Spectral dimension controlling the decay of the quantum first-detection probability

Abstract

We consider a quantum system that is initially localized at xin and that is repeatedly projectively probed with a fixed period τ at position x d . We ask for the probability that the system is detected in x d for the very first time, Fn, where n is the number of detection attempts. We relate the asymptotic decay and oscillations of Fn with the system's energy spectrum, which is assumed to be absolutely continuous. In particular Fn is determined by the Hamiltonian's measurement spectral density of states (MSDOS) f(E) that is closely related to the density of energy states (DOS). We find that Fn decays like a power law whose exponent is determined by the power law exponent dS of f(E) around its singularities E * . Our findings are analogous to the classical first passage theory of random walks. In contrast to the classical case, the decay of Fn is accompanied by oscillations with frequencies that are determined by the singularities E * . This gives rise to critical detection periods τc at which the oscillations disappear. In the ordinary case dS can be identified with the spectral dimension found in the DOS. Furthermore, the singularities E * are the van Hove singularities of the DOS in this case. We find that the asymptotic statistics of Fn depend crucially on the initial and detection state and can be wildly different for out-of-the-ordinary states, which is in sharp contrast to the classical theory. The properties of the first detection probabilities can alternatively be derived from the transition amplitudes. All our results are confirmed by numerical simulations of the tight-binding model, and of a free particle in continuous space both with a normal and with an anomalous dispersion relation. We provide explicit asymptotic formulae for the first detection probability in these models.

Uncertainty Relation between Detection Probability and Energy Fluctuations

“…As mentioned, more advanced methods, intended for larger systems, are discussed in [24]. A general strategy, to improve the results obtained here, is based on further collecting bright or dark states, instead of the two we used in Equations ( 7), (12) and (19). In principle, and in particular with a simple computer program, one may use more states with a Gram-Schmidt method, and gain tighter bounds than those found here.…”

“…However, destructive interference may divide the Hilbert space into two components called dark and bright, and this yields an inefficient search and an effect superficially similar to classical non-ergodicity, . More specifically, an observer performs repeated strong measurements, on a node other than the starting node, in an attempt to detect the particle [ 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ]. In the time intervals between the measurements the dynamics is unitary.…”

Dark states of quantum search cause imperfect detection

“…In Ref. [20] we showed that the denominator of ϕ(z) is a Cauchy transform of the so-called wrapped measurement spectral density of states, μ(θ ) :…”

“…Reference [20] discusses μ(λ) in depth for systems with a continuous spectrum. [Note that this reference defines ν(λ) in a slightly different way.]…”

The quantum Zeno and anti-Zeno effects with non-selective projective measurements

“…Variants of Eq. ( 1 ) have also been considered under the guise of quantum recurrence and the quantum first detection problem 55 – 58 .…”