### 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).

Original language | English (US) |
---|---|

Article number | 195319 |

Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Physical Review B - Condensed Matter and Materials Physics |

Volume | 70 |

Issue number | 19 |

DOIs | |

State | Published - Nov 2004 |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

**su(N,N) algebra and constants of motion for bosonic mean-field exciton equations.** / Yang, Z. S.; Kwong, Nai-Hang; Binder, Rudolf.

Research output: Contribution to journal › Article

*Physical Review B - Condensed Matter and Materials Physics*, vol. 70, no. 19, 195319, pp. 1-11. https://doi.org/10.1103/PhysRevB.70.195319

}

TY - JOUR

T1 - su(N,N) algebra and constants of motion for bosonic mean-field exciton equations

AU - Yang, Z. S.

AU - Kwong, Nai-Hang

AU - Binder, Rudolf

PY - 2004/11

Y1 - 2004/11

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=12344317996&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=12344317996&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.70.195319

DO - 10.1103/PhysRevB.70.195319

M3 - Article

AN - SCOPUS:12344317996

VL - 70

SP - 1

EP - 11

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 0163-1829

IS - 19

M1 - 195319

ER -