In this paper we compare the concept of the tunneling time introduced in quant-ph/0405028 with those of the phase and dwell times. As is shown, unlike the latter our definition of the transmission time coincides, in the limit of weak scattering potentials, with that for a free particle. This is valid for all values of the particle's momentum, including the case of however slow particles. All three times are also considered for a resonant tunneling. In all the cases the main feature to distinguish our concept from others is that the average starting point of transmitted (reflected) particles does not coincide with that of all particles. One has to stress here that there is no such an experiment which would give coordinates of all the three points, simultaneously. For measuring the position of the average starting point of transmitted particles we propose an experimental scheme based on the Larmor precession effect.
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…
The stationary phase method is often employed for computing tunneling {\em phase} times of analytically-continuous {\em gaussian} or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After reexamining the above barrier diffusion problem where we notice the wave packet collision necessarily leads to the possibility of multiple reflected and transmitted wave packets, we study the phase times for tunneling/reflec…
We develop a new variant of the wave-packet analysis and solve the tunneling time problem for one particle. Our approach suggests an individual asymptotic description of the quantum subensembles of transmitted and reflected particles both at the final and initial stage of tunneling. We find the initial states of both subensembles, which are non-orthogonal. The latter reflects ultimately the fact that at the initial stage of tunneling it is impossible to predict whether a particle will be transmitted through or reflected off the barrier. At the same time, in this case, one can say about the to-…
We study the tunneling time of Dirac fermions in graphene magnetic barrier through an electrostatic potential and a mass term. This latter generates an energy gap in the spectrum and therefore affects the proprieties of tunneling of the system. For clarification, we first start by deriving the eigenspinors solutions of Dirac equation and second connect them to the incident, reflected and transmitted beam waves. This connection allows us to obtain the corresponding phases shifts and consequently compute the group delay time in transmission and reflection. Our numerical results show that the gro…
Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak $τ_p$ and a width $Δτ$. Numerical results for other tunneling times, as the phase-delay time, fall within $Δτ$. The Büttiker traversal time is the closest to $τ_p$. Our results resemble calcul…
Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one - dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semi-classical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles trans…
We introduce a complex-extended continuum level density and apply it to one-dimensional scattering problems involving tunneling through finite-range potentials. We show that the real part of the density is proportional to a real "time shift" of the transmitted particle, while the imaginary part reflects the imaginary time of an instanton-like tunneling trajectory. We confirm these assumptions for several potentials using the complex scaling method. In particular, we show that stationary points of the potentials give rise to specific singularities of both real and imaginary densities which repr…
The transmission of wave packets through tunneling barriers is studied in detail by the method of quantum molecular dynamics. The distribution function of the times describing the arrival of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The behavior of the average coordinate of a packet, the average momentum, and their variances is investigated. It is found that under the barrier a part of the packet is reflected and a Gaussian barrier increases the average momentum of the transmitted packet and its variance in momentum…
Using a recently developed procedure - multiple wave packet decomposition - here we study the phase time formulation for tunneling/reflecting particles colliding with a potential barrier. To partially overcome the analytical difficulties which frequently arise when the stationary phase method is employed for deriving phase (tunneling) time expressions, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and an one-dimensional rectangular potential barrier. Summing the amplitudes of the reflected and transmitted waves - using a method we call m…
It is now common practice to solve the Schrödinger equation to estimate the tunneling current between two metal electrodes at specified potentials, or the transmission through a potential barrier by assuming an incident, reflected, and transmitted wave. However, we suggest that these methods may not be appropriate for nanoscale circuits. The electron man-free path may be as long as 68.2 nm in metallic elements so we consider the possibility that quantum effects may occur throughout a nanoscale circuit, including the connections. Analytical methods are presented for modeling the coherent transf…
Two-particle scattering probabilities in tunneling scenarios with exchange interaction are analyzed with quasi-particle wave packets. Two initial one-particle wave packets (with opposite central momentums) are spatially localized at each side of a barrier. After impinging upon a tunneling barrier, each wave packet splits into transmitted and reflected components. When the initial two-particle anti-symmetrical state is defined as a Slater determinant any type of (normalizable) one-particle wave packet, it is shown that the probability of detecting two (identically injected) electrons at the sam…
We consider exact time-dependent analytic solutions to the Schrödinger equation for tunneling in one dimension with cut off wave initial conditions at $t=0$. We obtain that as soon as $t \neq 0$ the transmitted probability density at any arbitrary distance rises instantaneously with time in a linear manner. Using a simple model we find that the above nonlocal effect of the time-dependent solution is suppressed by consideration of low-energy relativistic effects. Hence at a distance $x_0$ from the potential the probability density rises after a time $t_0=x_0/c$ restoring Einstein causality. Thi…
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…
Analytic solutions to the time-dependent Schrödinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $α$, we find that the probability density exhibits two evolving structures. One refers to the propagation of a {\it forerunner} related to a {\it time domain resonance} [Phys. Rev. A {\bf 64}, 0121907 (2001)], while the other consists of a semiclassical propagating wavefront. We find a regime where the {\it forerunners} are absent, correspondi…
The tunneling time problem earlier studied in Phys. Rev. Lett 108 170402 (2012) using a non-relativistic time-of-arrival (TOA) operator predicted that tunneling time is instantaneous. This raises the question on whether instantaneous tunneling time is a consequence of using a non-relativistic theory. Here, we extend the analysis by proposing a formalism on the construction of relativistic TOA-operators for spin-0 particles in the presence of an interaction potential via quantization. We then construct the corresponding barrier traversal time operator and impose the condition that the barrier h…
Tunneling is an important physical process. The observation that particles surmount a high mountain in spite of the fact that they don't have the necessary energy cannot be explained by classical physics. However, this so called tunneling became allowed by quantum mechanics. Experimental tunneling studies with different photonic barriers from microwave frequencies up to ultraviolet frequencies pointed towards a universal tunneling time (Haibel,Esposito). Experiments and calculations have shown that the tunneling time of opaque photonic barriers (optical mirrors, e.g.) equals approximately the …
We show that scattering a quantum particle on a one-dimensional potential barrier as well as scattering the electromagnetic wave on a quasi-one-dimensional layered structure (both represent scattering problems with one 'source' and two 'sinks') violate the superposition principle; the role of nonlinear elements is played here by the potential barrier and the layered structure, splitting the incident (probability and electromagnetic) wave into two parts (transmitted and reflected). This explains why all attempts to solve the tunneling time problem within the framework of the standard (linear) m…
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…
How much time does a tunneling particle spend in a barrier? A Larmor clock, one proposal to answer this question, measures the interaction between the particle and the barrier region using an auxiliary degree of freedom of the particle to clock the dwell time inside the barrier. We report on precise Larmor time measurements of ultra-cold $^{87}$Rb atoms tunneling through an optical barrier, which confirm longstanding predictions of tunneling times. We observe that atoms generally spend less time tunneling through higher barriers and that this time decreases for lower energy particles. For the …
Following our work [Phys. Rev. Lett. 125, 020401 (2020)], we discuss a semiclassical description of one-dimensional quantum tunneling through multibarrier potentials in terms of complex time. We start by defining a complex-extended continuum level density of unbound systems and show its relation to a complex time shift of the transmitted wave. While the real part of the level density and time shift describes the passage of the particle through classically allowed coordinate regions, the imaginary part is connected with an instanton-like picture of the tunneling through forbidden regions. We de…
Quantum states can be described equivalently by density matrices, Wigner functions or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient C…
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…
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 pack…
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…
It is shown that in the case of the one-particle one-dimensional scattering problem for a given time-independent potential, for each state of the whole quantum ensemble of identically prepared particles, there is an unique pair of (subensemble's) solutions to the Schrödinger equation, which, as we postulate, describe separately transmission and reflection: in the case of nonstationary states, for any instant of time, these functions are orthogonal and their sum describes the state of all particles; evolving with constant norms, one of them approaches at late times the transmitted wave packet a…
We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects pred…
We introduce a formalism for the calculation of the time of arrival t at a detector of particles traveling through interacting environments. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t. We interpret our results in terms of a Positive Operator Valued Measure. We then compute the probability distribution in times of arrival at a detector of those particles that, after their initial preparation, have undergone quantum tunneling or reflection due to the presence of potential barriers. We obtain the expected ret…
