Tải bản đầy đủ - 0 (trang)
II. THREE-PULSE MULTIDIMENSIONAL FEMTOSECOND OPTICAL SPECTROSCOPIES

II. THREE-PULSE MULTIDIMENSIONAL FEMTOSECOND OPTICAL SPECTROSCOPIES

Tải bản đầy đủ - 0trang

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



Copyright © 2001 by Taylor & Francis Group, LLC



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.



Copyright © 2001 by Taylor & Francis Group, LLC



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.



Copyright © 2001 by Taylor & Francis Group, LLC



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 ( D 0 for < 0 and  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



Tài liệu bạn tìm kiếm đã sẵn sàng tải về

II. THREE-PULSE MULTIDIMENSIONAL FEMTOSECOND OPTICAL SPECTROSCOPIES

Tải bản đầy đủ ngay(0 tr)

×