How Much Time Does a Measurement Take?

Abstract

We consider the problem of measurement using the Lindblad equation, which allows the introduction of time in the interaction between the measured system and the measurement apparatus. We use analytic results, valid for weak system-environment coupling, obtained for a two-level system in contact with a measurer (Markovian interaction) and a thermal bath (non-Markovian interaction), where the measured observable may or may not commute with the system-environment interaction.Analysing the behavior of the coherence, which tends to a value asymptotically close to zero, we obtain an expression for the time of measurement which depends only on the system-measurer coupling, and which does not depend on whether the observable commutes with the system-bath interaction. The behavior of the coherences in the case of strong system-environment coupling, found numerically, indicates that an increase in this coupling decreases the measurement time, thus allowing our expression to be considered the upper limit for the duration of the process.

Efficient finite-time measurements under thermal regimes

“…III shows the zcomponent solution and the simplified (T = 0 and ω 0 = 0) x -component solutions found on [24] for the phase-damping interaction; finally, on Sec. V we show the formula for upper limit for the measurement duration found on [25]. All the other results, however, are new.…”

“…In previous papers [18,24,25], our formalism was applied to a two-state system in contact with an environment via phase-damping interaction, in the case of an Ohmic spectral density. Under some restrictions -both the natural system frequency and the environmental temperature set to zero -chosen to simplify the problem and allow analytical solutions, we verified that [24], when the measurement does not commute with the system-environment interaction, (i) the more intense the system-environment interaction, the more marked is the decrease of the population -i.e., the larger is the measurement error -and (ii) the more intense the system-measurement apparatus interaction, the less marked is the rate of change of the populations.…”

“…In this section, we overview the model of noisy finite-time measurement that was developed and employed in our previous works [24,25]. This model assumes that the system interacts with the measurement apparatus through a Markovian interaction.…”

“…Both here and in our previous applications [18,24,25], we have considered S a 2-state system and B the environment, with corresponding Hamiltonianŝ…”

Understanding the pointer states

“…Arguably, the state vector becomes a more fitting description of the post-measurement system exactly because its ensemble approach does not predict the future of each system as completely as the state vector. [55] As a criterion to determine the instant when the classical world emerged from the quantum one, we will employ the moment when the quantum interferences between the possible measurement results disappear, and only classical indeterminacies remain [56]. In other words, we are measuring the length of time after which the system cannot be represent by a state vector, the decoherence time (in our notation, t 1 ).…”