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Physical Review Letters

Wave Equation for Dark Coherence in Three-Level Media

Joseph H. Eberly, V. V. Kozlov

Published Jun 4, 2002• 30 citations
Publisher

Abstract

We report the derivation of a wave equation for coherence in "dark state" two-photon-resonance spectroscopy. One of its consequences is a dark state area theorem. The dark area theorem is a single ordinary differential equation which is globally equivalent, in a way we describe, to the full set of five coupled nonlinear partial differential equations that govern space-time evolution of two-pulse coherence in a lambda medium. The predictions of the dark area theorem are open to test via laser spectroscopy in dilute vapors and inhomogeneously broadened solids.

DOI
10.1103/physrevlett.88.243604

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We report the derivation of a wave equation for coherence in "dark state" two-photon-resonance spectroscopy. One of its consequences is a dark state area theorem. The dark area theorem is a single ordinary differential equation which is globally equivalent, in a way we describe, to the full set of five coupled nonlinear partial differential equations that govern space-time evolution of two-pulse coherence in a lambda medium. The predictions of the dark area theorem are open to test via laser spectroscopy in dilute vapors and inhomogeneously broadened solids.
We analyze the effects of partial coherence of ground-state preparation on two-pulse propagation in a three-level $\ensuremath{\Lambda}$ medium, in contrast to previous treatments that have considered the cases of media whose ground states are characterized by probabilities (level populations) or by probability amplitudes (coherent pure states). We present analytic solutions of the Maxwell-Bloch equations, and we extend our analysis with numerical solutions to the same equations. We interpret these solutions in the bright and dark dressed-state basis, and show that they describe a population transfer between the bright and dark state. For mixed-state $\ensuremath{\Lambda}$ media with partial ground-state phase coherence the dark state can never be fully populated. This has implications for phase-coherent effects such as pulse matching, coherent population trapping, and electromagnetically induced transparency (EIT). We show that for partially phase-coherent three-level media, self-induced transparency dominates EIT and our results suggest a corresponding three-level area theorem.
We examine fully coherent two-pulse propagation in a $\ensuremath{\Lambda}$-type medium, under two-photon resonance conditions and including inhomogeneous broadening. We examine the effects of both short pulse preparation and medium preparation. We contrast the cases in which the two pulses have or have not matched envelopes, and media with and without ground state coherence. We find that an extended interpretation of the area theorem for single-pulse self-induced transparency is able to unify two-pulse propagation scenarios, including some aspects of electromagnetically induced transparency and stimulated Raman scattering. We present numerical solutions of both three-level and adiabatically reduced two-level density matrix equations and Maxwell's equations, and show that many features of the solutions are quickly interpreted with the aid of analytical solutions that we also provide for restricted cases of pulse shapes and preparation of the medium. In the limit of large one-photon detuning, we show that the adiabatic two-level equations commonly used to study stimulated Raman scattering are not reliable for pulse areas in the $2\ensuremath{\pi}$ range, which allows puzzling features of previous numerical work to be understood.
The authors develop a theoretical analysis which describes coherent nonlinear resonant effects where two optical pulses of different frequencies undergo two-photon self-induced transparency (two-photon SIT) in an inhomogeneously broadened three-level system. The field (Maxwell) and atomic (Schrodinger; density matrix representation) equations are coupled in a self-consistent manner by virtue of the resultant macroscopic polarizations at the two frequencies. This treatment is accomplished under circumstances where atomic coherence plays a predominant role. The resulting equations yield a new two-photon area theorem which gives first the intensity dependent, two-photon absorption law analogous to Beer's law for the one-photon transition. Secondly, a 2 pi Lorentzian pulse solution is derived which propagates without loss at the modified pulse velocity.
We consider the propagation of two laser pulses in a medium of three-level $\ensuremath{\Lambda}$ atoms with nonvanishing two-photon detuning and different oscillator strengths of the transitions. We show that the two-photon detuning and the derivative of the mixing angle are described by one and the same equation. We analyze the solution of this equation and its physical consequences. In particular, we obtain the conditions and the distance where the nonadiabaticity and the two-photon detuning may be reduced. We derive a simple expression for the maximum length to which excitation in the medium may reach.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
Exact and asymptotic line shape expressions are derived from the semiclassical density matrix representation describing a set of closed three-level $\ensuremath{\Lambda}$ atomic or molecular states including decoherences, relaxation rates, and light shifts. An accurate analysis of the exact steady-state dark-resonance profile describing the Autler-Townes doublet, the electromagnetically induced transparency or coherent population trapping resonance, and the Fano-Feshbach line shape leads to the linewidth expression of the two-photon Raman transition and frequency shifts associated to the clock transition. From an adiabatic analysis of the dynamical optical Bloch equations in the weak field limit, a pumping time required to efficiently trap a large number of atoms into a coherent superposition of long-lived states is established. For a highly asymmetrical configuration with different decay channels, a strong two-photon resonance based on a lower states population inversion is established when the driving continuous-wave laser fields are greatly unbalanced. When time separated resonant two-photon pulses are applied in the adiabatic pulsed regime for atomic or molecular clock engineering, where the first pulse is long enough to reach a coherent steady-state preparation and the second pulse is very short to avoid repumping into a new dark state, dark-resonance fringes mixing continuous-wave line shape properties and coherent Ramsey oscillations are created. Those fringes allow interrogation schemes bypassing the power broadening effect. Frequency shifts affecting the central clock fringe computed from asymptotic profiles and related to the Raman decoherence process exhibit nonlinear shapes with the three-level observable used for quantum measurement. We point out that different observables experience different shifts on the lower-state clock transition.