We apply a linearly polarized laser pulse of the form

*E* (

*t*) =

(

*t*)e

^{i[ (}^{t}^{)}^{t}^{+(}^{t}^{)]} to a simple two-level system with a |1

→ |2

transition, where |1

and |2

represent the ground and excited eigenlevels, respectively, of the field-free Hamiltonian. The laser carrier frequency or the centre frequency for pulsed lasers is ω. We have

(

*t*) and

(

*t*) as the instantaneous amplitude and phase. We can define the rate of change of the instantaneous phase,

, as the frequency sweep. If we expand the instantaneous phase function of

*E* (

*t*) as a Taylor series with constants

*b*_{n} , we have

To our knowledge, this approach of formulating a general expression for the continuously frequency modulated (‘chirped’) pulses is the first of its kind [

2,

5]. Establishing this generalization enables us to treat all possible chirped pulse cases by exploring the effects of each of the terms in

equation (1) initially for a simple two-level system and then extend it to the multilevel situation for a model five-level system of the anthracene molecule, which has been previously investigated with complicated shaped pulses [

4]. We use a density matrix approach by numerically integrating the Liouville equation

for a Hamiltonian in the rotating frequency modulated (FM) frame of reference. (

*t*) is a 2 × 2 density matrix whose diagonal elements represent populations in the ground and excited states and off-diagonal elements represent the coherent superposition of states. The Hamiltonian for the simple case of a two-level system under the influence of an applied laser field can be written in the FM frame for the

*N* -photon transition [

7] as

The time derivative of the phase function,

, appears as an additional resonance offset over and above the time-independent detuning Δ = ω

_{R} −

*N*ω, while the direction of the field in the orthogonal plane remains fixed. We define the multiphoton Rabi frequencies as complex conjugate pairs: Ω

_{1}(

*t*) =

*k*(μ

_{eff} (

*t*))

^{N}
*/* and

, where

*k* is a proportionality constant having dimensions of (energy)

^{(1−}^{N}^{)}, which in SI units would be joule

^{(1−}^{N}^{)}. For the |1

→ |2

transition, ω

_{R} = ω

_{2} − ω

_{1} is the single-photon resonance frequency. We have assumed that the transient dipole moments of the individual intermediate virtual states in the multiphoton ladder result in an effective transition dipole moment,

, which is a product of the individual

*N* virtual state dipole moments, μ

_{N} (i.e.,

). This approximation is particularly valid when intermediate virtual level dynamics for multiphoton interaction can be neglected [

6,

7].

Let us extend the two-level formalism to the multilevel situation involving IVR. In the conventional zeroth-order description of intramolecular dynamics, the system can be factored into an excited state that is radiatively coupled to the ground state, and nonradiatively to other bath states that are optically inactive (). These ‘dark’ states have no radiative transition moment for going from the ground state as determined by optical selection rules. They can belong to very different vibrational modes in the same electronic state as the ‘bright’ state, or to different electronic manifolds. These dark states can be coupled to the bright state through anharmonic or vibronic couplings. Energy flows through these couplings and the apparent bright state population disappears. Equivalently, the oscillator strength is distributed among many eigenstates. The general multilevel Hamiltonian in the FM frame for an

*N* -photon transition (

*N* 1), expressed in the zero-order basis set, is

where Ω

_{1}(

*t*) (and its complex conjugate,

) is the transition matrix element expressed in Rabi frequency units, for the transition between the ground state |0

and the excited state |1

. The background levels |2

*,* |3

*, . . .* are coupled to |1

through the matrix elements

*V*_{12},

*V*_{23} etc. Both the Rabi frequency Ω

_{1}(

*t*) and the detuning frequency

are time dependent (the time dependence is completely controlled by the experimenter). In general, the applied field would couple some of the dark states together, or would couple |1

to dark states, and thus the

*V*_{i j} terms would have both an intramolecular, time-independent component and a field-dependent component. As an alternative to

equation (2), the excited states’ submatrix containing the bright state |1

and the bath states |2

, |3

*, . . .* can be diagonalized to give the eigenstate representation containing a set of Δ′

_{i} as diagonal elements and the corresponding Ω′

_{i} as off-diagonal elements. The eigenvalues of such a time-dependent Hamiltonian representation are often referred to as the dressed states of the system. Such a representation corresponds closely to what is observed in conventional absorption spectroscopy. As long as the intensity of the field is very low (|Ω′

_{i}|《Δ′

_{i}) the oscillator strength going from the ground state (and hence the intensity of the transition, which is proportional to |Ω′

_{i}|

^{2}) is distributed over the eigenstates, and the spectrum mirrors the distribution of the dipole moment. On the other hand, a pulsed excitation creates a coherent superposition of the eigenstates within the pulse bandwidth. Physically, in fact, the presence of the dark states has been key to the loss of selectivity of excitation to a specified bright state.

Another common situation with short pulses is a ladder excitation situation where the individual excited states undergo dephasing through a coupled energy structure with states |0

, |1

, |2

etc in the zero-order basis as shown in . Such a model of IVR is often referred to as a tier model and is in common use for polyatomic molecules and in most rovibrational states [

8]; it can be represented by the following Hamiltonian:

A short pulse laser can optically couple the states |1

, |2

, |3

etc to the ground state |0

with respective transition matrix elements expressed in Rabi units as Ω

_{1}(

*t*), Ω

_{2}(

*t*), Ω

_{3}(

*t*) etc and their corresponding complex conjugates. The background levels |4

, |5

, |6

etc are coupled to the optically excited states |1

, |2

and |3

through the matrix elements

*V*_{14},

*V*_{15},

*V*_{24} etc. Such a molecular system can become useful for realizing qubits [

9] effectively if this large number of optically coupled states can be accessed simultaneously as has been the case for the atomic system of Rydberg states of caesium [

10]. However, the difficulty in extending this scheme to the molecular system starts at the very first step of initializing the qubits due to high decoherence of the possible qubit states as in the gedanken system Hamiltonian presented in

equation (3).