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arXiv:1903.10029 · Modern Physics Letters A

Enhancement of superluminal weak values under Lorentz boost

Abhishek Som, Sourin Das

Published Sep 16, 2020• 0 citations
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Abstract

The local group velocity defined as the weak value of the velocity operator in (1 + 1)-dimensional Klein-Gordon and Dirac theory is studied. As shown by Berry [J. Phys. A 45, 185308 (2012)], when the pre- and post-selected states for evaluating the weak value are chosen at random from an ensemble of available states, it gives rise to a universal probability distribution for the local group velocity which can have both subluminal and superluminal components. In this work, we explore the possibility of enhancement of the superluminal fraction of this total probability distribution by applying a Lorentz boost and show that it can indeed be enhanced both in the case of Klein–Gordon and Dirac theories.

DOI
10.1142/s021773232050279x

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The local group velocity defined as the weak value of the velocity operator in (1 + 1)-dimensional Klein-Gordon and Dirac theory is studied. As shown by Berry [J. Phys. A 45, 185308 (2012)], when the pre- and post-selected states for evaluating the weak value are chosen at random from an ensemble of available states, it gives rise to a universal probability distribution for the local group velocity which can have both subluminal and superluminal components. In this work, we explore the possibility of enhancement of the superluminal fraction of this total probability distribution by applying a Lorentz boost and show that it can indeed be enhanced both in the case of Klein–Gordon and Dirac theories.
For Klein?Gordon and Dirac waves representing massive quantum particles, the local group velocity v (weak value of the velocity operator) can exceed c. If the waves consist of superpositions of many plane waves, with different (but subluminal) group velocities u, the superluminal probability Psuper, i.e. that |v| > c for a randomly selected state, can be calculated explicitly. Psuper?depends on two parameters describing the distribution (power spectrum) of u in the superpositions, and lies between 0 and 1/2 for Klein?Gordon waves and 1?1/ and 1/2 for Dirac waves. Numerical simulations display the superluminal intervals in space and regions in spacetime, and support the theoretical predictions for Psuper.
Abstract Weak values characterize a quantum system in the period of time between preparation and measurement and may lie outside the eigenvalue spectrum of the measured operator. The probability of such “superweak" values for random quantum states has been calculated and applied to Klein–Gordon and Dirac waves, where the maximal probability for superluminal propagation was shown to be 1/2. In a recent paper, a different definition for the velocity of a relativistic quantum particle was proposed in terms of a ratio of two weak values. In this paper, we find the probability distribution of such ratios. With the new definition, the superluminal probability of photons is found to be bounded between 1−1/21-1/\sqrt{2}1−1/2​ and 1/21/\sqrt{2}1/2​ , while for general eigenvalue distributions the superluminal probability can take any value between 0 and 1.
We consider the tunnelling particle as a pre- and post-selected system and prove that the tunnelling time is the expectation value of the position of a ‘clock’ degree of freedom weakly coupled to it. Such a value, called a ‘weak value’, typically falls outside the eigenvalue spectrum of the operator. The appearance of unusual weak values has been associated with a unique interference structure called ‘superoscillations’ (band-limited functions which on a finite interval, approximate functions with spectra well outside their band). It is proposed that superoscillations play an important role in the interferences which give rise to superluminal effects. To demonstrate that, we consider a certain simple tunnelling barrier which allows a wave packet to travel in zero time and negligible distortion, a distance arbitrarily longer than the width of the wave packet. The peak is shown to result from a superoscillatory superposition at the tail. Similar reasoning applies to the dwell time. For this system, both the Wigner time (related to the group velocity) and a clock time correspond to superluminal velocities.
We consider a case when a weak value is introduced as a physical quantity rather than as an average of weak measurements. The case we treat is a time evolution of a particle by the 1 + 1-dimensional Dirac equation. In particular, in a spontaneous pair production via a supercritical step potential, a quantitative explanation can be given by a weak value for the group velocity of the particle. We also show that the condition for the pair production (supercriticality) corresponds to the condition when the weak value takes a strange value (superluminal velocity).
We show that the group velocity of a weak pulse can be manipulated by controlling the phases of two weak optical fields applied to a V-shaped three-level system. Such control can even cause the probe propagation to change from subluminal to superluminal. We consider two schemes: in the first, the excited states are coupled by decay-induced coherence, which is an inherent property of the medium, and in the second, quantum coherence is created by coupling the excited states to each other by a strong microwave field. We also discuss the group velocity reduction experienced by a single weak propagating probe due to decay-induced coherence.
While particles cannot travel faster than the speed of light, nor can information, this assumption has over the years been frequently questioned. Most recently, it has been argued [New J. Phys. 22, 033038 (2020)] that in a world with superluminal observers local determinism is impossible, linking the two pillars of physics-quantum theory and relativity-suggesting that the latter serves as the foundation for the former. Motivated by this approach, in this work, we extend the framework of quantum reference frames to incorporate superluminal Lorentz transformations. We apply this conceptual result to examine an apparent paradox where particles acquire negative energies after undergoing a superluminal Lorentz boost and propose a resolution within our framework. We also discuss Bell experiments under superluminal quantum reference frame transformations, showing that involved probabilities remain conserved.