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Journal of the Optical Society of America B

Dynamical symmetries of the vacuum field Jaynes–Cummings model

Gui-Yu Hu, P. K. Aravind

Published Sep 1, 1989• 20 citations
Publisher

Abstract

We examine the dynamics of the Jaynes–Cummings model when it is restricted to the following initial conditions: atom in an arbitrary (pure or mixed) state and field in the vacuum state. It is shown that the model exhibits two dynamical symmetries that are characterized by the Lie algebras SU(2) × SU(2) and SU(3). A pair of interacting spins in a magnetic field also possesses the former symmetry, whereas the latter symmetry naturally describes a three-level atom driven by two laser fields; thus close correspondences with these systems are established. A pseudospin treatment of the model is devised and used to carry out a systematic investigation of its dynamics. On the basis of this treatment we (a) obtain the constants of motion of the problem, (b) establish bounds on the atomic and field inversions for arbitrary model conditions, (c) expose a striking symmetry between field and atomic (i.e., dipole) squeezing, and (d) discuss the occurrence of nonclassical correlations between the atom and the cavity field.

DOI
10.1364/josab.6.001757

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We examine the dynamics of the Jaynes–Cummings model when it is restricted to the following initial conditions: atom in an arbitrary (pure or mixed) state and field in the vacuum state. It is shown that the model exhibits two dynamical symmetries that are characterized by the Lie algebras SU(2) × SU(2) and SU(3). A pair of interacting spins in a magnetic field also possesses the former symmetry, whereas the latter symmetry naturally describes a three-level atom driven by two laser fields; thus close correspondences with these systems are established. A pseudospin treatment of the model is devised and used to carry out a systematic investigation of its dynamics. On the basis of this treatment we (a) obtain the constants of motion of the problem, (b) establish bounds on the atomic and field inversions for arbitrary model conditions, (c) expose a striking symmetry between field and atomic (i.e., dipole) squeezing, and (d) discuss the occurrence of nonclassical correlations between the atom and the cavity field.
We use the standard Jaynes-Cummings model with the field initially in the vacuum state and the atom in an arbitrary superposition state to illustrate the concept of dynamic symmetry in quantum optics. By dynamic symmetry we mean a situation in which a Hamiltonian is expressible as a linear combination of operators obeying a Lie algebra. This circumstance permits a psuedospin (or coherence vector) treatment to be given of the dynamics of the system. We give two such interpretations of the Jaynes-Cummings model. In the first we regard the atom and the field as a pair of interacting two-level systems and formulate a SU(2)-SU(2) dynamic symmetry. In the second we interpret the composite atom-field system is a three-level system and formulate a SU(3) dynamic symmetry. The former model illustrates the ideas of Fano1 and the latter those of Hioe and Eberly.2 We give a complete pseudospin description from both points of view and also obtain all the constants of motion.
We study the dynamics of a two-level atom interacting with a single mode of a damped cavity at 0 K when the cavity is initially in the vacuum state and the atom enters it in an arbitrary (pure or mixed) state. A complete analytical solution of this simple model is presented. On the basis of this solution we firstly investigate the pseudo-spin dynamics of the atom and the cavity field, secondly give an illustration of the Araki-Lieb theorem concerning the von Neumann entropies of interacting quantum systems and thirdly demonstrate the generation of entangled states of the atom and cavity field that are of interest in connection with the Bell inequalities.
Dynamics of entanglement due to intensity-dependent interaction between a two-level atom and a single-mode electromagnetic field in a Kerr medium is studied. The form of the interaction is such that the Hamiltonian evolution is exactly solvable. The Hamiltonian is shown to be a deformed Jaynes-Cummings model admitting a closed, symmetric algebra. Dynamics of population inversion and atom-field entanglement are studied taking the initial state of the field to be either a coherent state or a squeezed vacuum. Analysis is extended to the case of a two-mode cavity field interacting with a two-level atom. For the two-mode case, the initial field is a pair coherent state or a two-mode squeezed vacuum. Effects due to nonlinearity, intensity-dependent interaction and detuning on the dynamics are discussed and compared with those of the single-mode case.
We study interaction between a pair of indistinguishable two-level atoms and the single-mode cavity field. It is supposed that the pair of two-level atoms is laser cooled and trapped into the ground vibrational state, in which vibrational quantum number ⟨nv⟩=0\langle n_v\rangle =0⟨nv​⟩=0 . Two Jaynes–Cummings models are investigated. One is the Jaynes–Cummings model with intensity-dependent coupling and the another is the two-photon Jaynes–Cummings model of a pair of indistinguishable two-level atoms. It should be noted that in the present model, at initial moment t=0t=0t=0 , in intensity-dependent Jaynes–Cummings model the cavity field is prepared in Holstein–Primakoff SU(1,1) CS, while in two-photon Jaynes–Cummings model it is prepared in the squeezed vacuum state. Moreover, at initial moment t=0t=0t=0 , pair of atoms is supposed to be in the first excited state ∣e1⟩\vert e_1\rangle ∣e1​⟩ in both models. By using exact analytical solutions for state-vectors of the coupled atom-field systems amplitude-squared squeezing of the quantized cavity field is examined as a function of the ∣ξ∣\vert \xi \vert ∣ξ∣ parameter. In this situation, in both models higher-order squeezing has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the second-order squeezing.
The atomic pseudospin operators describing a two-level atom, and obeying the SU(2) Lie algebra, can exhibit reduced quantum fluctuations (“squeezing”) in certain states of the atom. These states include both pure and mixed states of the atom. Parameterizing the atom by the spherical polar components S, θ, and ϕ of its pseudospin vector S, we determine the parameter ranges in which squeezing can occur and the magnitude of this squeezing. We illustrate our considerations by examining two models, namely, the coherent-state Jaynes–Cummings model and the optical Bloch equations, in which mixed-state SU(2) squeezing arises dynamically.