Exact Solution of Quantum Optical Models by Algebraic Bethe Ansatz Methods

  • R. K. Bullough
  • N. M. Bogoliubov
  • J. T. Timonen
  • A. V. Rybin
Conference paper


From long standing interests in solitons and integrable systems, e.g. SIT (1968– 74)1,2, “optical solitons” CQ04 (1977)3, we solve exactly, by algebraic Bettie ansatz (= quantum inverse) methods4, models of importance to quantum optics including the quantum Maxwell-Bloch envelope equations for plane-wave quantum self-induced transparency (SIT) in one space variable (x) and one time (t)2; and in the one tinte (t)5 a family of models surrounding and extending the Tavis-Cummings model6 of N 2-level atoms coupled to one cavity mode for ideal cavity (Q = ∞) QED. Additional Kerr type nonlinearities or Stark shifted levels can he incorporated into the Hamiltonian H of one of the most general models in the second case and H which we solve exactly, and from which revivals, evolution of photon statistics, etc., can be calculated, can be written in the form5
with.S ±, S z N-atom Dicke operators. When N = 1, (S z )2 = 1 and eqn. (1) is the.Jaynes-Cummings model with Kerr nonlinearity which is solved exactly. When γ = 0, H is the T-C model and when N = 1, 2, 3,..., results are important to the realised stochastic dynamics of the 85 Rb atom micromaser. In each case the problem is reduced to the solution of a set of polynomial equations typified by the form for eqn. (1) with γ = 0: and M is the total photon number while σ = 1, 2,..., K where K = min(N, M) + 1. Thus e.g. the “vacuum field Rabi splitting” given for M = 1 proves for eqn. (1) with γ ≠ 0 to be δE N,M +1 =2g[N+(2g)−20 − ω + γ − γN)2]1/2 for any N. Exact results also concern the quantum attractive nonlinear Schrödinger equation —i∂ø/∂t = ∂2ø/∂x 2 − 2cøø2(c < 0) which governs pulse propagation in optical fibres: c-number pulses arise as Lim n → ∞ on matrix elements 〈n, X′, t|ø(x)|n + 1, X, t〉, first found by Wadati (1984)7, but problems concerning the photon number n arise: these are that the attractive nonlinear Schrödinger equation (c < 0) has no stable ground state, although this can be stabilised by e.g. relativistic invariance whereby4 the quantum NLS model with c < 0 becomes the quantum sine-Gordon model which is exactly solved2,4.


Quantum Inverse Rabi Splitting Conformal Matter Minimal Connection Integrable Lattice Model 
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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • R. K. Bullough
    • 1
  • N. M. Bogoliubov
    • 2
    • 3
  • J. T. Timonen
    • 2
  • A. V. Rybin
    • 2
  1. 1.Department of MathsUMISTManchesterUK
  2. 2.Department of PhysicsUniversity of JyväskyläJyväskyläFinland
  3. 3.Steklov Mathematical InstituteSt. PetersburgRussia

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