Phase spaces, from classical to quantum mechanics, and back

Stefan Berceanu

N. Giovannini

H A Kastrup

The problem of quantizing phase and modulus associated with the polar coordinates of a 2-dimensional phase space dates back to the first years of quantum mechanics: The symplectic space P = { φ ∈ ( -π,+π],p > 0} has the global structure S 1 × R + and cannot be quantized in the conventional naive way. The appropriate method is the group theoretical quantization which here leads to the group 50(1,2) and the positive discrete series of its irreducible unitary representations. The basic classical observables p cosφ.p sinφ and p correspond to the self-adjoint operators K 1 , K 2 and K < of the SO ( 1,2) Lie algebra. The approach provides appropriate quantum observables for the phase space P and a promising basis for the description of quantum optical structures

Marcel Reginatto, A. D. Bermúdez Manjarres, Sebastian Ulbricht

Abstract Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used to develop a novel formulation of classical mechanics in Hilbert space. This formulation differs from the Koopman-von Neumann theory in many ways. Classical observables are represented by van Hove operators, which satisfy a commutation algebra isomorphic to the Poisson algebra of functions in phase space. Moreover, these operators are both observables and generators of transformations, which makes it unnecessary to introduce unobservable auxiliary operators as in the Koopman-von Neumann theory. In addition, for consistency with classical mechanics, a constraint must be imposed that fixes the phase of the wavefunction. The approach can be extended to hybrid mixed classical-quantum systems in Hilbert space. The formalism is applied to the measurement of a quantum two-level system (qubit) by a classical apparatus.

Manuel Gadella

D. Steck, Mark G. Raizen

The study of quantum tunneling dates back to the early days of quantum mechanics, and has been a major topic of investigation in both theory and experiment. In recent years, there has been significant effort to understand tunneling phenomena for systems that exhibit chaos in their classical counterpart [13,15,18,29,42]. In particular, tunneling was predicted for mixed systems where chaotic and stable regions are intermingled [14]. The tunneling in this case is between two stable regions (referred to as nonlinear resonances, or islands of stability) in the classical phase space. The classical transport between these islands is forbidden by “dynamical barriers” in phase space. In contrast, quantum tunneling can couple the two islands so that a wave packet oscillates coherently between the two symmetry-related regions. This phenomenon is called dynamical tunneling because it occurs in phase space, without a fixed potential barrier. In the case of a mixed phase space, the presence of the chaotic region can enhance the rate of dynamical tunneling, a phenomenon known as chaos-assisted tunneling [53]. In this chapter, we review our experimental observation of chaosassisted tunneling with ultracold cesium atoms [48-51]. We first give a brief overview of some basic concepts and of earlier work in order to provide a context for our results.