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arXiv:quant-ph/0002011 · Physical Review A

Time of arrival in the presence of interactions

Juan León, J. Julve, P. Pitanga, Fernando Jordán de Urries

Published May 3, 2000• 55 citations
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Abstract

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.

DOI
10.1103/physreva.61.062101

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Journal of Physics A Mathematical and Theoretical•Apr 21, 2011•7 citations
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.
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 model ideal arrival-time measurements for free quantum particles and for particles subject to an external interaction by means of a narrow and weak absorbing potential. This approach is related to the operational approach of measuring the first photon emitted from a two-level atom illuminated by a laser. By operator normalizing the resulting time-of-arrival distribution, a distribution is obtained which for freely moving particles not only recovers the axiomatically derived distribution of Kijowski for states with purely positive momenta but is also applicable to general momentum components. For particles interacting with a square barrier the mean arrival time and corresponding ``tunneling time'' obtained at the transmission side of the barrier become independent of the barrier width (Hartman effect) for arbitrarily wide barriers, i.e., without the transition to the ultraopaque, classical-like regime dominated by wave packet components above the barrier.
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.
The time of arrival at an arbitrary position in configuration space can be given as a function of the phase space variables for the Liouville integrable systems of classical mechanics, but only for them. We review the Jacobi-Lie transformation that explicitly implements this function of positions and momenta. We then discuss the recently developed quantum formalism for the time of arrival. We first analyze the case of free particles in one and three space dimensions. Then, we apply the quantum version of the Jacobi-Lie transformation to work out the time of arrival operator in the presence of interactions. We discuss the formalism and its interpretation. We finish by disclosing the presence (absence) of "instantaneous" tunneling for thin (thick) barriers.
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.
The propagator approach combined with the multiple-scattering theory is applied to the particle time of arrival (TOA) problem. This approach allows us to naturally include in the consideration the components of the particle initial wavefunction (defined at t = t0) corresponding to the positive (forward-moving term) and negative (backward-moving term) momenta. For a freely moving particle it is shown that the Allcock definition of the ideal total TOA probability disregards the backward-moving and interference terms entirely. In the presence of a measuring apparatus modeled by an imaginary step potential with the amplitude V0, the general expression for the TOA rate is obtained, the forward-moving component of which coincides with that obtained by Allcock. It is shown that when the initial particle wavefunction is well separated from the point of arrival and has a well-defined average momentum, the contribution of the backward-moving and interference terms are small and can be neglected. For a small V0, except the well-known convolution result by Allcock–Kijowski, the exponential form of the TOA rate follows at the double limit condition V0 → 0, t − t0 ~ /2V0 → ∞ (2V0(t − t0)/ is finite) while the backward-moving and interference terms vanish. We show that the Allcock result for the TOA rate is valid in the entire range of V0 including the Zeno case (V0 → ∞) and the normalized TOA rate can be introduced for all values of V0 as a probability distribution. The latter is illustrated for the Gaussian wave packet.