Quantum spins and their approach to the classical limit

Élliott H. Lieb

Élliott H. Lieb

Élliott H. Lieb

L. E. Ballentine

A classical spin, dynamically driven by a polarized beam, is shown to exhibit a complex variety of behaviors, incuding stable fixed points, bifurcations, limit cycles, and chaos. In the classical (large-spin) limit of the corresponding quantum model, the probability distributions for the spin components converge to their classical counterparts, even though the classical motions are qualitatively different for different ranges of parameters. However, the nature of the quantum-to-classical limit has some unexpected features, which are discussed.

L. Bakemeier, Andreas Alvermann, Holger Fehske

We study the quantum phase transition of the Dicke model in the classical oscillator limit, where it occurs already for finite spin length. In contrast to the classical spin limit, for which spin-oscillator entanglement diverges at the transition, entanglement in the classical oscillator limit remains small. We derive the quantum phase transition with identical critical behavior in the two classical limits and explain the differences with respect to quantum fluctuations around the mean-field ground state through an effective model for the oscillator degrees of freedom. With numerical data for the full quantum model we study convergence to the classical limits. We contrast the classical oscillator limit with the dual limit of a high-frequency oscillator, where the spin degrees of freedom are described by the Lipkin-Meshkov-Glick model. An alternative limit can be defined for the Rabi case of spin length one-half, in which spin frequency renormalization replaces the quantum phase transition.

L. E. Ballentine

Classical and quantum models of a localized spin driven by a polarized-spin beam have been shown to exhibit a rich variety of dynamical behaviors. In the classical model these include stable limit points, limit cycles, and chaos; however, in the quantum model the density matrix of the localized spin generally approaches a steady state. This paradox is resolved by considering the autocorrelation functions of the spin components, which reveal the expected quantum-classical correspondence. The noncommuting of the limits j\ensuremath{\rightarrow}\ensuremath{\infty} and t\ensuremath{\rightarrow}\ensuremath{\infty} is explained in this model by the limiting behavior of certain eigenvalue distributions, and the significance of these limits is discussed. The study also highlights the unresolved problem of interpreting the complex correlation functions of noncommuting operators.