A macroscopic clock model to solve the paradox of Schrödinger's cat.

We propose detecting the moment an atom emits a photon by means of a nearly classical macroscopic clock and discuss its viability. It is shown that what happens in such a measurement depends on the relation between the clock's accuracy and the width of the energy range available to the photon. Implications of the analysis for the long standing Schrödinger's cat problem are reported.

Speed-up and slow-down of a quantum particle.

We study non-relativistic propagation of Gaussian wave packets in one-dimensional Eckart potential, a barrier, or a well. In the picture used, the transmitted wave packet results from interference between the copies of the freely propagating state with different spatial shifts (delays), [Formula: see text], induced by the scattering potential. The Uncertainty Principle precludes relating the particle's final position to the delay experienced in the potential, except in the classical limit. Beyond this limit, even defining an effective range of the delay is shown to be an impracticable task, owing to the oscillatory nature of the corresponding amplitude distribution. Our examples include the classically allowed case, semiclassical tunnelling, delays induced in the presence of a virtual state, and scattering by a low barrier. The properties of the amplitude distribution of the delays, and its pole representation are studied in detail.

Tunnelling times, Larmor clock, and the elephant in the room.

A controversy surrounding the "tunnelling time problem" stems from the seeming inability of quantum mechanics to provide, in the usual way, a definition of the duration a particle is supposed to spend in a given region of space. For this reason, the problem is often approached from an "operational" angle. Typically, one tries to mimic, in a quantum case, an experiment which yields the desired result for a classical particle. One such approach is based on the use of a Larmor clock. We show that the difficulty with applying a non-perturbing Larmor clock in order to "time" a classically forbidden transition arises from the quantum Uncertainty Principle. We also demonstrate that for this reason a Larmor time (in fact, any Larmor time) cannot be interpreted as a physical time interval. We provide a theoretical description of the quantities measured by the clock.

Klein paradox for bosons, wave packets and negative tunnelling times.

We analyse a little known aspect of the Klein paradox. A Klein-Gordon boson appears to be able to cross a supercritical rectangular barrier without being reflected, while spending there a negative amount of time. The transmission mechanism is demonstrably acausal, yet an attempt to construct the corresponding causal solution of the Klein-Gordon equation fails. We relate the causal solution to a divergent multiple-reflections series, and show that the problem is remedied for a smooth barrier, where pair production at the energy equal to a half of the barrier's height is enhanced yet remains finite.

Paths, negative "probabilities", and the Leggett-Garg inequalities.

We present a path analysis of the condition under which the outcomes of previous observation affect the results of the measurements yet to be made. It is shown that this effect, also known as "signalling in time", occurs whenever the earlier measurements are set to destroy interference between two or more virtual paths. We also demonstrate that Feynman's negative "probabilities" provide for a more reliable witness of "signalling in time", than the Leggett-Garg inequalities, while both methods are frequently subject to failure.

Relative frequencies of constrained events in stochastic processes: An analytical approach.

The stochastic simulation algorithm (SSA) and the corresponding Monte Carlo (MC) method are among the most common approaches for studying stochastic processes. They relies on knowledge of interevent probability density functions (PDFs) and on information about dependencies between all possible events. Analytical representations of a PDF are difficult to specify in advance, in many real life applications. Knowing the shapes of PDFs, and using experimental data, different optimization schemes can be applied in order to evaluate probability density functions and, therefore, the properties of the studied system. Such methods, however, are computationally demanding, and often not feasible. We show that, in the case where experimentally accessed properties are directly related to the frequencies of events involved, it may be possible to replace the heavy Monte Carlo core of optimization schemes with an analytical solution. Such a replacement not only provides a more accurate estimation of the properties of the process, but also reduces the simulation time by a factor of order of the sample size (at least ≈10(4)). The proposed analytical approach is valid for any choice of PDF. The accuracy, computational efficiency, and advantages of the method over MC procedures are demonstrated in the exactly solvable case and in the evaluation of branching fractions in controlled radical polymerization (CRP) of acrylic monomers. This polymerization can be modeled by a constrained stochastic process. Constrained systems are quite common, and this makes the method useful for various applications.

Complex angular momentum theory of state-to-state integral cross sections: resonance effects in the F + HD → HF(v' = 3) + D reaction.

State-to-state reactive integral cross sections (ICSs) are often affected by quantum mechanical resonances, especially near a reactive threshold. An ICS is usually obtained by summing partial waves at a given value of energy. For this reason, the knowledge of pole positions and residues in the complex energy plane is not sufficient for a quantitative description of the patterns produced by resonance. Such description is available in terms of the poles of an S-matrix element in the complex plane of the total angular momentum. The approach was recently implemented in a computer code ICS_Regge, available in the public domain [Comput. Phys. Commun., 2014, 185, 2127]. In this paper, we employ the ICS_Regge package to analyse in detail, for the first time, the resonance patterns predicted for integral cross sections (ICSs) of the benchmark F + HD → HF(v' = 3) + D reaction. The v = 0, j = 0, Ω = 0 → v' = 3, j' = 0, 1, 2, and Ω' = 0, 1, 2 transitions are studied for collision energies from 58.54 to 197.54 meV. For these energies, we find several resonances, whose contributions to the ICS vary from symmetric and asymmetric Fano shapes to smooth sinusoidal Regge oscillations. Complex energies of metastable states and Regge pole positions and residues are found by Padé reconstruction of the scattering matrix elements. The accuracy of the ICS_Regge code, relation between complex energies and Regge poles, various types of Regge trajectories, and the origin of the J-shifting approximation are also discussed.

Quantum law of rare events for systems with bosonic symmetry.

In classical physics, the joint probability of a number of individually rare independent events is given by the Poisson distribution. It describes, for example, the unidirectional transfer of a population between the densely and sparsely populated states of a classical two-state system. We derive a quantum version of the law for a large number of noninteracting systems (particles) obeying Bose-Einstein statistics. The classical law is significantly modified by quantum interference, which allows, among other effects, for the counterflow of particles back into the densely populated state. The suggested observation of this classically forbidden counterflow effect can be achieved with modern laser-based techniques used for manipulating and trapping cold atoms.

Qubit residence time measurements with a Bose-Einstein condensate.

We show that an electrostatic qubit located near a Bose-Einstein condensate trapped in a symmetric double-well potential can be used to measure the duration the qubit has spent in one of its quantum states. The strong, medium, and weak measurement regimes are analyzed. The analogy between the residence and the traversal (tunnelling) times is highlighted.

On the origin of the forward peak and backward oscillations in the F + H(2)(v = 0) --> HF(v' = 2) + H reaction.

We present a semiclassical complex angular momentum (CAM) analysis of the forward scattering peak which occurs at a translational collision energy around 32 meV in the quantum mechanical calculations for the F + H(2)(v = 0, j = 0) --> HF(v' = 2, j' = 0) + H reaction on the Stark-Werner potential energy surface. The semiclassical CAM theory is modified to cover the forward and backward scattering angles. The peak is shown to result from constructive/destructive interference of the two Regge states associated with two resonances, one in the transition state region and the other in the exit channel van der Waals well. In addition, we demonstrate that the oscillations in the energy dependence of the backward differential cross section are caused by the interference between the direct backward scattering and the decay of the two resonance complexes returning to the backward direction after one full rotation.

Overlapping resonances and Regge oscillations in the state-to-state integral cross sections of the F+H2 reaction.

A Regge pole analysis is employed to explain the oscillatory patterns observed in numerical simulations of integral cross section for the F+H(2)(v=0,j=0)-->HF(v(')=2,j(')=0)+H reaction in the translational collision energy range 25-50 meV. In this range the integral cross section for the transition, affected by two overlapping resonances, shows nearly sinusoidal oscillations below 38 meV and a more structured oscillatory pattern at larger energies. The two types of oscillations are related to the two Regge trajectories which (pseudo) cross near the energy where the resonances are aligned. Simple estimates are given for the periods of the oscillations.

Interacting resonances in the F+H2 reaction revisited: complex terms, Riemann surfaces, and angular distributions.

We study the effect of overlapping resonances on the angular distributions of the reaction F+H2(v=0,j=0)-->HF(v=2,j=0)+H in the collision energy range from 5 to 65 meV, i.e., under the reaction barrier. Reactive scattering calculations were performed using the hyperquantization algorithm on the potential energy surface of Stark and Werner [J. Chem. Phys. 104, 6515 (1996)]. The positions of the Regge and complex energy poles are obtained by Pade reconstruction of the scattering matrix element. The Sturmian theory is invoked to relate the Regge and complex energy terms. For two interacting resonances, a two-sheet Riemann surface is contracted and inverted. The semiclassical complex angular momentum analysis is used to decompose the scattering amplitude into the direct and resonance contributions.