# Jaynes–Cummings model and quantum chaos

Prof. Marek Kuś accepted the invitation on 7 July 2011

**Jaynes–Cummings model and quantum chaos** refers to an important model of quantum optics in a regime of quantum chaos

**Short note and Related References
added by Scholarpedia Editor D.Shepelyansky in May 2020 **

The Jaynes-Cummings model [1] is the cornerstone system of quantum optics describing interactions of resonator photons with an atom, considered in a two-level approximation. The usual experimental conditions correspond to a weak coupling constant between photons and atom. In this regime the quantum evolution of the system is integrable demonstrating revival energy exchange between photons and atom [1,2,3,4]. Such revival behavior had been first observed in experiments with Rydberg atoms inside a superconducting cavity [5].

At strong coupling the dynamics may become nontrivial with the emergence of classical chaos when the dynamics of oscillator is treated as classical and two-level atom as quantum [6] (see also [7]). However, for one atom even with a strong coupling to a quantum photonic field the evolution, spectrum and eigenstates are still rather simple due to a total energy balance [8],[9]. The regime of quantum chaos with the level spacing statisitcs of Random Matrix Theory appears only when several two-level atoms are interacting with the quantized photonic field of resonator as it was shown in [10].

## Related References

- E.T.Jaynes, F.W.Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser", Proc. IEEE. 51(1): 89 (1963)
- J.J.Sanchez-Mondragon, N.B.Narozhny, J.H.Eberly, "Theory of spontaneous-emission line shape in an ideal cavity", Phys. Rev. Lett. 51: 550 (1983)
- L.Allen, J.H.Eberly, "Optical resonance and two-level atoms", Dover Publs. Inc., New York (1987)
- M.O.Scully, M.S.Zubairy, "Quantum optics", Cambridge University Press, Cambridge, England, (1997)
- G.Rempe, H.Walther, N.Klein, "Observation of quantum collapse and revival in a one-atom maser", Phys. Rev. Lett. 58(4): 353 (1987)
- P.I.Belobrov, G.M.Zaslavskii, G.Kh.Tartakovskii, "Stochastic breaking of bound states in a system of atoms interacting with a radiation field", Sov. Phys. JETP 44(5): 945 (1976)
- J.R.Ackerhalt, P.W.Milonni, M.-L.Shin, "Chaos in quantum optics", Physics Reports 128(4-5): 205 (1985)
- R.Graham, M.Hohnerbach, "Two-state system coupled to a boson mode: Quantum dynamics and classical approximations", Z. Phys. B Cond. Mat. 57: 233 (1984)
- M.Kus, "Statistical properties of the spectrum of the two-level system", Phys. Rev. Lett. 54: 1343 (1985)
- R.Graham, M.Hohnerbach, "Statistical spectral and dynamical properties of two-level systems", Phys. Rev. Lett. 57: 1378-1378 (1986)