N-Level Systems Interacting with Bosons: Semiclassical Limits

  • G. A. Raggio
Part of the NATO ASI Series book series (NSSB, volume 144)


We consider an N-level quantum system (S) coupled to a Boson quantum field. The latter is described by the symmetric Fock space F built upon a Hilbert space H. The unitary dynamical evolution of this whole is specified by the Hamiltonian
$$ {{\rm H}^\lambda } = 1 \otimes F + d\Gamma \left( h \right) \otimes 1 + \lambda Q\left( \xi \right) \otimes V, $$
on the tensor product FV, where V=ℂN is the Hilbert space of S. Here, F (a selfadjoint operator acting on V) is the free Hamiltonian for S, dΓ(h) is the second-quantization of a strictly positive selfadjoint operator h (defined on H) and describes the free evolution of the field, Q(\sd) is the field operator given by \( Q(f) = \left( {a(f) + a*(f)} \right)/\sqrt {{2,}} \), f∈HInline), where a(\sd) (resp. a*(.)) is the usual annihilation (resp. creation) operator defined on F and satisfying [a(f),a*(g)] = <f,g>1 (where <\sd,\sd> is the scalar product of H chosen linear in the second component), V is a selfadjoint operator acting on V, λ is the coupling constant, and finally \( \xi \in {\rm H} \) is such that Hλ gives rise to a selfadjoint operator.* This Hamiltonian has been studied and used very often in diverse physical contexts; a well known version thereof involving finitely many field-modes ** is
$$ {H^\lambda } = 1 \otimes F + \sum\limits_{j = 1}^K {{\omega _j}a_j^*{a_j}} \otimes 1 + \left( {\lambda /\sqrt 2 } \right)\sum\limits_{j = 1}^K {\left( {{\xi _j}a_j^* + {{\bar \xi }_j}{a_j}} \right) \otimes V,} $$
where aj (resp. \( a_J^{*} \)) is the familiar (harmonic oscillator) annihilation (resp. creation) operator associated with the j-th mode.


Coherent State Selfadjoint Operator Semiclassical Limit Quantum Statistical Property Markovian Master Equation 
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Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • G. A. Raggio
    • 1
  1. 1.Laboratorium für physikalische Chemie, ETH ZürichSwitzerland

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