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Submitted to J. Chem. Phys.

Energy relaxation of the amide-I mode in hydrogen-bounded peptide units : a route

to conformational change

Vincent Pouthier∗

Institut UTINAM, Universit´ e de Franche-Comt´ e,

UMR CNRS 6213, 25030 Besan¸ con cedex, France.

(Dated: November 7, 2007)

A one-site Davydov model involving a C=O group engaged in a hydrogen bond is introduced

to study the amide-I relaxation due to Fermi resonances with a bath of intramolecular normal

modes. In the amide-I ground state, the hydrogen bond behaves as a harmonic oscillator whose

eigenstates are phonon number states. By contrast, in the amide-I first excited state, the hydrogen

bond experiences a linear distortion so that the system eigenstates are superimpositions of number

states. By assuming the hydrogen bond in thermal equilibrium at biological temperature, it is

shown that the amide-I excitation favors the population of these excited states and the occurrence

of coherences. Due to the interaction with the bath, the vibron decays according to an exponential

or a biexponential law depending on whether the Fermi resonance is wide or narrow. Therefore, each

excited state relaxes over a set of number states according to specific pathways. The consequence is

twofold. First, the relaxation leads to a redistribution of the number state population which differs

form the initial Boltzmann distribution. Then, it allows for coherence transfers so that, although

the vibron has disappeared, the hydrogen keeps the memory of its initial distortion and it develops

free oscillations.

PACS numbers:

I.INTRODUCTION

In living systems, the energy released by the hydroly-

sis of adenosine triphophate (ATP) is a universal energy

source allowing many biological processes such as enzyme

catalysis, cell motion, active ion transport or muscle con-

traction. This biological activity results from the trans-

duction of the released energy into a mechanical work

which usually originates from a conformational change of

the proteins that perform the required task. Although in

some cases the final conformation has been determined,

the detailed mechanisms for both energy transduction

and conformational change remain mysterious.

To elucidate this puzzle, a recent attention has been

paid by Cruzeiro and co-workers to apply the so-called

vibrational excited states hypothesis [1–4]. It is based

on the fact that the first step in protein function is the

storage of the released energy in the vibrational excited

states of some groups of the protein. This idea, which can

be traced back to the seminal works of McClare [5] in the

1970s, has been improved by Davydov and co-workers to

explain bioenergy transport in α-helices (see for instance

Ref. [6–8]).

At the present time, both theoretical and experimental

evidences suggest that energy flow in α-helices may result

from a polaron mechanism [11–23]. The released energy

is responsible for the excitation of the amide-I mode of a

peptide group which mainly involves the stretching vibra-

tion of the C=O group. This resonant coupling is prob-

ably mediated by the intermediate vibrational excitation

∗Electronic address: vincent.pouthier@univ-fcomte.fr

of water. Due to the dipole-dipole coupling, the amide-I

vibration delocalizes along the helix leading to the occur-

rence of vibrational excitons called vibrons. The polaron

formation originates from the strong coupling between

the vibrons and the phonons describing the vibrations of

the hydrogen bonds which stabilize the helix backbone.

Since the vibron bandwidth is lower than the phonon

cutoff frequency, the quantum behavior of the phonons

plays a crucial role and the nonadiabatic strong coupling

regime is reached. The creation of a vibron is thus ac-

companied by a virtual cloud of phonons describing a

localized lattice distortion which follows instantaneously

the vibron. The vibron dressed by the lattice distortion

forms the small polaron.

Although the coherent nature of small polarons is cer-

tainly lost at biological temperature [24–27], the Davy-

dov model is suitable for explaining how the energy re-

leased at an active site can propagate to other regions of

the protein. Nevertheless, it is unable to describe the en-

ergy relaxation of the amide-I mode since it conserves the

number of amide-I quanta. As pointed out by Cruzeiro

and co-workers [2–4], this weak point of the model is

fundamental because the energy released by the decay of

the amide-I mode can be used by the protein to realize a

conformational change.

The vibrational energy relaxation in proteins is very

fast and the amide-I lifetime is typically about 1.5 ps.

Indeed, time resolved spectroscopy of myoglobin revealed

that this lifetime varies from 1.8 ps at 10 K to 1.3 ps at

310 K [28, 29]. Note that Austin and co-workers have

reported a lifetime of about 2.7 ps. Moreover, they ob-

served both a blue shifted peak and a red shifted self-

trapped state whose lifetime at low temperature is 15 ps

and 30 ps, respectively [30, 31]. Pump-probe and hole

burning spectroscopies applied to three small globular

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peptides showed that the amide-I relaxation of all inves-

tigated peptides occurs over a similar time scale of about

1.2 ps [32, 33]. In addition, the 2D spectroscopy of ala-

nine based 21-residue revealed a lifetime lower than 5 ps

at room temperature [34]. In the same way, the amide-

I mode decays after about 2 ps at 93 K in Acetanilide

(ACN), a molecular crystal whose structural properties

are similar to those of α-helices [35]. Similarly, time re-

solved spectroscopies were applied to N-methylacetamide

(NMA), a model compound for a single peptide unit (see

for instance Ref.[36]). It has been shown that the amide-

I mode decays according to a biexponential law which

exhibits a very fast energy transfer over a time scale of

about 0.2 ps.

These experimental features cannot be explained in

terms of a direct energy transfer between the amide-I

mode and the solvent or the surrounding. They clearly

suggest that the amide-I lifetime results from intramolec-

ular energy redistribution due to the anharmonic cou-

pling with intramolecular normal modes whose displace-

ments are strongly localized on the C=O group. This sce-

nario has been corroborated by recent theoretical anal-

ysis. Indeed, the low temperature molecular dynamics

simulation of myoglobin revealed that a fast energy trans-

fer between the amide-I mode and a lower frequency

mode is mediated by Fermi resonances [37]. Similarly,

the numerical simulation of the relaxation in NMA shows

that the energy redistribution involves the Fermi reso-

nance between the amide-I mode and either the overtone

of a high frequency mode or a combination band between

two high frequency modes [38].

In that context, the aim of the present paper is to ad-

dress a comprehensive theory to describe how the amide-I

relaxation, due to the coupling with a set of intramolec-

ular normal modes, can modify the quantum state of a

conformational degree of freedom of a protein. To pro-

ceed, we consider a one-site Davydov model in which a

single C=O group is engaged in a hydrogen bond. There-

fore, the hydrogen bond plays the role of the conforma-

tional degree of freedom since hydrogen bonding is a fun-

damental element in protein structure and function. For

instance, it stabilizes the helical structures of α-helices

and it participates in the formation of β-turns allowing

the reversal of polypeptide chains. Therefore, breaking a

single hydrogen bond may impair the stability of a pro-

tein. Although this model is rather unrealistic to describe

the dynamics of a real protein, it gives a simple approach

to clearly understand how energy relaxation provides a

route to a conformational change. A more realistic model

will be presented in forthcoming works.

Note that the present work has been inspirited by the

recent developments of Cruzeiro and co-workers [2–4].

Nevertheless, three points are fundamentally different.

First, we develop a formalism at biological temperature.

Then, instead of the semi-classical approach in which the

hydrogen bond is treated classically, we use a full quan-

tum description. Finally, in the spirit of the experimental

results, the amide-I decay results from a coupling with

a thermal bath. In a marked contrast, Cruzeiro and co-

workers have introduced an extra term in the Davydov

model to break its conservative nature. This term does

not characterize a relaxation mechanism and it corre-

sponds to a time independent linear perturbation of the

amide-I mode.

The paper is organized as follows. In section II, the

one-site Davydov model is described and the correspond-

ing Hamiltonians are defined. In Section III, the amide-I

relaxation is described in terms of the reduced density

matrix whose time evolution is characterized by a gen-

eralized master equation established in Sec.

equation is solved numerically in Section V where a de-

tailed analysis of the vibrational dynamics is performed.

Finally, these results are discussed and interpreted in Sec-

tion VI.

IV. This

II.MODEL AND HAMILTONIANS

In a general way, the dynamics of a protein exhibits

a tremendous complexity due to the rather large num-

ber of degrees of freedom. For instance, in α-helices, the

3D conformation is stabilized by the hydrogen bond be-

tween the carboxyl oxygen (CO) of an amino acid and

the amide hydrogen (NH) of a second amino acid that

is situated four residues ahead in the linear sequence

of the polypeptide chain. The helix is thus formed by

three spines of hydrogen-bonded peptide units connected

through covalent bonds. To overcome this difficulty, we

consider a simplified system involving only a single C=O

group engaged in a hydrogen bond. This system con-

tains a amide-I mode which behaves as a high frequency

oscillator described by the boson operators b†and b and

whose internal frequency and anharmonicity are labeled

ω0and A, respectively. This mode interacts with the low

frequency vibration of the hydrogen bond which is mod-

eled by a harmonic oscillator with frequency Ω and with

boson operators a†and a. The coupling, whose strength

is specified by the parameter ∆, accounts for the mod-

ulation of the amide-I frequency which depends linearly

on the coordinate of the hydrogen bond. In that context,

the Hamiltonian of the one-site Davydov model is defined

as (using the convention ? = 1)

HA= ω0b†b − Ab†b†bb + Ωa†a + ∆(a†+ a)b†b

A natural basis set to describe the dynamics of HAis

formed by the number states { |v,p? } in which v refers

to the number of amide-I quanta, also called vibrons, and

p defines the number of quanta connected to the vibra-

tion of the hydrogen bond. These latter quanta will be

called optical phonons so that the name vibron-phonon

system (VPS) will be used to describe the one-site Davy-

dov model. Therefore, for non vanishing ∆ values, HA

can be solved exactly by performing a Lang-Firsov trans-

formation [42].The corresponding eigenstates are ex-

pressed in terms of the number states as

(1)

|Ψvp? = θv|v,p?(2)

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where θ = exp(−(∆/Ω)(a†−a)) is the so-called dressing

operator. Its matrix elements are defined as [8]

?p|θ|¯ p? =

?

p!

¯ p!u¯ p−pe−u2

2L¯ p−p

p

[u2](3)

where u = ∆/Ω and Ln

Laguerre polynomials. The associated eigenenergies are

defined in terms of the small polaron binding energy

EB= ∆2/Ω as

m[x] stands for the generalized

ωvp= v(ω0− EB) − v(v − 1)(A + EB) + pΩ (4)

According to Eq.(2), the VPS eigenstates are phonon

number states when the amide-I mode is in its ground

state (v = 0). By contrast, when v > 0, the potential

energy of the hydrogen bond experiences a linear pertur-

bation. The corresponding eigenstates are linear super-

impositions of phonon number states and they describe

a distortion of the hydrogen bond.

In addition to the standard Davydov model, we in-

troduce a set of high frequency intramolecular normal

modes. Each mode is described by a harmonic oscillator

whose frequency is ωαand whose displacement is defined

in terms of the boson operators b†

modes form a thermal bath whose Hamiltonian is written

as

αand bα. These normal

HB=

?

α

ωαb†

αbα

(5)

The thermal bath contains the normal modes of the

biopolymer whose displacements are strongly localized on

the C=O group. These modes are thus responsible for the

amide-I relaxation which results from energy exchanges

mediated by the anharmonicity of the potential energy of

the system. Because the density of normal modes in the

frequency range of half the amide-I frequency is rather

large, even for small biomolecules [38, 40, 41], the anhar-

monicity facilitates the occurrence of Fermi resonances.

To mimic the effect of such resonances, we follow Gerber

et co-workers [39] and assume that the potential between

the amide-I and the thermal bath is represented by a sum

of separable pair coupling terms as

∆H =

?

α

χα(b†2

αb + b2

αb†)(6)

where χα denotes the strength of the coupling between

the αth bath mode and the amide-I vibration.

Finally, the full Hamiltonian H = HA+ HB+ ∆H

yields a rather simple model for the VPS dynamics. It

allows for an exact quantum treatment of both the amide-

I mode and the hydrogen bond vibration and it includes

the coupling with a bath of intramolecular normal modes.

In the next sections, this model Hamiltonian will be used

to characterize the energy relaxation of the amide-I mode

and the evolution of the hydrogen bond quantum state.

III.VIBRATIONAL DYNAMICS AND

REDUCED DENSITY MATRIX

In a general way, the vibrational dynamics of a molec-

ular system coupled to a complex bath is described by

the so-called reduced density matrix. In our previous

works, a special attention has been paid to characterize

the vibron dynamics, only [25–27]. The reduced density

matrix was determined by performing a trace over the hy-

drogen bond degrees of freedom so that the phonons were

assumed to belong to the thermal bath. In the present

paper, a fully different approach is used since we intend

to understand the way the quantum state of the hydro-

gen bond is modified when the amide-I mode relaxes.

Therefore, the partial trace is done over the intramolec-

ular normal modes, only, and all the information that is

desired on the hydrogen bond can be extracted form the

reduced density matrix.

In that context, by performing a trace over the thermal

bath, the representation of the reduced density matrix in

the VPS eigenstate basis is written as

σv¯ v

p¯ p(t) = ?Ψvp|TrB[e−iHtρeiHt]|Ψ¯ v¯ p?

where ρ denotes the initial density matrix of the whole

system ” VPS + thermal bath ”.

Without any perturbation, the system is in thermal

equilibrium at the biological temperature T. Since ω0≈

1660 cm−1, the vibrons cannot be thermally excited.

However, this is no longer the case for the other degrees

of freedom whose true eigenstates are not well defined.

As a result, a statistical average is required so that the

optical phonons and the thermal bath are described by

using standard Boltzmann distributions ρphand ρB, re-

spectively.

To study the amide-I relaxation, we consider that the

energy released by the hydrolysis of ATP acts as an ex-

ternal source which brings the system in a state out of

equilibrium. This source is resonantly coupled with the

amide-I mode, only, and it does not affect significantly

the remaining degrees of freedom. Moreover, it is sup-

posed to act during a very short time scale so that the

population of the amide-I mode increases adiabatically

without any change in the quantum states of both the op-

tical phonons and the thermal bath. Consequently, the

released energy allows for the occurrence of v0 vibrons

on the amide-I mode. Note that since an ATP molecule

releases 0.49 eV under normal physiological conditions,

the vibron number is typically equal to v0= 1 or v0= 2.

Therefore, the whole system is prepared in an initial state

specified by the full density matrix

(7)

ρ = ρA⊗ ρB

(8)

where ρA= |v0??v0|⊗ρphdefines the VPS density matrix.

Because ρA does not commute with HA, it does not

refer to a statistical mixture of the VPS eigenstates. It

characterizes the presence of v0amide-I quanta, the hy-

drogen bond being described by a statistical mixture of

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number states. In the VPS eigenstate basis, the con-

sequence is twofold. First, the initial population of the

eigenstates with v0 vibrons differs from the Boltzmann

distribution. Then, ρAexhibits non vanishing coherences

which result from the fact that |Ψv0p? is a superimposi-

tion of the number states |v0¯ p?. Consequently, ρA has

diagonal and non diagonal matrix elements defined as

(see Eq.3)

?Ψv0p|ρA|Ψv0¯ p? =

?

p!

¯ p!(1 − w)¯ p−p+1wp(−u)¯ p−p

× e−u2(1−w)L¯ p−p

where u = v0∆/Ω and w = exp(−Ω/kBT).

From the knowledge of the reduced density matrix, we

can compute in principle the average value of any ob-

servable required to characterize the VPS dynamics. In

that context, to get information about the decay of the

amide-I mode, we first define < v(t) >= Trσ(t)b†b as the

average vibron number at time t. It is written as

p

[−u(1 − w)2/w](9)

< v(t) >=

?

vp

vσvv

pp(t) (10)

Similarly, the average phonon number at time t is de-

fined as < p(t) >= Trσ(t)a†a. To calculate < p(t) >,

let us remind that, depending of the vibron number, the

VPS eigenstates characterize a distortion of the hydro-

gen bond. The boson operators, a†and a, experience a

translation so that the phonon number matrix elements

are expressed as

?Ψvp|a†a|Ψ¯ v¯ p? = δv¯ v?p|

?

a†−v∆

Ω

??

a −v∆

Ω

?

|¯ p? (11)

By performing a trace according to Eq.(7), < p(t) > is

thus defined as

< p(t) >=

?

(12)

?

vp

p +

?v∆

Ω

?2?

σvv

pp(t) −2v∆√p + 1

Ω

ℜσvv

pp+1(t)

Finally, to describe the modification of the distortion

of the hydrogen bond, let us introduce the reduced hy-

drogen bond coordinate as < u(t) >= Trσ(t)(a†+ a).

After evaluating the phonon operators in the eigenstate

basis, it is expressed as

< u(t) >=

?

vp

−2v∆

Ω

σvv

pp(t) + 2?p + 1ℜσvv

pp+1(t) (13)

Eqs.(10), (12) and (13) show that the reduced density

matrix is the central object of the present study.

knowledge allows us to characterize the dynamics of both

the amide-I mode and the hydrogen bond. Nevertheless,

these equations reveal that only particular elements are

required. Indeed, the desired observables only depend on

Its

the population of the eigenstates σvv

coherences σvv

pp+1(t). In the following of the text, these

elements will be defined as

pp(t) and on specific

Pvp(t) = σvv

Qvp(t) = σvv

pp(t)

pp+1(t)(14)

In that context, the next section is devoted to the deriva-

tion of a Generalized Master Equation (GME) to study

the time evolution of Pvp(t) and Qvp(t).

IV. MASTER EQUATION

In this section, a standard projector method is used

to derive a GME for the reduced density matrix (see for

instance Refs. [43–47]). To proceed, two simplifying as-

sumptions are invoked. First, the coupling between the

amide-I mode and the thermal bath is supposed to be

sufficiently weak to perform a second order perturbation

theory with respect to ∆H. Second, we assume that the

bath exhibits correlations over a very short time scale so

that the Markovian limit is used.

Within these assumptions, the derived GME mixes in

a complicated manner both diagonal and non diagonal

matrix elements. To overcome this difficulty, we apply

the secular approximation which allows us to separate

the dynamics of matrix elements whose time evolution is

rather different [45]. Indeed, the dynamics of σv¯ v

typically governed by the energy difference |ωvp− ω¯ v¯ p|.

Since the VPS does not exhibit degenerate eigenstates,

the populations are clearly uncoupled with the coher-

ences whose smallest frequency is equal to Ω. Moreover,

the VPS supports two kinds of coherences. First, the

coherences involving states referring to the same vibron

number (v = ¯ v) evolve typically according to a frequency

of about a few times the phonon frequency (|p − ¯ p|Ω).

By contrast, coherences between states characterized by

a different vibron number (v ?= ¯ v) involve high frequen-

cies about a few times the amide-I frequency. As a re-

sult, these two kinds of coherences evolve almost inde-

pendently.

In that context, after some algebraic manipulations,

the GME yields two independent equations to character-

ize the time evolution of both the populations Pvp(t) and

the coherences Qvp(t). The dynamics of the populations

is governed by a standard Pauli Master equation written

as

dPvp(t)

dt

¯ v¯ p

p¯ p(t) is

=

?

W¯ v¯ p→vpP¯ v¯ p(t) − Wvp→¯ v¯ pPvp(t)(15)

where Wvp→¯ v¯ p denotes the rate for the transition from

|Ψvp? to |Ψ¯ v¯ p? due to the coupling with the thermal bath.

It is defined as

Wvp→¯ v¯ p = vδ¯ vv−1|?¯ p|θ|p?|2g1

?ωvp− ω¯ v¯ p

2

?ω¯ v¯ p− ωvp

?

(16)

+ (v + 1)δ¯ vv+1|?¯ p|θ†|p?|2g0

2

?

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where gi(ω) is the coupling distribution expressed in

terms of the Bose number pB(ω) = (exp(ω/kBT) − 1)−1

as

gi(ω) =

?

α

2πχ2

αδ(ω − ωα)(pB(ω) + i)2

(17)

Similarly, the time evolution of the coherences Qvp(t)

is controlled by the equation

dQvp(t)

dt

= (iΩ − Γvp)Qvp(t) +

?

¯ v¯ p

Tv¯ v

p¯ pQ¯ v¯ p(t)(18)

where Γvp= (γvp+γvp+1)/2 is expressed in terms of the

full relaxation rate γvpdefined as

γvp=

?

¯ v¯ p

Wvp→¯ v¯ p

(19)

In Eq.(18), Tv¯ v

diated by the thermal bath. It is defined as

p¯ pdescribes the transfer of coherences me-

Tv¯ v

p¯ p = vδ¯ vv−1?¯ p + 1|θ|p + 1??p|θ†|¯ p?g0

?ωvp− ω¯ v¯ p

?ω¯ v¯ p− ωvp

2

?

(20)

+ (v + 1)δ¯ vv+1?¯ p + 1|θ†|p + 1??p|θ|¯ p?g1

2

?

Eqs.(15) and (18) show that the bath strongly modifies

the dynamics of both the amide-I mode and the hydrogen

bond. Indeed, it is responsible for incoherent transitions

between eigenstates involving a different vibron number.

These transitions favor a redistribution of the popula-

tions of the eigenstates which is described by the Pauli

Master equation Eq.(15). Consequently, a vibron can be

absorbed or emitted by the bath according to the transi-

tion rates Eq.(16). However, during such a mechanism,

the phonon number fluctuates since the potential energy

experiences by the hydrogen bond depends on the vibron

number. For instance, if the VPS is initially in the state

|Ψ1p?, the hydrogen bond is described by the superim-

position of number states θ|p?. The absorption of the

vibron projects the amide-I in its ground state in which

the hydrogen bond reaches a number state. From quan-

tum mechanics, this final number state is not well defined

since the hydrogen bond can reach all the number states

|¯ p? which belong to the initial superimposition θ|p?. Such

a process can only be characterized by the corresponding

quantum probability |?¯ p|θp?|2which is a measure of the

overlap between the initial state and the final state.

Similarly, the influence of the thermal bath on the co-

herences is twofold. First, since the bath induces tran-

sitions, each eigenstate has a finite lifetime. Therefore,

any coherence will be destroyed in the long time limit due

to the decay of the eigenstates involved in a superimpo-

sition. The bath is thus responsible for the well-known

dephasing mechanism which is described by the dephas-

ing constant Γvp. Then, the thermal bath allows for a

transfer of coherences. To understand this feature more

clearly, let us consider the situation in which the initial

state is a superimposition of |Ψ1p? and |Ψ1p+1?. Under

the coupling with the bath, these two eigenstates can, for

instance, decay in the ground state of the amide-I mode.

During this process, the hydrogen bond has a non vanish-

ing probability to reach a state formed by the superimpo-

sition of two number states |¯ p? and |¯ p + 1?. As shown in

Eq.(20), the corresponding probability involves the prod-

uct between the overlaps ?¯ p|θ|p? and ?¯ p+1|θ|p+1?. Con-

sequently, this mechanism clearly shows that the initial

coherence Q1p(t) can be partially transfered to the co-

herence Q0¯ p(t).

As shown in Eqs.(16) and (20), both incoherent transi-

tions and coherence transfers result from the competition

between two mechanisms. The first mechanism originates

from the overlap between the final state and the initial

state of the hydrogen bond during the vibron relaxation.

It involves the matrix elements of the dressing opera-

tor and it participates in the selection of the eigenstates

which are coupled due to the interaction with the thermal

bath. The second mechanism corresponds to the ability

of the bath to exchange energy with the amide-I mode via

the Fermi resonances. It is characterized by the coupling

distributions gi(ω), i = 0,1 (Eq.(17)).

measures the ability of the bath at thermal equilibrium

to supply the energy 2ω to the amide-I mode, whereas

g1(ω) accounts for its ability to absorb the energy 2ω.

At this step, the knowledge of the transition rates and

of the coherence transfers allows us to solve numerically

the GME Eqs.(15) and (18). The corresponding solutions

can thus be used to evaluate the different observables de-

fined in Eqs.(10), (12) and (13). This numerical proce-

dure, illustrated in the following section, will provide a

complete understanding of the VPS dynamics.

Indeed, g0(ω)

V. NUMERICAL RESULTS

In this section, the previous formalism is applied to

study the vibrational dynamics of the VPS for v0= 1. To

proceed, typical values for the parameters of the Davydov

model are introduced [16]. The frequency and the anhar-

monicity of the amide-I mode are fixed to ω0 = 1660

cm−1and A = 8.0 cm−1, respectively.

phonon frequency is equal to Ω = 50 cm−1and the

temperature is fixed to T = 310 K. Finally, the vibron-

phonon coupling ∆ will be considered as a free parameter

varying form 0 to 30 cm−1.

To characterize the interaction with the thermal bath,

the coupling distribution gi(ω) is modeled by a gaussian

law centered around ωc= ω0/2 and written as

The optical

gi(ω) = (pB(ω) + i)

G

√παexp

?

−

?ω − ωc

α

?2?

(21)

where G = 400 cm−2and where α defines the width of the

Fermi resonance. The coupling distribution is illustrated

in Fig. 1 for the transition between the ground state and