Coherence and Quantum Optics VII pp 571-573 | Cite as

# Exact Solution of Quantum Optical Models by Algebraic Bethe Ansatz Methods

Conference paper

## Abstract

From long standing interests in solitons and integrable systems, e.g. SIT (1968– 74)
with.

^{1,2}, “optical solitons” CQ04 (1977)^{3}, we solve exactly, by algebraic Bettie ansatz (= quantum inverse) methods^{4}, 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 model^{6}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 form^{5}(1)

*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}*R*b 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 δ*E*_{ N,M }_{+1}=2*g*[*N*+(2*g*)^{−2}(ω_{0}− ω + γ − γ*N*)^{2}]^{1/2}for any*N*. Exact results also concern the quantum attractive nonlinear Schrödinger equation —*i*∂ø/∂*t*= ∂^{2}ø/∂*x*^{2}− 2*c*ø^{†}ø^{2}(*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 whereby^{4}the quantum NLS model with*c < 0*becomes the quantum sine-Gordon model which is exactly solved^{2,4}.## Keywords

Quantum Inverse Rabi Splitting Conformal Matter Minimal Connection Integrable Lattice Model
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## References

- 1.References in: A.I. Maimstov, A.M. Basharov, S.O. Elyutin and Yu. M. Sklyarov,“Present State of Self-induced Transparency Theory”,
*Physics Reports***191**:1 (1990).CrossRefGoogle Scholar - 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 - 3.R.K. Bullough and P.J. Caudrey, Optical solitons and their spill-wave analogues in 1íe, in: “Coherence and Quantum Optics IV”, L. Mandel and E. Wolf eds., Plenum Press, New York (1978).Google Scholar
- 4.R.K. Bullough and J. Timonen, Quantum and classical integrable models and statistical mechanics, in: “Proc. 7th Physics Summer School in Statistical Mechanics and Field Theory” (Canberra, Jan. 10-28, 1994 ) Vladimir V. Bazhanov ed., World Scientific Publ. Co. Pte. Ltd, Singapore (1995). To appear.Google Scholar
- 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
- 6.M. Tavis and F.W. Cummings,
*Phys. Rev*. 170:379 (1968); 188: 692 (1969).CrossRefGoogle Scholar - 7.M. Wadachi and Masa-aki Sagagimi,
*J. Phys. Soc. Japan*53: 1933 (1984).MathSciNetCrossRefGoogle Scholar - 8.R.K. Bullough and P.J. Caudrey, Solitons and the Korteweg-de Vries equation: integrable systems 1834-1995,
*Acta Applicandae Mathematicae*39: 193 (1995).MathSciNetCrossRefMATHGoogle Scholar

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