The mechanism of superluminal traversal time through a potential well or potential barrier is investigated from the viewpoint of interference between multiple finite wave packets, due to the multiple reflections inside the well or barrier. In the case of potential-well traveling that is classically allowed, each of the successively transmitted constituents is delayed by a subluminal time. When the thickness of the well is much smaller in comparision with a characteristic length of the incident wave packet, the reshaped wave packet in transmission maintains the profile of the incident wave packet. In the case of potential-barrier tunneling that is classically forbidden, though each of the successively transmitted constituents is delayed by a time that is independent of the barrier thickness, the interference between multiple transmitted constituents explains the barrier-thickness dependence of the traversal time for thin barriers and its barrier-thickness independence for thick barriers. This manifests the nature of Hartman effect.
We consider the quantum traversal time of an incident wave packet across a potential well using the theory of quantum time of arrival (TOA)-operators. This is done by constructing the corresponding TOA-operator across a potential well via quantization. The expectation value of the potential well TOA-operator is compared to the free particle case for the same incident wave packet. The comparison yields a closed-form expression of the quantum well traversal time which explicitly shows the classical contributions of the positive and negative momentum components of the incident wave packet and a p…
Apparently 'superluminal' transmission, e.g., in quantum tunnelling and its variants, occurs via a subtle interference mechanism which allows reconstruction of the entire spacial shape of a wave packet from its front tail. It is unlikely that the effect could be described adequately in simpler terms.
A phenomenological model for a measurement of barrier traversal times for particles is proposed. Two idealized detectors for passage and arrival provide entrance and exit times for the barrier traversal. The averaged traversal time is computed over the ensemble of particles detected twice, before and after the barrier. The Hartman effect can still be found when passage detectors that conserve the momentum distribution of the incident packet are used.
We study the use of the Faraday effect as a quantum clock for measuring traversal times of evanescent photons through magneto-refractive structures. The Faraday effect acts both as a phase-shifter and as a filter for circular polarizations. Only measurements based on the Faraday phase-shift properties are relevant to the traversal time measurements. The Faraday polarization filtering may cause the loss of non-local (Einstein-Podolsky-Rosen) two-photon correlations, but this loss can be avoided without sacrificing the clock accuracy. We show that a mechanism of destructive interference between …
We examine the passage of ultra-cold two-level atoms through the potential produced by the vacuum of the cavity field. The peak of the transmitted wave packet generally occurs at the instant given by the expression for phase time even if that instant is earlier than the instant at which incident wave packet's peak enters the cavity and thus the phase time can be considered as the appropriate measure of the time required for the atom to traverse the cavity. We show that the phase tunneling time for ultra-cold atoms could be both super - and sub - classical time and we show how this behaviour ca…
This article investigates some solutions of the time-dependent free Dirac equation. Visualizations of these solutions immediately reveal strange phenomena that are caused by the interference of positive- and negative-energy waves. The effects discussed here include the Zitterbewegung, the opposite direction of momentum and velocity in negative-energy wave packets, and the superluminal propagation of the wave packet's local maxima.
We show that the group delay in tunneling is not a traversal time but a lifetime of stored energy or stored probability escaping through both ends of the barrier. Because it is a lifetime associated with both forward (transmitted) and backward (reflected) fluxes, it cannot be used to define a group velocity for forward transit in cases where a wavepacket is mostly reflected. For photonic tunneling barriers the group delay is identical to the dwell time which is also a property of an entire wave function with reflected and transmitted components. Theoretical predictions and experimental reports…
In quantum tunnelling, what appears an infinitely fast barrier traversal can be explained in terms of an Aharonov-like weak measurement of the tunnelling time, in which the role of the pointer is played by the particle's own coordinate. A relativistic wavepacket is shown to be reshaped through a series of subluminal shifts which together produce an anomalous 'superluminal' result.
We study the phenomenon of one-dimensional non-resonant tunnelling through two successive potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time is INDEPENDENT not only of the barrier widths (the so-called "Hartman effect"), but also of the R-width: so that the effective velocity in the region R, between the two barriers, can be regarded as infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunnelling experiment herein cons…
We analyse in detail the reshaping mechanism leading to apparently "superluminal" advancement of a wave packet traversing a classically forbidden region. In the coordinate representation, a barrier is shown to act as an effective beamsplitter, recombining envelopes of the freely propagating pulse with various spacial shifts. Causality ensures that none of the constituent envelopes are advanced with respect to free propagation, yet the resulting pulse is advanced due to a peculiar interference effect, similar to the one responsible for "anomalous" values which occur in Aharonov's "weak measurem…
It is proposed that superoscillations play an important role in the interferences which give rise to superluminal effects. To exemplify that, we consider a toy model which allows for a wave packet to travel, in zero time and negligible distortion a distance arbitrarily larger 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.
We obtain the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation regime when the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation originated from the study based on non-relativistic dynamics of tunneling is overcome. The treatment of the problem suggests revealing ins…
Drawing from the optimal transport theory adapted to the relativistic setting we formulate the principle of a causal flow of probability and apply it in the wave packet formalism. We demonstrate that whereas the Dirac system is causal, the relativistic-Schrödinger Hamiltonian impels a superluminal evolution of probabilities. We quantify the causality breakdown in the latter system and argue that, in contrast to the popular viewpoint, it is not related to the localisation properties of the states.
The general and explicit relation between the phase time and the dwell time for quantum tunneling or scattering is investigated. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, here we demonstrate that these two distinct transit time definitions give connected results where, however, the phase time (group delay) accurately describes the exact position of the scattered particles. The analytical difficulties that arise when the stationary phase method is employed for obtaining phase (traversal) times are all overcome. Multiple wave packet decomp…
We study the unconventional transmission properties of a wave-packet through a PT-symmetric potential region, as describing actual electromagnetic wave propagation along a waveguide filled with gain and loss media. The non-trivial behavior of the transmission probability manifests in the giant amplification of the incident electromagnetic signal of given wavelengths for well-defined configurations, depending on the gain/loss contrast. Maximum transmission peaks are related to spectral singularities and a strict correlation exists between the "resonant" wavelengths and the gain/loss contrast. T…
The time of passage of the transmitted wave packet in a tunneling collision of a quantum particle with a square potential barrier becomes independent of the barrier width in a range of barrier thickness. This is the Hartman effect, which has been frequently associated with ``superluminality''. A fundamental limitation on the effect is set by non-relativistic ``causality conditions''. We demonstrate first that the causality conditions impose more restrictive bounds on the negative time delays (time advancements) when no bound states are present. These restrictive bounds are in agreement with a …
This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. We notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. We investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstr…
Based on the Dirac equation, the behavior of relativistic electrons which tunnel a potential barrier of height V0 for incoming energies between V0 and V0+m is studied by using the wave packet formalism. The choice of this incoming energy zone guarantees that only electrons participate to the tunneling process. In the opaque limit, as shown in a previous analysis, the transit velocity is proportional to the barrier width and for relativistic potential tends to a constant value greater than $c$. In this paper, a new numerical study shows a very surprising result: superluminal transmissions are a…
We investigate ultrafast dynamics of the lowest singlet excited electronic state in liquid nitrobenzene using Ultrafast Transient Polarization Spectroscopy (UTPS), extending the well-known technique of Optical-Kerr Effect (OKE) spectroscopy to excited electronic states. The third-order non-linear response of the excited molecular ensemble is highly sensitive to details of excited state character and geometries and is measured using two femtosecond pulses following a third femtosecond pulse that populates the S1 excited state. By measuring this response as a function of time delays between the …
Motivated by recent time domain experiments on ultrafast atom ionization, we analyze the transients and timescales that characterize, besides the relatively long lifetime, the decay by tunneling of a localized state. While the tunneling starts immediately, some time is required for the outgoing flux to develop. This short-term behavior depends strongly on the initial state. For the initial state tightly localized so that the initial transients are dominated by over-the-barrier motion, the timescale for the flux propagation through the barrier is close to the Büttiker-Landauer traversal time. T…
We study the tunneling zone solutions of a one-dimensional electrostatic potential for the relativistic (Dirac to Klein-Gordon) wave equation when the incoming wave packet exhibits the possibility of being almost totally transmitted through the barrier. The transmission probabilities, the phase times and the dwell times for the proposed relativistic dynamics are obtained and the conditions for the occurrence of accelerated tunneling transmission are all quantified. We show that, in some limiting cases, the analytical difficulties that arise when the stationary phase method is employed for obta…
A relativistic version of the (consistent or decoherent) histories approach to quantum theory is developed on the basis of earlier work by Hartle, and used to discuss relativistic forms of the paradoxes of spherical wave packet collapse, Bohm's formulation of Einstein-Podolsky-Rosen, and Hardy's paradox. It is argued that wave function collapse is not needed for introducing probabilities into relativistic quantum mechanics, and in any case should never be thought of as a physical process. Alternative approaches to stochastic time dependence can be used to construct a physical picture of the me…
Tunnelling is one of the most paradigmatic and evocative phenomena of quantum physics, underlying processes such as photosynthesis and nuclear fusion, as well as devices ranging from SQUID magnetometers to superconducting qubits for quantum computers. The question of how long a particle takes to tunnel, however, has remained controversial since the first attempts to calculate it, which relied on the group delay. It is now well understood that this delay (the arrival time of the transmitted wave packet peak at the far side of the barrier) can be smaller than the barrier thickness divided by the…
