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Journal of Modern Optics

Two-pulse Optical Echoes by Selective Mechanisms

Alan V. Durrant, T.B. Smith

Published Feb 1, 1989• 1 citation
Publisher

Abstract

The possibility of a class of optical echo-producing mechanisms is suggested that does not invoke the phase-reversal action of the second pulse. The mechanisms act selectively on the radiating atomic dipole moments, or on their radiation fields, with a periodic dependence on Δτ, where Δ is the angular frequency of the radiator relative to the central angular frequency of the distribution and τ the interpulse time. In the new formulation, the customary expression for the field radiated by an atomic dipole following a two-pulse excitation sequence is multiplied by a factor periodic in Δτ. It is shown that a sequence of echo pulses is predicted when the average over the radiators is calculated. One such mechanism is proposed: the selective absorption or amplification of the dephased free-induction decay field as they propagate through the sample.

DOI
10.1080/09500348914550301

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Physical Review A•Jun 1, 1989•2 citations
The possibility of a class of optical echo-producing mechanisms is suggested that does not invoke the phase-reversal action of the second pulse. The mechanisms act selectively on the radiating atomic dipole moments, or on their radiation fields, with a periodic dependence on Δτ, where Δ is the angular frequency of the radiator relative to the central angular frequency of the distribution and τ the interpulse time. In the new formulation, the customary expression for the field radiated by an atomic dipole following a two-pulse excitation sequence is multiplied by a factor periodic in Δτ. It is shown that a sequence of echo pulses is predicted when the average over the radiators is calculated. One such mechanism is proposed: the selective absorption or amplification of the dephased free-induction decay field as they propagate through the sample.
The decay in the hyperfine coherence of optically trapped atoms due to interactions with the environment, is only partly suppressed by ``echo'' spectroscopy [M. F. Andersen et al. Phys. Rev. Lett. 90, 023001 (2003)], primarily due to dynamical (time-dependent) dephasing mechanisms. We demonstrate here an improved pulse sequence, for which the decay of coherence is reduced by a factor of 2.52.52.5 beyond the reduction offered by the ``echo'' spectroscopy. This reduction occurs when each dark period in-between pulses is shorter than the time scale over which substantial dephasing develops. The coherence time is then limited by mixing to other vibrational levels in the trap and, to a lesser extent, the lifetime of the internal states of the atoms.
The primary objective of this paper is to provide a formalism to predict and characterize the echoes which may be produced in a gaseous sample by an arbitrary sequence of one- or two-frequency laser excitation pulses which successively transfer atomic population between two or more atomic energy levels. The resulting echoes can occur at times which depend on the relative frequencies of the excitation pulses as well as on the times at which the pulses were applied. Using an idealized model of the excitation pulses and the atoms, the authors derive a simple transformation equation which provides the density matrix of a single atom after a general excitation sequence. Equations are derived which utilize the single-atom density matrix to predict the properties of the echoes (if any) which result. The means by which the sample of atoms "remembers" the information necessary to produce different types of echoes is discussed, and it is then shown how a study of the echo can provide information concerning the atomic relaxation processes which tend to destroy this information. It is observed that trilevel echoes can be used to determine the relaxation characteristics of superposition states between energy levels coupled by two-photon transitions. In the case of certain other echoes (such as the three-excitation-pulse stimulated echo) the authors discuss the heretofore unappreciated fact that the sample remembers the "echo information" via a nonthermal velocity distribution of the atoms in one specific atomic state. In these cases a study of the echo behavior can provide information pertaining to the relaxation processes affecting only the one atomic state. The formalism developed is used to predict the properties of several echo effects. First, to provide a connection with previous work, the well-known photon echo is briefly discussed. Then important new properties of the three-excitation-pulse-stimulated echo are described. Finally, three different types of trilevel echo are analyzed. Each of the trilevel echoes discussed has widespread applicability in the study of relaxation processes.
We numerically investigate optical dephasing from as many as 500 interacting two-level absorbers by making the semiclassical approximation on the radiation–matter interaction which includes retarded dipole–dipole coupling and superradiant coupling. We calculate the superradiant decay rate following single pulse optical excitation, the photon echo decay and the fluorescence. By carrying out an expansion to the second order in time we demonstrate a fundamental difference between the homogeneous free-induction decay and photon echo decay which is due to radiation damping. We show that an echo sequence is capable of reversing superradiant damping in some cases. Specific computations for pentacene doped mixed molecular crystals, where ‘‘domains’’ of resonant impurities are likely to exist, are presented and compared with experimental results.
The first observation of optical rotary echoes, the optical analog of rotary spin echoes, is reported. Rotary echoes are produced in a quantum-mechanical two-level system which is driven resonantly (nutation effect) by a coherent field that suffers a sudden phase retardation. The initial nutation transient dephases, because of an inhomogeneity in either the driving field or the transition frequency of the sample, and then rephases to form an echo following the phase-shifting pulse. Hence, optical rotary and photon echoes are similar processes---the former being an interference in nutation and the latter an interference in free precession. Perturbative solutions of the Bloch equations are derived for the case where inhomogeneous dephasing arises from Doppler broadening and the Rabi frequency exceeds the homogeneous damping rate. Observations of the optical rotary echo in I2{\mathrm{I}}_{2}I2​ vapor are facilitated by the technique of laser frequency switching, which generates precise phase shifts in the optical field. The measurements corroborate detailed theoretical predictions.
Large-amplitude oscillations are predicted in the free-induction decay (FID) of atomic or spin two-level quantum systems that are coherently prepared by a resonant electromagnetic field in the form of a square amplitude pulse of duration TTT. The effect should be detectable in nuclear magnetic resonance (NMR) or in optical resonance experiments when the transition line shape is inhomogeneously broadened (${T}_{2}^{*}\ensuremath{\ll}{T}_{2}$) and the applied field is intense enough that the pulse area $\ensuremath{\chi}T\ensuremath{\ge}2\ensuremath{\pi}$, where $\ensuremath{\chi}$ is the Rabi frequency. Physically, the effect is due to Rabi oscillations which are generated in the preparative stage and are then reproduced because of atomic memory in the radiative period which follows. The atomic polarization derived from the Bloch equations assumes the form of an integral over the inhomogeneous line shape, and can be evaluated numerically for arbitrary linewidth $\ensuremath{\sigma}$ or analytically when $\ensuremath{\sigma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The characteristics of the oscillation are unusual since the oscillation frequency, which is of order $\ensuremath{\chi}$, increases with time while the oscillation envelope vanishes identically for times $t\ensuremath{\ge}2T$. Coherent emission is therefore confined to one pulse width TTT immediately following the pulse. Previous NMR and infrared FID experiments with small-area pulses ($\ensuremath{\chi}T\ensuremath{\le}4\ensuremath{\pi}$), interpreted erroneously in terms of an edge echo effect, support the oscillatory FID behavior described here. For the case $\ensuremath{\chi}T>4\ensuremath{\pi}$, an experimental test would be even more decisive, as the oscillations are more pronounced.
The interaction between an intense radiation field and atoms that undergo collisions in an atomic vapor is studied. With the assumption that velocity-changing collisions are state independent and weak, we obtain an analytic expression for the time-dependent density-matrix elements by solving a quantum-mechanical transport equation using a dressed-state approach. Our analytical calculation is used to predict the form of the signal in an extended-pulse photon echo. The results show good agreement with an experiment carried out by Yodh et al. [Phys. Rev. Lett. 53, 659 (1984)]. It is found that the decay rate of the echo signal decreases rapidly with increasing duration of the second pulse in the pulse excitation sequence. A comparison between theory and experiment is presented and a physical interpretation of the results is given.