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IEEE Lasers and Electro-Optics Society Annual Meeting (LEOS)

Applications of nonperturbative nonlinear optics

Paul L. Kelley, David A. Coppeta, P. J. Harshman, O. Blum, T. Kenneth Gustafson

Published Dec 17, 2002• 0 citations
Publisher

Abstract

The Liouville equation for the density matrix can be recast so that the external field terms which couple density matrix elements are second order in the field. This approach is shown to have a profound effect on the analysis of strong field resonant nonlinear optical problems. First, in the case where parity or appropriate field restrictions apply, there is a reduction of the number of coupled equations which must be solved by at least a factor of two. Second, this reformulation naturally leads to radiative renormalization terms which are directly related to saturation, Stark shifts and Rabi splittings. The renormalization is equivalent to a Dyson equation analysis. We use this simple formalism to obtain exact solutions to a number of nonlinear optical problems including cases where there are two or more strong, resonant fields. These problems include: 1) two levels, two strong fields, 2) three levels, two strong fields, and 3) four levels, three and four strong fields. Discussed in detail are the two level case and the three and four level cases where weak fields couple the ground state to other states. We use these results to discuss the physics of optically induced transparency, population beats, optical switching in quantum wells, and fully resonant coherent anti-Stokes Raman scattering (CARS).

DOI
10.1109/leos.1994.586332

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The Liouville equation for the density matrix can be recast so that the external field terms which couple density matrix elements are second order in the field. This approach is shown to have a profound effect on the analysis of strong field resonant nonlinear optical problems. First, in the case where parity or appropriate field restrictions apply, there is a reduction of the number of coupled equations which must be solved by at least a factor of two. Second, this reformulation naturally leads to radiative renormalization terms which are directly related to saturation, Stark shifts and Rabi splittings. The renormalization is equivalent to a Dyson equation analysis. We use this simple formalism to obtain exact solutions to a number of nonlinear optical problems including cases where there are two or more strong, resonant fields. These problems include: 1) two levels, two strong fields, 2) three levels, two strong fields, and 3) four levels, three and four strong fields. Discussed in detail are the two level case and the three and four level cases where weak fields couple the ground state to other states. We use these results to discuss the physics of optically induced transparency, population beats, optical switching in quantum wells, and fully resonant coherent anti-Stokes Raman scattering (CARS).
The Liouville equation for the density matrix can be recast so that the external-field terms which couple density-matrix elements are second order in the field. This approach is shown to have a profound effect on the steady-state analysis of strong-field resonant nonlinear optical problems. First, in the case where parity or appropriate field restrictions apply, there is a reduction of the number of coupled equations which must be solved by at least a factor of 2. Second, this reformulation naturally leads to radiative-renormalization terms which are directly related to saturation, Stark shifts, and Rabi splittings. We can use this simple formalism to obtain solutions to a number of resonant nonlinear problems including cases where there are two or more strong fields. The renormalization described here is equivalent to a Dyson-equation analysis.
A diagrammatic density matrix technique is developed and applied to study nonlinear optical processes involving responses from weak probe fields interacting with a system under strong excitation. These diagrams consist of lowest-order double-Feynman diagrams which represent the density matrix perturbation terms describing the system response to the weak probe fields1,2 combined with dressed propagators which represent the density matrix elements of the system under strong excitation. The dressed propagators are described by a set of linear equations which, in most cases, may be derived in inspection. The methodology is similar to the bare-atom approach.2 The advantages of using this diagrammatic technique are that (1) it allows one to select and express analytically the dominant and appropriate density matrix elements (diagrams and dressed propagators); (2) analytical expressions may be constructed directly from the diagrams and thus lengthy calculations needed to obtain closed-form solutions are greatly reduced; (3) it provides a physical interpretation for the density matrix solutions and nonlinear optical processes; and (4) it may be generalized to more complicated, multilevel systems. We demonstrate the application of this technique to describe gain saturation, Stark splitting, and Raman scattering.
We have formulated expressions providing the number of terms (Y-G diagrams)1contributing to an n-th order susceptibility of a non-linear optical process. Obtaining an accurate count of these terms is important for diverse reasons: a) To accurately specify the number of terms needed to evaluate the nonlinear optical susceptibility either analytically or numerically, b) To help develop sequences of diagrammatic contributions which can be summed to provide analytical results for the strong interaction limit. An example is the two-level system for semiclassical and quantized fields, c) To treat two or more processes (terms) which could give rise to interference, resulting, for example, in extra resonances due to a lack of complete coherence, d) To generalize results from quantum electrodynamics to situations for which multiple photon interactions and zero point fluctuations associated with non-vacuum states need to be considered. We shall discuss the number of possible diagrams in terms of three factors, which we call a permutation factor relating to the number of possible frequencies that can be assigned to each interaction vertex, a commutation factor relating to interaction vertex assignment on the two sides of the diagram, and a state factor relating to the number of diagrams generated from an enumeration of specific material state assignments between interaction vertices. We will discuss the application of these results to several situations of interest including second harmonic generation, degenerate four-wave mixing, and resonant CARS (coherent anti-Stokes Raman scattering) in a four-level system. The relevance of the present results to experimental results which have been obtained will be discussed along with future directions which need to be addressed including the summability of diagrams under specific approximations, the calculation of expectation values, and the nonperturbative representation of processes.
Calculations are made to determine the response (power absorbed, magnetization, etc.) of a multilevel quantum system under the influence of any number of classically described driving fields (well defined in amplitude and phase). A nonperturbative theory is developed to account for those fields that are in resonance with the system. The nonresonant effects, such as virtual energy shifts, are included in the theory. We limit ourselves to a discussion of quantum systems that are in weak contact with a thermal reservoir. The procedure is the following A projection-operator formalism is developed which separates the density matrix of a system into two parts. One part is related to those portions of the driving fields that interact quite strongly with the system. The other part is the nonresonant or weakly interacting portion. An exact equation of motion, which includes the effects of the nonresonant part, is derived for the strongly interacting part of the density matrix. It is then shown that when relaxation can be represented by phenomenological decay constants, a unitary transformation can be found which reduces the equation of motion for the strongly interacting part of the density matrix so that an exact solution is possible. This solution is shown to give the major effects of the driving fields and can be used as the zeroth-order solution in a perturbation treatment of the whole density matrix. Frequency shifts such as the Bloch-Siegert shift and shifts due to couplings between the levels of interest (resonance levels) and other levels (nonresonance levels) all arise in the same natural way in our treatment. Our results in many instances are the extensions of the well-known results for a two-level quantum-mechanical systemm to an NNN-level one, and include resonant multiple quantum effects. As an example, a straightforward application of the method is carried out in order to compute the shifts in the resonance frequencies of a four-level system, driven by two oscillating fields connecting three of the levels, due to the counter-rotating components of the driving fields and the presence of the fourth level, which is not coupled to the other levels by the driving fields.
A nonperturbative theoretical study of four-wave-mixing processes in four-level systems is presented. Using the density-matrix formalism, numerical results are obtained for two arbitrarily intense pump fields with the third field a weak probe. The fourth wave, radiated by the system, is obtained analytically for some special cases. We find, for different frequency detunings, Rabi frequencies, and relaxation rates that the line shapes exhibit: (i) dynamic Stark splitting and shifting, and merging of resonances; (ii) power broadening with two-photon antiresonance behavior resulting from a destructive interference between different contributions to the optical polarization; and (iii) extra resonances induced by field intensities and saturation processes.