Quantum Mechanics of an Oscillator with Infinitesimal Frequency

S. G. Rajeev

Abstract The variational principle is formulated in first order form, deriving Hamilton's equations. Poisson brackets are found as generating infinitesimal canonical transformations. Symmetries are found to lead to conservation laws. A glimpse of the star product of quantum mechanics is given: the classical Poisson bracket is its infinitesimal approximation. In an exercise, the quantum harmonic oscillator is solved in the star product formalism.

Zhihao Xiao, T. Fuse, Sahel Ashhab +5 more

We study a qubit-oscillator system, with a time-dependent coupling coefficient, and present a fast scheme for generating entangled Schr\"odinger-cat states with large mean photon numbers and also a scheme that protects the cat states against dephasing caused by the nonlinearity in the system. We focus on the case where the qubit frequency is small compared to the oscillator frequency. We first present the exact quantum state evolution in the limit of infinitesimal qubit frequency. We then analyze the first-order effect of the nonzero qubit frequency. Our scheme works for a wide range of coupling strength values, including the recently achieved deep-strong-coupling regime.

Jeffrey Barrett, Isaac Goldbring

Abstract The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

Thomas A. Garrity