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Physical Review A

Semiclassical treatment of multiphoton interactions: ac Stark shifts

Thomas A. DeTemple, Samuel J. Petuchowski, Ho Kyoon Chung

Published Dec 1, 1980• 8 citations
Publisher

Abstract

The extraction of ac Stark shifts and splittings from a semiclassical treatment of multiple wave-atom interactions is considered. Within the rotating-wave and quasistatic approximations, we show that the information can be extracted from the complex Beer's coefficient sensed by a weak tunable probe wave, whose lower level is fully populated and unconnected to the strong fields. This situation leads to a great simplification of the density-matrix equations of motion, from which algorithms are derived for the algebraic construction of the Beer's coefficients appropriate to other cases not treated. Both level-shifted resonance and line-strength information are contained in the Beer's coefficient, many examples of which are given, some for the first time. Generalized sum rules for the transitions are derived along with a connection of this approach to other approaches including dressing transformations.

DOI
10.1103/physreva.22.2636

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The extraction of ac Stark shifts and splittings from a semiclassical treatment of multiple wave-atom interactions is considered. Within the rotating-wave and quasistatic approximations, we show that the information can be extracted from the complex Beer's coefficient sensed by a weak tunable probe wave, whose lower level is fully populated and unconnected to the strong fields. This situation leads to a great simplification of the density-matrix equations of motion, from which algorithms are derived for the algebraic construction of the Beer's coefficients appropriate to other cases not treated. Both level-shifted resonance and line-strength information are contained in the Beer's coefficient, many examples of which are given, some for the first time. Generalized sum rules for the transitions are derived along with a connection of this approach to other approaches including dressing transformations.
A signal-flow-graph representation of Schr\"odinger's equation appropriate to the analysis of multiphoton interactions in the semiclassical limit is described. Within the rotating-wave and quasistatic approximations the equations of motion for the density-matrix elements can be recast into a system of algebraic equations on the basis of which the signal flow graphs are constructed. By introducing selected graph manipulations, the graph can be simplified and direct extraction of information is facilitated by using standard graph algebra. Illustrations are given for typical cases involving three waves and three- or four-level systems for serial, parallel, and parametric interactions. Also, the construction of an ac Stark graph, which is specifically used for the determination of ac Stark shifts, is illustrated. We show that perturbation results for virtual transitions as well as for real transitions can be obtained directly from the graph.
A semiclassical treatment is given for the serial, near-resonant interaction of three waves in seven equivalent four-level atomic or molecular systems. Explicit equations are presented for the complex susceptibilties for the three waves, valid for $t\ensuremath{\gg}{T}_{2}$, which contain implicit ac Stark-shift and -splitting information, and can be separated into one-, two-, and three-photon contributions. For the case of a weak third or terminal wave, equations for the degree of ac Stark shifting and splitting are found and indicate that the three-photon or triple-resonance contribution is immune to the optical Stark effect if the remaining two waves are tuned to at least two-photon resonance conditions. The conditions for Doppler-free triple resonance are outlined, and it is shown that an equivalent ac Stark broadening may exist but may also be minimized by an appropriate choice of field intensities.
A semiclassical density matrix treatment is developed for the parallel near-resonant interaction of three waves in a four-level atomic or molecular system. Explicit solutions are developed and contain implicit AC Stark-shift and splitting information. The role of splitting and AC Stark-shifts is discussed for a system of two competing emitted waves starting from a common optically pumped level; a situation encountered in many optically pumped lasers. Application of the results to a Raman-Raman competition situation in ammonia is illustrated.
A nonperturbative semiclassical theory is presented for the treatment of four-wave mixing (FWM) in a four-level system. This treatment is based on a transformation to a generalized rotating frame followed by a numerical diagonalization of the resulting time-independent Hamiltonian. For strong fields, all resonant frequencies are shifted from the atomic (molecular) frequencies. Lines are split up in a way similar to the ac Stark effect for a two-level system, and for even stronger fields, the effect of stirring is observed for the first time in four-wave mixing. In addition, a new extra resonance appears which is induced by the saturating field. This field-induced resonance is the new member in the family of extra resonances (pressure induced, or stochastic fluctuation induced), and it should be observed whenever saturating fields interact with a multilevel atomic or molecular system.
A gas of two-level atoms is strongly saturated by a standing wave field. The same transition is probed by a weak tunable field. The observed probe absorption displays features due to both population modifications and coherences induced by the strong field. The probe spectrum is evaluated and interpreted in terms of coherent and incoherent multiphoton processes, power shifts and Stark splittings. Specific standing wave effects are identified and discussed. The relations to previous treatments are clarified.
For pt.I see Chem. Phys. Lett. vol.96, p.464 (1983). The non-linear multiphoton excitation dynamics of two-level systems under the influence of two strong linearly polarised monochromatic fields is studied exactly for the first time along the semiclassical many-mode Floquet theory. The present approach relaxes the restrictions of the conventional generalised rotating-wave approximation (GRWA) and allows correct treatment of various non-linear features such as power broadening, resonance shift, asymmetry in absorption lineshape, etc. By extending the nearly degenerate perturbative method of Salwen (1955), approximate analytical expressions for transition probabilities are obtained for multiphoton resonance processes for which (n+1) photons of one field are absorbed, whereas n photons of the other field are emitted, where n is an arbitrary integer. Detailed comparison of the analytical, the GRWA and the numerically exact results is given. Also presented is a case study of the time evolution of spin 1/2 systems in multi-quantum NMR transitions driven by pulsed bichromatic radiofrequency fields.