Narrow transitions induced by broad band pulses  |g> |f> Loss of spectral resolution

Perturbation solution to Schrödinger Equation We solve the Schrödinger equation using perturbation theory. The time dependent Schrödinger equation: The interaction of the atom with the electric field

Perturbation solution to Schrödinger Equation In the perturbative regime we can write H as: Where is a varying parameter which characterizes the strength of the interaction. We now seek a solution to Schrödinger equation in the form: We require that all terms proportional to N satisfy the Schrödinger equation:

Perturbation solution to Schrödinger Equation We assume that initially the atom is in the ground state so the solution for the zero's order is: We represent the Nth order contribution to the wavefunction as:

Perturbation solution to Schrödinger Equation We get a set of equations: We multiply by u m (r) and integrate

Perturbation solution to Schrödinger Equation This equation relates the amplitude of the Nth order to the amplitude of the N-1 order by a time integration.

Perturbation solution to Schrödinger Equation First order: we include only one state -

Perturbation solution to Schrödinger Equation The transient absorption is dictated by all frequency components

Perturbation solution to Schrödinger Equation Second order:

Perturbation solution to Schrödinger Equation At t If  mg >>  0,  mg =  +  ’

Perturbation solution to Schrödinger Equation At t E(t) is real, therefore E(-  )= E(  )* If  mg <<  0,  mg =  -  ’

Perturbation solution to Schrödinger Equation IfIntermediate levels are far detuned

Two photon nonresonant transition All the paths are in phase Transform limited pulses maximize the two photon absorption Antisymmetric phase maintains the efficiency

Two photon nonresonant transition Transform limited pulses maximize the two photon absorption Spectral phase Temporal envelope

Nonresonant TPA Control Experimental results Antisymmetric phase has no effect on transition probability Specific spectral phase mask can annihilate the absorption rate  Selective excitation Step location/bandwidth Meshulach & Silberberg, Nature, 396, 239 (1998),Phys. Rev. A 60, 1287 (1999)

Controlling The Spectrum of E 2  Transformed limited pulse Shaped pulse 

Raman Transition Transform limited pulses maximize the transition rate Periodic phase functions maintain the efficiency

Raman Transition Spectral phase Temporal envelope

CARS spectroscopy t Modulated spectral phase function Fourier transform Ba(NO 3 ) 2 (1048 cm -1 ) Diamond (1333 cm -1 ) Toluene (788, 1001, 1210 cm -1 ) lexan Spectrocopy in the fingerprint region Reduced nonresonant background Spectral resolution ~ 30 cm -1, 70 times the pulse band width N. Dudovich, D. Oron and Y. Silberberg, J. Chem. Phys. 118, 9208 (2003).

Narrow transitions induced by broad band pulses: weak fields One photon transition: Nth photon transition Two photon transition Raman transition The transition is excited by a single frequency component of E N

Two Photon Resonant Transition If there is a single intermediate state: The transition is not maximized by a transform limited pulse There is a destructive interference between frequencies below and above the resonance On resonantOff resonant

Enhancement of resonant TPA amplitude shaping N. Dudovich, B. Dayan, S. M. Gallagher Faeder and Y. Silberberg, Phys. Rev. Lett., 86, 47 (2001) Higher Cutoff Wavelength [nm] Fluorescence Intensity [a.u.] 0.5 Pulse power [a.u.] Time [fs] Intensity [a.u.] I(t) Eliminate all frequency components that contribute destructively blocker

Enhancement of resonant TPA phase shaping Invert the sign around the resonance to induce constructive interference instead of destructive one  phase step N. Dudovich, B. Dayan, S. M. Gallagher Faeder and Y. Silberberg, Phys. Rev. Lett., 86, 47 (2001).

Two photon absorption Two degenerate non- interfering paths Angular momentum control

Px Angular momentum control E x E x transitions Two degenerate orthogonal states can be separately controlled ExEx ExEx ExEx E-E- PxP+ E x E + transitions P+P+ Px

Four wave mixing Assuming all intermediate levels are detuned,

Four wave mixing

11 22 33  11 22 33  11 22 33  The polarization is maximized by a transform limited pulse The response is instantaneous  =  1 +  2 -  3  =  1 -  2 +  3  = -  1 +  2 +  3

 Coherent Anti-Stokes Raman Scattering (CARS) In a CARS process a pump and a Stokes photon coherently excite a vibrational level. A probe photon interacts with the excited level to emit a signal photon. Large, directional and coherent signal (compare to Raman scattering). Attractive for microscopy applications -provides a vibrational imaging with 3D sectioning capability.

Four wave mixing Assuming all intermediate levels are detuned, including one resonant level:

Four wave mixing The response is not instantaneous – the nonlinear polarization can be enhanced

Four wave mixing Multiplex CARS We can use the broad pulse to pump and a narrow probe to map the excitation Can we probe with a broad band probe?

Four wave mixing If E(  ) is transform limited then:

Four wave mixing If there are several vibrational states: Loss of spectral resolution

Four wave mixing  g +  gate Resonance enhancement around  g +  g

Extracted Raman spectra Transform limited pulse Phase-shaped pulse The resolution is dictated by the phase gate width (25 cm -1 )

Narrow transitions induced by broad band pulses: weak fields One photon transition: Nth photon transition Two photon transition Raman transition The transition is excited by a single frequency component of E N

Strong field coherent control Two photon transition The transition depends on many orders of E N

Strong field coherent control The transition cannot be analyzed in a perturbative manner. We cannot ignore coupling to all other levels in the system. Adiabatic approach: If the transition rate is faster than the variation of the interaction we can find the new stationary states (dressed states) and then change them adiabatically with the laser field. Adaptive search of the optimal solution We have a high degree of control: we can shape the pulse using N free parameters.

Coherent control: Using shaped ultrashort pulses to control the reaction Can an ultrashort pulse cause a molecule to vibrate in such a way as to break the bond of our choice?

Strong field coherent control – adaptive algorithm

C O CH 3 C O + C O + Manipulating the dissociation yields in acetophenone Different pulse shapes can optimize different photo-fragments. Levis and coworkers Generation Ratio: C 7 H 5 O/C 6 H 5 Normalized ion intensity and ratio

The absolute phase Different absolute phases for a four-cycle pulse Different absolute phases for a single-cycle pulse With a pulse shaper we manipulate the envelope of the pulse, however for short pulses the absolute phase becomes important

Ultrashort pulses LL/v g When the group velocity is different than the phase velocity the absolute phase changes with time

Ultrashort pulses

If a pulse has more than one octave, it has both f and 2 f for some frequency, f. Interfering them in a SHG crystal yields two contributions at 2 f : that from the original beam and the SH of f. Simply measuring the spectrum is performing spectral interferometry yields a fringe phase: Stabilizing the absolute phase