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Physical Review

Further Observations of Multiple-Quantum Transitions. Saturation Effects in Radio-Frequency Transitions

Polykarp Kusch

Published Jan 15, 1956• 49 citations
Publisher

Abstract

Multiple-quantum as well as the single-quantum transitions, $\ensuremath{\Delta}F=0$, $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1, 2, 3, 4$, have been observed in ${\mathrm{K}}^{39}$. In the neighborhood of the rf amplitude, required to give a maximum transition probability, the lines have the behavior predicted for transitions induced by a rectangular rf pulse. For higher rf amplitudes, the line width becomes constant or else decreases after having reached a maximum width, the line frequency is modified less rapidly than predicted and anomalous transition probabilities occur. All these effects are markedly dependent on the details of construction of the rf circuit in which transitions are induced. It is suggested that these effects may arise from the end regions of the circuit through an adiabatic traverse of the circuit by the atom.

DOI
10.1103/physrev.101.627

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We develop a detailed theory to calculate multiple-quantum transition probabilities. It is shown that the quantum electrodynamical (QED) approach must be used in order to explain quantitatively the experimental observations of Kusch. The anomalous behavior of shifts in resonance frequency and saturation of widths of transition probabilities can be explained on the basis of a QED renormalization principle. In this paper, we work out the detailed calculations for the case of a double-quantum transition quantitatively. The agreement between QED calculation and Kusch's observation is remarkable. We conclude that the renormalization principle of QED is important even for low-frequency and finite-intensity fields.
In an electron-spin two-level system a bichromatic radiation field, consisting of a transverse microwave field and a radio frequency field oriented parallel to the static magnetic field, can induce multiple photon transitions of the type ${\ensuremath{\sigma}}_{\text{mw}}^{+}+k{\ensuremath{\pi}}_{\text{rf}}$, with the photons ${\ensuremath{\sigma}}_{\text{mw}}^{+}$ and ${\ensuremath{\pi}}_{\text{rf}}$ absorbed from the microwave and radio frequency fields, and kkk being an integer. The two-level spin system may become transparent under such a bichromatic radiation field. This phenomenon directly depends on the presence of the $\ensuremath{\pi}$ photons of the longitudinal radio frequency field. An analytical description of the multiple photon processes and the resulting transparency effects, based on a toggling-frame approach, is given. The effective field amplitudes of the multiple photon transitions are found to have a Bessel-function-like dependence on the radio frequency amplitude. The behavior of the magnetization vector during a bichromatic pulse is illustrated by using classical equations of motion. The theoretically predicted effects are verified experimentally. As an example for an application of the investigated type of electromagnetically induced transparency, a one-pulse echo experiment is described, where the free evolution period is created artificially.
Power broadening-the broadening of the spectral line profile of a two-state quantum transition as the amplitude of the driving field increases-is a well-known and thoroughly examined phenomenon in spectroscopy. It typically occurs in continuous-wave driving when the intensity of the radiation field increases beyond the saturation intensity of the transition. In pulsed-field excitation, linear power broadening occurs for a pulse of rectangular temporal shape. Pulses with smooth shapes are known to exhibit much less power broadening, e.g., logarithmic for a Gaussian pulse shape. It has been predicted, but never experimentally verified, that pulse shapes which vanish in time as ∼|t|^{-λ} should exhibit the opposite effect-power narrowing-in which the postpulse transition line width decreases as the amplitude of the driving pulse increases. In this Letter, power narrowing is demonstrated for a class of powers-of-Lorentzian pulse shapes on the IBM Quantum processor ibmq_manila. Reduction of the line width by a factor of over 10 is observed when increasing the pulse area from π to 7π, in a complete reversal of the power broadening paradigm. Moreover, thorough study is conducted on the truncation of the pulse wings which introduces a (small) power-broadened term which prevents power narrowing from reaching extreme values. In the absence of other power broadening mechanisms, Lorentzian pulses truncated at sufficiently small values can achieve as narrow line profiles as desired.
The response of an overdamped current biased Josephson junction to a pulsed rf drive has been investigated by both numerical and analog simulations. The rf signal was a periodic rectangular pulse of varying amplitude and duration. A considerable increase of the width of the rf-induced steps with respect to those of an equivalent junction driven by a sinusoidal signal has been obtained. In the limiting case of a pulse repetition frequency much lower than the junction RC cutoff frequency, all steps have a maximum width which is independent of the step order n and increases with decreasing pulse duration.