Quantum ohmic dissipation : coherence vs. incoherence and symmetry-breaking. a simple dynamical approach

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Quantum ohmic dissipation : coherence vs. incoherence and symmetry-breaking. a simple dynamical approach

C. Aslangul, N. Pottier, D. Saint-James

To cite this version:

C. Aslangul, N. Pottier, D. Saint-James. Quantum ohmic dissipation : coherence vs. incoherence and symmetry-breaking. a simple dynamical approach. Journal de Physique, 1985, 46 (12), pp.2031-2045.

�10.1051/jphys:0198500460120203100�. �jpa-00210151�

Quantum ohmic dissipation : coherence

^vs.

incoherence and

symmetry-breaking. A simple dynamical approach

C.

Aslangul (1),

^{N. Pottier}

(1)

^{and D.} Saint-James

(2)

(1)

Groupe ^de

Physique

des Solides de l’Ecole Normale

Supérieure

(*), Université Paris VII, 2 place Jussieu, 75251 Paris Cedex 05, ^France

(2)

Laboratoire de

Physique Statistique,

Collège ^de France, 11 place ^Marcelin Berthelot,

75231 Paris Cedex 05, ^France

(Reçu ^{le 7 mai} 1985, accepte ^sous

forme difinitive

le 22 août 1985)

Résumé. ²⁰¹⁴ La

dynamique

^d’une

particule

située dans un double

puits

^de

potentiel

^et couplée à ^{un bain de}

phonons

suivant une loi de

dissipation

^«

ohmique

^{» est étudiée à l’aide de méthodes} usuelles de

mécanique statistique quantique.

Nous complétons ^{les résultats}

déjà

^{connus sur la} brisure de

symétrie

^{à T = 0 de la manière suivante :}

(i) ^en déterminant le comportement

critique

^du paramètre d’ordre, ^{on montre} que la brisure de

symétrie

^n’est complète que pour ^un

couplage

infini ;

(ii) ^la

dynamique

^{suit une} loi de Kohlrausch

généralisée

de la forme

exp( 2014 atb),

où les deux paramètres ^{a et b}

sont calculés exactement; ^un ralentissement critique ^{est très} clairement mis en évidence.

A température finie, la brisure de

symétrie

^{ne se}

produit

plus, ^bien

qu’un

comportement précurseur ^{de la tran-}

sition se manifeste lorsque ^{T ~ 0+.}

Ces résultats sont

généralisés à

^une

particule

^{dans un} potentiel ^à puits multiples : ^en présence ^de dissipation ohmique, ^un

phénomène

^de localisation peut ^se

produire

^{à T = 0.}

En dernier lieu, ^nous examinons d’autres lois de

dissipation

(non

ohmiques)

^{et nous montrons} que seul le modèle

ohmique

^est

capable

^de

produire

^{une telle} transition, bien que, là encore, ^un comportement précurseur ^existe

pour des lois de

dissipation quasi ohmiques.

Abstract ²⁰¹⁴ By

using

^standard

quantum-statistical

^{methods, we}

study

^the

dynamics

^{of a}

particle

^{in a double-}

well

potential, coupled dissipatively,

^{in an «} ohmic » way, ^{to a} bath of phonons.

We refine

previous

^{results about the T = 0} symmetry

breaking :

(i) the critical behaviour of the order parameter is exhibited and the symmetry ^{is shown to be} only

fully

^broken

for infinite

coupling;

(ii) ^the

dynamics

^{follows a}

generalized

Kohlrausch law of the type

exp(- atb),

where ^both parameters a ^{and b}

are exactly calculated; ^{a critical}

slowing

^down is very

clearly displayed

At finite temperature, ^the symmetry

breaking

^does

disappear,

although ^a precursor behaviour of the transition exists when T ~ 0+.

These results are extended to a

particle

^{in a}

multiple-well

potential ^{for which a} localization

phenomenon

^is

seen to occur at T ⁼ 0.

Finally,

^we investigate ^other

dissipation

^laws (i.e. ^non ohmic) ^{and we show} that the ohmic model is

unique by

its

ability

^to

produce

^{such a} transition, although ^a precursor behaviour can be obtained for near-ohmic

dissipation

laws.

Classification

Physics

^Abstracts

03.b5 ^- 05.30 ^- 74.50 ^- 71.50

1. Introduction

The

question

^{which we intend to} discuss in the _pre- sent paper is that of the influence of

dissipation

^on

(*) Laboratoire associe au C.N.R.S.

the fundamental quantum mechanism of quantum coherence. The

underlying

^{idea is to} decide whether

dissipation destroys phase

^coherence.

This

question

^is

evidently

^{related to the}

general problem

^of relaxation in

quantum systems,

^for

which,

as it is well

known,

the lines of attack fall into two

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460120203100

main

categories [1] :

^either

people

have tried new

schemes of

quantization

^or

they

have used standard quantum

mechanics, considering

the evolution of the

(small)

system ^and of the reservoir as a whole.

As underlined in reference

[1],

the first type ^of

approach

^contains either controversial results or

obscurities. In the second type ^of

approach,

^known

as the

system-plus-reservoir approach,

^and

pioneered by Senitzky [2],

the interaction of the

system

of interest with the reservoir is

explicitly

^{taken into account}

Irreversible behaviour of the small

system

^arises when the reservoir is considered as a continuum of

modes,

^thus

rejecting

^{Poincar6 times to}

infinity.

The new aspect ^of

dissipation

^{which we would like}

to

study

here is the

following :

^since

1982,

it has been shown

by

several authors

[3, 4] that,

^{for a certain}

type

^of

dissipation (the

^{so-called « ohmic »}

dissipa- tion),

^a

spontaneous symmetry breaking

may appear in some

dissipative quantum

systems when the dissi-

pation

^becomes

greater

^{than a} critical value.

This is all the more

interesting

as, in these « ohmic »

dissipative quantum

systems, ^the

Heisenberg

repre- sentation of a certain

coordinate q

^{follows a classical}

equation

^{of motion}

which is

generally

^{believed to be}

obeyed by

^macros-

copic

quantum ^variables

[5] (M

^{is the mass of the}

particle

^{submitted to a}

potential Y(q) ;

q is a

pheno- menological damping

^{constant or friction} coefficient and

F(t)

^{is a}

fluctuating force).

^These

systems

^have

been the centre of recent interest in the context of

testing

^the

applicability

^of quantum ^{mechanics to}

macroscopic

^variables

[6].

Following

references

[3, 4],

^{we shall}

mainly study

here a

particle

^{in a} double-well

potential, coupled dissipatively (in

^{an « ohmic »}

way)

^to its environment This

problem

^{can be} very

clearly

^settled

down,

^but

it cannot be

given

^{an exact} solution. Thus

approximate

treatments have been

proposed

^{in the} literature.

Most of them use either

path-integral Feynman

^for-

mulation in

imaginary

time and instanton

techniques,

or renormalization group

approachs [3, 4, 7].

^Two

other

approaches

^must

equally

^be

quoted :

^{that of}

reference

[8],

^{in which a} variational method of dis-

playing

the transition is

proposed,

and that of refe-

rence

[9]

^{in which a} linear response calculation is

handled

As for the real-time

dynamics,

^{it has not}

been

investigated

ⁱⁿ

detail, except

^for references

[9, 10]

outside the critical

region.

Our _purpose in this _paper is thus to refine and to

supplement previous

treatments of this

problem,

^and

that will be attained

by using

^standard quantum- mechanical

theory. Indeed, by using

^standard

methods,

we recover

qualitatively

^and

quantitatively nearly

^all

previous results,

^{and in}

particular

^the spontaneous

symmetry breaking

^{which occurs at zero} temperature

[11].

^The

improvements

^over

previous

^treatments

are the

following :

^{we are able to}

give

^a

description

of the real-time

dynamics

^{of the}

particle,

^{for zero}

as well as for finite temperature; ^{in the}

vicinity

^{of the}

transition

point,

^{we find a critical}

slowing

^down.

Another

advantage

^{of our} methods is that the

approxi-

mations involved are more

transparent

^{from a}

physi-

sical

point

^{of view. Moreover it will be shown that}

the order parameter is continuous and reaches its maximum value

only

for infinite

coupling;

^the par- ticle is then

fully

^localized.

The _paper is

organized

^{as follows. In section}

2,

we describe the model under

study;

in section

3,

we

present

^an

approximate method,

based on a

master

equation formalism,

^which

yields

^{the real-}

time

dynamics

^{in the case of} intermediate or strong

coupling.

^{In section}

4,

^we present ^another

approxi-

mation

scheme,

^{based on a one-boson}

assumption,

which

yields

the real-time

dynamics

^{in the weak cou-}

pling

^case. Section 5 contains a

comparison

^between

our results and

previous

^{results on} the real-time

dyna-

mics

[9, 10]. Then,

in section

6,

^we discuss the

speci- ficity

of the ohmic

dissipation

^{model. In section}

7,

we show how the results on the

particle

^{in a double-}

well

potential

^{can be extended to a}

particle

^{in a}

multiple

^well.

Finally,

^{in section}

8,

^we present ^our conclusions.

2. The model

2 .1 THE DAMPED PARTICLE IN THE SYMMETRIC DOUBLE- WELL. ^- As indicated in the

Introduction, following

references

[3, 4],

^{we consider a}

particle

^{of mass M}

moving

^{in a}

symmetric

double-well

potential V(q) = V( - q)

with its minima located at q ⁼

± q°

₂ ^and

separated by

^{a local maximum at q = 0.}

This

particle

^{is in contact with a}

dissipative medium,

which is

generally

^modelled

by

^a

phonon

^reservoir

of bandwidth

liwe

^,

[5].

^As

previously indicated,

^we

shall

mostly

be interested here in the ^case of« ohmic »

dissipation,

in which the coordinate of the

particle

evolves with time

according

^to the classical

equation

of motion

(1).

The

problem

^{is the}

following :

^{at time t =}

t°,

^a

particle

is located in one of the two

wells;

^the

tempe-

rature is

supposed

^{to be low}

enough

^{so that thermal}

activation over the

potential

^{barrier can be}

neglected

The

question

arises : how does its _average

position

evolve with time ?

Evidently,

^a

purely

^classical par-

ticles, initially

^{located in one of the two}

wells,

^remains

localized on the same side for an infinite time. On the

contrary,

^a

purely quantum-mechanical particle (wi-

thout

bath), initially

located in one of the two

wells,

can tunnel

through

^the

potential barrier,

^{and thus}

oscillates back and forth between the two wells. This is the

phenomenon

^{of «}

quantum

^{coherence ». Such}

a

particle

^{can be said to be} delocalized : its _average

position

^is

equal

^{to zero. Let us now consider a} quan-

tum-mechanical

particle coupled

^{to a}

bath,

^{in such a}

way ^so that its

position

operator ^{in the}

Heisenberg picture obeys

^the

equation

^{of motion}

(1).

^In this situa-

tion, Chakravarty [3]

^and

Bray

^{and Moore}

[4]

^have

found a transition : for a weak

coupling

^{with the}

bath,

^the

regime

^{is akin to the} pure

quantum regime

and the

particle undergoes

^a

damped

oscillation between the two

wells;

^for

higher

values of the cou-

pling,

^{the mean} coordinate of the

particle

^relaxes

towards zero without

oscillating,

but when the

coupl- ing

^{to the bath becomes}

greater

^{than a critical}

value,

a

spontaneous

symmetry

breaking

^{occurs, which here}

signifies

^{that the mean} coordinate of the

particle eventually

relaxes towards a non-zero

value,

^indicat-

ing

^a localization in one of the two wells. This

regime

thus bears a resemblance to the classical

regime

^and

corresponds

^{to a}

complete

^{loss of}

phase

^coherence

[12, 13].

In the

system-plus-reservoir approach,

^one

begins by writing

down the total Hamiltonian for this _pro- blem.

Following

Caldeira and

Leggett [5],

^{the dissi-}

pative

interaction of the

system

with its environment is modelled

by

^{a linear}

coupling

^{with a} bath of har- monic oscillators. In obvious notations :

The last term has been added in order to cancel

spurious unphysical divergences;

^{it ensures that there}

is not shift in the

potential

^{due to the}

coupling [5].

It is easy ^to derive from

equation (2)

^the

equations

of motion for the coordinate

q(t)

^{and the momen-}

tum

p(t)

^{of the}

particle

^{in the}

Heisenberg picture.

As we

already

^{shown in a}

previous

paper about the

damped

^free

particle [14],

^once

p(t)

^is

eliminated,

^the

dynamical equation

^for

q(t)

^{reads as follows :}

where the interaction with the bath manifests

itself,

as it is

well-known, by

^a

dissipative

^{retarded term and}

by

^a

fluctuating

^force term, ^both

being

^related

by

^the

fluctuation-dissipation

^{theorem. Note that}

equation (3),

which has been established without _any

approxi- mation,

contains the exact

dynamics

^{of the}

particle.

Let us now discuss the conditions under which it can

be

given

the form of the classical

equation

^{of motion}

(1).

2.2 THE OHMIC DISSIPATION MODEL. ^- As

usual,

^the

bath is described in the continuous limit

by

^its

density

of modes

p(c~). Following

^{the same line of}

reasoning

as in reference

[14]

^{in the case of the}

damped

^free

particle,

^{a classical}

equation

of motion of the

type

JOURNAL DE PHYSIQUE. ^- T. 46, ^No 12, DtcaoRE 1985

of

equation (1)

^for

q(t)

is obtained

by setting

^down

where

f,((olco,,)

^{is some} cut-off function such that

£(0)

⁼

1,

^and

decreasing

^{on a}

frequency

range of order co,,. When the choice

(4)

^is

made,

^{it can be shown}

[14]

^that

K(t)

^{is a} memory kernel

essentially given by

the Fourier transform of the cut-off function

fc(ro/wc)

^and

that,

^for temperatures ^{such that}

kø

T >

liwc, F(t)

^{is a}

rapidly fluctuating

^{force of}

correlation time of order -

coc

^{1. The} constraint

(4) imposed

^{on the bath}

density

of modes and ^on the

coupling

^{constant in order to obtain a}

classical-like equation

of motion will from now on be referred to as the ohmic

dissipation

^{model. Let us}

emphasize that,

ⁱⁿ accordance with the

fluctuation-dissipation theorem,

^not

only

^the

friction,

but also the

fluctuating

force term have a determined form.

2.3 REDUCTION TO A SPIN-BOSON HAMILTONIAN

[4]. -

Let us consider the

particle

in the double-well and let

S~o

^{denote the}

frequency

of small oscillations around one of the

minima,

^and

Yo

the barrier

height

We shall limit the

study

^{to the}

low-temperature

situation

kø

^T

V~,

in which thermal activation

over the

potential

^{barrier can be}

neglected Moreover,

we shall _suppose that this barrier is

sufficiently high

and

large (Yo > hlmq’)

^{for the}

splitting

coo between the

symmetric

^and

antisymmetric

^states

(tunnelling frequency)

^{to be} much lower than

00. Finally,

^we

shall _suppose that the

temperature

is such that

kB T ~~o :

^the system ^{can then be}

safely

^truncated

to

only

^one level in each well.

After this

truncation,

^{we are left with an}

effectively

two-level

system coupled

^{to a bath of}

phonons;

this ensemble ^can be described

by

^the

spin-boson

Hamiltonian

(6x,

ay

and a,,

^are the standard Pauli

matrices) :

where the

operator §

qo Uz

corresponds

^{to the}

posi-

tion operator q ^{of the}

particle ;

^{in order to be}

complete,

we have now to

identify

^the

coupling

^{term of the}

spin-

boson Hamiltonian

Hsb (5)

^{with the}

coupling

^term

of the Hamiltonian

(2).

^{This can be done}

by identifying

The ohmic constraint

(4)

then takes the form

where the dimensionless

parameter

^a is related to the friction

coefficient by

The cut-off

frequency

We has to be chosen such that

c~~ >

wo and

cor >> kB Tlh. Clearly,

^{with these} assump-

tions,

^one

expects

^{that the}

precise

^form

of fr

^{has no}

bearing

^{on the}

dynamics

^{of the}

system

in the time domain

c~~ t >

^{1 and} may be chosen at convenience.

To conclude this

section,

^{let us now comment}

briefly

^on the reduction of the Hamiltonian

(2)

^to

the

spin-boson

Hamiltonian

(5). Clearly,

^the operator

2’ qo u%

^possesses

only

^{two discrete}

eigenvalues

±

q°

2 ^{Z -} 2

whereas the

genuine position operator q

^{has a conti-}

nuous

spectrum.

The kinetic and

potential

energy have been contracted into the _{u x} term, ^and

since,

as

well-known,

the Pauli matrices

algebra

^is

comple- tely

different from the

algebra

^{of the}

position

^and

momentum

operators (in particular

^{it is}

impossible

to find an

operator having

^{the same} commutation relation

with a.

^{than p with}

q),

^this amounts to carry out a kind of

coarse-graining

^{in which some infor-}

mation has been

lost;

^one may

loosely speak

^{of a}

reduction of the Hamiltonian.

Conversely,

the results obtained on the

dynamics

^of a. ^can be cast in terms

of q,

^{but one cannot}

expect u%(t)

^to

obey

any classical- like

equation

of motion of

type (1), which,

^as

explained above,

^{is obtained for a}

quantum system coupled

^{to a}

bath when

writing

^{down the}

equations

^{of motion}

for both

q(t)

^and

p(t).

In the

following

^two

sections,

^{we shall}

successively analyse

^{the cases of weak} and moderate or strong

coupling.

3. Intermediate or

strong coupling

^case.

3.1 PARTIAL DIAGONALIZATION OF THE HAMILTONIAN.

- As

pointed

^out

by Silbey

and Harris

[8]

^and

by Zwerger [9],

^the

unitary

transformation

diagonalizes

^the Hamiltonian

Hgb(5)

^when Wo ⁼ 0.

The transformed

Hamiltonian n

⁼

SHsb S -1

^{reads :}

The

operators Bt

^{are defined as}

Formally, J7

divides into an interaction term

and a bath term.

In what

follows,

^{we shall be} interested in the time evolution of the _average value of _6z, defined in the

Schrodinger picture

^as

(p(t)

is the full

(i.e. system plus bath) density

^matrix

and the trace operator ^acts in the whole

space).

The basic remark is the

following :

^the

unitary

transformation S leaves _{u %}

invariant, and,

^conse-

quently,

^one

gets

with

Since a,,

^{is a pure}

spin

operator,

equation (14)

^takes

the form

where the trace

operator

^{acts in the}

spin

space and

p,(t)

is the reduced

spin density

^matrix

(the

^trace

operator

^acts in the bath

space).

Similarly,

^{the time}

derivative åz(t) >

^is

given by

By setting

one

gets

The time evolution

of p

has thus been calculated

by using 17

^{instead of}

H,

^this

being legitimate

^as

long

^{as one is} interested in

quantities

^{such as} a.

which are invariant under the

unitary

^trans-

formation S.

3.2 MASTER EQUATION FOR THE REDUCED SPIN DENSITY MATRIX

ps(t).

^-

According

^{to the}

general damping

theory [15],

^{an evolution}

equation

^{for the reduced}

spin density

^matrix

ps(t)

^{can be deduced from the}

Liouville

equation

^for

p(t) by using

for instance standard

projection operators techniques.

^{One assu-}

mes, ^as

usual,

that the initial

density

^matrix

p(to

⁼

0)

is

factorized,

i.e. that

where pb

^{is the}

equilibrium

^bath

density

^matrix.

The time rate of

change « Op,10t >>

^then

comprises only

^{two terms : one is a} proper evolution term, and the other one is a relaxation term, ^{due to the} interaction with the bath. As

pointed

^out

by

^several

authors

[16-19],

this relaxation term can be

given

two

forms,

either the usual time-convolution or

retarded

form,

^{or a «} time-convolutionless »

(i.e.

^non

retarded)

^one.

Clearly

both forms are exact as

long

as no

supplementary approximations

^are

made;

however,

^{in what}

follows,

^we would like to consider the interaction between the

system

^{and the bath} in

H (the Hint term)

^{as a}

perturbation.

The calculation will be restricted to the Born

approximation (1).

It has been

argued [16-19] that,

^{from the}

point

^of

view of a

systematic expansion,

the time-convolution- less formalism

prevails

^over the usual convolution one, since in the latter

formalism,

^{a certain amount of} rearrangement ^{of the terms} is necessary ^to obtain a

coherent

expansion

^{to a}

given

^order. This is the fundamental reason

why,

besides its

greater practical simplicity,

^we

preferred

the time-convolutionless for- malism.

Following

^van

Kampen [16]

and Hashitsume et al.

[18],

^as

long

^{as two} time scales can

clearly

^be

delineated,

in other

words,

^as

long

^{as it exists a short time}

scaler,,

characteristic of the relaxation

kernel,

^{and a}

long

time scale _T., characteristic of the operator ^{of interest}

(here

^the

density

^matrix

Ps(t)),

^{one can}

write,

^for

times t > T,, ^a time-convolutionless evolution _equa- tion for

p,(t)

which takes the form of a

systematic expansion

^with

respect

^{to the}

perturbation

where (’" ~b

stands for

Trb

p~’" The time-convolu- tionless

equation

^for

ps(t)

^{takes the}

form,

^{in the Born}

approximation

where

Hi’n,(- t’)

is written in the interaction

representation.

^It is easy ^to

give equation (22)

^{a more}

explicit

form

by detailing Hinc

^and

H:nt( - t’).

^{One obtains}

where the relaxation functions

0 + + (t), ql - + (t), 0 + - (t)

^and

0 - - (t)

^{are defined as}

Remember that the

operators Bt(t)

^are

computed

ⁱⁿ

the

non-interacting

scheme for the bath.

Equations (22)

^to

(24)

constitute

general

relaxation

equations;

(1)

^This evidently

implies

^{that the}

perturbation

^series properly exists, ^an assumption which, ^as

pointed

^out

by

V. Hakim, ^could be dubious in the close

vicinity

^{of a = 1}

(private

communication).

note that

they

^are valid whatever the

precise

^{form of}

the

dissipation,

^{ohmic or non ohmic. Let us now}

examine the consequences of the ohmic constraint

(7)

on

the

relaxation

equation

^{of the reduced}

spin

^den-

sity

^matrix

Ps(t).

3.3 TIME EVOLUTION OF THE REDUCED SPIN DENSITY MATRIX

ps(t)

IN THE OHMIC MODEL. ^- At

first,

^let

us calculate the _average value

Since b] and b,,

^{are boson} operators, ^one gets

When the ohmic constraint

(7)

^is

imposed,

^the

exponent ⁱⁿ

equation (26)

^{is seen to}

diverge,

^for

T = 0,

as well as for finite temperature, ^{so that}

( Bt )b

^is

equal

^{to zero} in this model. Let us however

emphasize

here that the ohmic model is not the

only

^{one in}

which

( Bt ~b

^{= 0.}

Any

^{model of}

dissipation

ⁱⁿ

which

p(w) ~ 1 G(w) 12 ’" c~8

^with

6

^{1 at zero tem-}

perature

^or

6

^{2 at finite}

temperature

will share this

property.

^This

point, and,

^more

generally,

^the

question

^{of the}

specificity

of the ohmic

dissipation model,

^{will be} discussed later on in detail

(see

^Sect.

6).

Thus,

^{in the ohmic}

model,

the relaxation

equation (23)

of the reduced

spin density

^matrix

ps(t) considerably simplifies.

^{It is then}

straightforward

^to deduce from

equation (19)

^the

equation

of motion of

( Qz(t) ~.

This will be achieved in the

following paragraph.

3.4 SPIN DYNAMICS. ^- In order to obtain ^a detailed

expression,

^{it is} necessary ^to calculate the relaxation function

§’s, which,

in the ohmic

model,

^are

just time-integrals

of the correlation functions of the operators

B=f:(t), expressed

^{in the}

non-interacting

scheme.

By using

the commutation relations between the Pauli matrices and the

symmetry properties

^of

the

§’s,

^{one gets a closed}

equation for ( uz(t) > :

where

A(t)

^{is defined as}

with

and

Equations (27)

^to

(30)

constitute our basic _equa-

tions ; they

^contain

explicitly

^{all the}

dynamics

^{of the}

spin.

^{Note that the}

exponential

time behaviour found

by Chakravarty

^and

Leggett ([l0a])

^is

simply

^obtained

from

equations (27)

^and

(28) by rejecting t

^to

infinity

in the

integral (28);

^{in our} context, this

corresponds

to a short _memory

approximation. Using

^the expo- nential cut-off introduced

by

^{the same authors :}

in the ohmic constraint

(7),

^{one can calculate}

simply

the functions

A1(t)

^and

A2(t) :

For kB T liwc, expression (32b)

^for

A2(t)

^{reduces to}

1 is the « temperature ^{time », as defined}

by

Equation (28)

^then

yields

The

explicit dynamics of U z( t)

^{can now be}

found

by integrating equation (27).

^{Let us} present and

physically

discuss the results we obtain for

( uz(t) ).

This will be done

successively

^{for zero and}

for finite temperature.

3.4.1

Zero-temperature

^case. - When the

tempe-

rature is

equal

^{to zero,}

expression (35)

^for

~1(t)

^reduces

to

and the solution of

equation (27)

^is

easily

^obtained

in a closed form :

(Note

that for the

particular

^{values a =}

1/2

^and

a ⁼ 1 the

preceeding expression

^of

Z(t)

is still valid

provided

the limits are

correctly taken.)

In the

meaningful

^{time domain}

c~~ t > 1, L(t)

^takes

the

simpler

^{form :}

This

expression

shows that :

(i)

^As

long

^{as a}

1, Z( + oo)

^is

equal

^{to zero. In}

terms of the

particle evolving

in the double-well

potential,

^{this means that its} average

position

^at

infinite time is

equal

^{to zero.}

Therefore,

^{in this}

regime,

the

particle

^{can be said to be}

delocalized,

^{albeit no}

oscillation does _appear.

However,

since for

(X = 0,

the situation

evidently

^{reduces to the} pure quantum situation in which the

particle

oscillates back and forth between the two

wells,

^{it is}

likely

^{that the} present calculation is no more valid for too low values of the

coupling;

this will be discussed later.

(ii)

^{When a becomes} greater ^than

1, Z( + oo)

^takes

a finite

value, given by

The _average

position

^at infinite time of the

particle evolving

in the double-well

potential

is then different from zero. The

particle

^{is thus}

partly

^{localized on}

the side where it was at time t ⁼ 0.

The

degree

^of

localization,

^{which can be charac-}

terized

by

the value of

E( + oo),

^{is a continuous}

function of a, which grows very

rapidly (i.e.

^{on a}

region

of width -

(c~o/c~~)2)

^{towards 1 as soon as}

a exceeds 1.

Therefore,

^this

system

^does

display

^a

continuous

symmetry breaking

^at the value a ⁼ 1 of the

coupling; however,

^the symmetry ^is

fully

broken

(i.e. Z( + oo)

⁼

1) only

for infinite

coupling,

which seems a

physically

sensible result

(see Fig. 1).

Fig. ^{1. -} Variation of E( + oo) ^{as a} function of the inverse coupling ^constant

03B1-1(w0/wc

^{= 0. 1).} The inset is an enlarg-

ment of the critical region.

The

symmetry breaking

^{can be described in the}

phase

transition

language,

the inverse

coupling

para- meter

a-1 playing

the role of the temperature, ^and

E( + cc) acting

^as the order parameter. ^This

phase

transition is of infinite order

(see Eq. (39)),

^{and the}

width of the critical

region

^{is of order}

(wOlwe)2,

^i.e.

very ^narrow. This is to be

compared

with the result of references

[3, 4]

^{in which a} discontinuous transi- tion at a ⁼ 1 is found. Note however

that,

for we -. oo,

E( + oo)

^{in our} calculation will also present ^a

jump

at a ⁼ 1.

The

time-dependence of E(t)

^reveals very

interesting

features. The full

time-dependence

^of

Z(t)

^is

given by equation (37); however,

^{it is} sufficient to

analyse

formula

(38)

^which

corresponds

^{to the}

meaningful

domain

We t

> 1. The

following

characteristics appear :

(i)

^As

long

^{as a}

1/2, E(t)

relaxes towards zero more

rapidly

^{than a} pure

exponential,

^i.e.

Z(t) - exp( - atb),

^b > 1.

(ii)

^{For a =}

1/2, Z(t)

^{has a}

purely exponential

behaviour.

(iii)

^When

1/2

^a

1, ~(t)

^{has a stretched} expo- nential form : in other

words,

it evolves

according

to the so-called Kohlrausch law

Z(t) - exp( - atb),

0 b 1. Such laws are believed to occur in a wide range of

phenomena

^and

materials,

^{and to contain}

the essential

physics

of relaxation in

complex, strongly interacting

^materials

[20, 21].

^{This law} appears here

as a consequence of the ohmic

dissipation model,

and manifests itself in a certain _range of values of the

coupling

^constant.

(iv)

^{As a}

approaches

^{1 from}

below,

the relaxation of

Z(t)

^{towards zero} is slower and slower. At

a=1,

the relaxation is seen to take the

powerlaw

^form

(1 + (o2 t2)- C02/2(02 ;

^since

~o/~c ~ 1,

^{it is}

extremely slow;

^{this can be viewed as a critical}

slowing

^down.

(v)

^{For a} >

1, ~(t)

^{starts from 1 and} goes uni-

formly

very

slowly

^{to its final}

value % 1 ; Z(t)

^behaves

as

exp(- atb),

^{b 0.}

So in all _cases, one can say that the relaxation of

Z(t)

^{follows for}

we t

> 1 ^{a «}

generalized >>

^Kohlrausch

law;

^the

corresponding

curves are

plotted

^on

figure

^2.

To

give

^an idea of the

extremely

slow character of the evolution of

Z(t)

^{when a} >

1,

^{let us indicate}

some orders of

magnitude.

^For

instance,

^when

a ⁼

1.02, E( + oo)

⁼ 0.786 and

E(we t

⁼

1010) =

0.865. If c~~ ~ 1013

S- 1,

^{which is a}

typical phonon

Fig. ^{2. -} Variation of E(t) ^{at T = 0 as a} function of t for various values of

a(mo/m~

^{= 0.1).}

Actually,

for this value of

wo/w~,

^{the curve a = 0.1 is}

slightly

outside the region

of

validity

^{and is} only

given

here for the sake of illustration.

frequency,

^this

corresponds

^{to a}

macroscopic

^time

t ^~

10- 3

_s, for which

L(t)

^{is still}

relatively

^{far from}

its final value. This behaviour is

clearly

apparent ^on

figure 3,

^which

displays

^the variations of

E(t)

^for

a ⁼ 1.02 and a ⁼ 0.9.

3.4.2 Finite temperature ^{case. - In this} case,

A(t)

is

given by equation (35)

and then it is not easy ^to derive from

equation (27)

^a closed form

expression

for

~(t).

^{Thus one must in}

general

^{resort to a nume-}

rical

integration.

However,

^an

asymptotic formula,

valid for times

t >

i/2

^{a, can be obtained.}

Indeed, provided

^that

is

finite,

^the upper bound of the

integral

ⁱⁿ

equation (35)

can be extended to

infinity

^{when t} >

1/2

^{a. One thus}

gets in this time domain

or

The rate

A( (0)

^can

easily

be calculated :

where r denotes the standard Euler’s _gamma func- tion.

Besides,

^{one can}

verify, by directly expanding

the

integral expression for ( uz(t»

^{which can be}

deduced from

equation (35),

^{that the constant in}

equation (41)

^{is indeed}

equal

^to

( 7~(0)).

^{This cal-}

culation thus shows

that,

^{at finite}

temperature,

uz(t)

^{relaxes towards zero in an}

exponential

manner as soon as t >

z/2

^a.

Moreover, equation (35)

^also proves that the effect of

temperature

^on the time evolution of

( ~(0 )

^is

negligible

^as

long

^as t

r/2

cx; in other

words,

^on

this time _range, the behaviour of

( 6z(t) ~

^{is the same}

as at zero temperature.

Fig. ^{3. -} Variation of f(t) ^{at T =} 0 as a function of t for

a ⁼ 1.02 and a ⁼

0.9(wo jcv~

= 0.1 ). ^The asymptotic ^value

for a ⁼ 1.02 is equal ^{to 0.786.}

Stricto _sensu, the

symmetry breaking disappears

^at

finite

temperature :

whatever the value of a

(greater

or lower than

1, provided

^our calculation should be

valid,

^a condition which

only

excludes low values of _a, as it will be seen

later,

^and

perhaps

^the

vicinity

of

1), E(t)

goes ^{to zero as t -~ oo.} The

particle

^{in the}

double-well

potential

^is

always

delocalized. However, the « relaxation

time »,

^{as defined}

by

has for limit value as To

0,

^{either 0}

(when

^a

1/2),

or oo

(when

^a >

1/2).

^{This is related to the corres-}

ponding

zero-temperature ^behaviour

of ( a_,(t) >

^as

mentioned above in 3.4. l. For T >

0,

^{one could}

naively expect

^{that the}

decay

^time

of ~ a_,(t) >

^should

be of the order Of T. However,

equations (41)

^and

(43)

show that this time is renormalized

by

^{a factor}

,

which,

^{for a} >

1,

. -, ,

becomes

extremely large

when T -+ 0. The

spin

^thus

displays

^a behaviour which is a precursor of the zero-

temperature transition at a ⁼ 1. It can be seen

that,

^{at finite}

temperature,

the relaxation of

E(t)

takes

place

^{on a}

microscopic

^{time even when a} > 1.

Let us consider for instance the case a Z ^{1. When} We ^{T =}

10,

^{which for} We ^~

1013 s-1 corresponds

^to

a

temperature

^{T - 2.4}

K,

^one

gets

we

TR

⁼

103

^or

TR ~ 10-’o

^s. Thus it is

only

^for

extremely

^low

values of the

temperature

^that

TR

would take macro-

scopic values;

^for

instance,

^{for T - 2.4 x 10-’}

K,

one would get

TR

⁼

10- 3

^s.

The curves

representing uz(t) >

^{as a} function of

t for different values of a are

plotted

^on

figure

^4.

These ^curves have been obtained

by

^numerical

integration

^of

equation (27),

^with

~1(t) given by

equa- tion

(35). By comparison

^{with the}

corresponding

T ⁼ 0 curves

(Fig. 2),

^{two trends are} observed : for

relatively high

values of the

coupling, uz(t) >

closely

^{follows the}

exponential

^{law as}

given by equation (41),

^{but for}

relatively

^{small a}

values,

^the

dynamics

appears ^to be

quite

^{similar to the}

dynamics

at T ⁼ 0. These two trends will be confirmed

by

^a

Fig. ^{4. -} Variation of E(t) ^{at finite} temperature

(we

^{z =} 10)

as a function of t for various values of

rx(wolwe

⁼ 0.1).

more detailed

quantitative analysis,

which will be done in the

following paragraph.

3 . S CONDITIONS OF VALIDITY OF THE TREATMENT. ^-

Let us now

precise

the conditions under which the convolutionless

equation (27)

and the results which

were derived from it in the two

preceding paragraphs

are valid. Our discussion will follow the arguments

developed by

^van

Kampen [16]

^and

by

^Hashitsume

et al.

[18],

and will be carried out

successively

^for

zero and for finite temperature.

3.5.1

Zero-temperature

^{case. - When T =}

0,

^an

exact

analytical

solution of

equation (27)

^{can be found}

(Eqs. (37)

^and

(38)).

Thus it is sufficient to discuss the conditions of

validity

of the convolutionless

equation

itself. As

already indicated,

^this

equation

^{is valid as}

long

^{as two} time scales can be

clearly

delineated : the short _{one, ’rc’} which characterizes the relaxation

kernel,

^{and the}

long

one, zm, which characterizes the operator of interest

(here a,,(t) >).

Since the two functions to be considered are not

just exponential functions,

^{it is not so}

simple

^to

assign

^{to them} well-defined time

scales,

^{all the more}

as

they

possess different time

dependences. However,

one can

roughly

^define ’rc ^as the time after which the relaxation kernel

(1 + W2 t2)-, (see Eq. (36))

^has

decreased

by

^{a factor}

lie,

^which

yields :

One must have _i~ im, where im is linked ^to

z uz(t).

^{For a} >

1,

this condition is

automatically satisfied,

since the time scale

of ~ U z( t) )

^is

extremely long,

^as indicated above. For a

1, equation (38) approximately yields :

Note that

T- 1

is the so-called renormalized tun-

neling frequency appearing

ⁱⁿ

equation (15)

^{of refe-}

rence

[10a].

^The

separation

of time scales

implies

^{that :}

and,

^for

w~lc~o > 1,

^{this can} be rewritten as :

The

separation

^{of the two} time scales also means

that the renormalized

tunnelling frequency

^{has to be}

small as

compared

^{to the bare} one; in other

words,

the effective

tunneling

time Tm ^must be great ⁱⁿ

comparison

with the relaxation time of the bath as

measured

by

T,.

Our calculation thus has a wide _range of

applica- bility since, according

^to

equation (47), only

^values

of a near zero are excluded. When the condition

(47)

is

satisfied,

^the convolutionless

equation (27)

^{and its}

solution are valid for

times t >

_T,, ^as

long

^{as the}

expansion

parameter Wo T, is

small,

^{i.e. :}

Note

that,

^for

high

^a

values,

^the

expansion parameter

reduces to

woll

^We

(an effectively

^small

quantity

ⁱⁿ

the

model), whereas,

^{for small a}

values,

^{it is}

approxi- mately equal

^to

(wolwe) el/2a,

this latter

quantity being

indeed small as

long

^{as the}

separation

^{of time}

scales,

^as

given by equation (47),

^{is achieved.}

3.5.2 Finite temperature ^{case. -} The discussion of the

validity

of the convolutionless

equation (27)

^at

finite temperature ^{is more}

involved; indeed,

^{the solu-}

tion of

equation (27)

^{has been obtained}

through

^a

numerical

integration,

which renders _uneasy the discussion of the

separation

of time scales.

However,

as shown

above,

in certain cases,

analytical approxi-

mations of the solution can be

given,

which then

simplifies

the discussion.

Let us first define the short time scale at finite tem-

perature. The relaxation kernel

(see Eq. (35))

^{is the}

product

^{of two}

functions,

^{one of which}

being

^the

relaxation kernel at T ⁼

0, (1 + COC2 t 2) - a,

^{of time}

scale,r, (Eq. (45)),

and the other one,

(tlt)/sinh (tlt))2a,

decreasing approximately

^{on a} time scale of the order

T/2

^{a. One can thus}

roughly

define the short time scale at finite temperature ^{as min}

(~~, r/2 a).

Since for times t >

r/2 a, ( a_,(t) > approximately

relaxes

according

^{to an}

exponential

^{law of time}

constant

TR (see Eq. (41)),

^this

approximation

^would

be

meaningful

^for

nearly

the whole time domain

provided

^that

Condition

(49)

is fulfilled in the

region

^{above the}

solid curve 1 of the

(a, T) plane (see Fig. 5).

^{In this}

region,

the condition of

separation

of time scales is

automatically realized,

^{which in turn} insures the

validity

of the convolutionless

equation

^itself.

As

explained

ⁱⁿ

paragraph 4,

^the

~ehaviour

^of

( U z( t) >

in the time _range t ;5

i/2

^{a is the same as}

at zero temperature.

Therefore,

^as

long

^as

z Uz(t) >

^{is well}

represented by

^its zero