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Applied Physics Letters

Population inversion by nonadiabatic frequency chirping

Paul Horwitz

Published Mar 15, 1975• 28 citations
Publisher

Abstract

A calculation is presented of the asymptotic final state of a two−level system, initially in the ground state, in the presence of coherent constant−intensity radiation, the frequency of which is linearly swept through the resonance line. The resulting very simple formula approaches the appropriate limits in the sudden and adiabatic approximations.

DOI
10.1063/1.88167

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Molecular π pulses: Population inversion with positively chirped short pulses

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Population Inversion in Optical Adiabatic Rapid Passage with Phase Relaxation

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The transition probability $P$ in the optical adiabatic rapid passage is calculated for a two-level model with dephasing. In the limit $\frac{{T}_{2}}{{\ensuremath{\tau}}_{p}}\ensuremath{\rightarrow}0$, where ${T}_{2}$ is the dephasing time and ${\ensuremath{\tau}}_{p}$ is the time scale of passage through the resonance region, $P$ is given by $P=\frac{[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\pi}{\ensuremath{\chi}}^{2}}{|\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}|})]}{2}$ in which $\ensuremath{\chi}$ is the Rabi flopping frequency at resonance and $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}$ is the time derivative of the offset angular frequency. Generally, $P$ takes its maximum value at an intermediate $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}$ for fixed $\ensuremath{\chi}$ and is bounded by the above formula and by $P=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\pi}{\ensuremath{\chi}}^{2}}{2|\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}|})$, which gives the asymptotic value in the limit $\frac{{T}_{2}}{{\ensuremath{\tau}}_{p}}\ensuremath{\rightarrow}\ensuremath{\infty}$.

Physical Review Letters•May 11, 1987•39 citations
A calculation is presented of the asymptotic final state of a two−level system, initially in the ground state, in the presence of coherent constant−intensity radiation, the frequency of which is linearly swept through the resonance line. The resulting very simple formula approaches the appropriate limits in the sudden and adiabatic approximations.
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.
Doppler-free population inversion induced by coherent two-photon resonance is calculated analytically for a two-photon analog of the hyperbolic-secant pulse and numerically for a Gaussian, linearly chirped pulse. The numerical calculations are carried out for a two-photon resonance of a chirped ruby laser line with the $6{S}_{\frac{1}{2}}\ensuremath{\rightarrow}9{D}_{\frac{3}{2}}$ transition in Cs. Complete inversion is found when the fractional change in frequency during the pulse lies between 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ and 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$. The coherent two-photon resonances are described by optical Bloch equations which are derived from a multiple-time-scale perturbation theory.
We describe a class of continuously phase-modulated radiation pulses that result in coherent population inversion on resonance as well as over a large range of transition frequencies and radiation field strengths. This is a population-inversion analogy to self-induced transparency. Simulations of the inversion properties of the modulated inversion pulse (MIP) are presented. It is shown that the inversion behavior can be explained by treating the MIP as a highly efficient adiabatic sweep. Criteria for establishing adiabaticity are discussed in detail. Finally, a method is presented for generating a sequence of phase-shifted radio-frequency pulses, from the continuously modulated pulse, which can be implemented on modern NMR and coherent optical spectrometers; experimental confirmation is given.
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.
Detailed theoretical analysis and numerical simulation indicate that nearly complete electronic population inversion of molecular systems can be achieved with intense positively chirped broadband laser pulses. To provide a simple physical picture, a two-level model is used to examine the condition for the so-called π pulses and a four-level model is designed to demonstrate for molecular systems the correlation between the sign of the chirp and the excited state population. The proposed molecular π pulse is the combined result of vibrational coherence in the femtosecond regime and adiabatic inversion in the picosecond regime. Numerical results for a displaced oscillator, for LiH and for I2, show that the proposed molecular π pulse scheme is robust with respect to changes in field parameters such as the linear positive chirp rate, field intensity, bandwidth, and carrier frequency, and is stable with respect to thermal and condensed phase conditions including molecular rotation, rovibronic coupling, and electronic dephasing.
The transition probability PPP in the optical adiabatic rapid passage is calculated for a two-level model with dephasing. In the limit $\frac{{T}_{2}}{{\ensuremath{\tau}}_{p}}\ensuremath{\rightarrow}0$, where T2{T}_{2}T2​ is the dephasing time and ${\ensuremath{\tau}}_{p}$ is the time scale of passage through the resonance region, PPP is given by $P=\frac{[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\pi}{\ensuremath{\chi}}^{2}}{|\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}|})]}{2}$ in which $\ensuremath{\chi}$ is the Rabi flopping frequency at resonance and $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}$ is the time derivative of the offset angular frequency. Generally, PPP takes its maximum value at an intermediate $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}$ for fixed $\ensuremath{\chi}$ and is bounded by the above formula and by $P=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\pi}{\ensuremath{\chi}}^{2}}{2|\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\omega}}|})$, which gives the asymptotic value in the limit $\frac{{T}_{2}}{{\ensuremath{\tau}}_{p}}\ensuremath{\rightarrow}\ensuremath{\infty}$.