We study the contribution to the transition probability of a multiple-photon process for which resonance occurs of the part of the interaction operator quadratic in the vector potential. We consider only the case in which this vector potential is constant within the region of space to which the system is confined. Under this assumption we show that the ratio of the contribution from the quadratic term in the vector potential to that from the linear term is proportional to the square of the energy difference between the initial state of the whole system (material system plus radiation field) and the intermediate state for which resonance occurs. As this energy difference is negligible near resonance, the quadratic term in the vector potential (or double-photon contributions) can be neglected when calculating transition probabilities for multiple-photon processes near resonances in the dipole approximation.
We study the contribution to the transition probability of a multiple-photon process for which resonance occurs of the part of the interaction operator quadratic in the vector potential. We consider only the case in which this vector potential is constant within the region of space to which the system is confined. Under this assumption we show that the ratio of the contribution from the quadratic term in the vector potential to that from the linear term is proportional to the square of the energy difference between the initial state of the whole system (material system plus radiation field) and the intermediate state for which resonance occurs. As this energy difference is negligible near resonance, the quadratic term in the vector potential (or double-photon contributions) can be neglected when calculating transition probabilities for multiple-photon processes near resonances in the dipole approximation.
The transition regimes of solitons in four-photon resonant processes in the case of two-photon absorption of the fundamental radiation are numerically investigated. The standard system of equations for the amplitudes of probability of finding the system in state with certain energy is used to derive the expression for the induced polarization in the nonlinear medium. As for the equations for the amplitudes of the optical pulses, the general case is considered in which both the amplitudes and phases are space-time dependent. We focus on the finite difference methods and the case of simultaneously propagating solitons at all frequencies of the interacting waves (simultons). The obtained results indicate that upon certain threshold conditions all interacting pulses become the solitons of Lorentzian shape. The numerical analysis has also shown that the soliton amplitudes significantly depend on the ratio between the nonlinear polarizability at the fundamental frequency ω0 and that of combination of ω0 and the trigger-field frequency ω1(2ω0 + ω1). In the second part of the paper, we apply the method of phase planes to show that at typical values of parameters, the solitons are stable.
The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\gamma}}_{2}\ensuremath{\ll}\ensuremath{\Omega}$, ${\ensuremath{\gamma}}_{1}={\ensuremath{\gamma}}_{2}$, and $\frac{{\ensuremath{\gamma}}_{2}{k}^{2}{\ensuremath{\epsilon}}^{4}}{{\ensuremath{\Omega}}^{2}}$, ${\ensuremath{\gamma}}_{1}\ensuremath{\ll}\ensuremath{\Omega}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase relaxation widths, respectively, and $\ensuremath{\Omega}$ is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than ${\ensuremath{\gamma}}_{1}^{\ensuremath{-}1}$. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to ${\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$ and only important at early times and a term proportional to $2{\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$.
A pair of nonidentical two-level atoms, separated by a fixed distance R, interact through photon exchange. The system is described by a state vector which is assumed to be a superposition of four "essential states": (1) the first atom is excited, the second one is in the ground state, and no photon is present, (2) the first atom is in its ground state, the second one is excited, and no photon is present, (3) both atoms are in their ground states and a photon is present, and (4) both atoms are excited and a photon is also present. The system is initially in state (1). The probabilities of each atom being excited are calculated for both the minimally-coupled interaction and the multipolar interaction in the electric dipole approximation. For the minimally-coupled interaction Hamiltonian, the second atom has a probability of being instantaneously excited, so the interaction is not retarded. For the multipolar interaction Hamiltonian, the second atom is not excited before the retardation time, which agrees with special relativity. For the minimally-coupled interaction the nonphysical result occurs because the unperturbed Hamiltonian is not the energy operator in the Coulomb gauge. For the multipolar Hamiltonian in the electric dipole approximation the unperturbed Hamiltonian is the energy operator. An active view of unitary transformations in nonrelativistic quantum electrodynamics is used to derive transformation laws for the potentials of the electromagnetic field and the static Coulomb potential. For a specific choice of unitary transformation the transformation laws for the potentials are used in the minimally-coupled second-quantized Hamiltonian to obtain the multipolar Hamiltonian, which is expressed in terms of the quantized electric and magnetic fields.
A theory of resonance two-photon absorption, which describes this effect as a combination of two-stage and two-photon processes, is developed in a unified mathematical formalism. Expressions for the transition probabilities and kinetic equations are derived, the absorption coefficient is determined for different and identical frequencies, and the shift of the maximum of the resonance two-photon absorption line is obtained. Investigations are made of the intensity dependences of these expressions and of the kinetics of resonance two-photon transitions. The results are compared with available experimental data.
Two pairs of energy levels of a molecule are considered. Optical transitions among each of these two pairs are coupled by the molecular binary collision. The case where the collision keeps the velocity of the molecule constant while reorienting it and maintaining the rotational energy is considered. Since the velocity is maintained constant, the molecules undergoing both the optical transitions belong to the same velocity group. This condition may be satisfied by tuning the molecular level structure by a varying Stark field. In short, we are considering coupling of two optical transitions due to the velocity-maintaining collisions. -- An expression for the energy absorbed in the molecular system when the collision - induced double resonance occurs is found. This leads to the expression for the signal absorption in any molecular system in general. The treatment begins with the phenomenological equation of motion for the velocity - dependent density matrix of the two level molecular system. This is obtained by adding the phenomenological terms to the usual equation of motion in the Heisenberg picture. A two level molecular system can be represented by a 2X2 density matrix. Since the system under consideration consists of two pairs of levels, two 2X2 density matrices are dealt with. The effect of collisions is taken care of by introducing the collision kernel in the equation of motion. The equation of motion for the diagonal and the off-diagonal elements are obtained from the equation of motion for the general density matrix. The expression for the difference in the diagonal elements is obtained in terms of the difference in the off - diagonal elements and vice versa. From these two equations, an expression for the difference in the off - diagonal elements is obtained in terms of measurable quantities. This difference along with the expectation value of the component of the dipole moment between the molecular energy states of interest is introduced in an expression for the energy absorbed in the molecular system. This expression involves the time derivative of the polarisation induced in the molecules by the field. Since the molecular system exhibits dynamic behaviour, the energy expression is time averaged. Also, since different molecular velocity groups of the Maxwell velocity distribution curve are involved in the collision, the energy expression is velocity averaged over the Maxwellian velocity range. -- Two cases are considered. First, energy expression is found in the general case, i.e., when the velocity changes during the collision. Secondly, energy is calculated when the velocity is maintained constant during the collision. This expression shows that the energy depends linearly and quadratically upon the collision constants. This is shown to be a general expression for the particular energy expression obtained by Shoemaker et al. -- Numerical evaluation for the signal absorption in the molecule methyl fluoride is done. The value is found to be ∆I = 1.323 X 10⁻² ergs/sec cm². The contribution due to the linear terms in the collision constants is 1.15 X 10⁻² ergs/sec cm² and the contribution due to the quadratic terms is 1.73 X 10⁻³ ergs/sec cm². The meaning of this is that the quadratic contribution is 13.1% of the total contribution, which is a considerable proportion. This contribution will be larger for a molecule with larger collision constants. This fact argues in favour of the general expression obtained.