Nonlinear Schrödinger equation and two-level atoms

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Abstract
Starting with the same form of atomic nonlinearity, Weinberg [Ann. Phys. (NY) 194, 336 (1989)] and Wódkiewicz and Scully [Phys. Rev. A 42, 5111 (1990)] obtained contradictory results concerning an evolution of the atomic inversion w in a two-level atom in Weinberg’s nonlinear quantum mechanics: If the atom is initially in a ground state then either the evolution of w (1) can be linear if one uses a nonlinear generalization of the Jaynes-Cummings Hamiltonian, or (2) is always nonlinear if one uses the nonlinear Bloch equations derived from the nonlinear atomic Hamiltonian function. It is shown that the difference is rooted in inequivalent descriptions of the composite ‘‘atom-plus-field’’ system. The linear evolution of w results from a ‘‘faster-than-light communication’’ between the atom and the field. If one applies a description without the ‘‘faster-than-light telegraph’’ then the calculations based on a suitably modified Jaynes-Cummings Hamiltonian lead to the same dynamics of w as is found in semiclassical calculations based on Bloch equations. It is shown also that a nonlinear quantum mechanics based on a nonlinear Schrödinger equation does not possess a natural probability interpretation.
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