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physica status solidi (b)

Sum Rules for Nonlinear Optical Susceptibilities

F. Bassani, Sandro Scandolo

Published Sep 1, 1992• 9 citations
Publisher

Abstract

Abstract New sum rules are obtained for the dispersive and dissipative parts of the nonlinear susceptibilities, applicable to pump and probe experimental situations. A three‐level model shows how they clarify the nonlinear lineshapes. A detailed calculation on atomic hydrogen shows how the sum rules allow a global estimate of the magnitude of nonlinear effects.

DOI
10.1002/pssb.2221730126

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Nonlinear sum rules: The three-level and the anharmonic-oscillator models

Sandro Scandolo, F. Bassani

Recently derived sum rules for nonlinear optics are shown to be relevant to a number of experiments in atomic and solid-state physics. To interpret the experiments on atomic transitions and exciton transitions in solids, we analyze the three-level model and examine the modification of the optical spectrum induced on a probe light by the presence of an additional intense light beam. We show that the nonlinear sum rules explain the known experimental data and predict anomalous asymmetry in the nonlinear line shape. We also consider the cases of vibrational states of diatomic molecules and of optical modes of polar crystals and treat them with the model of the anharmonic oscillator. The nonlinear sum rules are shown to be obeyed, thus clarifying the processes of nonlinear absorption.

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Pump and Probe nonlinear processes: new modified sum rules from a simple oscillator model

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The nonlinear oscillator model is useful to basically understand the most important properties of nonlinear optical processes. It has been shown to give the correct asymptotic behaviour and to provide the general feature of harmonic generation to all orders, in particular dispersion relations and sum rules. We investigate the properties of Pump and Probe processes using this model, and study those cases where general theorems based on the holomorphic character of the Kubo response function cannot be applied. We show that it is possible to derive new sum rules and new Kramers-Kronig relations for the two lowest moments of the real and imaginary part of the third order susceptibility and that new specific contributions become relevant as the intensity of the probe increases. Since the analitic properties of the susceptibility functions depend only upon the time causality of the system we are confident that these results are not model depemdent and therefore have a general validity, provided one substitutes to the equilibrium values of the potential derivatives the density matrix expectation values of the corresponding operators.

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In earlier chapters we have considered in detail the susceptibility formalism of nonlinear optics, which is perhaps the most familiar approach and has a wide range of application. Starting from the constitutive relations of Chapter 2, the susceptibility formalism is quite general. In many practical applications, a particular phenomenon can be described accurately by a single order of nonlinearity, and the susceptibility then provides a useful and convenient description. However, this is not always the case. Some of the most interesting phenomena in nonlinear optics involve close resonance with the transition frequencies of the medium, and perhaps also the use of very intense optical fields. As remarked in §4.5, in these circumstances the resonant susceptibilities display mathematical divergences which are clearly unphysical. Strictly, these divergences occur only because higher-order nonlinearities have been neglected. Successive orders of nonlinearity take into account such effects as saturation, power broadening and level shifts (optical Stark effect). For intense fields or very close resonance, the contributions from several orders of nonlinearity may be comparable in magnitude. Therefore, despite its generality, the susceptibility formalism does not necessarily provide the most practical approach for the description of resonant processes. This chapter is concerned with the problem of deriving alternative and more manageable descriptions of resonant nonlinear processes, illustrated by examples. For much of the chapter, the nonlinear medium is treated as a two-level system (described in §6.2).

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f -Sum Rules for Dissipative Systems

Xinxin Yang, Bohao Wu, Yu Chen +1 more

The f-sum rules set general constraints on the response of a quantum many-body system to an external probe and hold significant relevance in the realm of various spectroscopy measurements. In practical conditions, a system unavoidably couples with the environment and acquires effective dissipation. In this Letter, we derive and prove a set of f-sum rules for dissipative systems. Within the framework of linear response theory, we obtain the system response in both linear order of probe field and dissipation parameter. We formulate and prove one first-order and two second-order dissipative f-sum rules. These rules are validated numerically for some example models, and the realization schemes are proposed. In addition, the potential applications are discussed for two interacting many-body systems.

Physical Review Letters•Sep 4, 2024•2 citations
Abstract New sum rules are obtained for the dispersive and dissipative parts of the nonlinear susceptibilities, applicable to pump and probe experimental situations. A three‐level model shows how they clarify the nonlinear lineshapes. A detailed calculation on atomic hydrogen shows how the sum rules allow a global estimate of the magnitude of nonlinear effects.
Recently derived sum rules for nonlinear optics are shown to be relevant to a number of experiments in atomic and solid-state physics. To interpret the experiments on atomic transitions and exciton transitions in solids, we analyze the three-level model and examine the modification of the optical spectrum induced on a probe light by the presence of an additional intense light beam. We show that the nonlinear sum rules explain the known experimental data and predict anomalous asymmetry in the nonlinear line shape. We also consider the cases of vibrational states of diatomic molecules and of optical modes of polar crystals and treat them with the model of the anharmonic oscillator. The nonlinear sum rules are shown to be obeyed, thus clarifying the processes of nonlinear absorption.
The nonlinear oscillator model is useful to basically understand the most important properties of nonlinear optical processes. It has been shown to give the correct asymptotic behaviour and to provide the general feature of harmonic generation to all orders, in particular dispersion relations and sum rules. We investigate the properties of Pump and Probe processes using this model, and study those cases where general theorems based on the holomorphic character of the Kubo response function cannot be applied. We show that it is possible to derive new sum rules and new Kramers-Kronig relations for the two lowest moments of the real and imaginary part of the third order susceptibility and that new specific contributions become relevant as the intensity of the probe increases. Since the analitic properties of the susceptibility functions depend only upon the time causality of the system we are confident that these results are not model depemdent and therefore have a general validity, provided one substitutes to the equilibrium values of the potential derivatives the density matrix expectation values of the corresponding operators.
It is explicitly shown, for optical processes arbitrarily comprising two-, three- or four-photon interactions, that the sum over all matter states of any optical susceptibility is exactly zero. The result remains true even in frequency regions where damping is prominent. Using a quantum electrodynamical framework to render the photonic nature of the fundamental interactions, the result emerges in the form of a traceless operator in Hilbert space. The generality of the sum rule and its significance as a thermodynamic limit are discussed, and the applicability to real systems is assessed.
In earlier chapters we have considered in detail the susceptibility formalism of nonlinear optics, which is perhaps the most familiar approach and has a wide range of application. Starting from the constitutive relations of Chapter 2, the susceptibility formalism is quite general. In many practical applications, a particular phenomenon can be described accurately by a single order of nonlinearity, and the susceptibility then provides a useful and convenient description. However, this is not always the case. Some of the most interesting phenomena in nonlinear optics involve close resonance with the transition frequencies of the medium, and perhaps also the use of very intense optical fields. As remarked in §4.5, in these circumstances the resonant susceptibilities display mathematical divergences which are clearly unphysical. Strictly, these divergences occur only because higher-order nonlinearities have been neglected. Successive orders of nonlinearity take into account such effects as saturation, power broadening and level shifts (optical Stark effect). For intense fields or very close resonance, the contributions from several orders of nonlinearity may be comparable in magnitude. Therefore, despite its generality, the susceptibility formalism does not necessarily provide the most practical approach for the description of resonant processes. This chapter is concerned with the problem of deriving alternative and more manageable descriptions of resonant nonlinear processes, illustrated by examples. For much of the chapter, the nonlinear medium is treated as a two-level system (described in §6.2).
The f-sum rules set general constraints on the response of a quantum many-body system to an external probe and hold significant relevance in the realm of various spectroscopy measurements. In practical conditions, a system unavoidably couples with the environment and acquires effective dissipation. In this Letter, we derive and prove a set of f-sum rules for dissipative systems. Within the framework of linear response theory, we obtain the system response in both linear order of probe field and dissipation parameter. We formulate and prove one first-order and two second-order dissipative f-sum rules. These rules are validated numerically for some example models, and the realization schemes are proposed. In addition, the potential applications are discussed for two interacting many-body systems.