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Physical Review

Effect of a Sinusoidal Excitation Amplitude on the Performance of an Atomic-Beam Spectrometer

J.H. Shirley

Published Aug 5, 1967• 7 citations
Publisher

Abstract

A theoretical analysis has been made of the transition probabilities for a Rabi-type atomic-beam spectrometer in which the exciting field amplitude seen by the atoms has a sinusoidal rather than a rectangular envelope. The time-dependent Schr\"odinger equation was integrated numerically and a velocity average of the transition probabilities performed. The results indicate that the linewidth increases as the fourth root of the excitation power and that frequency shifts due to coupling of the exciting field with other atomic states are reduced by an order of magnitude at excitation powers much above optimum.

DOI
10.1103/physrev.160.95

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A theoretical analysis has been made of the transition probabilities for a Rabi-type atomic-beam spectrometer in which the exciting field amplitude seen by the atoms has a sinusoidal rather than a rectangular envelope. The time-dependent Schr"odinger equation was integrated numerically and a velocity average of the transition probabilities performed. The results indicate that the linewidth increases as the fourth root of the excitation power and that frequency shifts due to coupling of the exciting field with other atomic states are reduced by an order of magnitude at excitation powers much above optimum.
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.
When atoms interact with a phase-modulated field, the probability of finding the atom in the excited-state oscillates at the second harmonic of the modulation frequency, $2{\ensuremath{\omega}}_{m}.$ The amplitude of this oscillating probability is a resonant function of the Rabi frequency \ensuremath{\Omega}, and this is termed a \ensuremath{\beta} Rabi resonance. In this work, we examine the line shape of the \ensuremath{\beta} Rabi resonance both theoretically and experimentally. We find that a small-signal theory of the \ensuremath{\beta}-Rabi-resonance condition captures much of the line shape's character, and, in particular, that the resonance's ``line Q'' (i.e., $2\ensuremath{\delta}{\ensuremath{\Omega}}_{1/2}/\ensuremath{\Omega})$ is proportional to the modulation frequency. This result can be applied to the atomic candle, where \ensuremath{\beta} Rabi resonances are employed to stabilize field strength. Considering our results in the context of developing an optical atomic candle, we find that a free-running diode laser's intensity noise could be improved by orders of magnitude using the atomic candle concept.
Abstract We consider the interaction of atomic hydrogen, in its ground state, with an electromagnetic pulse whose duration is fixed in terms of the number of optical cycles. We study the probability of excitation of the atom in the static field limit i.e. for field frequencies going to zero. Despite the fact that the well-known Born–Fock adiabatic theorem is valid only for a system whose energy spectrum is discrete, we show that it is still possible to use this theorem to derive, in the low frequency limit, an analytical formula which gives the probability of transition to any excited state of the atom as a function of the field intensity, the carrier envelope phase and the number of optical cycles within the pulse. The results for the probability of excitation to low-lying excited states, obtained with this formula, agree with those we get by solving the time-dependent Schrödinger equation. The domain of validity is discussed in detail.
The phase and the rotational averaging of single and multi-photon transition probabilities for the interaction of an atom or molecule with an applied sinusoidal field is discussed. It is shown that while the correct interval for carrying out the phase average is 2π in general, a phase interval of π is satisfactory under certain specified conditions. Illustrative two-level examples indicate that if these conditions are not satisfied, the two choices of phase interval can lead to significant differences in the phase averaged temporal transition probabilities which are subsequently essentially removed upon long-time averaging. A numerical scheme for evaluating rotational averaged transition probabilities is developed and illustrated for model calculations involving two-level molecular systems. The examples are chosen to illustrate the effects of rotational averaging, both when a molecule possesses non-zero diagonal dipole matrix elements (‘permanent dipole moments’) and when it does not, on single and multi-photon spectra. Often resonance profiles are sharpened, Bloch Siegert shifts and resonance maxima reduced, and the structure due to diagonal dipole matrix elemnts eliminated, upon rotational averaging. A recently developed rotating wave or Rabi type approximation is used to explain the effects of diagonal dipole matrix elements for both the fixed orientation and rotationally averaged spectra.
The effect of externally produced Stark splitting upon fine-structure-level probabilities is investigated in order to describe periodic intensity variations previously found in hydrogen lines. Coupled differential equations for the probability amplitudes of mixed levels are solved exactly in terms of initial amplitudes; effects due to spontaneous transition probabilities and Lamb shift are included. In general, the time-dependent line intensity is a combination of exponential decays and sinusoidal oscillations. The frequencies of oscillations are discussed in detail, and the effects of different initial conditions upon the intensities are considered for several special cases.