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Physical Review A

Multilevel inversion schemes in and beyond the adiabatic limit

J. Oreg, G. Hazak, Joseph H. Eberly

Published Nov 1, 1985• 51 citations
Publisher

Abstract

A systematic investigation of N-level inversion schemes, with use of ``chirped frequency lasers,'' in and beyond the adiabatic limit is presented. (1) It is shown that, contrary to intuition, the favorable scheme for adiabatic inversion requires that, during the process, resonances occur in reversed order, i.e., first the (N-1)-N transition and then backward along the atomic-level ladder down to the transition 1-2. (2) A new multilevel inversion process at fast rates (far beyond the adiabatic limit) is found. Surprisingly, the process occurs for pulse area equal to \ensuremath{\pi} in spite of large (time-dependent) detunings. Full analytic explanation of this generalized \ensuremath{\pi} pulse inversion is presented. General new properties of the multilevel adiabatic approximation are also derived and discussed.

DOI
10.1103/physreva.32.2776

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Quantum Electronics and Laser Science Conference•Apr 24, 1989•0 citations
A systematic investigation of N-level inversion schemes, with use of ``chirped frequency lasers,'' in and beyond the adiabatic limit is presented. (1) It is shown that, contrary to intuition, the favorable scheme for adiabatic inversion requires that, during the process, resonances occur in reversed order, i.e., first the (N-1)-N transition and then backward along the atomic-level ladder down to the transition 1-2. (2) A new multilevel inversion process at fast rates (far beyond the adiabatic limit) is found. Surprisingly, the process occurs for pulse area equal to \ensuremath{\pi} in spite of large (time-dependent) detunings. Full analytic explanation of this generalized \ensuremath{\pi} pulse inversion is presented. General new properties of the multilevel adiabatic approximation are also derived and discussed.
We derive and discuss the behavior of several multilevel models relevant to the problem of molecular multiphoton excitation under the action of a quasimonochromatic laser field, the amplitude of which rises slowly and falls rapidly on a certain internal time scale. Our method is suitable for the numerical solution of problems involving the adiabatic excitation of large multilevel systems. We find adiabatic inversion to be a general characteristic of the models considered. The fact that adiabatic inversion is accompanied by marked qualitative and quantitative changes in the expectation value of the dipole operator makes our results useful for understanding the propagation of laser pulses. An explicit expression that is especially convenient for numerical evaluation is presented for determining the limiting internal time scale, which is determined by the energy levels and transition moments of the system and the laser frequency. We illustrate the criterion for adiabatic inversion by applying it to a particular multilevel model. From a density-matrix treatment we conclude that collisional damping destroys the coherence required for adiabatic inversion. On the basis of our results we believe that the adiabatic approximation may be appropriate for some experiments of current interest.
A numerical analysis is presented of several potential pulsed-laser applications of adiabatic population inversion in atoms or molecules. A simple numerical method is demonstrated for evaluating adiabatic processes in multilevel systems for both swept-frequency and resonant, time-delayed pulses. Adiabatic inversion with time-delayed pulses is generalized to systems with any number of levels. The effect of Doppler broadening on time-delayed adiabatic pumping of a three-level system is investigated.
Frequency swept pulses are used to invert population over a broad band of transition frequencies without the need for a precise calibration of the pulse amplitude. As long as the adiabatic approximation is valid, population is adiabatically inverted at each individual transition frequency of the swept range. Even though this picture fails to be true when the adiabatic approximation breaks down, population inversion can still be achieved for an appreciable range of transition frequencies and field amplitudes. We discuss population inversion in an ensemble of two-level systems by a frequency-swept pulse, the so-called constant adiabaticity pulse, without invoking the adiabatic approximation. The equations of motion are integrated by a seminumerical method to analyze population inversion in the regime where usually the adiabatic approximation is applied. The effects of resonance offset and variable pulse amplitude are described by an average Hamiltonian expansion to discuss pulse performance beyond the validity of the adiabatic approximation. As a function of the adiabaticity parameter (reciprocal of the pulse area), the inversion bandwidth increases in a stepwise fashion due to the consecutive cancellation of average Hamiltonians. The first inversion over a finite range of transition frequencies and pulse amplitudes is shown to occur for an adiabaticity parameter of 1/15.
The problem of achieving population inversion adiabatically in an NNN-level system using one or more laser fields whose detunings and/or amplitudes are continuously varied is studied analytically and numerically. The SU(N)\mathrm{SU}(N)SU(N) coherence vector picture is shown to suggest unexpected inversion procedures and also to give a generalized interpretation of adiabatic following. It is shown that the (${N}^{2}\ensuremath{-}1$)-dimensional SU(N)\mathrm{SU}(N)SU(N) space contains an ($N\ensuremath{-}1$)--dimensional steady-state subspace $\mathit{\ensuremath{\Gamma}}(t)$ whose orthonormal basis vectors ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{1},\dots{}, {\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{N\ensuremath{-}1}$ are given explicitly in terms of the Hamiltonian matrix elements. The motion of the system can be interpreted as a "generalized precession" of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ about $\mathit{\ensuremath{\Gamma}}$. Multilevel adiabatic following occurs when the angle $\ensuremath{\chi}(t)$ between the coherence vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ and its projection onto $\mathit{\ensuremath{\Gamma}}$ is very small. The multiple dimension of $\mathit{\ensuremath{\Gamma}}$ is shown to provide a variety of paths for adiabatic inversion. The adiabatic solution is obtained by solving $N\ensuremath{-}1$ simple equations for the directional cosines of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ on ${\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}}_{i}$. The adiabatic solution and time scale and the state taken up by the atomic variable are discussed analytically and numerically for a three-level system.
Enhanced population transfer from the lowest to the highest level in the three-state ladder 5s->5p->5d of the rubidium atom by frequency swept ultra short laser pulses is observed if the frequency sweep first makes the pulse resonant with the 5s->5p transition, and later with the 5p->5d transition. With this intuitive scheme it is possible to achieve complete population inversion via the intermediate level by two successive adiabatic passages. Complete inversion is also obtained by a {open_quote}counterintuitive{close_quote} direction of the frequency-sweep, in which case population of the intermediate level during the pulse is strongly reduced. The atomic system can be seen as a model system for multistep vibrational excitation in an anharmonic molecular potential.
We derive two types of perturbative approximation for the response of collisionally broadened two-level atoms to a time-dependent field. The perturbative approximations for the inversion and polarization are valid to any desired order in the derivatives of the field. Comparison with a numerical solution of the full density-matrix equations of motion demonstrates that the new approximations significantly improve upon the adiabatic-following and collisional steady-state approximations for time-dependent pulses in the appropriate damping regime. Application of the approximations to pulse propagation, sideband instability, and population pulsations is discussed. The collisional steady-state and adiabatic-following approximations find extensive use in many areas of nonlinear optics. The deficiencies of these approximations are well known: The collisional steady-state approxirnation is valid only for monochromatic (cw) radiation and the adiabatic-following approximation is valid only in the absence of damping and only if the product of the detuning and a characteristic time is much larger than one. In spite of their manifest faults, the collisional steady-state and adiabatic-following approximations are often exceptionally good approximations, in the appropriate damping regime, for the local response of a vapor of two-level atoms to a slowly varying field. However, timedependent nonlinear processes modify a propagating field zuch that it varies less slowly and the neglected portions of the atomic response become more important. ' Therefore improved approximations for the damped atomic response to a time-dependent field are desirable. In this paper we present two improved perturbative approximations for the inversion and polarization of collisionally broadened two-level atoms induced by a time-dependent field. The adiabatic-following approximation for two-level systems was derived phenomenologically by Grischkowsky, who used it to explain certain experimental results of near-resonance laser-pulse propagation. Motivated by the work of Grischkowsky, Crisp provided an analytic derivation for the adiabatic-following approximation and obtained a quadrature for the error. The essential feature of Crisp's derivation is the expansion of the kernel of the integral representation of the off-diagonal density-matrix element p2, in a Fourier series with integration proceeding term by term. Both the adiabatic-following approximation and the collisional steady-state approximation were obtained in first order. Both Crisp and Grischkowsky used the adiabatic inversion to derive a correction to the adiabatic-following approximation for the polarization. We have shown how Crisp's method can be extended to obtain corrections to the adiabatic inversion and to improve corrections to the polarization in the absence of damping. In this paper we show how the integrated Fourier series can be used to obtain two types of perturbative approximation for the inversion and polarization of collisionally broadened twolevel systems that are valid to any desired order in the derivatives of the field. The two types of perturbative approximation to the inversion are (i) a formal solution of the rate equation for the inversion which is obtained from the density-matrix equations for the populations; and (ii) an approximation obtained using the approximation (p22 — p») —I — 4p2, P, 2, valid for weak damping. The approximations for the polarization are obtained by substituting either (i) or (ii) for the inversion in the integrated-Fourier-series expansion of p2, . The approximation obtained using the first type of approximate inversion converges most rapidly for strong damping and reduces to the collisional steady-state approximation when the magnitude and phase of the field are constant. The approximation obtained using the second type of approxirnate inversion converges most rapidly for weak damping and in first order reduces to the adiabaticfollowing approximation for zero damping. Comparison with a numerical solution of the density-matrix equations demonstrates that retention of higher-order derivatives by retaining more terms in the integrated Fourier series results in a significant improvement in accuracy for a time-dependent field. However, Rabi oscillations, which can be an important component of the nonlinear response, ' cannot be obtained by finite-order approxi