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Laser Physics Letters

Dynamics of two-level systems irradiated by sub-one-cycle laser pulses

N V Bordyug, В. П. Крайнов

Published Apr 27, 2007• 11 citations
Publisher

Abstract

Transitions in a two-level system irradiated by linearly polarized cosine and sine waveform sub-one-cycle pulses are considered. The probabilities of transition after the end of laser pulse are derived numerically as functions of two dimensionless variables: E τ and F τ, where τ is the pulse duration, E is the energy distance between considered levels, and F is the dipole matrix element of the transition (all quantities are presented in atomic units). We derive also the dipole moment of a two-level system after the pulse termination as a function of the same dimensionless parameters. There is no significant dependence of dipole moment and inversion population on the cosine, or sine waveform.

DOI
10.1002/lapl.200710041

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Transitions in a two-level system irradiated by linearly polarized cosine and sine waveform sub-one-cycle pulses are considered. The probabilities of transition after the end of laser pulse are derived numerically as functions of two dimensionless variables: E τ and F τ, where τ is the pulse duration, E is the energy distance between considered levels, and F is the dipole matrix element of the transition (all quantities are presented in atomic units). We derive also the dipole moment of a two-level system after the pulse termination as a function of the same dimensionless parameters. There is no significant dependence of dipole moment and inversion population on the cosine, or sine waveform.
We consider the process of transition of a two-level system to the excited state with subsequent photon emission in the presence of a laser pulse with a high degree of unipolarity. Within the framework of quantum electrodynamics, we obtain analytical expressions for the differential probability of the process depending on the temporal scales of the problem: laser pulse duration, excited-state lifetime, inverse transition frequency, and inverse frequency of the photon emitted. Besides, we calculate the total absorption probability by integrating over the three-dimensional photon momentum and summing over polarizations. We compare the results obtained for unipolar and bipolar (many-cycle) pulses.
The absolute laser phase dependence of the time-dependent populations of the molecular states, including the steady-state (long time) populations of the states, associated with the interaction of a molecule with a pulsed laser is investigated using illustrative two-level examples. One-photon transitions, including the effects of permanent dipoles, are discussed as a function of the pulse duration, intensity, and (absolute) laser phase, for selected laser frequencies. The effects of laser phase can be large, depending on the values of the pulse duration for a given frequency and intensity. The effects of permanent dipoles, relative to no permanent dipoles, are significant for large laser field strengths ε0. When the laser-molecule coupling parameter b=μ12ε0/E21⩾0.2, where μ12 and E21 are the transition dipole and energy difference between the ground and excited states, respectively, the dynamics of the pulse-molecule interaction are (strongly) phase dependent, independent of pulse duration, whereas the corresponding steady-state populations of the molecular states may or may not be phase-dependent depending on the pulse duration. Analytical rotating wave approximations for pulsed laser-molecule interactions are useful for interpreting the dynamics and the steady-state results as a function of field strength and pulse duration, including the effects of permanent dipole moments. The results reported in this paper are based on molecular parameters associated with an S0→S1 electronic transition in a dipolar molecule. However, they are presented in reduced form and therefore can be scaled to other regions of the electromagnetic spectrum. Short, intense pulses at or beyond the limits of current laser technology will often be required for the types of absolute laser phase effects of this paper to be appreciable for electronic excitations. The discussion, in the UV-VIS, also suffers from the use of a two-level model and from the requirement of field intensities that can be beyond the Keldysh limit. For other spectral regions, these absolute laser phase effects will be much more readily applicable.
Abstract Mathematical modeling of the excitation dynamics of an isolated two-level particle under bichromatic irradiation is carried out on basis of solutions of the Schrödinger equation for the probability amplitudes of the states of the composite system of the particle and an interaction two-mode quantized radiation field, with the photons for each of which in a superposition state at the initial time. An analytical expression describing the change in the time of the population of the excited state of a particle in the adsorption of one of the irradiation photons in the simplest case when the irradiation is a sequence of identical pulses of a sinusoidal form is obtained. The obtained time dependence is compared with the time dependence of the diagonal matrix element of the statistical operator of the particle in the excited state found on the basis of the numerical solution of the system of the optical Bloch equations, with the corresponding irradiation described by the classical theory. Examples of significant differences in the type of these compared time dependences for a number of characteristic cases of irradiation are given.
The problem of a two-level atom interacting with a non-monochromatic radiation pulse is studied in the resonant approximation for arbitrary detunings, and without the resonant approximation. By using the time-dependent theory of perturbation the analytical expressions for the transition probability and induced dipole moment of the atom are obtained. It is shown that the upper level population of the atom exhibits oscillations that are absent in the resonant approximation. The temporal shape of the pulse is shown to affect significantly the atomic transition probability. The behaviour of the transition probability and dipole moment spectrum for arbitrary pulse shapes, phases and detunings is investigated. The interference between the spectral components arising in nonlinear processes is shown to play a considerable role in the formation of a dipole moment spectrum.
We consider the population inversion in a two-level system generated by a sub-one-cycle pulse excitation. Specifically, we explore the effect that the time derivative of the pulse envelope has on the Rabi dynamics. Our analysis is based on a combination of analytical, perturbative, and nonperturbative treatments and is complemented by numerical simulations. We find a shortening of the Rabi inversion period and show that complete inversion is unobtainable under resonant, ultrashort pulse conditions. The impact of nonresonant and carrier-envelope phase-dependent effects on the dynamics of two-level and multilevel systems is studied numerically, and conditions for complete population inversion are derived.
The maximal population transfer in a two-level system induced by an ultrashort, linearly polarized laser pulse and the optimal pulse shape for maximal population transfer depend critically on the pulse duration and phase. In the case of temporally symmetric pulses, half-cycle pulses are optimal and are capable of inverting the population completely. For antisymmetric pulses, bimodal pulses are optimal but cannot invert the population completely. Deviations from the Rabi formula for the pulse parameters do not occur until the pulses become shorter than a few optical cycles.