Adiabatic passage in a three-state system with non-Markovian relaxation:
  The role of excited-state absorption and two-exciton processes
Adiabatic passage in a three-state system with
non-Markovian relaxation: The role of excited-state
absorption and two-exciton processes.
B. D. Fainberg1,2∗and V. A.Gorbunov1
1Faculty of Sciences, Physics Department, Holon Institute of Technology,
52 Golomb St., Holon 58102, Israel
2Raymond and Beverly Sackler Faculty of Exact Sciences,
School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel
August 8, 2021
Abstract
The influence of excited-state absorption (ESA) and two-exciton processes on a co-
herent population transfer with intense ultrashort chirped pulses in molecular systems
in solution has been studied. An unified treatment of adiabatic rapid passage (ARP)
in such systems has been developed using a three-state electronic system with relax-
ation treated as a diffusion on electronic potential energy surfaces. We have shown
that ESA has a profound effect on coherent population transfer in large molecules that
∗Corresponding author. E-mail: fainberg@hit.ac.il
http://arxiv.org/abs/0704.1959v1
necessitates a more accurate interpretation of experimental data. A simple and phys-
ically clear model for ARP in molecules with three electronic states in solution has
been developed by extending the Landau-Zener calculations putting in a third level
to random crossing of levels. A method for quantum control of two-exciton states in
molecular complexes has been proposed.
1 Introduction.
The possibility of the optical control of molecular dynamics using properly tailored pulses
has been the subject of intensive studies in the last few years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Chirped pulses can selectively excite coherent
wave packet motion either on the ground electronic potential energy surface of a molecule or
on the excited electronic potential energy surface due to the intrapulse pump-dump process
[1, 5, 11, 12]. In addition, they are very efficient for achieving optical population transfer
between molecular electronic states. Total electronic population inversion can be achieved
using coherent light-matter interactions like adiabatic rapid passage (ARP) in a two- or three-
state system [24, 25], which is based on sweeping the pulse frequency through a resonance.
Since the overwhelming majority of chemical reactions are carried out in liquid solution,
adiabatic passage in molecules in solution was studied for two-state electronic system (ARP)
in Refs.[26, 27, 28], and for stimulated Raman adiabatic passage (STIRAP) configuration
in Refs.[29, 30]. It has been shown in Ref.[26] that relaxation does not hinder a coherent
population transfer for positive chirped pulses and moderate detuning of the central pulse
frequency with respect to the frequency of Franck-Condon transition.
However, a two electronic state model for molecular systems is of limited utility. Indeed,
excited-state absorption (ESA) occurs for majority of complex organic molecules [31, 23].
Even a molecular dimer consisting from two-level chromophores has an additional excited
state corresponding to two-exciton excitation. An unified treatment of ARP in such sys-
tems can be developed using three-state electronic system interacting with reservoir (the
vibrational subsystems of a molecule (chromophores) and a solvent).
More often than not ESA in complex organic molecules corresponds to a transition from
the first excited singlet state S1 to a higher singlet state Sn (n > 1), which relaxes back
to S1 very fast [32, 33, 34, 31]. Therefore, it would look as if ESA does not influence on
population transfer S0 → S1 from the ground state S0. However, in the presence of ESA an
exciting pulse interacts with both S0 → S1 and S1 → Sn transitions. It is well known that
coherent optical interactions occurring in adjacent optical transitions in a three-state system
markedly affect each other. The examples are STIRAP, lasing without inversion, coherent
trapping, electromagnetically induced transparency and others. (For textbook treatments of
these effects see, for example, [35]). Therefore, one would expect an appreciable change of
a population transfer S0 → S1 with chirped pulses in the presence of excited state absorp-
tion in the coherent regime when the chirp rate in the frequency domain is not large and,
consequently, the pulse is rather short.
Our objective is to answer the following questions: “How do ESA and two-exciton pro-
cesses influence on a coherent population transfer in molecular systems in solution? What
is the potential of chirped pulses for selective excitation of the single and two-exciton states
and their selective spectroscopy?”
In addition, the three-state system under discussion enables us to consider STIRAP as
well. Therefore, we shall also briefly concern slowing down the pure dephasing on STIRAP
in intense fields when relaxation is non-Markovian.
The outline of the paper is as follows. In Sec.2 we present equations for the density matrix
of a three-state molecular system under the action of shaped pulses when the interaction
with a dissipative environment can be described as the Gaussian-Markovian modulation (so
called the total model). In Sec.3 we formulate a number of approaches to this model that
enables us, first, to clarify the underlying physics and, second, to understand the validity of
the results obtained by the total model. The ESA effects on ARP in complex molecules are
considered in Sec.4. In Sec.5 we study population transfer in molecular dimers with taking
into account two-exciton processes. In Sec.6 we consider slowing down the pure dephasing
on STIRAP in strong fields when the system-bath interaction is not weak (non-Markovian
relaxation). We summarize our results in Sec.7. In the Appendix we extend calculations
of two-photon excitation of a quantum ladder system by a chirped pulse [36] to non-zero
two-photon detuning.
2 Basic equations
Let us consider a molecular system with three electronic states n = 1, 2 and 3 in a solvent
described by the Hamiltonian
|n〉 [En +Wn(Q)] 〈n| (1)
where E3 > E2 > E1, En is the energy of state n,Wn(Q) is the adiabatic Hamiltonian of
reservoir R (the vibrational subsystems of a molecular system and a solvent interacting with
the three-level electron system under consideration in state n).
The molecular system is affected by two shaped pulses of carrier frequencies ω1 and ω2
E(t) =
i=1,2
Ei(t) + c.c. =
i=1,2
~Ei (t) exp[−iωit+ iϕi (t)] + c.c. (2)
which are resonant to optical transitions 1 → 2 and 2 → 3, respectively (ladder configura-
tion). Here Ei (t) and ϕi (t) describe the change of the pulse amplitude and phase, respec-
tively, in a time t. The instantaneous pulse frequencies are ωi (t) = ωi − dϕidt .
The influence of the vibrational subsystems of a solute and a solvent on the electronic
transition can be described as a modulation of this transition by low frequency (LF) vibra-
tions {ωs} [37, 38]. In accordance with the Franck-Condon principle, an electronic tran-
sition takes place at a fixed nuclear configuration. Therefore, for example, the quantity
u(Q) = W2(Q) −W1(Q) − 〈W2(Q) −W1(Q)〉1 is the disturbance of nuclear motion under
electronic transition 1 → 2. Here 〈〉n ≡ TrR (...ρRn) denotes the trace operation over the
reservoir variables in the electronic state n,
ρRn = exp (−βWn) /TrR exp (−βWn) , β = 1/kBT .
The relaxation of electronic transition 1 → 2 stimulated by LF vibrations is described by
the correlation function K(t) = 〈u(0)u(t)〉 of the corresponding vibrational disturbance with
characteristic attenuation time τs [12, 38]. We suppose that ~ωs ≪ kBT . Thus {ωs} is an al-
most classical system and operators Wn are assumed to be stochastic functions of time in the
Heisenberg representation. The quantity u can be considered as a stochastic Gaussian vari-
able. We consider the Gaussian-Markovian process when K(t)/K(0) ≡ S(t) = exp(−|t|/τs).
The corresponding Fokker-Planck operator Lj = τ
+ (q − dj) ∂∂q + 1
describes the
diffusion in the effective parabolic potential
Uj (q) = Ej +
ω̃2 (q − dj)2 (3)
of electronic state j where τ−1s = D̃nβω̃
2 and D̃ is the diffusion coefficient. Going to a di-
mensionless generalized coordinate x = qω̃
β, one can obtain the equations for the elements
of the density matrix ρij(x, t) by the generalization of the equations of Ref.[26]. Switching
to the system that rotates with instantaneous frequency
ρ̃12(x, t) = ρ12(x, t) exp[−i(ω1t− ϕ1(t))], ρ̃23(x, t) = ρ23(x, t) exp[−i(ω2t− ϕ2(t))],
ρ̃13(x, t) = ρ13(x, t) exp{−i[(ω1 + ω2)t− (ϕ1(t) + ϕ2(t))], (4)
we get
ρ11(x, t) = Im[Ω1ρ̃12(x, t)] + L1ρ11(x, t)
ρ22(x, t) = − Im[Ω1ρ̃12(x, t) + Ω∗2ρ̃32(x, t)] + L2ρ22(x, t) + 2Γ32ρ33(x, t)
ρ33(x, t) = − Im[Ω2ρ̃23(x, t)] + (L3 − 2Γ32)ρ33(x, t) (5)
ρ̃12(x, t) = i
ω21 − ω1(t)− (~β)−1x2x
ρ̃12(x, t) +
Ω∗1[ρ22(x, t)− ρ11(x, t)]−
Ω2ρ̃13(x, t) + L12ρ̃12(x, t) (6)
ρ̃13(x, t) = i
ω31 − ω1(t)− ω2(t)− (~β)−1x3x
ρ̃13(x, t) +
Ω∗1ρ̃23(x, t)−
Ω∗2ρ̃12(x, t) + (L13 − Γ32)ρ̃13(x, t) (7)
ρ̃23(x, t) = i
(ω31 − ω21)− ω2(t)− (~β)−1 (x3 − x2)x
ρ̃23(x, t) +
Ω∗2 (t) [ρ33(x, t)− ρ22(x, t)]+
Ω1ρ̃13(x, t) + (L23 − Γ32)ρ̃23(x, t) (8)
where Ω1 = D21E1/~ and Ω2 = D32E2/~ are the Rabi frequencies for transitions 1 → 2
and 2 → 3, respectively. Here ωi1 = ωeli1 + x2i /(2~β) is the frequency of Franck-Condon
transition 1 → i, ωelij = (Ei − Ej)/~ is the frequency of purely electronic transition j → i,
Dij are matrix elements of the dipole moment operator, 2Γ32 is a probability of nonradiative
transition 3 → 2 for the excited state absorption problem (see below); |xj | = (~βω1jst )1/2 is a
dimensionless shift between the potential surfaces of states 1 and j (x1 = 0), which is related
to the corresponding Stokes shift ω
st of the equilibrium absorption and luminescence spectra
for transition 1 → j. The last magnitude can be written as ω1jst = ~βσ
2s where σ
2s denotes
the LF vibration contribution to a second central moment of an absorption spectrum for
transition 1 → j. The terms
Lj = τ
+ (x− xj)
on the right-hand side of Eqs.(5) describe the diffusion in the corresponding effective parabolic
potential
Uj(x) = Ej +
(x− xj)2 (j = 1, 2, 3), (10)
Lij = (Li + Lj)/2.
The partial density matrix of the system ρ̃ij (x, t) describes the system distribution with a
given value of x at time t. The complete density matrix averaged over the stochastic process
which modulates the system energy levels, is obtained by integration of ρ̃ij (x, t) over the
generalized coordinate x:
〈ρ̃〉ij (t) =
ρ̃ij (x, t) dx (11)
where diagonal quantities 〈ρ〉jj (t) are nothing more nor less than the populations of the
electronic states: 〈ρ〉jj (t) ≡ nj , n1 + n2 + n3 = 1.
We solve coupled Eqs.(5)-(8), using a basis set expansion with eigenfunctions of diffusion
operator L13, similar to Ref. [26].
The solutions, corresponding to the procedure described in this section, are termed the
total model for short, bearing in mind that they take into account all the relaxations (diffu-
sions) related to populations and electronic coherences between all the electronic states.
3 Approximate models
In this section we describe a number of approaches to the total model (Eqs.(5)-(8)).
3.1 System with frozen nuclear motion
For pulses much shorter than τs one can ignore all the terms ∼ Li, Lij on the right-hand
sides of Eqs.(5)-(8). It means that our system can be described as an ensemble of inde-
pendent three-level systems with different transition frequencies corresponding to a pure
inhomogeneously broadened electronic transitions. In this case the density matrix equations
can be integrated independently for each x. After this the result must be averaged over x.
Solutions of the undamped equations for the density matrix are interesting from the point
of view of evaluation of the greatest possible population of excited states due to coherent
effects, because these solutions ignore all the irreversible relaxations destructing coherence.
In addition, a comparison between the latter solutions and calculations for the total model
enables us to clarify the role of relaxation in the chirp dependence of population transfer
(see Sec.4 below). The approach under discussion in this section is termed ”relaxation-free”
model for short.
3.2 Semiclassical (Lax) approximation
For broad electronic transitions satisfying the ”slow modulation” limit, we have σ
s ≫ 1,
where σ
2s is the LF vibration contribution to a second central moment of an absorption
spectrum for transition i → j. In the last case electronic dephasing is fast, and one can
use a semiclassical (short time) approximation [39]. This limit is also known as the case of
appreciable Stokes losses because the perturbation of the nuclear system under electronic
excitation i → j (a quantity Wj−Wi) is large. Then one can ignore the last term Lij ρ̃ij(x, t)
on the right-hand side of the corresponding equation for the nondiagonal element of the
density matrix [26, 40, 12, 41] that describes relaxation (diffusion) of ρ̃ij(x, t) (Eqs.(6) and
(8)). The solutions, which correspond to disregarding terms Lij ρ̃ij(x, t) for broad electronic
transitions i → j are termed ”partial relaxation” model for short [26]. It is worthy to note
that the ”partial relaxation” model offers a particular advantage over the total model. The
point is that the first can be derived not assuming the standard adiabatic elimination of
the momentum p for the non-diagonal density matrix [41], which is incorrect in the ”slow
modulation” limit [42]. This issue is quite important in the light of the limits imposed on
Eqs.(6) and (8) for nondiagonal elements of the density matrix [43, 44].
Indeed, in the Wigner representation [45, 46, 47] equation for ρ̃12 may be written in the
rotating frame as (see Eq.(6))
ρ̃W12(q, p, t) = i[(U2 (q)− U1 (q))/~− ω1(t)]ρ̃W12(q, p, t)−
Ω2ρ̃W13(q, p, t)+
Ω∗1[ρW22(q, p, t)− ρW11(q, p, t)] + LFP12ρ̃W12(q, p, t) (12)
Eq.(12) has been derived for harmonic potentials, Eq.(3), by generalization of equations of
Refs.[48, 49, 42, 41] where
LFP12 = −p
+ γp+
(U1 (q) + U2 (q))
is the Fokker-Planck operator for overdamped Brownian oscillator with attenuation constant
In the case of appreciable Stokes losses when the perturbation of the nuclear system
under electronic excitation 1 → 2 (a quantity (U2 (q)−U1 (q))/~−ωel21) is large, the quantity
ρ̃W12(q, p, t) oscillates fast due to the first term on the right-hand side of Eq.(12) (see also
Ref.[42]). Therefore, to the first approximation, on can neglect changes of ρ̃W12(q, p, t) due
to the last term on the right-hand side of Eq.(12). Neglecting this term, integrating both
side of Eq.(12) over momentum, and bearing in mind that
ρ̃ij(q, t) =
ρ̃Wij(q, p, t)dp (13)
and x = qω̃
β, we get
ρ̃12(x, t) = i[ω21−ω1(t)−(~β)−1x2x]ρ̃12(x, t)+
Ω∗1[ρ22(x, t)−ρ11(x, t)]−
Ω2ρ̃13(x, t) (14)
that is nothing more nor less Eq.(6) without the last term L12ρ̃12(x, t) on the right-hand
side. As a matter of fact, a derivation of Eq.(14) does not involve the assumption that the
momentum is instantly equilibrated. The same can be done with Eq.(8) for ρ̃23.
4 Adiabatic population transfer in the presence of excited-
state absorption
We shall study the ESA effects on ARP in complex molecules by the example of Coumarin
153 in liquid solution [31]. In the frequency domain, the electric field can be written as
|E(ω̃)| exp[iΦ(ω̃)] and the phase term Φ(ω̃) can be expanded in a Taylor series Φ(ω̃) =
Φ(ω) + (1/2)Φ′′(ω)(ω̃ − ω)2 + ... We shall consider linear chirped pulses of the form
E(t) = E0 exp[−
(δ2 − iµ)(t− t0)2] (15)
where the parameters δ and µ are determined by the formulae [11, 12]:
δ2 = 2{τ 2p0 + [2Φ′′ (ω) /τp0]
2}−1, µ = −4Φ′′ (ω)
τ 4p0 + 4Φ
′′2 (ω)
, (16)
τp0 = tp0/
2 ln 2, tp0 is the pulse duration of the corresponding transform-limited pulse. Fig.1
shows populations of electronic states after the completion of the one pulse action as functions
-5 -4 -3 -2 -1 0 1 2 3 4 5
�"(�)  (x10
Figure 1: Populations of electronic states after the completion of the pulse action as functions
of Φ′′(ν) in a three-state system. Calculations without decay of the upper state 3 into state
2: n1 (dotted line), n2 (solid line), n3 (dashed line). Line with hollow circles - n2 in the
model with fast decay 3 → 2 Γ32=10 ps−1. For comparison we also show n2 for a two-state
system (line with squares). Total relaxation model with diffusion of all matrix elements.
of the chirp rate in the frequency domain Φ′′(ν) = 4π2Φ′′ (ω). For the molecule under
consideration a two-photon resonance occurs at the doubled frequency of the Franck-Condon
transition 1 → 2. Absorption spectrum corresponding to transition 1 → 3 is rather narrow
that means x3 = 0. The values of parameters for Fig.1 were as follows: the pulse duration
of the transform-limited (non-chirped) pulse tp0 = 10 fs, ω
st = 2686 cm
−1, D12 = D32 = 6
D [31], τs = 70 fs, the saturation parameter, which is proportional to the pulse energy
[26], Q′ ≡
π|D12Emax|2tp/(2~2
2σ122s ) = 5; the one-photon resonance for Franck-Condon
transition 1 → 2 occurs at the pulse maximum, i.e. ω = ω21.
Fig.2 contrasts calculations using the total model (Fig.1) with those of the partial relax-
ation model. The latter includes both diffusion of all the diagonal elements of the density
matrix and one off-diagonal element ρ13. The point is that transition 1 → 3 occurs with-
out changing the state of vibrational subsystems of a molecule and a solvent, and therefore
-5 -4 -3 -2 -1 0 1 2 3 4 5
�"(�)  (x10
Figure 2: Populations of electronic states n1 (dotted lines), n2 (solid lines) and n3 (dashed
lines) after the completion of the pulse action calculated without decay of the upper state 3
into state 2 as functions of Φ′′(ν). The partial relaxation and the total models - lines with
and without hollow circles, respectively. All the parameters are identical to those of Fig.1.
can not be described in a semiclassical (short time) approximation. Fig.2 shows a good
agreement between calculation results for the models under consideration.
One can see from Fig.1, first, that population n2 for a molecule with a fast decay 3 → 2,
which closely resembles experimental data [11] for LD6901, is distinctly different from that
of a two-state system for |Φ′′(ν)| < 15 · 103 fs2 when the excited pulse is rather short. This
means that the excited state absorption has a profound effect on coherent population transfer
in complex molecules. Second, n3 strongly decreases when |Φ′′(ν)| increases.
To understand these results, we will consider first two transitions separately. One can
obtain the following criterion for the adiabaticity of one transition in the absence of relax-
ation: Q′ >> 1 where Q′ is the saturation parameter. It conforms to the value of Q′ = 5
used in our calculations. The condition Q′ >> 1 follows from the adiabatic criterion for a
two-level system:
1According to Ref.[23], LD690 shows ESA.
-5 -4 -3 -2 -1 0 1 2 3 4 5
�”(�� �x104 fs2) 
Figure 3: Populations of electronic states n1 (dotted line), n2 (solid line), n3 (dashed line)
and n2 + n3 (line with hollow circles) after the completion of the pulse action as functions
of Φ′′(ν) for the relaxation-free model τs → ∞. Other parameters are identical to those of
Fig.1. In the case under consideration the combined population n2 + n3 does not depend on
Φ′′(ν).
dω(t)
≪ |Ω1,2(t)|2 (17)
where Ω1,2(t) = |D21,32E(t)|/~ are the Rabi frequencies for transitions 1 → 2 and 2 → 3,
respectively. Adiabatic criterion Eq.(17) was fulfilled in our simulations for both transitions
1 → 2 and 2 → 3 at any Φ′′(ν). However, Fig.1 shows that n3 strongly decreases when
|Φ′′(ν)| increases. To clarify the reasons for strong decreasing n3 it is instructive to carry
out the corresponding calculations for the relaxation-free model of Sec.3.1 shown in Fig.3.
In this case excitation of state 3 with a transform-limited pulse is slightly more effective as
compared to a strongly chirped pulse of the same energy. The point is that a two-photon
resonance occurs for a number of spectral components of a transform-limited pulse and only
at the maximum of a strongly chirped pulse. However, Fig.3 does not show strong decreasing
the population of state 3 when Φ′′(ν) increases. This means that relaxation is responsible
for strong decreasing n3 as a function of Φ
′′(ν) in spite of the fact that relaxation does not
destroy ARP when the Rabi frequencies exceed the reciprocal irreversible dephasing time
(T ′)−1 [27]
Ω1,2 >> 1/T
′ (18)
The last condition was fulfilled in our simulations at least for |Φ′′(ν)| . 104 fs2.
To clarify this issue, we shall consider a population transfer between randomly fluctuating
levels.
4.1 Population transfer between randomly fluctuating levels
The picture of randomly fluctuating levels [27] offers a simple and physically clear explanation
of numerical results [26] obtained for population transfer in a two-state system. Here we shall
generalize the Landau-Zener (LZ) calculations putting in a third level [50] to random crossing
of levels.
Let us write the Schrödinger equations for the amplitudes of states a1,2,3 for the system
under consideration. Switching to new variables ãk:
ak = ãk exp
, (19)
we obtain in the rotating wave approximation
(U1 − U2)/~+ ω1(t) −Ω1/2 0
−Ω1/2 0 −Ω2/2
0 −Ω2/2 (U3 − U2)/~− ω2(t)
Throughout this section effective parabolic potentials (10) are considered as functions of gen-
eralized coordinate α = x
σ122s−ω12st : Uj(α) = Ej+ ~2ω12st {α+
ω12st [
ω12st+(−1)sgn(xj)
st ]}2.
(U3 − U2)/~− ω2(t) = [(ωel32 + ω12st /2)− ω2] + α + µ2t (21)
for x3 = x1 (that corresponds to Coumarin 153), and
(U1 − U2)/~+ ω1(t) = [ω1 − (ωel21 − ω12st /2)] + α− µ1t, (22)
for linear chirped pulses ω1,2 (t) = ω1,2 − µ1,2t.
Let us define instantaneous crossings of state 2 with photonic repetitions 1′ and 3′ of
states 1 and 3, respectively. They are determined by the conditions that quantities Eqs.(21)
and (22) are equal to zero:
α12(t) = (ω
21 − ω12st /2)− ω1 + µ1t ≡ α12(0) + µ1t (23)
α23(t) = ω2 − (ωel32 + ω12st /2)− µ2t ≡ α23(0)− µ2t
Near the intersection points one can consider α as a linear function of time. For small t,
α(t) ≈ α12(0)+ α̇t. Let α12(0) = α23(0), i.e. states 2, 1′ and 3′ cross at the same point when
t = 0. This means
ωel21 + ω
32 = ω1 + ω2 (24)
i.e. the two-photon resonance occurs for t = 0. Then Eqs.(20) take the following form
(α̇− µ1)t −Ω1/2 0
−Ω1/2 0 −Ω2/2
0 −Ω2/2 (α̇ + µ2)t
that can be reduced to Eqs.(2) of Ref.[50]. Using the solution obtained in [50] and consid-
ering identical chirps when µ1 = µ2 ≡ µ, we get for the initial condition |a1(−∞)|2 = 1,
|a2,3(−∞)|2 = 0
|a3(∞)|2 =
(1− P )(1−Q) for − |µ| < α̇ < |µ|
P (1− P )(1−Q) for both α̇ > −µ when µ < 0, and α̇ < −µ when µ > 0
Q(1− P )(1−Q) for both α̇ < µ when µ < 0, and α̇ > µ when µ > 0
where
P = exp
4|α̇− µ|
, Q = exp
4|α̇ + µ|
Similar to Ref.[27], we consider α as a stochastic Gaussian variable. Consequently, we
must average Eqs.(26) over random crossing of levels described by Gaussian random noise
induced by intra- and intermolecular fluctuations. It can be easily done for a differentiable
(non-Markovian) Gaussian process [27], bearing in mind an independence of α and α̇ from
each other for such processes. Therefore, we shall consider in this section a differentiable
(non-Markovian) Gaussian noise, as opposed to previuous sections. In addition, we consider
a slow modulation limit when σ122sτ
s >> 1. Averaging Eqs.(26), we obtain the following
expression for the population of state 3 when µ > 0
∫ −|µ|
P (1− P )(1−Q)f(α, α̇)dα̇ +
∫ |µ|
(1− P )(1−Q)f(α, α̇)dα̇
Q(1 − P )(1−Q)f(α, α̇)dα̇] (28)
Here f(α, α̇) is the joint probability density for α and its derivative α̇:
f(α, α̇) =
σ122s (−k̈(0))
2k̈(0)
, (29)
k̈(0) is the second derivative of the correlation function k(t) =< α(0)α(t) >= σ122s exp(−|t|/τs)
of the energetic fluctuations evaluated at zero. Eq.(28) is written for µ > 0 (negatively
chirped pulse). One can easily show that n3 is symmetrical with respect to the chirp sign.
The point is that a simple stochastic model of this Section misses any chromophore’s effects
on bath, in particular the dynamical Stokes shift (see Ref.[51] for details). This is opposite
to the models of previous sections, which do describe the dynamical Stokes by the drift term
(the second term on the right-hand side of Eq.(9)).
Integrating Eq.(28) with respect to α and entering a dimensionless variable y = α̇/ |µ| ,
we get
P (1− P )(1−Q) exp
(1− P )(1−Q)] exp
Q(1− P )(1−Q) exp
dy] (30)
where
P = exp
2|y − 1|
and Q = exp
2|y + 1|
, (31)
2 |µ|
> 0, ξ = −
k̈(0)
> 0 (32)
are dimensionless parameters.
When adiabatic criterion Eq.(17) is satisfied, parameter κ is much larger than 1 since
|dω(t)/dt| = |µ| for a linear chirped pulse. Then the integrals on the right-hand side of
Eq.(30) can be evaluated by the method of Laplace, similar to Ref.[27]. The result is espe-
cially simple for strong interaction, Eq.(18), where the irreversible dephasing time of transi-
tions 1 → 2 and 2 → 3 is given by [27] T ′ = 1/[−k̈(0)]1/4. Then, as one can see also from
Eqs.(30) and (31), the main contribution to n3 is given by
dy = erf
|µ|T ′2√
Since erf(1.5) = 0.966, we obtain that relaxation does not hinder a population transfer to
state 3 when
|µ|T ′2 ≥ 2 (34)
For strongly chirped pulses [52], µ|T ′2/
2 ≈ 2
2π2T ′2/|Φ′′(ν)|.
Eq.(34) expresses an extra criterion for coherent population transfer to those we have
obtained before for a two-level system [27], Eqs.(17) and (18). New criterion (34) implies
conservation of the “counter-movement” of the “photonic repetitions” of states 1 and 3,
in spite of random crossing of levels. Condition (34) is exemplified by Fig.4. In addition,
Fig.4 shows an excellent agreement of simple formula (33) with numerical calculations. It
is worthy to note that condition (18) was fulfilled in our simulations, though in the last
case T ′ = (τs/σ2s)
1/3 is determined independently of k̈(0) [53], which does not exist for the
Gaussian-Markovian process.
4.2 Influence of excited-state absorption when detuning from two-
photon resonance occurs
For Coumarin 153 in liquid solution considered above a two-photon resonance occurs at
the doubled frequency of the Franck-Condon transition 1 → 2. In this section we consider
populations of electronic states when the condition for two-photon resonance is violated.
Figs. 5 and 6 show populations of electronic states for the total model after the completion
15 20 25 30
T', fs
Figure 4: Population of state 3 as a function of the irreversible dephasing time T ′ for Φ′′(ν) =
104 fs2 calculated by Eq.(33) (solid line with circles) and numerical solution of Eqs.(5)-(8)
(dashed line with squares). n3,no relaxation ≡ n3(T ′ → ∞). Other parameters are identical to
those of Fig.1.
-5 -4 -3 -2 -1 0 1 2 3 4 5
�"(	)  (x10
Figure 5: Populations of electronic states after the completion of the pulse action as functions
of Φ′′(ν) in a three-state system. The frequency of purely electronic transition 3 → 2, ωel32,
decreases by ω12st /4 with the conservation of x3 = 0. Calculations without decay of the upper
state 3 into state 2: n1 (dotted line), n2 (solid line), n3 (dashed line). The corresponding
populations in the model with fast decay 3 → 2 Γ32 = 10 ps−1 are shown by the same lines
with hollow circles.
-5 -4 -3 -2 -1 0 1 2 3 4 5
"(�)  (x10
Figure 6: Populations of electronic states after the completion of the pulse action as functions
of Φ′′(ν) in a three-state system. Equilibrium position of state 3 is offset to the right by
x3 = x2/2 and down so that frequencies of Franck-Condon transitions 1 → 2 and 2 → 3 are
equal: ω21 = ω32. Calculations without decay of the upper state 3 into state 2: n1 (dotted
line), n2 (solid line), n3 (dashed line). The corresponding populations in the model with fast
decay 3 → 2 Γ32 = 10 ps−1 are shown by the same lines with hollow circles.
of the pulse action as functions of Φ′′(ν) for the same values of parameters as for Fig.1 with
the only difference concerning the position of state 3. The frequency of purely electronic
transition 3 → 2 ωel32 decreases by ω12st /4 with the conservation of x3 = 0 for Fig. 5. Equilib-
rium position of state 3 is offset to the right by x3 = x2/2 and down so that frequencies of
Franck-Condon transitions 1 → 2 and 2 → 3 are equal: ω21 = ω32 for Fig. 6.
One can see from Figs.1, 5 and 6, first, that population n1 and, as a consequence, n2+n3
depend only slightly on the occurrence of fast decay 3 → 2. Second, populations n2 and
n3 in the absence of fast decay 3 → 2 are very sensitive to the violation of the two-photon
resonance condition. However, a behavior of n2, when fast decay 3 → 2 occurs, and n1
as functions of Φ′′(ν) is very similar for the figures under discussion, regadless of the two-
photon resonance condition. Experimental measurements commonly correspond to n2 and
are carried out under the fast decay 3 → 2 conditions. Thus the behavior of n2 for fast decay
3 → 2 shown in Figs.1, 5 and 6 is rather versatile.
5 Population transfer in the presence of two-exciton
processes. Selective excitation of single and two-
exciton states with chirped pulses
Consider a dimer of chromophores each with two electronic states described by the Frenkel
exciton Hamiltonian [54, 55, 56] and excited with electromagnetic field Eq.(2). The Hamil-
tonian of the dimer is given by
m=1,2
~Ω̄mB
mBm + ~J(B
1 B2 +B
2 B1) +Hbath +Heb −
m=1,2
Dm ·E(t)(B+m +Bm) (35)
where B+m = |m〉〈0| (Bm = |0〉〈m|) are exciton creation (annihilation) operators associated
with the chromophore m, which satisfy the commutation rules [Bn, B
m] = δnm(1−2B+mBm),
δnm is the Kroenecker delta; |0〉 and |m〉 denote the ground state and a state corresponding
to the excitation of chromophore m, respectively. Dm is the transition dipole moment of
molecule m, Hbath represents a bath and Heb its coupling with the exciton system. We
assume that the bath is harmonic and that the coupling is linear in the nuclear coordinates
Heb = −~
mBn (36)
where αmn represent collective bath coordinates. ~Ω̄1(~Ω̄2) and ~J are the exciton energy
of 1 (2) chromophore and their coupling energy at the equilibrium nuclear coordinate of the
ground electronic state. One can consider αmn as diagonal: αmn = αmδnm on the assumption
that the electronic coupling constant fluctuation amplitude is negligibly smaller than the site
energy fluctuation amplitude [55].
Diagonalizing the electronic Hamiltonian
m=1,2
~Ω̄mB
mBm + ~J(B
1 B2 +B
2 B1) (37)
by unitary transformation [57]
U−1 =
cos θ sin θ
− sin θ cos θ
where
tan 2θ =
Ω̄1 − Ω̄2
, 0 < θ < π/2, (39)
one can get the eigenstates for the one-exciton states |ei〉 and the transition dipole moments
Dei (i = 1, 2) corresponding to the transitions between the ground and single-exciton states
= U−1
A1 cos θ + A2 sin θ
−A1 sin θ + A2 cos θ
Here aei = |ei〉, Dei and Ai = B+i |0〉, Di; D1 and D2 are the site transition moments. The
two one-exciton energies are given by
~Ω̄e1 = ~Ω̄1 cos
2 θ + ~Ω̄2 sin
2 θ + ~J sin 2θ, (41)
~Ω̄e2 = ~Ω̄1 sin
2 θ + ~Ω̄2 cos
2 θ − ~J sin 2θ,
The two-exciton state wavefunction and its energy are as following
|e3〉 = B+1 B+2 |0〉 ≡ B+e3 |0〉 (42)
~Ω̄e3 = ~Ω̄1 + ~Ω̄2 (43)
The transition dipole moments between the single-exciton and two-exciton states are given
De1e3 = D1 sin θ +D2 cos θ, De2e3 = D1 cos θ −D2 sin θ (44)
However, the transition between the ground and two-exciton states is not allowed.
In the eigenstate representation, the Hamiltonian of Eq.(35) is rewritten as
i=1,2,3
~(Ω̄ei − αei)B+eiBei − ~
i,j=1,2
i 6=j
αeiejB
Bej +Hbath−
i=1,2
[Dei(B
+Bei) +Deie3(B
Be3 +B
Bei)] · E(t) (45)
Here the interaction with the bath is given by
αe1 αe1e2
αe2e1 αe2
= U−1HebU = ~
α1 cos
2 θ + α2 sin
2 θ 1
(α2 − α1) sin 2θ
(α2 − α1) sin 2θ α1 sin2 θ + α2 cos2 θ
αe3 = α1 + α2, (47)
for the single-exciton and two-exciton states, respectively. Eqs.(46) and (47) define the
fluctuating parts of the single-exciton and two-exciton state transition frequencies.
Consider various correlation functions. Assuming that baths acting on different chro-
mophores are uncorrelated
〈αm(t)αn(0)〉 = 0 for m 6= n (48)
and that the site energy fluctuation correlation functions are identical for the two monomers
[55, 56], we get
〈αe1(t)αe1(0)〉 = 〈αe2(t)αe2(0)〉 = ~−2K(t)(cos4 θ + sin4 θ),
〈αe3(t)αe3(0)〉 = 2~−2K(t) (49)
where K(t) = ~−2〈α1(t)α1(0)〉 = ~−2〈α2(t)α2(0)〉 ≡ ~−2〈ᾱ(t)ᾱ(0)〉. The further calculations
simplify considerably if the off-diagonal part of the interaction with the bath in the exciton
representation αe1e2 = αe2e1 in Eq.(46) can be neglected. This approximation is discussed in
Refs.[55, 58].
The correlation function K(t) can be represented as the Fourier transform of the power
spectrum Φ(ω) of ~α1(= ~α2) [59]
K(t) =
dωΦ(ω) exp(iωt)
where
Φ(−ω) = Φ(ω) exp(−β~ω) (50)
Using Eq.(50), the real and imaginary parts of K(t) = K ′(t) + iK ′′(t) can be written as
K ′(t) =
dωΦ(ω)[1 + exp(−β~ω)] cosωt
K ′′(t) =
dωΦ(ω)[1− exp(−β~ω)] sinωt
In the high temperature limit one get
K ′(t) = 2
dωΦ(ω) cosωt
K ′′(t) = ~β
dωΦ(ω)ω sinωt
where K(0) = K ′(0) = 2
dωΦ(ω) = ~2σ2 = ~ωStβ
−1; σ2 and ωSt are a second cen-
tral moment and the Stokes shift of the equilibrium absorption and luminescence spectra,
respectively, for each monomer.
Similar to Sec.2, we will consider ᾱ = −u/~ as a stochastic Gaussian variable with the
correlation function 〈ᾱ(t)ᾱ(0)〉 = σ2 exp(−|t|/τs) corresponding to the Gaussian-Markovian
process. In this case the Fokker–Planck operators for the excited state of each monomer has
the following form
Lm = τ
+ (x− xm)
where x = qω̃
β = ᾱ/
σ2 is a dimensionless generalized coordinate. Bearing in mind
Eqs.(49), the Fokker–Planck operators for the eigenstates |j〉 = |0〉,|ei〉 of the exciton Hamil-
tonian can be written by Eq.(9) where x0 = 0, xe1 = xe2 = xm(cos
4 θ+sin4 θ) and xe3 = 2xm.
The corresponding transition frequencies at the equilibrium nuclear coordinate of the ground
electronic state are defined by Eqs.(41) and (43).
Consider a homodimer complex consisting of identical molecules with Ω̄1 = Ω̄2 ≡ Ω̄ and
D1 = D2 ≡ D. For this case, using Eqs.(39), (40), (41), (43) and (44), we obtain θ = π/4,
-5 -4 -3 -2 -1 0 1 2 3 4 5
�"(
)  (x104 fs2)
Figure 7: Populations of the ground (dotted line), single- (solid line) and two-exciton (dashed
line) states of a homodimer complex after the completion of the pulse action as functions of
Φ”(ν) for J = −300 cm−1 (J < 0 - J-aggregate), Q′ = 2.9, tp0 = 10 fs, τs = 100 fs. The
partial relaxation and the total models - lines with and without hollow circles, respectively.
~Ω̄e1,2 = ~(Ω̄± J), ~Ω̄e3 = 2~Ω̄ (52)
De1 = De1e3 =
2D, De2 = De2e3 = 0
We thus need to consider only three states: |0〉, |e1〉 and |e3〉, since state |e2〉 is not excited
with light. Letting |1〉, |2〉 and |3〉 represent |0〉, |e1〉 and |e3〉, respectively, we arrive at
a three-state system considered above where ω21 = Ω̄ + J, ω31 = 2Ω̄, D21 = D32 =
x1 = 0, x2 =
xm, x3 = 2xm.
Fig.7 shows populations of single and two-exciton states after the excitation with a linear
chirped pulse, Eqs.(15) and (16), as functions of Φ′′(ν). Here the one-photon resonance for
Franck-Condon transition 1 → 2 occurs at the pulse maximum, i.e. ω = ω21 = Ω̄ + J , and
the Stokes shift of the equilibrium absorption and luminescence spectra for each monomer
is equal to ωmonst = 400 cm
−1. Fig.7 also contrasts calculations using the total model (lines
without hollow circles) with those of the partial relaxation model when only diagonal matrix
elements of the density matrix undergo diffusion (lines with hollow circles). Fig.7 shows a
good agreement between the calculation results for both models.
Furthermore, one can see strong suppressing the population of the two-exciton state for
negatively chirped (NC) pulse excitation. As a matter of fact, one can suppress or enhance
two-exciton processes using positively or NC pulses. Our calculations (see table below) show
twofold benefits of NC pulse excitation (Φ′′ = −104 fs2) with respect to the transform limited
pulse (Φ′′ = 0) of the same duration (tp = 71 fs) and energy tuned to one-exciton transition:
the population transfer to the single exciton state is larger, and that to the two-exciton state
is smaller.
Populations after the
completion of pulse action
Transform limited
pulse (Φ′′ = 0, tp = 71fs)
NC pulse (Φ′′ = −104fs2, tp = 71fs)
n2 0.317 0.573
n3 0.208 0.057
It is worthy to note good selective properties of chirped pulses, bearing in mind strong
overlapping Franck-Condon transitions 1 → 2, ω21, and 2 → 3, ω32. Really, the corresponding
frequencies differ by ω32 − ω21 = −2J − 34ω
st for the model under consideration that
comes to ω32 − ω21 = 300 cm−1 for the used values of parameters. On the other hand,
the bandwidth of the absorption spectrum at half maximum for transition 2 → 3 comes to
∆ω = 2
2 ln 2σ232s ≈ 1024 cm−1 that is larger than ω32 − ω21. Here σ232s = (~β)−1ω23st is
the LF vibration contribution to a second central moment of an absorption spectrum for
transition 2 → 3 and ω23st = (~β)−1(x3 − x2)2 = 94ω
st is the corresponding Stokes shift.
This issue can be understandable in terms of the competition between sequential and
direct paths in a two-photon transition [36]. Consider a three-level atomic ladder system
in the absence of relaxation with close transition frequencies ω21 ≈ ω32 where ω21 can be
associated with one-exciton excitation and frequency ω31 - with two-exciton excitation.The
system is affected by one phase modulated pulse of carrier frequency ω, Eqs.(2), (15) and
(16). In the Appendix we have calculated the excited-state amplitude a3 due to two-photon
transition 1 → 3 involving a nearly resonant intermediate level 2 for such system. Amplitude
a3 = aTP +aS consists of two contributions. The first one aTP corresponds exactly to that of
the nonresonant two-photon transition. This contribution aTP ∼ 1/|Φ′′(ω)|, and it is small
for strongly chirped pulses [52]
2|Φ′′(ω)| ≫ τ 2p0 (53)
This result has a clear physical meaning. The point is that the phase structure (chirp) of the
pulse determines the temporal ordering of its different frequency components. For a strongly
chirped pulse when a pulse duration is much larger than that of the corresponding transform-
limited one, one can ascribe to different instants of time the corresponding frequencies [52].
As a matter of fact, in the case under consideration different frequency components of the
field are determined via values of the instantaneous pulse frequency ω(t) for different instants
of time. Therefore, only a small part of the whole pulse spectrum directly excites the two-
photon resonance.
The second contribution is given by [36]
aS = −
D32D21π
E(ω21)E(ω32){1− sgn[(ω21 − ω32)Φ′′ (ω)]} (54)
where E(ω̃) is the Fourier transform of the positive frequency components of the field ampli-
tude E (t) exp[iϕi (t)]. The consideration of the Appendix enables us to extend the results of
Ref.[36] to non-zero two-photon detuning Ω2 = ω31 − 2ω 6= 0. Eq.(54) describes a sequential
process, the contribution of which is a steplike function. This process can be suppressed
when the pulse frequencies arrive in counter-intuitive order (ω32 before ω21) that occurs in
our simulations of a J-aggregate for NC excitation. Fig.7 and the table above show that the
selective properties of chirped pulses under discussion are conserved on strong field excita-
tion and for broad transitions. The selective excitation of single and two-exciton states can
be used for preparation of initial states for nonlinear spectroscopy based on pulse shaping
[60, 61].
6 Strong interaction and STIRAP
The three-state system under discussion enables us to consider STIRAP as well. STIRAP in
molecules in solution was studied in Refs.[29], where the solvent fluctuations were represented
as a Gaussian random process, and in Ref.[30], where the system-bath coupling was taken to
be weak in the sense that the relaxation times were long in comparison to the bath correlation
time, τc. Intense fields were shown in Ref.[30] to effectively slow down the dephasing when the
energetic distance between the dressed (adiabatic) states exceeds 1/τc. The point of the last
paper is that in contrast to usual undressed states, which intersect, the dressed (adiabatic)
states do not intersect. Therefore, the spectral density of the relaxation induced noise,
which has a maximum at zero frequency, strongly diminishes for frequencies corresponding
to the light-induced gap between dressed states, resulting in suppressing pure dephasing
between the dressed states. In this section we show that this conclusion holds also for non-
Markovian relaxation when the system-bath interaction is not weak and, therefore, can not
be characterized only by τc.
In the rotating wave approximation the Schrödinger equations for STIRAP in Λ-configuration
can be written as follows
U ′1 −~Ω1/2 0
−~Ω1/2 U2 −~Ω2/2
0 −~Ω2/2 U ′3
where U ′1 = U1 + ~ω1 and U
3 = U3 + ~ω2 are ”photonic replications” of effective parabolic
potentials U1(x) and U3(x) (Eq.(10)), respectively. We consider the two-photon resonance
condition when ω1 − ω2 = (E3 −E1)/~ and x1 = x3 = 0 that would appear reasonable when
|1〉 and |3〉 are different vibrational levels of the same electronic state. Then U ′1 = U ′3.
Adiabatic states Uad corresponding to Eq.(55) can be found by equation
U ′1 − Uad −~Ω1/2 0
−~Ω1/2 U2 − Uad −~Ω2/2
0 −~Ω2/2 U ′3 − Uad
This gives the following adiabatic states
Uad0 = U
1 = U
Uad± =
(U2 + U
(U2 − U ′1)2 + ~2(Ω21 + Ω22) (56)
One can see that initial U ′1 and final U
3 diabatic states coincide with one of adiabatis states
Uad0 . For strong interaction the last will be well separated from other adiabatic states U
± due
to avoided crossing. Therefore, during STIRAP the system will remain in the same adiabatic
state Uad0 , which is U
1 for t = −∞ and U ′3 for t = +∞. Its evolution due to relaxation
stimulated by LF vibrations can be described by the corresponding Fokker-Planck operator
Lad0 = L1,3 = τ
+ x ∂
describing diffusion in adiabatic potential Uad0 = U
1 = U
This means that during transition 1 → 3 the system motion along a generalized coordinate
x does not change. In other words, such a transition will not be accompanied by pure
dephasing. This conclusion is a generalization of the previous result [30] relative to slowing
down the dephasing in strong fields, which was obtained for weak system-bath interaction,
to non-Markovian relaxation.
7 Conclusion
In this work we have studied the influence of ESA and two-exciton processes on a coherent
population transfer with intense ultrashort chirped pulses in molecular systems in solution.
An unified treatment of ARP in such systems has been developed using a three-state elec-
tronic system with relaxation treated as a diffusion on electronic potential energy surfaces.
We believe that such a simple model properly describes the main relaxation processes related
to overdamped motions occurring in large molecules in solutions.
Our calculations show that even with fast relaxation of a higher singlet state Sn (n > 1)
back to S1, ESA has a profound effect on coherent population transfer in complex molecules
that necessitates a more accurate interpretation of the corresponding experimental data. In
the absence of Sn → S1 relaxation, the population of state |3〉, n3, strongly decreases when
the chirp rate in the frequency domain |Φ′′(ν)| increases. In order to appreciate the physical
mechanism for such behavior, an approach to the total model - the relaxation-free model -
was invoked. A comparison between the total model behavior and that of the relaxation-free
model has shown that relaxation is responsible for strong decreasing n3 as a function of
Φ′′(ν) in spite of meeting adiabatic criteria for both transitions 1 → 2 and 2 → 3 separately.
By this means usual criteria for ARP in a two-state system must be revised for a three-state
system.
To clarify this issue, we have developed a simple and physically clear model for ARP with
a linear chirped pulse in molecules with three electronic states in solution. The relaxation
effects were considered in the framework of the LZ calculations putting in a third level
generalized for random crossing of levels. The model has enabled us to obtain a simple
formula for n3, Eq.(33), which is in excellent agreement with numerical calculations. In
addition, the model gives us an extra criterion for coherent population transfer to those we
have obtained before for a two-state system [27]. New criterion, Eq.(34), implies conservation
of the “counter-movement” of the “photonic repetitions” of states 1 and 3, in spite of random
crossing of levels.
Furthermore, we also applied our model to a molecular dimer consisting from two-level
chromophores. A strong suppressing of two-exciton state population for NC pulse excitation
of a J-aggregate has been demonstrated. We have shown that one can suppress or enhance
two-exciton processes using positively or NC pulses. As a matter of fact, a method for
quantum control of two-exciton states has been proposed. Our calculations show good
selective properties of chirped pulses in spite of strong overlapping transitions related to the
excitation of single- and two-exciton states.
In the light of the limits [43, 44] imposed on Eqs.(6) and (8) for nondiagonal elements
of the density matrix for the total model, we used a semiclassical (Lax) approximation
(Eq.(14)) (the partial relaxation model). The latter offers a particular advantage over the
total model. The point is that the partial relaxation model can be derived not assuming
the standard adiabatic elimination of the momentum for the non-diagonal density matrix,
which is incorrect in the ”slow modulation” limit [42]. A good agreement between calcu-
lation results for the partial relaxation and the total models in the slow modulation limit
(see Figs.2 and 7) shows that a specific form of the relaxation term in the equations for
nondiagonal elements of the density matrix ρ̃12(x, t) and ρ̃23(x, t) is unimportant. By this
means the limits imposed on the last equation [43, 44] are of no practical importance for the
problem under consideration in the slow modulation limit. This issue can be explained as
follows. Our previous simulations [26] show that in spite of a quite different behavior of the
coherences (nondiagonal density matrix elements) for the partial relaxation and the total
models, their population wave packets ρjj (x, t) behave much like. Since we are interested
in the populations of the electronic states nj =
ρjj (x, t) dx only, which are integrals of
ρjj (x, t) over x, the distinctions between the two models under discussion become minimal.
In conclusion, we have also demonstrated slowing down the pure dephasing on STIRAP
in strong fields when the system-bath interaction is not weak (non-Markovian relaxation).
Acknowledgement
This work was supported by the Ministry of absorption of Israel.
Appendix
Consider a three-level system E1 < E2 < E3 with close transition frequencies ω21 ≈
ω32 where ω21 can be associated with a single-exciton excitation and frequency ω31 - with
two-exciton excitation. The system is affected by one phase modulated pulse of carrier
frequency ω, Eq.(2). The excited-state amplitude for a two-photon transition involving a
nearly resonant intermediate level, can be written as [62, 36]
a3 = −
D32D21
E(ω21)E(ω32) +
E(Ω + ω)E(Ω2 − Ω + ω)
Ω− (ω21 − ω)
where E(ω̃) is the Fourier transform of the positive frequency components of the field am-
plitude E (t) exp[iϕi (t)], Ω = ω̃ − ω, P is the principal Cauchy value, Ω2 = ω31 − 2ω is the
two-photon detuning. For linear chirped excitation, Eqs.(15) and (16), E(ω̃) is given by
E(ω̃) =
πE0τp0 exp{−
Ω2[τ 2p0/2− iΦ′′(ω)]} (58)
Using Eq.(58) and introducing a new variable z = Ω− Ω2/2, Eq.(57) can be written as
a3 = −
D32D21π
2(E0τp0)2
{exp[−1
(δ2 + (Ω2 − δ)2)(τ 2p0/2− iΦ′′(ω))]
+ exp[−
Ω22(τ
p0/2− iΦ′′(ω))]
exp[−z2(τ 2p0/2− iΦ′′(ω))]
z − (δ − Ω2/2)
} (59)
where δ = ω21−ω is one-photon detuning. The integral on the right-hand side of Eq.(59) can
be evaluated for strongly chirped pulses [52], Eq.(53), when a pulse duration is much larger
than that of the corresponding transform-limited one. In this case two frequency ranges give
main contributions to the integral. The first one results from the method of stationary phase
[63], and it is localized near the two-photon resonance z = ω − ω31/2 = 0 in the small range
∆ω ∼ 1/
|Φ′′(ω)|. In this case only a small part ∆ω ∼ 1/
|Φ′′ (ω) | of the whole pulse
spectrum ∆ωpulse = 4/τp0 directly excites the two-photon resonance, and the corresponding
contribution ∼ 1/
|Φ′′(ω)| is small due to Eq.(53).
The second contribution to the integral is located near z = δ−Ω2/2 and it is due to the
pole at the real axes. This contribution is given by Eq.(54) of Sec.5.
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	Introduction.
	Basic equations
	Approximate models
	System with frozen nuclear motion
	Semiclassical (Lax) approximation
	Adiabatic population transfer in the presence of excited-state absorption
	Population transfer between randomly fluctuating levels
	Influence of excited-state absorption when detuning from two-photon resonance occurs
	Population transfer in the presence of two-exciton processes. Selective excitation of single and two-exciton states with chirped pulses
	Strong interaction and STIRAP
	Conclusion
