Quantum Molecular Dynamics                        Winter  2002 

Project 4
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I. Stimulated Raman Adiabatic Passage (STIRAP)

Program name:
stirap.m

Stimulated Raman Adibatic Passage (STIRAP)
What program does:
Calculates the evolution of a wavefunction in a 3-level system under the
influence of a Stimulated Raman Adiabatic Passage (STIRAP) mechanism (the Stokes pulse
precedes the pump pulse).  The Hamiltonian is given by eq. 11.38 in the text.

The program calculates this by two different methods:
(i) via solution of the time-dependent Schrodinger equation in differential
form;
(ii) via solution of the time-dependent Schrodinger equation in integral form,
\psi (t+\Delta t) = e^{-iHt/\hbar} \psi(t).  This propagator is calculated by
diagonalizing the Hamiltonian (including the instantaneous field).
This numerical method is based on an adiabatic assumption, i.e. that the field
is changing slowly.

Assignments:
a.  Make the field weaker and weaker.  Is there a point at which the
adiabatic propagation method fails?  Is there a point at which the STIRAP
mechanism fails?  Check this against the formula for the validity of the
adiabatic assumption, $\Omega \tau \ge 1$, where 
$\Omega = (\Oemga_S^2 + \Omega_P^2)^{1/2}$, and $\tau$ is the time for the
entire STIRAP process.

References:
Section 11.3, Bergmann et al., Rev. Mod. Phys. 70, 1003 (1998).
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II. Local-in-Time Optimization

Program name:
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Assignment:
write programs to:

a. Calculate the STIRAP pulse sequence by the local optimization method.
Specifically, use program (i) in stirap.m, but instead of using a
predetermined \Omega_P, \Omega_S, let the program calculate 
Omega_P, \Omega_S, at each time step using eqs. 11.44-11.45, i.e. choosing
the pulses such that at each instant in time the population in level 2 is held 
constant by an instantaneous adjustment of the intensity of the pump and Stokes pulses.

b. Calculate a pulse sequence to produce vibrational "excitation without demolition".
Start with program rabi.m (see supplementary material). 
Adjusting the phase of the light instantaneously to prevent excited electronic state
population transfer, while choosing the amplitude to increase the ground state
energy content.

In both part a. and b. you will need to start with a 'seed' pulse, a weak 
resonant pulse of any sort to get the process going.

References:
Section 11.5, Malinovsky and Tannor, Phys. Rev. A 56, 4929 (1997).
Kosloff et al., Phys. Rev. Lett. 69, 2172 (1992).
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Optimization via Optimal Control Theory (OCT)

Program name:
optimize.m

What program does:
a. Calculates an optimal pulse sequence to transfer the population from level
1 to level 3 in a three level system, using the Krotov method.  The optimization 
algorithm represents an iterative solution of eqs. 12.8-12.12 until self-consistency 
is achieved.

Assignment:
(i) Study the dependence of the final yield as a function of iteration number.
(ii) Change the penalty function, \lambda, and see how this affects (i).
(iii) try a variety of intial guesses and see how this affects (i).

References:
Section 12.2, Peirce et al. Phys. Rev. A37, 4950 (1988);
Tannor et al., in Time Dependent Quantum Molecular Dynamics, Plenum, 1992.
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First, Second, and Third order Perturbation Theory with Femtosecond Pulses

Program names:
(a) i2_interfere.m
(b) i2_pump_probe.m
(c) i2_cars.m

What programs do:
a. Calculates the first order amplitude on the excited electronic state,
for an arbitrary shaped excitation pulse.
b. Calculates the second order amplitude on the ground electronic state,
for an arbitrary shaped excitation (pump) and stimulation (dump) pulse.
c. Calculates the third order amplitude on the excited electronic state,
for an arbitrary shaped excitation (pump) and stimulation (dump) pulse,
and reexcitation (probe) pulse.

Assignment:
(a) use i2_interfere.m to check the dependence of wavepacket interferometry
on the three parameters we discussed in class, i.e. frequency of the carrier
frequency, relative phase of the two pulses, and time delay.
(b) use i2_pump_probe.m to explore the dependence of the second order
wavefunction on the time delay and frequency of the second pulse.  Map out
the total population and the vibrational level that is most population as a 
function of these two parameters.
(c) calculate the overlap of the third order amplitude with the ground
vibrational state as a function of time for different choices of time delay
between pulses 1 and 2, and 2 and 3.

References in text:
Section 9.1, 9.2
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Chirped Pulses

i2_chirp.m
What program does:
a. Calculates the first and second order amplitude on the excited and ground
electronic states, respectively, for chirped excitation.

Assignment:
a. Start with a weak field.  Investigate how positive vs. negative chirp
affects the shape of the excited state wavepacket in the vicinity of the
first encounter with the outer turning point.
b. Change to a strong field.  Investigate how positive vs. negative chirp
affects the total population on the excited vs. the ground state.

References:
weak fields:  Wilson et al.
Strong fields: Ruhman and Kosloff, Cao and Wilson
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First and Second Order Pertubation Theory with Phase Masks.

program name:
mask.m

What program does:
Calculates the first and second order amplitude in a two level system
where the complete freedom of the phase profile is used.

Assignment:
a) Show that if the phase profile is antisymmetric, centered at a
frequency which is 1/2 the resonance transition frequency, there two photon
transition is identical to that produced by a transformed limited pulse
(0 phase everywhere)  for all such asymmetric phase profiles.  
b) Show that if the phase profile is symmetric, of the form a cos (b \Omega +
\phi), there are zeros in the two photon absorption as a function of a.

What program does:
Calculates the first and second order amplitude in a three level system
where the complete freedom of the phase profile is used.

a) Show that by inverting the phase of the frequencies above and below the 
resonance, that it is possible to increase the two-photon absorption by
more than a factor of five over a transform limited pulse.  Give a
theoretical explanation for this result.

References:
Two-photon dark states: Meshulach and Silberberg, Nature, 396, 239 (1998); 
Two-photon absorption in the present of a resonance, Dudovich et al, Phys.
Rev. Lett. 86, 47 (2001).
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