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Physical Review A

Nonperturbative analysis of the two-level atom: Applications to multiphoton excitation

R. E. Duvall, E. Valeo, Carl R. Oberman

Published Jun 1, 1988• 24 citations

Abstract

Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system, ${\mathrm{Cd}}^{12+}$, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.

DOI: 10.1103/physreva.37.4685

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Australian Journal of Physics•Oct 1, 1995•0 citations

Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system,

{\mathrm{Cd}}^{12+}

, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.

The multiple-linear-time-scale method is used to construct a perturbation theory for

N

-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/

{\mathrm{cm}}^{2}

) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/

{\mathrm{cm}}^{2}

), the approximate solution is valid for many thousands of Rabi cycles.

The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order

{10}^{8}

W/

{\mathrm{cm}}^{2}

) is determined by the intensity-dependent level width (

6f

configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order

{10}^{12}

W/

{\mathrm{cm}}^{2}

) are determined by both the intensity-dependent widths and shifts a

\stackrel{\mathrm{`}}{\mathrm{s}}

well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.

Multiphoton spectroscopy The subject of multiphoton excitation spectroscopy began in 1931 when Göppert-Mayer [450] wrote a theoretical paper in which she calculated the transition rate for an atom in the presence of two photons rather than just one. At the time, the process seemed rather exotic, and it was reassuring that the calculated rate was so low as to guarantee that it could not readily be observed in the laboratory with conventional sources. This conclusion was reassuring because it implies that a simple perturbative theory (one photon per transition is the weak-field limit) is adequate for most purposes. The subject came to life with the advent of lasers, when it became easy to create intense beams of light. Since the probability of excitation by two photons grows as the square of the photon density, whereas the probability of single-photon excitation grows only linearly with photon density, two-photon transitions gain in relative strength with increasing intensity despite the small value of the rate coefficient. The development of multiphoton spectroscopy has followed that of lasers: as the available power has increased, so has the number of photons involved in individual transitions. More significantly, it has become apparent that the physics of the interaction between radiation and matter is not the same at high laser powers as under weak illumination, i.e. that there is a qualitative change which sets in at strong laser fields. This is normally expressed by saying that perturbative approximations break down.

The multiphoton resonant excitation of three-level atoms in a strong laser field is investigated. The creation of various superposition states is shown. For implementation of these effects either the mean dipole moment in the excited states of an atom or, in three-level systems, a transition dipole moment between excited states (with energies close enough to each other) must be nonzero. The time evolution of the system is found using a nonperturbative resonant approach. Our calculations for the hydrogen atom suggest that by the appropriate optical pulses with moderate strong intensities one can achieve various superposition states by multiphoton resonant excitation.

Perturbation theory of multiphoton ionisation due to a classical electromagnetic field is modified to allow for intermediate resonances with bound states. Complex energies, generally associated with resonances, do not enter into this formalism. For a monochromatic field of frequency u, a constant ionisation rate can then be defined unambiguously and only such continuum states are excited which correspond to the absorption of energy in integral multiples of hu. As an application, differential and total cross sections for the two-photon ionisation of hydrogen, for frequencies below the n = 3 resonance region, are obtained in closed form. Existing data for generalised cross sections, calculated numerically using the complex coordinate method, are in good agreement with the present results. Finally, the mean fractional ionisation resulting from a pulse of finite duration is estimated on the basis of the associated power spectrum. For short pulses, the time dependence of ionisation exhibits a departure from that expected of a time-independent rate.