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Journal of Physics A Mathematical and Theoretical

Quantum first time-of-arrival operators

Eric A. Galapon, Anthony Villanueva

Published Oct 15, 2008• 31 citations
Publisher

Abstract

This paper undertakes the problem of constructing first time-of-arrival operators for arbitrary everywhere analytic potentials in the space of operators spanned by the Bender–Dunne basis (Bender and Dunne 1989 Phys. Rev. D 40 3504). The operators are conjugate with their respective system Hamiltonians and reduce to their corresponding classical time-of-arrival expressions for infinitesimal ℏ. The supraquantized time-of-arrival operators (Galapon 2004 J. Math. Phys. 45 3180) and the confined time-of-arrival operators (Galapon et al 2004 Phys. Rev. Lett. 93 180406, Galapon 2006 J. Mod. Phys. A 21 6351) are then shown to be two distinct representations of the time-of-arrival operators constructed in the Bender–Dunne space.

DOI
10.1088/1751-8113/41/45/455302

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This paper undertakes the problem of constructing first time-of-arrival operators for arbitrary everywhere analytic potentials in the space of operators spanned by the Bender–Dunne basis (Bender and Dunne 1989 Phys. Rev. D 40 3504). The operators are conjugate with their respective system Hamiltonians and reduce to their corresponding classical time-of-arrival expressions for infinitesimal ℏ. The supraquantized time-of-arrival operators (Galapon 2004 J. Math. Phys. 45 3180) and the confined time-of-arrival operators (Galapon et al 2004 Phys. Rev. Lett. 93 180406, Galapon 2006 J. Mod. Phys. A 21 6351) are then shown to be two distinct representations of the time-of-arrival operators constructed in the Bender–Dunne space.
The constructions of two classes of self-adjoint and compact first time of arrival operators for confined systems under arbitrary, everywhere continuous potential are detailed, extending in the interacting case the concept of confined quantum time of arrival operators first developed for the free particle. One class is the quantized confined time of arrival operators and another is the class of algebra preserving confined time of arrival operators. The former is the projection of the quantization of the classical time of arrival, and is constructed by solving the quantization problem of the multiple-valued and non-everywhere-real-valued classical time of arrival for arbitrary potential. The later arises to address the nonconjugacy with the Hamiltonian of the entire class of the quantized confined time of arrival operators, and is constructed by solving the obstruction to quantization present in Euclidean space. These two sets of operators coincide for linear systems; but differ for nonlinear systems, with the former as the leading term of the latter. The confined time of arrival operators for potentials representative of linear and nonlinear systems are numerically investigated and demonstrated to have the same dynamical behaviors as those of the free confined time of arrival operators. In particular, the eigenfunctions evolve according to Schrödinger equation such that the corresponding probabilities of locating the quantum particle in the neighborhood of the arrival point are maximum at their respective eigenvalues.
We give full account of our recent report in E. A. Galapon, R. Caballar, and R. Bahague, Phys. Rev. Lett. 93, 180406 (2004), where it is shown that formulating the free quantum time of arrival problem in a segment of the real line suggests rephrasing the quantum time of arrival problem to finding a complete set of states that evolve to unitarily arrive at a given point at a definite time. For a spatially confined particle, here it is shown explicitly that the problem admits a solution in the form of an eigenvalue problem of a class of compact and self-adjoint time of arrival operators derived by a quantization of the classical time of arrival. The eigenfunctions of these operators are numerically demonstrated to unitarily arrive at the origin at their respective eigenvalues.
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.
We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the basis elements.