II. THREE-PULSE MULTIDIMENSIONAL FEMTOSECOND OPTICAL SPECTROSCOPIES
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354
Piryatinski et al.
where the total ﬁeld E r, t consists of three applied ﬁelds:
3
E r,
D
[Ej
exp ikj r
iωj
C EŁj
ikj r C iωj ]
exp
(2)
jD1
where ωj , kj , and Ej is the frequency, wave vector, and envelope of the jth
ﬁeld, respectively. Even though a polypeptide is typically small compared
to the optical wavelength, the sample is larger than the wavelength, and
the optical signal is subjected to the phase-matching condition (18). This
implies that the four-wave mixing signal may only be generated in the
directions deﬁned by the wave vectors ks D šk1 š k2 š k3 . For resonant
techniques the signal is dominated by contributions that satisfy the rotating
wave approximation (RWA). All other terms are highly oscillatory and may
be neglected. The three columns in Fig. 1(B) show the Feynman diagrams
corresponding to the three wavevectors which survive the RWA for the
present model.
We next survey the possible detection modes in a three pulse
experiment. The excitation pulses come at times 1 D t t1 t2 t3 ,
t2 t3 , and 3 D t t3 , where t1 and t2 are delay times between
2 Dt
the pulses and t3 is the delay time between the last excitation pulse and
actual time when the polarization is measured (Fig. 1A). Equation (1) can
be recast in the following form:
1
P3 t D
1
dt3
0
ðE t
1
dt2
0
t2
dt1 R t3 , t2 , t1 E t
t3 E t
3, 2, 1
t2
t3
t3
where R t3 , t2 , t1 Á 6R t, t t3 , t
order response function is given by
R t,
t1
0
Á i3 TrfPˆ t [Pˆ
3
3
t2
t3 , t
, [Pˆ
2
, [Pˆ
t1
1
t2
t3 . The third-
, ]]]g,
(4)
and
R t3 , t2 , t1 Á i3 TrfPˆ t3 C t2 C t1 [Pˆ t2 C t1 , [Pˆ t1 , [Pˆ 0 , ]]]g
(5)
where Pˆ
are the polarization operators in the Heisenberg representation
and is the equilibrium density matrix.
Eq. (5) has eight terms coming from the three commutators. For each
choice of a wavevector, only some of these terms survive the RWA. Let us
consider an impulsive technique involving short pulses and denote the relevant terms of R by R0 . Various detection modes that probe different projections of the third-order response function R t3 , t2 , t1 may be employed. The
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IR Spectroscopy in Polypeptides
355
time-integrated homodyne signal is given by
Shom t2 , t1 D
dt3 jR0 t3 , t2 , t1 j2
(6)
The time-gated homodyne signal is given by
STG t3 , t2 , t1 D jR0 t3 , t2 , t1 j2
(7)
Frequency dispersed homodyne detection gives
SFD ω3 , t2 , t1 D jR ω3 , t2 , t1 j2
(8)
where
R ω3 , t2 , t1 D
dt3 R0 t3 , t2 , t1 exp iω3 t3
(9)
The time-gated heterodyne signal obtained in a phase-locked measurement
gives the real and the imaginary parts of the response function:
SR t3 , t2 , t1 D Re R0 t3 , t2 , t1 ,
SI t3 , t2 , t1 D Im R0 t3 , t2 , t1 .
10
The most detailed detection is given by the Wigner spectrogram, which is
bilinear in the response function (45–49):
SRAM ω3 ; t3 , t2 , t1 D
d R0 t3 C , t2 , t1 R0 t3
, t2 , t1 exp 2iω3
11
The time-gated (frequency-dispersed) signals are obtained by integrating
the spectrogram over frequency (time), i.e.
STG t3 , t2 , t1 D
dt3 SRAM ω3 ; t3 , t2 , t1
(12)
and
SFD ω3 , t2 , t1 D
dt3 SRAM ω3 ; t3 , t2 , t1
(13)
In the next section we present a closed form expression for the vibrational
response function using the Frenkel-exciton model.
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356
Piryatinski et al.
III. THE THIRD-ORDER RESPONSE OF VIBRATIONAL
EXCITONS
We model the amide band as a system of N interacting localized vibrations.
For the sake of third-order spectroscopies, we only need to consider the
lowest three levels of each peptide group with energies 0, m , 0m m D
1, . . . , N . The matrix elements of the dipole operator corresponding to
the 0-1 and 1-2 transitions are denoted m and 0m , respectively, and their
ratio is Äm Á 0m / m . To introduce the vibrational Frenkel exciton model,
we deﬁne the exciton-oscillator operators (17,39,50):
Bˆ †n D j1inn h0j C Än j2inn h1j
with the commutation relations
2 ˆ† ˆ
ˆ †n ] D υmn [1
ˆ m, B
2 Äm
Bm Bm ]
[B
(14)
(15)
The polarization operator that describes the coupling to the driving ﬁeld
E t is then given by
Pˆ D
n
ˆ †n
Bˆ n C B
(16)
n
The Hamiltonian can be represented in the form (17,39,50)
gn † 2
ˆ †m Bˆ n C
hmn B
HD
Bˆ n Bˆ n 2 C Hb E t Pˆ
2
mn
n
(17)
where hmn D υmn m C Jmn , with Jmn being the hopping matrix, and gm Á
2
2Äm2 [m C 2 Äm
m ] is an exciton-exciton interaction energy, where
0
m Á m 2 m is the vibrational anharmonicity. This Hamiltonian
describes excitons as oscillator (quasiparticle) degrees of freedom.Ł Hb
represents a bath Hamiltonian. We shall not specify it and merely require
that it conserves the number of excitons. The bath induces relaxation
kernels. The structure of the ﬁnal expression is independent of the speciﬁc
properties of the bath; the latter only affects the microscopic expression for
the relaxation kernels (51).
The response function has been calculated in Ref. 39. It depends on
the following three Green functions:
Gmn
Ł
ÁÂ
TrfBˆ m
Bˆ †n 0 g
18
Both the commutation relations [Equation (15)] and the Hamiltonian
[Equation (17)] contain higher order products of Bˆ †n and Bˆ n . These higher terms
do not contribute to the third-order optical response and were neglected.
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IR Spectroscopy in Polypeptides
2
Gmn,kl
ÁÂ
p
Gmk,ln
ÁÂ
TrfBˆ m
Bˆ n
TrfBˆ m
Bˆ †k
357
Bˆ †k 0 Bˆ †l 0 g
19
g
20
Bˆ †n
0 Bˆ l 0
ˆ [Bˆ † ] are the annihilawhere is the equilibrium density matrix, B
tion [creation] operators in the Heisenberg representation, and Â
is the
Heavyside step function (Â D 0 for < 0 and Â D 1 for > 0).
Gmn
is the one-exciton Green function. G 2 represents the time
evolution of the two exciton states and satisﬁes the Bethe-Salpeter equation:
00
2
D Gmk
Gmn,kl
C Gml
Gnl
C
Gnk
d
pqp0 q0
00
Gmp
00
Gnq
pq,p0 q0
00
0
00
d
Gp0 k
0
0
0
0
Gg0 l
0
where is the exciton-exciton scattering matrix. The response function will
be expressed in terms of rather than G 2 . The third Green function G p
represents the evolution of the exciton density matrix hBC
n Bm i (populations
and coherences). We have recast it in the form:
p
G†nl
Gmk,ln D Gmk
C Gmn,kl
The ﬁrst term represents the factorized evolution in terms of exciton amplitude hBC Bi D hBC ihBi. G is the irreducible (unfactorized) part of the Green
function. The response function will be expressed in terms of G rather
than G p .
In summary, the response function depends on the exciton Green
function G, the exciton-exciton scattering matrix , and the irreducible
part G of the density matrix Green function. It has two components:
R t,
3, 2, 1
D Rc t,
3, 2, 1
C Ri t,
3, 2, 1
(21)
The coherent part is
Rc t;
3, 2, 1
D
1
i3
n
m1
m2
m3
2
G†m3 m00
perm
ð
1
1
d 0 Gn0 m1
ð n00 m00 ,n0 m0
00
0
1
0
Gm 0 m 2
C c.c.
Copyright © 2001 by Taylor & Francis Group, LLC
0
1
00
d
00
3
Gnn00 t
00
22