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Physical Review B

su(N,N) algebra and constants of motion for bosonic mean-field exciton equations

Zhenshan S. Yang, N. H. Kwong, Rolf H. Binder

Published Nov 15, 2004• 2 citations
Publisher

Abstract

The ultrafast (picosecond) coherent dynamics of exciton systems in semiconductors can be approximately described by bosonic mean-field equations. These equations are nonlinear and therefore difficult to solve analytically. It is thus important to study the general dynamical properties of these equations, such as the underlying symmetry and corresponding conservation laws. It is shown in this paper that, for an $N$-species exciton system (e.g., heavy-hole and light-hole excitons), a mean-field Hamiltonian (including the coupling to external fields and fermionic corrections) can be formulated which is a member of the $su(N,N)$ algebra. As a consequence, the equations of motion for the center-of-mass momentum dependent exciton distribution and the coherent biexciton amplitude can be cast into a form similar to that of the optical Bloch vector in two-level atoms that belong to the algebra $su(2)$ [or, more generally, $N$-level atoms with algebra $su(N)$]. It is shown that the analog to the Bloch sphere in $N$-level atoms is an unbounded hypersurface (generalized hyperboloid) that constrains the motion of the exciton distribution and coherent biexciton amplitude. Further constants of motions that constrain the motion on the hypersurface are found from an $su(N,N)$ generalization to the Hioe-Eberly method in $su(N)$ systems ($N$-level atoms) [F. Hioe and J. Eberly, Phys. Rev. Lett. 47, 838 (1981)].

DOI
10.1103/physrevb.70.195319

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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.
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The availability of femtosecond pulses in the optical-frequency regime allows the study of coherent effects in the interaction between semiconductors and light. Because the properties of the exciting pulses become important ingredients in spectroscopy, the theoretical analysis must treat the complete time dependence of the system. For this purpose we use the semiconductor Bloch equations (SBE),1-3 which are the coupled equations of motion for the interband polarization and the electron and hole populations for an interacting electron-hole system in the random-phase approximation. The equations describe the electron-hole Coulomb attraction leading to excitonic features; they contain phasespace filling and band-gap renormalization effects, as well as the dipole interaction with the applied light field.
Summary form only given. The optical properties of semiconductor quantum wells (QW) and, in particular, the coherent dynamics of excitons following resonant excitation with ultrafast laser pulses have attracted much attention in recent years. It is generally accepted that there is a transfer of coherence between the optical field and the QW that disappears in a characteristic time T/sub 2/ (picoseconds for GaAs) after the laser is turned off. However, the questions as to how the coherence is actually induced and that of the nature of the coherent state of the solid are poorly understood. In this work we address these points by re-examining the longstanding problem of the (classical vs. quantum) nature of the ubiquitous beats associated with the light-hole (LX) heavy-hole (HX) excitons, which are observed in transient optical experiments on QW. To this end, we consider the coherent behavior of excitons using the simplest albeit non-trivial model where they are treated as non-interacting bosons. Our results bear directly on studies for which nonlinear effects are not important and are not aimed directly at explaining the fourwave-mixing (FWM) experiments that dominate the field. Nevertheless, since nonlinear effects are, typically, weak compared with harmonic contributions, the free boson picture provides in all cases the correct lowest order wavefunction of the photoexcited solid.