by Nicola Vona (31.10.2012)

**My research work is based on the idea of particle trajectory. I am applying it to several problems, principally in Quantum Mechanics.**

Trajectories are usually considered a concept quite alien to quantum theory, where everything is fuzzy and undetermined. Nevertheless, a recent, important experiment (Kocsis et al., 2011) showed that with present technology it is possible to go beyond the orthodox paradigm for quantum measurements, experimentally detecting the trajectories followed by the particles in a double slit setting. These trajectories were previously regarded as undetectable. This experiment does not, of course, disprove Quantum Mechanics, indeed the term *detect* assumes here a new meaning: the trajectories were not measured following a single particle during its evolution, but rather revealed by a statistical analysis conducted on a large set of particles.

With my work, I am analyzing further experimental situations where the concept of trajectory allows for a deeper analysis than the orthodox formalism. In this regard, I am collaborating with two experimental groups, one in Heidelberg (DE) and one in Frascati (IT). The most promising case that we are considering is the measurement of arrival times in time-of-flight settings.

All the predictions obtained from the orthodox formalism are calculated through the wave function at a given *time*, so it is particularly difficult to estimate *when* a particle will arrive at the detector. Despite this difficulty, after several efforts, a paradigm to calculate arrival time statistics with the orthodox formalism was found (see Muga and Leavens, 2000). On the contrary, supplementing Quantum Mechanics with a trajectory structure (Dürr and Teufel, 2009) the notion of arrival time becomes straightforward: a particle reaches the detector simply when its trajectory reaches the detector. The resulting prediction for measurements of arrival time differs from the orthodox one, therefore we are studying how to conduct an experiment, able to discern this difference.

For free particles, a situation that shows a substantial difference is that in which the motion of the particles is mostly due to the so called spread of the wave-packet, rather than to their momentum (cf. fig. 1); this is realized when the uncertainty on the particle velocity is much bigger than the average velocity itself, so that the probabilities of positive and negative velocities are comparable.

(a) Position distribution at t = 0, 5, and 10; the red line represents the detector.

(b) Momentum distribution.

(c) Arrival time distribution; K is the orthodox prediction, J the trajectory-based one.

Also of interest is the fact that the paradigm based on the trajectories predicts a breakdown of any detector-independent model as soon as some trajectories cross the detector multiple times. Conversely, the orthodox paradigm can be applied to any state. This circumstance can be investigated in a double slit setting (cf. fig. 2).

(b) Momentum distribution.

(c) Arrival time distribution; K is the orthodox pre- diction, J the trajectory-based one; note the green area where J gives a negative probability, sign of a breakdown of the model.

(d) Detail of the trajectories in the region where the multiple crossings occur.

I am currently studying the experimental realizability of both the described cases, and of their eventual combination.

References:

D. Dürr and S. Teufel. Bohmian mechanics: the physics and mathematics of quantum theory. Fundamental Theories of Physics. Springer, 2009.

S. Kocsis, B. Braverman, S. Ravets, M.J. Stevens, R.P. Mirin, L. Krister Shalm, and A.M. Steinberg. Observing the average trajectories of single photons in a two-slit interferometer. Science, 332(6034):1170–1173, 2011.

J.G. Muga and C.R. Leavens. Arrival time in quantum mechanics. Physics Reports, 338(4):353 – 438, 2000.