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Physical Review A

Accidental degeneracy beats: A distinct type of beat phenomenon in nonlinear optical spectroscopy

Kusai A. Merchant, David E. Thompson, M. D. Fayer

Published Jan 17, 2002• 22 citations
Publisher

Abstract

A type of beat in nonlinear optical spectroscopy that is distinct from quantum beats (QB's) and polarization beats, is described. Like a quantum beat, this beat, which we refer to as an accidental degeneracy beat (ADB), can only be seen in multilevel systems. However, unlike quantum beats, which are the result of intramolecular interferences, ADB's are interferences between different subensembles of molecules in the sample. They require multilevel systems with spectral overlap. ADB's can appear as separate frequencies or as phase and amplitude contributions with the same frequency as that of quantum beats. A procedure for distinguishing between quantum beats and ADB's is outlined, and criteria under which ADB's are expected to be observed are delineated. Calculations of the spectrally resolved stimulated vibrational echo signal from an inhomogeneously broadened coupled anharmonic oscillator system are presented to illustrate the differences between the two types of beats. ADB's carry information about the anharmonicity of a system, while QB's carry information about intramolecular correspondence of transition frequencies in a multilevel system.

DOI
10.1103/physreva.65.023817

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A type of beat in nonlinear optical spectroscopy that is distinct from quantum beats (QB's) and polarization beats, is described. Like a quantum beat, this beat, which we refer to as an accidental degeneracy beat (ADB), can only be seen in multilevel systems. However, unlike quantum beats, which are the result of intramolecular interferences, ADB's are interferences between different subensembles of molecules in the sample. They require multilevel systems with spectral overlap. ADB's can appear as separate frequencies or as phase and amplitude contributions with the same frequency as that of quantum beats. A procedure for distinguishing between quantum beats and ADB's is outlined, and criteria under which ADB's are expected to be observed are delineated. Calculations of the spectrally resolved stimulated vibrational echo signal from an inhomogeneously broadened coupled anharmonic oscillator system are presented to illustrate the differences between the two types of beats. ADB's carry information about the anharmonicity of a system, while QB's carry information about intramolecular correspondence of transition frequencies in a multilevel system.
We analyze theoretically the time-resolved signal detected in three-pulse, photon echo experiments where several independent two-level systems are involved, in a case when the time delay T between the two last pulses is much longer than delay $\ensuremath{\tau}$ between the two first pulses. A spectral diffusion process on a long time scale is included in the calculation of the $\ensuremath{\tau}$-dependent response signal. If spectral diffusion occurs between the different systems, the signal exhibits quantum beats instead of the polarization interferences observed without spectral diffusion. This analysis provides different insights on dynamical processes and spectroscopy, as demonstrated in examples of experimental results obtained on vibrational dynamics of molecules embedded in matrices.
We analyze theoretically the quantum beat and the polarization interference phenomena observed as oscillations in pulsed four-wave mixing. We show that resolution of the signal with respect to the delay between the incoming pulses as well as the detection frequency provides valuable information on the nature of the oscillations. Most importantly, it is very simple to distinguish between polarization interference from two independent optical transitions and quantum beats from a three-level system. In case of polarization interference, the phase of the oscillations shifts \ensuremath{\pi} when the detected frequency passes any of the resonances. In case of quantum beat, the phase has little dependence on the detected frequency near the resonance.
Quantum beats in the radiation field of "three-level" atoms are analyzed by means of a previously developed boson-second-quantization (BSQ) formalism. Two types of atoms are considered: type I, in which beats are due to transitions from each of two closely spaced upper levels to a single lower level, and type II, in which beats are due to transitions between a single upper level and each of two closely spaced lower levels. This classification is motivated by the claim found in the literature that semiclassical theory and quantum theory give qualitatively different results concerning the existence of beats in the case of type-II atoms. The terminology associated with beats is made precise by distinguishing between amplitude modulation at the beat frequency of the transition frequency field EEE, and the existence of a spectral component at the beat frequency in E2{E}^{2}E2. With respect to the former, no qualitative difference is found in the predictions from semiclassical theory and quantum theory; with respect to the latter, results that appear paradoxical from an intuitive viewpoint are discussed. It is shown that for the case in which the atoms are all in the same coherent superposition of energy states, with all levels occupied, semiclassical and quantum-mechanical results become equivalent as the number of atoms increases. Reference is made to the disturbance caused by the measurement processes in the case of a single atom. The essential difference in spontaneous emission beats between type-I and type-II atoms [with the ground state(s) unoccupied] is explained by the fact that the phase of the beat amplitude may be determined by preparation of the initial state in type-I atoms but may not be determined in type-II atoms.
We show that time-resolved quantum beat spectroscopy in the four-wave mixing (FWM) geometry is a unique tool to investigate the electronic multi-level system associated with spectrally adjacent excitonic resonances. We have studied heavy- and light-hole excitons, excitons associated with different and spatially separated quantum wells, and excitons associated with spatially separated growth islands within the same quantum well. Though the linear spectra are qualitatively similar in all three cases, the respective beating of the time-resolved FWM-signal shows a very different dynamic behaviour in each case. We discuss appropriate electronic multi-level schemes and show that exciton transitions within different quantum wells can be described by independent twolevel systems. In contrast, excitonic transitions associated with different and spatially separated growth islands having different quantum well thickness are not independent but coupled.
In recent years, transient four-wave mixing (FWM) techniques have been used to investigate the dephasing times of non-equilibrium carriers in semiconductor heterostructures by monitoring the decay of the macroscopic polarization. Initially, this decay was measured by temporally-integrating the scattered FWM signal as a function of time delay between two (or more) pump pulses. More recently, the macroscopic polarization decay has been measured by time-resolving the FWM signal by cross correlating it with an ultrashort laser pulse via frequency up-conversion in a nonlinear crystal. When excitons with slightly different energies are excited, oscillations (or beats) have been observed both in the time-integrated FWM signal and in the time-resolved FWM signal. Such beats have been observed, for example, between light- and heavy-hole excitons and between excitons in wells of different widths. Both the spectral behavior1 of the time-integrated FWM signal and the temporal behavior2 of the time-resolved signal have been shown to allow the distinction between the polarization interference associated with two independent oscillators and the quantum beats associated with two coupled oscillators which share a common level. In addition, it has been demonstrated3 that the quantum beats produced by incident pulses with parallel polarizations are exactly out of phase (by π) with the beats produced by orthogonally polarized pulses. In each case1-3, the period and phase of the beats and the polarization of FWM signal were explained by using a six-level model, which excluded many-body effects, to describe the J=1/2 conduction and J=3/2 valence states. Within the last few months, however, it has been shown that a complete description of the polarization selection rules for the FWM signal requires excitation-induced dephasing4 or disorder-induced coupling of the σ + and σ – excitonic transitions5.