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arXiv:quant-ph/9912109 · Physical Review A

Time of arrival through interacting environments: Tunneling processes

Ken-Ichi Aoki, Atsushi Horikoshi, Etsuko Nakamura

Published Jun 29, 2000• 24 citations
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Abstract

We discuss the propagation of wave packets through interacting environments. Such environments generally modify the dispersion relation or shape of the wave function. To study such effects in detail, we define the distribution function ${P}_{X}(T),$ which describes the arrival time T of a packet at a detector located at point X. We calculate ${P}_{X}(T)$ for wave packets traveling through a tunneling barrier and find that our results actually explain recent experiments. We compare our results with Nelson's stochastic interpretation of quantum mechanics and resolve a paradox previously apparent in Nelson's viewpoint about the tunneling time.

DOI
10.1103/physreva.62.022101

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We discuss the propagation of wave packets through interacting environments. Such environments generally modify the dispersion relation or shape of the wave function. To study such effects in detail, we define the distribution function PX(T),{P}_{X}(T),PX​(T), which describes the arrival time T of a packet at a detector located at point X. We calculate PX(T){P}_{X}(T)PX​(T) for wave packets traveling through a tunneling barrier and find that our results actually explain recent experiments. We compare our results with Nelson's stochastic interpretation of quantum mechanics and resolve a paradox previously apparent in Nelson's viewpoint about the tunneling time.
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 retardation or advancement for transmitted wave packets, and non-foreseen double bump structures for some cases of reflection.
We introduce a formalism for the calculation of the time of arrival t at a space point for particles traveling through interacting media. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t in the context of the positive-operator-valued measures. We then compute the probability distribution in the times of arrival at a point for particles that have undergone reflection, transmission or tunneling off finite potential barriers. For narrow Gaussian initial wave packets we obtain multimodal time distribution of the reflected packets and a combination of the Hartman effect with unexpected retardation in tunneling. We also employ explicitly our formalism to deal with arrivals in the interaction region for the step and linear potentials.
The transmission of wave packets through barriers by tunneling is studied in detail by the method of quantum molecular dynamics. The distribution of the arrival times of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The average position and average momentum of the packet and their spread are investigated. It is found that below the barrier a part of the packet is reflected, and a Gaussian barrier increases the average momentum of the transmitted packet and its spread in momentum space.
We consider a scattering process inside a dispersive and absorptive medium, and derive an analytical expression for the scattering-time parameter that characterizes the duration of this process. We consider a microwave wave packet that scatters from a dielectric inhomogeneity inside an otherwise homogeneous waveguide. The waveguide is filled with a dielectric medium with a finite inhomogeneity region in the direction of the wave-packet propagation. The permittivity of the entire medium is described by a complex function of the frequency. Hence both absorption and dispersion affect the propagation of the wave packet. We define the scattering-time parameter to be the difference in the ``average arrival time'' between the free and actual propagation. We show that this time difference (in the limit of a narrow-band wave packet) is given by a linear combination of the real and imaginary parts of the ``complex tunneling time.'' In the special case where there is no dissipation or attenuation outside the scatterer, we recover the well-known result that the scattering-time parameter is identical to the real part of the complex tunneling time. The contribution of the imaginary part of the complex tunneling time to the scattering-time parameter can, therefore, be attributed to absorption and attenuation outside the scatterer. We further show that the real and imaginary-time parameters satisfy dispersion relations. Causality in this process is briefly discussed.
An interesting analytical expression is obtained for the coordinate probability distribution of a wave packet transmitted through a potential barrier. In the large-time asymptotics the distribution is given in terms of the transmission amplitude and the phase-space quasidistribution of the initial (not necessarily pure) state. The distortion of a (broad) wave packet due to the barrier penetration is determined by two timelike parameters, given by the energy derivative of the complex transmission amplitude. The resulting change of the packet shape reveals a dependence on the detector position and incorporates a possible speed-up effect and a negative dispersion.
We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.