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Journal of the Optical Society of America B

Excitation of an atom by a train of short pulses

Richard J. Temkin

Published May 1, 1993• 33 citations
Publisher

Abstract

A general, analytic theory is presented of the excitation of a two-level atom by a series of short optical laser pulses. The atom is assumed to be isolated (collisionless) with radiative decay from upper to lower state at rate γ. The laser pulses are assumed to have a duration tp such that γtp ≪ 1 (the transient regime), and the spacing between pulses tr is such that tr ≫ tp. In this regime the atomic density matrix is calculated as a function of time during each optical pulse. An analytic expression for the atomic state (Bloch vector) after n optical pulses is also derived. For large n the atomic state converges to a repeating (steady state) cycle of pump excitation followed by radiative decay. The excited-state population in the steady state can be large but only if the spacing between pulses tr is small so that γtr ≲ 0.5. These results indicate that lasers that produce trains of pulses, such as mode-locked conventional lasers and free-electron lasers, can be effective in optical pumping of atoms.

DOI
10.1364/josab.10.000830

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A general, analytic theory is presented of the excitation of a two-level atom by a series of short optical laser pulses. The atom is assumed to be isolated (collisionless) with radiative decay from upper to lower state at rate γ. The laser pulses are assumed to have a duration tp such that γtp ≪ 1 (the transient regime), and the spacing between pulses tr is such that tr ≫ tp. In this regime the atomic density matrix is calculated as a function of time during each optical pulse. An analytic expression for the atomic state (Bloch vector) after n optical pulses is also derived. For large n the atomic state converges to a repeating (steady state) cycle of pump excitation followed by radiative decay. The excited-state population in the steady state can be large but only if the spacing between pulses tr is small so that γtr ≲ 0.5. These results indicate that lasers that produce trains of pulses, such as mode-locked conventional lasers and free-electron lasers, can be effective in optical pumping of atoms.
An assembly of ground-state two-level atoms with a Maxwellian velocity distribution is considered to be illuminated by a near-resonant train of Gaussian pulses whose pulse separation is of the same order of magnitude as the natural decay time of the upper state by spontaneous emission. The equations of motion of the density matrix, including spontaneous decay, are numerically integrated over the duration of one pulse and are analytically integrated between pulses. In this way, the transition matrix is derived. The time evolution of the system is then followed by successive multiplications of the state vector by the transition matrix. Steady state is attained after a few pulses. Because the pulse train has side bands separated by the pulse repetition frequency, corresponding velocity groups are preferentially excited. The response (defined as the time-average spontaneous emission of the assembly) is found to be a quasi-sinusoidal function of the number of Rabi flops per pulse. Frequency modulation of the pulse train leads to smaller variation in the response and, consequently, to considerably higher response.
We study the dynamics of three-level $\ensuremath{\Lambda}$-type atoms driven by a coherent train of short, nonoverlapping laser pulses. We derive analytical nonperturbative expressions for the density matrix by approximating the pulses by a $\ensuremath{\delta}$ function. We demonstrate that, depending on the train parameters, several scenarios of system dynamics can be realized. We show the possibility of driving Raman transitions between the two ground states of the $\ensuremath{\Lambda}$ system, avoiding populating excited states by using pulses with effective area equal to $2\ensuremath{\pi}$. The number of $2\ensuremath{\pi}$ pulses needed to transfer the entire population from one ground state to another depends on the ratio between the Rabi frequencies of two allowed transitions. In the case of equal Rabi frequencies, the system can be transferred from one ground state to another with a single $2\ensuremath{\pi}$ pulse. When the total pulse area differs from $2\ensuremath{\pi}$ and the two-photon resonance condition is fulfilled, the system evolves into a ``dark'' state and becomes transparent to subsequent pulses. The third possible scenario is the quasi-steady-state regime when neither the total single-pulse area is equal to $2\ensuremath{\pi}$ nor is the two-photon resonance condition fulfilled. In this regime the radiative-decay-induced drop in the population following a given pulse is fully restored by the subsequent pulse. We derive an analytical expression for the density matrix in the quasi-steady-state regime. We analyze the dependence of the postpulse excited-state population in the quasi-steady-state regime on the train parameters. We find the optimal values for the train parameters corresponding to the maximum of the excited-state population. The maximum of the excited-state population in the steady-state regime is reached at an effective single-pulse area equal to $\ensuremath{\pi}$ and is equal to 2/32/32/3 in the limiting case when its radiative lifetime is much shorter than the pulse repetition period.
We present an analytical expression for the response of a two-level atom to a frequency-modulated optically coherent pulse train. The optical beam has sinusoidal frequency modulation and is chopped to have a square-wave envelope. We assume that the laser pulses are short compared with the atomic-decay time, the pulse-repetition time, and the modulation period. With this short-pulse assumption we are able to use a method similar to Temkin’s [J. Opt. Soc. Am. B10, 830–839 (1993)] and solve the optical Bloch equations in closed form.
This paper describes a model and provides the mathematical formulation for describing the spike pulsing phenomenon observed in optically pumped three-level lasers. The model is based on the pumping of electrons from a ground state to an excited pump band from which they relax very rapidly to the excited laser level. With population inversion then achieved, laser action starts and repetitive pulses with decreasing amplitude are generated; the pulses then damp out and the laser output reaches a steady state as long as sufficient pump radiation energy is available. The equations describing the system in terms of the population inversion and photon density are nonlinear. However, approximations are made which permit the required relations between the transition rates to be established for such action to progress. The time variation of the population inversion and the photon density at the laser frequency are obtained for both the pumping time interval and the stimulated emission laser region.
We describe a two-level model for the mesospheric sodium resonance fluorescence excited by a train of pulses. For pulse durations that are short compared with the 16-ns natural lifetime, population-rate equations are inadequate and must be replaced by density-matrix (Bloch) equations. We briefly contrast these two approaches and discuss several issues associated with pulse-train excitation. Analytical approximations to averages over atomic velocities and the transverse spatial profile of the laser are described. Estimates of return photon numbers for trains of pulses, ranging in duration from tens of picoseconds to half a nanosecond, are made and these estimates are compared with those of more conventional long-pulse lidar schemes. Roughly similar photon numbers are predicted for both long and short pulses whenever their average intensities are comparable. For average intensities, less than ~ 10 W/cm(2), trains of asymptotically equal to 0.5-ns pulses yield greater photon returns than trains of asymptotically equal to 30-ps pulses. For larger intensities the reverse can be true.
A theory of stepwise pulsed-laser excitation of cascade Zeeman quantum beats has been developed for a four-level system. The laser pulses are treated in the broad-line approximation and the full optical-pumping equations for the atomic-density matrix of each level have been obtained. These can be used to describe saturation and light shift effects. The weak-pumping theory solutions to the optical pumping equations have been used to obtain expressions for the cascade quantum-beat intensity versus time curves. Larmor precession, spontaneous-emission decay and the populating of the final cascade level during the excitation pulse are all allowed for. A Wiener-Levy phase diffusion model with slowly varying amplitudes is used for the laser pulses, and the final theoretical expressions for the quantum beats depend on the time-dependent intensities of the laser pulses as well as on their polarisations. Hyperfine effects have been ignored. The theory applies to both coincident pulses and to pulses with a time delay between them. An illustrative application to the case of two very short but coincident pulses (during which Larmor precession, spontaneous emission etc. can be ignored) is given. In an accompanying paper the theoretical expressions are further evaluated and successfully applied to the interpretation of cascade Zeeman quantum beat experiments in Fe I.