Exact Solution of Quantum Optical Models by Algebraic Bethe Ansatz Methods
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 form5with.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)−2 (ω0 − 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 − 2cø†ø2(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.
KeywordsQuantum Inverse Rabi Splitting Conformal Matter Minimal Connection Integrable Lattice Model
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- 2.N.M. Bogoliubov, A.V. Rybin, R.K. Bullough and J. Timonen, The Maxwell-Bloch system on a lattice, Phys. Rev. A. To appear August 1995.Google Scholar
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- 5.N.M. Bogoliubov, R.K. Bullough and J. Timonen, Exact solution of generalised Tavis- Cummings models in quantum optics. To be published.Google Scholar
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