Nonlinear wavepacket interferometry for polyatomic molecules

N. F.

 

Scherer

,

A. J.

 

Ruggiero

,

M.

 

Du

, and

G. R.

 

Fleming

,  , ();

N. F.

Scherer

,

R. J.

Carlson

,

A.

Matro

,

M.

Du

,

A. J.

Ruggiero

,

V.

Romero-Rochin

,

J. A.

Cina

,

G. R.

Fleming

, and

S. A.

Rice

, , ().

N. F.

Scherer

,

A.

Matro

,

L. D.

Ziegler

,

M.

Du

,

R. J.

Carlson

,

J. A.

Cina

, and

G. R.

Fleming

, , ().

See also

H.

 

Metiu

and

V.

 

Engel

,  , ();

R.

Bavli

,

V.

Engel

, and

H.

Metiu

, , ().

See E. J. Heller's chapter in Advances in Classical Trajectory Methods, edited by W. L. Hase (JAI Press, Greenwich, 1992), Vol. I, p. 165.

See Secs. III.B, IV.D, and V of Ref. 2.

J. A.

Cina

and

R. A.

Harris

, , ().

J. A.

Cina

and

R. A.

Harris

, , ().

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), Vol. I, Complement $Gv.$

D. W.

Pratt

, , ().

D. J.

Tannor

and

S. A.

Rice

, , ().

A. H.

Zewail

, , ().

D.

Leibfried

,

T.

Pfau

, and

C.

Monroe

, 1998, , and references therein.

D. T.

 

Smithey

,

M.

 

Beck

,

M. G.

 

Raymer

, and

A.

 

Faridani

,  , ();

K.

Vogel

and

H.

Risken

, , ().

M. G. Raymer, A. C. Funk, and D. F. McAlister, in Quantum Communication, Computing, and Measurement 2, edited by P. Kumar, G. Mauro D’Ariano, and O. Hirota (Kluwer Academic/Plenum, New York, 2000), p. 147.

T. C.

Weinacht

,

J.

Ahn

, and

P. H.

Bucksbaum

, , ().

T. J.

Dunn

,

I. A.

Walmsley

, and

S.

Mukamel

, , ().

L. W.

Ungar

and

J. A.

Cina

, , ().

M.

Cho

,

N. F.

Scherer

,

G. R.

Fleming

, and

S.

Mukamel

, , ().

W. P.

de Boeij

,

M. S.

Pschenichnikov

, and

D. A.

Wiersma

, , ().

Shaul Mukamel's comprehensive book, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995), gives theoretical treatments of a wide variety of nonlinear optical methods for molecular spectroscopy and details some of their added capabilities relative to linear measurements. Of particular relevance here are Chaps. 5, 10, and 11.

If the pulses are so short (less than a few optical cycles) and the system dynamics so rapid (competitive with the electronic transition frequency) that the rotating wave approximation breaks down, one could turn to an alternative approach to electronic interference spectroscopy that has recently been developed; see

A. W.

Albrecht

,

J. D.

Hybl

,

S. M.

Gallagher Faeder

, and

D. M.

Jonas

, , (). In that work a method for clocking intrapulse-pair delays by spectral interferometry rather than controlling them, as in the original phase-locking scheme of Scherer and co-workers (Refs. 1 and 2), is demonstrated. This new technique can operate even when the undersampling rate, $(1/2π)ΩL,$ of conventional phase locking is inadequate.

The rotating wave approximation—like the similarly motivated Born–Oppenheimer approximation—can be surprisingly resilient, however. In the theoretical studies of Ref. 23 we looked for but did not find significant deviations from the rotating wave approximation for pulses as short as 2.8 fs intensity-FWHM resonant with the $B←X$ transition of $I2$⁠.

Y.-C.

Shen

and

J. A.

Cina

, , ().

J.

 

Cao

and

K. R.

 

Wilson

,  , ();

J.

Cao

,

J.

Che

, and

K. R.

Wilson

, , ().

S.

Dilthey

,

S.

Hahn

, and

G.

Stock

, , ().

The discrimination against the A- and B-term three-pulse reference packets relative to those of the C- and D-terms could be made to operate in the opposite way by shifting the carrier frequency for pulses 3 and 4 well to the red of the absorption maximum.

We should be aware, however, that this limit begins to militate against spectral elimination of the A- and B-term overlaps.

We use Gaussian envelopes $Aj(t)=Ej exp(−t2/2s2)$ for pulses 3 and 4, where s is the pulse duration, and phase functions $Φ3=φ$ and $Φ4=φ+φd+Ωdtd.$ $φd$ is the relative optical phase between pulses 3 and 4 at the locking frequency $Ωd,$ and φ is an arbitrary overall phase for those two pulses, which does not affect the interference signal [see Eq. (2)].

W. M.

Kwok

and

D. L.

Phillips

, , ().

See also

D. L.

Phillips

, , ().

J.

Zhang

.

E. J.

Heller

,

D.

Huber

,

D. G.

Imre

, and

D. J.

Tannor

, , ().

S. E.

Novick

,

S. J.

Harris

,

K. C.

Janda

, and

W.

Klemperer

, , ().

T. Humble and J. Cina (work in progress).

Shen and Cina (Ref. 23) recently obtained approximate closed-form expressions for the effect of frequency-chirp on the position and momentum moments of a short-pulse excited wavepacket that will be useful in shaping the preparation pulses.

V.

Blanchet

,

M. A.

Bouchène

, and

B.

Girard

, , ().

Equations (3.8) and Eq. (3.13) of Ref. 2 give the interference contribution to photon absorption in terms of the vibronic transition energies and Franck–Condon factors; these equations would apply to either fluorescence excitation or transmission loss.

The essential equivalence of fluorescence excitation and transmission loss as measures of the interference contribution to photon absorption has been emphasized recently by Ref. 21. That paper is not entirely accurate in portraying the findings of Refs. 1 and 2, however. It ascribes to that work an assertion that phase-locked absorption experiments measure contributions to the optical free-induction decay from “nonresonant, virtually excited transitions with Bohr frequencies which may lie either inside or outside the pulse spectrum” (Ref. 21). But the existence of such contributions to the optical free induction decay induced by the first pulse in a linear wavepacket interferometry experiment is easily disproved, as only vibronic transitions within the pulse spectrum make persisting contributions to the pulse-induced dipole moment. Reference 2 emphasized that while in-phase and in-quadrature interference signals obtained with pulses short compared to the inverse absorption linewidth can be combined to yield the complete $e←g$ linear response function [as in Eq. (3.16) of Ref. 2] this is not strictly possible for longer pulses: “If the laser pulses are not arbitrarily abrupt, one can no longer write down an expression relating $χ(td)$ to the interference population; the finite spectral range of the pulses prohibits the complete determination of the system's response.” The claim inaccurately attributed to the original analysis would also contradict expressions (3.24) and (3.23) of paper 2 for the experimentally derived dispersive and absorptive susceptibility components, both of which contain pulse-envelope-modulated vibronic transition moments. That these two quantities are related by a (formally reciprocal) Kramers–Kronig transformation is evident from their parallel structure with the complete $e←g$ susceptibilities [(3.20) and (3.19) or Ref. 2]. Each of the susceptibility components assembled from the measured phase-locked transients according to the prescriptions (3.24) and (3.23) has the same form as the corresponding component of the full susceptibility, but with envelope-modulated vibronic transition moments replacing the ordinary Franck–Condon factors.

I.

 

Pastirk

,

E. J.

 

Brown

,

B. I.

 

Grimberg

,

V. V.

 

Lozovoy

, and

M.

 

Dantus

,  , ();

E. J.

 

Brown

,

Q.

 

Zhang

, and

M.

 

Dantus

,  , ();

and

I.

 

Pastirk

,

V. V.

 

Lozovoy

,

B. I.

 

Grimberg

,

E. J.

 

Brown

, and

M.

 

Dantus

,  , ().

The effects of pulse timing on homodyne-detected degenerate four-wave mixing signals—proportional to the square of the phase-matched component of the third-order induced polarization—are detailed by V. V. Lozovoy, I. Pastirk, E. J. Brown, B. I. Grimberg, and M. Dantus, Int. Rev. Phys. Chem. (in press).

The A–D overlaps in the present theory correspond in simple ways to the $R1–R4$ nonlinear response functions, respectively, that contribute to the third-order polarization in Mukamel's analysis (Ref. 20). Despite the widespread application of this useful formalism, the possibility to spectrally select C- and D-type terms (corresponding to $R3$ and $R4)$ over A- and B-terms $(R1$ and $R2)$⁠, or vice versa, does not seem to have been noted previously. But some spectral-selection arguments are made in Refs. 38 (see Fig. 5 of the last article listed, for example).

The arguments in the Appendix are reminiscent of conditions for vibrational mode suppression in photo-echo experiments identified by

C. J.

Bardeen

and

C. V.

Shank

, , ().

It is worth noting, however, that if nonlinear wavepacket interferometry experiments were used to monitor excited-state intramolecular vibrational wavefunctions of solvated species, inhomogeneous broadening could help suppress the D- (and A-) terms relative to C ( and B); see Chap. 10 of Ref. 20.

See also J. A. Cina and R. A. Harris, Ultrafast Phenomena IX, edited by W. Knox and P. Barbara (Springer-Verlag, Berlin, 1994), p. 486;

C. S.

 

Maierle

and

R. A.

 

Harris

,  , ().

For an up-to-date survey of work addressing the preparation and measurement of molecular superposition states, see C. S. Maierle, Ph.D. dissertation, University of California at Berkeley, 1999.

Forthcoming numerical calculations by R. P. Duarte-Zamorano and V. Romero-Rochín explore the effects of nonzero pulse duration in the model of Ref. 6 and quantify the limits of validity of various approximations used in that study.

J.A.

Cina

, , (), and references therein.