

− 1

q²dpp共t兲†Qˆ,关Qˆ,␳^ˆ共t兲兴‡ + 1

q²dpq共t兲†Pˆ,关Qˆ,␳^ˆ共t兲兴‡, 共41兲 with关·, ·兴+the anticommutator. The drift and diffusion coef- ficients fpq共t兲, fpp共t兲, dpp共t兲, and dpq共t兲 can be found in 关19兴;

they can all be expressed in terms of the quantity A共t兲

⬅A00共t兲. To get an explicit result for that amplitude we adopt Ullersma’s spectral function,

␥共␻兲 = 2

␲

␬␣²␻²

␣²⁺␻^{2, 共42兲} where␣^and␬are the decay rate of the autocorrelator of the bath coupling agent and an overall coupling strength, both now of the dimension of a frequency. For that choice the amplitude in question takes the form

A共t兲 = 2⌫

␭²+⍀²+⌫²− 2␭⌫关e^−␭t− e^−⌫tcos共⍀t兲兴 + ␭²+⍀²−⌫²

␭²+⍀²+⌫²− 2␭⌫

1

⍀e^−⌫tsin共⍀t兲. 共43兲 Here, the three rates 共⌫,⍀,␭兲 control the exact dynamics;

they are connected to the three model parameters共␻0,␬^,␣兲 by the characteristic equations

␭ =␣− 2⌫,

␻0

2+␣␬⁼⍀²+⌫²+ 2␭⌫,

␻0

2=共⍀²+⌫²兲共␭/␣兲. 共44兲

The coupling between the central oscillator and bath is thus seen to shift the unperturbed frequency as␻0→⍀+i⌫ and the unperturbed bath decay rate as␣→␭.

We should mention that共the diffusion coefficients兲 dpp共t兲 and dpq共t兲, in contrast to 共the drift coefficients兲 fpq共t兲 and fpp共t兲, also depend on the temperature.

As a final comment on the exact solution of the model we would like to add that, due to the initial condition共43兲, we

have具Pˆn典=具Qˆn典=0 and therefore get the mean displacement of the central oscillator from共37兲 as

具Qˆ共t兲典 = A˙共t兲具Qˆ共0兲典 + A共t兲具Pˆ共0兲典. 共45兲 Turning to the Markovian limit we assume that environ- ment correlations decay fast relative to the time scales of the central oscillator. In technical terms, we require

␣,␭ Ⰷ 兩⌫ + i⍀兩. 共46兲

That Markovian limit does not imply weak coupling. The characteristic equations共44兲 now become

␭ =␣^,

␻02+␣␬⁼⍀²+⌫²+ 2␣⌫,

␻0

2=共⍀²+⌫²兲, 共47兲

and entail the explicit results

⌫ =␬

2, ⍀²=␻02

−␬²

4, ␭ =␣. 共48兲

The master equation now reads, for times tⰇ␣^−1,

␳^ˆ˙共t兲 = − i

2q关Pˆ²+␻0

2Qˆ²,␳^ˆ共t兲兴 − i

q⌫†Qˆ,关Pˆ,␳^ˆ共t兲兴+‡

− 2

q²⌫具Pˆ²典eq†Qˆ,关Qˆ,␳^ˆ共t兲兴‡ + 1

q²共␻02具Qˆ²典eq

−具Pˆ²典eq兲†Pˆ,关Qˆ,␳^ˆ共t兲兴‡. 共49兲 The exact expressions for the stationary second moments 具Qˆ²典eqand具Pˆ²典eqare lengthy and can be found in关19兴; they are completely characterized by the three rates共⌫,⍀,␭兲 and by the temperature.

In the Markovian limit under study, the amplitude A共t兲 in 共43兲 also simplifies to

A共t兲 = 1

⍀e^−⌫tsin共⍀t兲, t Ⰷ 1/␣. 共50兲 We can now see that overdamping arises when⍀ becomes a pure imaginary number or equivalently when ␻0⬍⌫. The transition between normal damping and overdamping occurs at⍀=0, for the critical coupling

␬c= 2␻0. 共51兲 That critical coupling will have to be compared with the one obtained perturbatively.

C. Perturbative treatment

In order to compare exact and perturbative results we now look at the Redfield master equation for the QBM Hamil- tonian关18兴

␳^ˆ˙共t兲 = − i

2q关Pˆ²+共⍀p2+⌫p2兲Qˆ²,␳^ˆ共t兲兴 − i

q⌫p†Qˆ,关Pˆ,␳^ˆ共t兲兴+‡

− 2

q²⌫p具Pˆ²典eq†Qˆ,关Qˆ,␳^ˆ共t兲兴‡ + 1 q²共共⍀p

2+⌫p 2兲具Qˆ²典eq

−具Pˆ²典eq兲†Pˆ,关Qˆ,␳^ˆ共t兲兴‡, 共52兲 where

⌫p= 1 q

冕

0

t

dtsin␻0t

␻0

D共t兲,

共⍀p 2+⌫p

2兲 =␻0 2+

冕

0

⬁

d␻␥共␻兲

␻^{2 +} 2 q

冕

0

t

dt cos␻0t D共t兲,

2⌫p具P²典eq=

冕

0 t

dt cos␻0t C共t兲,

共⍀2p+⌫p2兲具Q²典eq−具P²典eq=

冕

0 t

dtsin␻0t

␻0

C共t兲. 共53兲

The Markovian approximation consists in taking the upper bounds of the time integrals of共53兲 to infinity and is justified when the free motion of the central oscillator共characterized by the frequency␻0兲 is much slower than the characteristic decay rate␣ of the correlation function of the environment 共␻0/␣→0兲. Again using Ullersma’s spectral function 共42兲 we get the foregoing rates as

⌫p= ␬␣² 2共␣²⁺␻02兲=␬

2+O

冉

^␻␣⁰²²

冊

^{, 共54兲}

共⍀p2+⌫p2兲 =␻02+␬␣⁻ ␬␣³

␣²+␻0

2=␻02+O

冉

^␬␻␣⁰²

冊

^,

具P²典eq= q␻0

2 coth␤^q␻0

2 =

␤→01

␤^,

共⍀p2+⌫2p兲具Q²典eq−具P²典eq =

␤→0

,

␬␣

␤共␣²⁺␻0 2兲= ␬

␤␣

冠

^{1 +O}

冉

^␻␣⁰²²

冊冡

^.

To be consistent with the Markovian assumption, all terms of order␻0/␣or smaller should be disregarded. When using the lowest-order master equation共52兲 we recover the mean dis- placement具Q共t兲典 of the rigorous treatment; in fact, we even get coinciding results for the nonperturbative and the pertur- bative rates in the Markovian limit ␻0/␣→0, i.e., ⌫=⌫p

=␬^{/ 2 and} ⍀=⍀p=␻02−␬²/ 4. In particular, therefore, the transition to overdamping occurs at the same critical value of the coupling, given by Eq.共51兲. We conclude that the over- damping regime in the QBM model in the Markovian limit can be described by second-order perturbation theory and therefore is a weak-coupling overdamping. It is worth men-

tioning that this result was anticipated by Cohen-Tannoudji in关21兴.

We finally note that for strong overdamping the slowest decay rate of the QBM model reads

s = −␬ 2+␬

2

冑

^{1 −}

冉

^␬␬^c

冊

^{2= −␻}␬^20+O

冉

^␻␬³⁰⁴

冊

^{, 共55兲}

in obvious similarity to the corresponding limit for the other models studied above关see 共16兲 and 共31兲兴.

V. CONCLUSION

For four different models, made of a system weakly inter- acting with its environment, we have studied the transition from normal damping to overdamping. Normal damping has the slowest relaxation rates that increase with growing cou- pling strength and is characterized by exponentially damped oscillations. In the overdamped regime the smallest relax- ation rates decrease with growing coupling and the dynamics displays nonoscillatory exponential decay. The critical value of the coupling at which the transition from normal damping to overdamping occurs can often be made sufficiently small 共by tuning model parameters兲 in order to be described by weak-coupling master equations such as the Redfield equa- tion. One way to make the critical coupling small is to de- crease the bare frequencies of the system, but other param- eters like the temperature or the system size can also enter the game. The compatibility of weak coupling and over- damping is counter to intuitive and widely spread expecta- tions.

ACKNOWLEDGMENTS

M.E. thanks Professor P. Gaspard for support and encour- agement in this research. M.E. is supported by the “Ministère de la Culture, de l’Enseignement Supérieur et de la Recherche du Grand-Duché de Luxembourg.” F.H. thanks Pierre Gaspard for his hospitality at the Université Libre de Bruxelles which made this work possible.

APPENDIX A: HARMONIC OSCILLATOR ENVIRONMENTS

We briefly recall some properties of the equilibrium auto- correlator of the environment coupling agent B,

␣共t兲 = 具Bˆ共t兲Bˆ典 = C共t兲 + iD共t兲 = TrB␳^ˆBeqe^−iHˆBt/qBˆ e^iHˆBt/qBˆ , 共A1兲

␳^ˆBeq= e^−␤HˆB/Z_B. 共A2兲 The real and imaginary parts of␣共t兲 obey C共t兲=C共−t兲 and D共t兲=−D共−t兲. Their Fourier transforms 关defined as ␣^˜共␻兲

=共1兲/共2␲兲兰_−⬁^⬁ dt e^i␻t␣共t兲=C˜共␻兲+iD˜ 共␻兲兴 are related by the fluctuation-dissipation theorem

C˜ 共␻兲 = 2iE_␤共␻兲

q␻ ^{D˜ 共}␻兲, 共A3兲 where

E_␤共␻兲 =q␻

2 coth␤^q␻

2 共A4兲

is the thermal energy of an oscillation with frequency␻^{. As} a consequence, we can write our correlator as

␣共t兲 =

冕

0

⬁

d␻qJ共␻兲

冉

^coth␤2^q␻cos␻^{t − i sin}␻^t

冊

^,

共A5兲 thus introducing the spectral strength J共␻兲 of the environ- ment often used in the literature,

J共␻兲 =2i

qD˜ 共␻兲, ␻⬎ 0. 共A6兲 It is, in fact, customary to use that spectral strength only for positive frequencies; an extension to real frequencies could be to require J to be odd in␻^.

For an oscillator bath with

Hˆ

B=1 2

兺

n=1 N

共Pˆn 2+␻n

2Qˆ

n

2兲, Bˆ =

兺

n=1 N

⑀nQˆ

n

the correlator becomes

␣共t兲 =

兺

n=1 N

⑀n 2TrB

e^−␤HˆB ZB

Qˆ

n共t兲Qˆn共0兲

=

兺

n=1 N

q⑀n2

2␻n

冉

^coth␤q2^␻ncos␻nt − i sin␻nt

冊

=

冕

0

⬁

d␻␥共␻兲q

2␻

冉

^{coth␤q2␻cos␻t − i sin␻t}

冊

^,

共A7兲 and has the Fourier transform

␣

˜共␻兲 =␥共兩␻兩兲q

4␻

冉

^{coth␤q2␻+ 1}

冊

^{. 共A8兲}

A comparison of the general form共A5兲 with the oscillator- bath form 共A7兲 of the correlator ␣共t兲 shows that the two spectral strengths J共␻兲 and ␥共␻兲 共which are both common currency兲 are related as

J共␻兲 =␥共␻兲

2␻ ^, ␻⬎ 0. 共A9兲

Ullersma’s choice关13兴 共also called Drude strength兲

␥共␻兲 = 2

␲

␬␣²␻²

␣²+␻^{2 共A10兲}

corresponds to an Ohmic environment because at small fre- quencies J共␻兲⬃␬␻^/␲^.

At high temperatures, the real part of the environment correlator is given by

C共t兲 =^␤→0

冕

0

⬁

d␻␥共␻兲

␤␻^2cos␻^{t =}␬␣

␤ ^{e−␣兩t兩. 共A11兲} The imaginary part of the environment correlation function is independent of temperature and reads

D共t兲 = −

冕

0

⬁

d␻␥共␻兲

2␻ ^sin␻^{t = −q}␬␣²

2 e^{−␣兩t兩}sgn共t兲.

APPENDIX B: ULLERSMA’S HAMILTONIAN Ullersma 关13兴, Haake and Reibold’s 关19兴 work with a modified Hamiltonian of the oscillator model,

Hˆ =1

2共Pˆ²+␻0 2Qˆ²兲 +1

2

兺

n=1 N

共Pˆn 2+␻n

2Qˆ

n 2兲 + Qˆ

兺

n=1 N

⑀nQˆ

n. The potential-energy part

V共Qˆ,兵Qˆn其兲 =1

2

冉

^␻02Qˆ2+

^兺

^{n=1N ␻n2Qˆn2}

冊

^{+ Qˆ}

^兺

^{n=1N ⑀nQˆn}

has a minimum of the potential created by the other har- monic oscillators on the central oscillator given by

冏

^⳵V共Qˆ⳵^0Qˆ,兵Qˆn ^n其兲

冏

Qˆ n=Qˆ

n共min兲

=␻n 2Qˆ

n共min兲 +⑀nQˆ

0= 0.

The central oscillator thus “feels” the potential V共Qˆ0,兵Qˆn共min兲其兲 =

冉

^␻202−n=1

^兺

^{N 2⑀␻n2n2}

冊

^Qˆ02.

Clearly, then, positivity is not manifest; rather, in order to have bound states, we have to impose the condition

␻0 2−

兺

n=1 N ⑀n2

␻n2=␻0

2−␬␣艌 0. 共B1兲

Ullersma’s Hamiltonian can be mapped onto the QBM Hamiltonian by renormalizing the frequency␻0 as ␻₀²→␻₀² +兺_n=1^N 共⑀²兲/共␻n2兲=␻02+␬␣. That mapping was extensively used above in transcribing the rigorous results of Ref.关19兴 to the dynamics generated by the QBR Hamiltonian.

Needless to say, we could have based our study of the transition from normal damping to overdamping on Ullers- ma’s model. Only one subtlety about that alternative treat- ment is worth being mentioned here. To leading order in

␻0/␣the critical value␬cof the coupling turns out to coin- cide with the border␬max=␻0

2/␣ to positivity loss following from共B1兲.

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