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Journal of Physics B Atomic and Molecular Physics

Interaction of a decaying atom with a linearly polarized oscillating field

Mark P. Silverman, F. M. Pipkin

Published Oct 1, 1972• 43 citations
Publisher

Abstract

The Schrodinger equation for the interaction of a decaying two-level atom with an external oscillating field is separated via a series of matrix transformations into a time independent and a time dependent part; the former yields the radiofrequency Stark shift for decaying states, the latter is normally negligible for atomic beam resonance experiments. In the limit of stable states, or for the case of two levels of equal lifetime, the shift reduces to that of the Bloch-Siegert effect. Exact analytical expressions for the state amplitudes are obtained for the case when the time dependent matrix can be neglected; the resulting transition probabilities are investigated as functions of frequency, time, and interaction strength and are compared with the rotating field theory and the exact numerical solutions.

DOI
10.1088/0022-3700/5/10/013

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The Schrodinger equation for the interaction of a decaying two-level atom with an external oscillating field is separated via a series of matrix transformations into a time independent and a time dependent part; the former yields the radiofrequency Stark shift for decaying states, the latter is normally negligible for atomic beam resonance experiments. In the limit of stable states, or for the case of two levels of equal lifetime, the shift reduces to that of the Bloch-Siegert effect. Exact analytical expressions for the state amplitudes are obtained for the case when the time dependent matrix can be neglected; the resulting transition probabilities are investigated as functions of frequency, time, and interaction strength and are compared with the rotating field theory and the exact numerical solutions.
The time-dependent Schr\"odinger equation is solved formally for an atomic or molecular system which is subjected simultaneously to a rotating magnetic field of constant amplitude and angular velocity and to a constant magnetic field along the axis of rotation. The method yields the transition probabilities in terms of the solutions to an eigenvalue problem. This eigenvalue problem is solved both for (a) a normal Zeeman effect and for (b) the case where a transition from a given level is isolated in frequency from other transitions from the same level. Case (a) is exactly soluble and yields a solution which is shown to be the same as that of Bloch and Rabi, but is in a form which is more convenient for integration over the velocity distribution. Case (b) must be solved by an approximate method which results in a prediction of multiple quantum transitions as observed by Kusch.
The time evolution of a two-level system interacting with a classical monochromatic radiation field is calculated by Fourier transforming the equations of motion. This can be done exactly if one uses the convolution of the product of two Fourier transformations. The method produces recurrence relationships for the transforms. The time-dependent amplitudes of the states and the transition probability are evaluated by using two such recurrence relationships. The time-averaged probability shows characteristic resonances at odd multiples of the ratio of the natural frequency of the system to the frequency of the radiation field. The peaks of the resonances are displaced by the Bloch-Siegert shift. At these resonances the system is completely saturated with the flipping frequency dependent on the strength of the radiation field. The time dependence and the frequency spectrum of the dipole moment is determined at the main resonance and at the three-quantum resonance. The spectrum contains the zero-frequency terms which are responsible for the coherence resonances.
The problem of the interaction of a quantum system having discrete states, with a classical oscillating field, is reexamined as a problem in the solution of the time-dependent Schrodinger equation with a periodic Hamiltonian. A method is presented for approximating the time-dependent Hamiltonian with a time-independent one in the weak oscillating field case. With the aid of Floquet's theorem the problem is exactly converted to one with a time-independent Hamiltonian represented by an infinite matrix. The approximation of only two states then permits finding the resonance line shape by perturbation theory for both single and multiple quantum transitions with equal ease. The simple case of only two states connected by an off diagonal sinusoidal perturbation is studied in detail, and a complete description of the average transition probability is found for the strong oscillating field case. A few more complex cases are discussed. A deeper understanding of the analysis is obtained by examining the theory with the oscillating field quantized. Experimental verification of the theory could best be obtained by the methods of atomic beam spectroscopy at radio frequencies.
The Bloch-Siegert-type shifts and multiple-quantum resonance shifts of a spin-one system subjected to an oscillating radio-frequency applied at right angles to a DC field have been calculated to sixth order in V, the amplitude of the oscillating field, for the single-quantum resonance, and to fourth order in V for the higher resonances. The level crossing criterion appropriate to two-level systems cannot be used here, and a resonance condition based on a previously derived analogue of Shirley's equation for spin-one systems is used instead. The calculation uses degenerate continued-fraction perturbation theory. Numerical calculations were performed in order to obtain the shape of the resonances, and partially check the analytic results. The atomic levels were assumed to be completely stable; the corrections due to finite atomic lifetimes are expected to be small.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-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/cm2{\mathrm{cm}}^{2}cm2) 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/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.