On the Relationship Between Complex Potentials and Strings of Projection Operators

It is of interest in a variety of contexts, and in particular in the arrival time problem, to consider the quantum state obtained through unitary evolution of an initial state regularly interspersed with periodic projections onto the positive $x$-axis (pulsed measurements). Echanobe, del Campo and Muga have given a compelling but heuristic argument that the state thus obtained is approximately equivalent to the state obtained by evolving in the presence of a certain complex potential of step-function form. In this paper, with the help of the path decomposition expansion of the associated propagators, we give a detailed derivation of this approximate equivalence. The propagator for the complex potential is known so the bulk of the derivation consists of an approximate evaluation of the propagator for the free particle interspersed with periodic position projections. This approximate equivalence may be used to show that to produce significant reflection, the projections must act at time spacing less than 1/E, where E is the energy scale of the initial state.Comment: 29 pages, LaTex, 4 figures. Substantial revision

Analytic Examples, Measurement Models and Classical Limit of Quantum Backflow

We investigate the backflow effect in elementary quantum mechanics - the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We compute the current and flux for states consisting of superpositions of gaussian wave packets. These are experimentally realizable but the amount of backflow is small. Inspired by the numerical results of Penz et al (M.Penz, G.Gr\"ubl, S.Kreidl and P.Wagner, J.Phys. A39, 423 (2006)), we find two non-trivial wave functions whose current at any time may be computed analytically and which have periods of significant backflow, in one case with a backwards flux equal to about 70 percent of the maximum possible backflow, a dimensionless number $c_{bm} \approx 0.04$, discovered by Bracken and Melloy (A.J.Bracken and G.F.Melloy, J.Phys. A27, 2197 (1994)). This number has the unusual property of being independent of $\hbar$ (and also of all other parameters of the model), despite corresponding to an obviously quantum-mechanical effect, and we shed some light on this surprising property by considering the classical limit of backflow. We discuss some specific measurement models in which backflow may be identified in certain measurable probabilities.Comment: 33 pages, 14 figures. Minor revisions. Published versio

A review of the decoherent histories approach to the arrival time problem in quantum theory

We review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expressed in terms of a POVM in this case. We turn finally to the question of decoherence of the corresponding histories, and we show that this can be achieved for simple states in the case of a free particle, and for general states for a particle coupled to an environment.Comment: 10 pages. To appear in DICE 2010 conference proceeding

Quantum Arrival Time For Open Systems

We extend previous work on the arrival time problem in quantum mechanics, in the framework of decoherent histories, to the case of a particle coupled to an environment. The usual arrival time probabilities are related to the probability current, so we explore the properties of the current for general open systems that can be written in terms of a master equation of Lindblad form. We specialise to the case of quantum Brownian motion, and show that after a time of order the localisation time the current becomes positive. We show that the arrival time probabilities can then be written in terms of a POVM, which we compute. We perform a decoherent histories analysis including the effects of the environment and show that time of arrival probabilities are decoherent for a generic state after a time much greater than the localisation time, but that there is a fundamental limitation on the accuracy, $\delta t$, with which they can be specified which obeys $E\delta t>>\hbar$. We confirm that the arrival time probabilities computed in this way agree with those computed via the current, provided there is decoherence. We thus find that the decoherent histories formulation of quantum mechanics provides a consistent explanation for the emergence of the probability current as the classical arrival time distribution, and a systematic rule for deciding when probabilities may be assigned.Comment: 30 pages, 1 figure. Published versio

Arrival Times, Complex Potentials and Decoherent Histories

We address a number of aspects of the arrival time problem defined using a complex potential of step function form. We concentrate on the limit of a weak potential, in which the resulting arrival time distribution function is closely related to the quantum-mechanical current. We first consider the analagous classical arrival time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. In the quantum case, we review the path decomposition expansion (PDX), in which the propagator is factored across a surface of constant time, so is very useful for potentials of step function form. We use the PDX to derive the usual scattering wave functions and the arrival time distribution function. This method gives a direct and geometrically appealing account of known results (but also points the way to how they can be extended to more general complex potentials). We use these results to carry out a decoherent histories analysis of the arrival time problem, taking advantage of a recently demonstrated connection between pulsed measurements and complex potentials. We obtain very simple and plausible expressions for the class operators (describing the amplitudes for crossing the origin during intervals of time) and show that decoherence of histories is obtained for a wide class of initial states (such as simple wave packets and superpositions of wave packets). We find that the decoherent histories approach gives results with a sensible classical limit that are fully compatible with standard results on the arrival time problem. We also find some interesting connections between backflow and decoherence.Comment: 43 page