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arXiv:1309.1710 · Physical Review A

Operational approach to indirectly measuring the tunneling time

Yunjin Choi, Andrew N. Jordan

Published Nov 25, 2013• 17 citations
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Abstract

The tunneling time through an arbitrary bounded one-dimensional barrier is investigated using the dwell-time operator. We relate the tunneling time to the conditioned average of the dwell-time operator because of the natural postselection in the case of successful tunneling. We discuss an indirect measurement by timing the particle and show that we are able to reconstruct the conditioned average value of the dwell-time operator by applying the contextual values formalism for generalized measurements based on the physics of Larmor precession. The experimentally measurable tunneling time in the weak interaction limit is given by the weak value of the dwell-time operator plus a measurement-context-dependent disturbance term. We show how the expectation value and higher moments of the dwell-time operator can be extracted from measurement data of the particle's spin.

DOI
10.1103/physreva.88.052128

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The tunneling time through an arbitrary bounded one-dimensional barrier is investigated using the dwell-time operator. We relate the tunneling time to the conditioned average of the dwell-time operator because of the natural postselection in the case of successful tunneling. We discuss an indirect measurement by timing the particle and show that we are able to reconstruct the conditioned average value of the dwell-time operator by applying the contextual values formalism for generalized measurements based on the physics of Larmor precession. The experimentally measurable tunneling time in the weak interaction limit is given by the weak value of the dwell-time operator plus a measurement-context-dependent disturbance term. We show how the expectation value and higher moments of the dwell-time operator can be extracted from measurement data of the particle's spin.
Baz' and Rybachenko have proposed the use of the Larmor precession as a clock to measure the time it takes a particle to traverse a barrier. An applied magnetic field is confined to the barrier. The spin of the incident particles is polarized perpendicular to this field. The extent of the Larmor precession occurring during transmission is used as a measurement of the time spent traversing the barrier. However, the particles tunneling through an opaque barrier also acquire a spin component parallel to the field since particles with spin parallel to the field have a higher transmission probability than particles with spin antiparallel to the field. Similar effects are actually used to polarize electrons and neutrons. An interpretation of this experiment compares the results with an approach which determines the traversal time by studying transmission of particles through a time-modulated barrier. This leads to three characteristic times describing the interaction of particles with a barrier. A dwell time measures the average time interval during which a particle interacts with the barrier whether it is reflected or transmitted at the end of its stay, a traversal time measures the time interval during which a particle interacts with the barrier if it is finally transmitted, and a reflection time measures the interaction time of a reflected particle.
Abstract Different approaches for considering barrier crossing times are analyzed, with special emphasis on recent experiments which attempt to measure what is commonly referred to as the Larmor tunneling time. We show that that these experiments cannot reveal the Larmor time, due to the finite energy width of the incident particles. The Larmor time, which measures changes in spin polarization, is classified together with other measurements such as the Buttiker–Landauer oscillating barrier time as indirect measurements of interaction times of scattered particles. In contrast, we present a direct quantum mechanical measure of a barrier crossing time taken to be the difference between the mean flight time for a particle transmitted through a potential barrier incident on a screen and the time it would take to reach the same screen without the barrier. These metrics are asymptotic, in the sense that they infer a time from a measurement after the scattering event is over, whereas other measures like the dwell time are local. Some time measures are well-defined only for incident states which are monochromatic in energy, others are well-defined also for incident wavepackets whose incident energy width is finite. In this paper we compare the different approaches to conclude that only the flight time can be used to answer the provocative (but ultimately ill-posed) question: how much time does it take to tunnel through a barrier?
The Larmor precession of a relativistic neutral spin-12\frac{1}{2}21​ particle in a uniform constant magnetic field confined to the region of a one-dimensional arbitrary potential barrier is investigated. The spin precession serves as a clock to measure the time spent by a quantum particle traversing a potential barrier. With the help of a general spin coherent state, it is explicitly shown that the precession time is equal to the dwell time.
We show that a useful concept of the time spent by a quantum-mechanical particle in a given spatial region can be expressed in terms of a Hermitian sojourn time operator. Mean values of this operator give the mean sojourn time (dwell time) of the particle. We show that the sojourn time operators occur in a natural way in first-order perturbation theory for the barrier perturbed by any finite-range potential. The perturbed S operator, and also the effect of the perturbation on the change of observables due to the scattering, are, to a first-order approximation, fully expressible in terms of sojourn time operators. We anticipate that this result can be generalized to the case when the translational motion of the tunneling particle is weakly coupled to additional degrees of freedom. As a specific example of such a system, we explicitly reconsider the Larmor clock, originally proposed as a device measuring interaction times. We show that the change of the spin components, to a first-order approximation, is indeed fully expressible in terms of a sojourn time operator. In particular, when set as a clock, the system measures mean values of the sojourn time operator or real parts of some of its matrix elements.
The Larmor precession of a neutral spinning particle in a magnetic field confined to the region of a one-dimensional rectangular barrier is investigated for both a nonrelativistic and a relativistic incoming particle. The spin precession serves as a clock to measure the time spent by a quantum particle traversing a potential barrier. With the help of a general spin coherent state it is explicitly shown that the precession time is equal to the dwell time in both the nonrelativistic and relativistic cases. We also present a numerical estimation of the precession time showing an apparent superluminal tunneling.