Connection between quantum-mechanical and classical time evolution via a dynamical invariant

Monitoring the quantum-classical transition in thermally seeded parametric down-conversion by intensity measurements

“…the dynamical suppression of quantum interference effects, cannot be the unique criterion to define a classical limit [12], which should emerge from an operational approach suitably linked to measurement schemes [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. To this aim here we address the bipartite system made by the pair of field modes obtained from parametric downconversion (PDC) as a convenient physical system to analyze the quantum-classical transition in the continuous variable regime.…”

Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics

“…Since x and p are purely time-dependent quantities,x andp can replace x and p in equations (3), (4) since these equations only contain derivatives with respect to space, not time. So, x and p in (27) would be replaced byx andp which would lead to the result obtained in [16] showing the connection between the exponent of the time-dependent Wigner function and the dynamical Ermakov invariant that is connected with the parametersẑ andû of the time-dependent kernel K(x, x , t, t ) and has been defined in equation (19) (for details see also [11]). In the quantum mechanical phase space picture according to Wigner, this results not only in changing initial position-and momentum-uncertainties into their values at time t but, also, an additional contribution occurs from the time-change of x 2 , or α 2 , respectively, expressed by the term proportional to [x,p] + , orαα, respectively, in the exponent of W (x, p, t).…”

“…So, considering time-dependent problems in quantum mechanics in terms of the time-dependent Schrödinger equation or equivalent formulations, one finds that not only classical position and momentum change in time (in a way that can be described by canonical transformations) but, also the typical quantum mechanical degrees of freedom, like position-and momentum-uncertainties, may be time-dependent (corresponding, e.g., to wave packets with time-dependent width). For certain problems with exact analytical solutions in form of Gaussian wave packets, it has been shown (see, e.g., [11] and references cited therein) how the transition from initial position and time (in configuration space) to any later position and time can be achieved with the help of a time-dependent kernel (or propagator) K(x, x , t, t ) according to…”

“…(see, e.g., [11]), is given by the product of two delta functions where now, however, x is replaced by…”

“…for the variable λ (for details see [11,13]). The complex quantity λ(t) can also be expressed in polar coordinates as…”