The Jaynes-Cummings Model

The applet on this page uses the Bloch sphere and the Q-function to visualize the evolution of a coupled quantum two-level system and harmonic oscillator, with the Hamiltonian

                (         )
H  = ˉhΔ σz + ˉhg- a†σ + aσ†
     2        2

This is very similar to the Hamiltonian used in the “Bloch Equations” applet, except that the classical field Ω has been replaced by a harmonic oscillator annihilation operator a (and similarly Ω* has been replaced by a creation operator a), times a “coupling constant” -2g. The harmonic operator represents a single mode of the quantized electromagnetic field, which we take to start in a coherent state ∣α 〉 , where α is a dimensionless (complex) amplitude such that α2 equals the average number of photons in the field. All these parameters can be set using the input fields below. (As usual, “Theta” and “Phi” are used to specify the initial state of the two-level system on the Bloch sphere.)

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It is not recommended to use a very large value for α as this may result in overflow errors.

The time evolution of the two-level system (for instance, a two-level atom) is depicted in the Bloch sphere as for the other applets in this site. A comparison with the “Bloch Equations” applet can be made if Ω in that applet is set equal to a constant, Ω = -2αg. The differences between the two cases (classical, externally prescribed field versus quantized, dynamically changing field) become apparent after a short time.

Note that for some initial conditions the trajectory collapses to near the center of the Bloch sphere whereas for some others it stays near the surface of the sphere, in an “almost pure” state, for a long time. The collapse is associated with a splitting of the Q function into two peaks, and with atom-field entanglement. This behavior is explained in detail in the paper by J. Gea-Banacloche, Phys. Rev. A 44, 5913 (1991) (see also J. Gea-Banacloche, Opt. Commun 88, 531 (1992), for a study of the case with nonzero detuning Δ). For sufficiently long times, a partial “revival” of the state purity is observed. (The collapse and revival phenomenon in the JCM was first observed by J. H. Eberly, N. B. Narozhny and J. J. Sánchez Mondragón, Phys. Rev. Lett. 44, 1323 (1980).)

The Q-function (sometimes called the Husimi function) can be thought of as a sort of joint probability distribution for the position and momentum of an oscillator. It is related to, but generally easier to calculate than, the Wigner distribution. A number of Web sites show Wigner distributions for various states of the quantized radiation field; see here and here for examples and further explanations.

The JCM was first introduced by E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).

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