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Journal of the Optical Society of America B

Third-order optical response of intermediate excitons with fractional nonlinear statistics

Vladimir Chernyak, Shaul Mukamel

Published Jun 1, 1996• 48 citations
Publisher

Abstract

The interplay of exciton statistics and Coulomb interactions in the optical response of semiconductors is studied by derivation of an effective Hamiltonian written in terms of exciton operators, starting with a two-band model. Statistical effects are incorporated through the nonboson commutation relations of exciton operators, which contain an exciton-size-dependent parameter q that may vary from q = 0 (boson statistics) through q ≪ 1 (Wannier excitons) to q = 2 (Frenkel excitons or Pauli statistics). A unified Green’s function expression for χ(3) that applies to excitons of an arbitrary nature is derived.

DOI
10.1364/josab.13.001302

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The interplay of exciton statistics and Coulomb interactions in the optical response of semiconductors is studied by derivation of an effective Hamiltonian written in terms of exciton operators, starting with a two-band model. Statistical effects are incorporated through the nonboson commutation relations of exciton operators, which contain an exciton-size-dependent parameter q that may vary from q = 0 (boson statistics) through q ≪ 1 (Wannier excitons) to q = 2 (Frenkel excitons or Pauli statistics). A unified Green’s function expression for χ(3) that applies to excitons of an arbitrary nature is derived.
We here propose a new procedure to derive a general expression of the third-order susceptibility valid for any semiconductor Hamiltonian. It relies on our theory for composite exciton many-body effects which is quite appropriate to this problem since all optical nonlinearities in semiconductors come from exchange scatterings with the virtual excitons coupled to photons. Our expression of χ (3) is sample volume free, as physically expected. It also contains a Pauli term (exchange without Coulomb) which is missed by all bosonized exciton approaches. This term, dominant at large detuning, is a direct manifestation of the exciton composite nature.
The electron-hole pair operator commutation relations for a semiconductor \nare explicitly derived showing that the pairs are quasi-bosons \nwith statistics intermediate between bosons and fermions. These pair \noperators are transformed into exciton operators, that incorporate the \nintra-pair Coulomb interaction, and it is found that the commutation \nrelations remain quasi-bosonic. The exciton dynamics written in terms \nof the exciton operators are derived in the Heisenberg picture and \ncompared with the traditionally used semiconductor Bloch equations.
We develop a heirarchy of equations that describes the electrodynamics of the semiconductor band edge. An exciton basis that incorporates intrapair Coulomb interaction is used. The excitons are treated as quasibosons that satisfy commutation relations intermediate between bosonic and fermionic. Their nonlinear dynamics are a consequence of phase-space filling and an effective exciton-exciton interaction that depends only on their center-of-mass wave vectors and internal quantum numbers. The resulting expansion in powers of the optical field is analogous to Axt and Stahl's dynamically controlled truncation, but takes a simpler form. If damping is omitted or if all interband and intraband dephasing time constants are the same, we find a set of relationships among the interband and intraband correlation functions which in the free pair basis give the semiconductor Bloch equations.
The enhancement of /spl chi//sup (3)/(-/spl omega/;/spl omega/,-/spl omega/,/spl omega/) and switching rate has been confirmed under resonant pumping of coherent excitons. We propose theoretically the following nonlinear optical responses of the exciton: (1) Superradiance of Frenkel excitons and excitonic n-strings; (2) Frenkel and Wannier excitons show different characteristics of the third-order optical responses /spl chi//sup (3)/(-/spl omega/;/spl omega/,-/spl omega/,/spl omega/); (3) Generation of phase-conjugated wave shows singular spectrum near mobility edge of the exciton in a bulk crystal.>
Summary form only given. Large-radius excitons in semiconductors suffer strong Coulomb interaction on close approach. This is a distinctive difference in comparison with two-level systems in quantum optics. A mean field theory of interacting excitons in semiconductors leads to purely repulsive interaction, and does not account for spin-dependent polarization mixing and the formation of excitonic molecules (biexcitons). We have developed a theory for the nonlinear optical response, which can account for both. The coherent polarization is driven by exciton-exciton correlation, which can be expressed in terms of spin-dependent retarded memory functions. The effective equation is a generalized Gross-Pitaevskii equation.
Summary form only given. While the conceptual realization of "3-level effects" in semiconductors is usually based on one-particle states given by the band structure of the system, we propose the realization of "3-level effects" in semiconductors based on many-particle states. Specifically, we study the quantum coherences related to excitons and biexcitons. Our analysis is based on the dynamics-controlled trunction (DCT) formalism and includes quantum correlations and quantum coherences within the 3rd-order nonlinear optical response. Within the /spl chi//sup (3)/ regime, only one-pair (excitonic) and two-pair (two-exciton) quantum coherences occur. We restrict the analysis to 1s-hh and 1s-lh excitons which dominate the nonlinear response. Results are shown for ZnSe quantum wells.