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Physical Review A

Timing in quantum measurements of position and momentum

Michael Bußhardt, Matthias Freyberger

Published Oct 28, 2010• 14 citations
Publisher

Abstract

The prototype for a simultaneous measurement of two conjugate variables was originally introduced by Arthurs and Kelly in 1965. It relies on coupling the quantum particle to be probed to two additional systems, which serve as measurement pointers. In this contribution we investigate an extended scheme to measure position and momentum of a massive particle. By considering an explictly time-dependent coupling we can quantify the timing of the measurement. We investigate how the noise resulting from such a measurement process depends on the interaction strength and the size of the pointers. In particular, we focus on the question of which measurement timing minimizes the corresponding uncertainty product.

DOI
10.1103/physreva.82.042117

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The prototype for a simultaneous measurement of two conjugate variables was originally introduced by Arthurs and Kelly in 1965. It relies on coupling the quantum particle to be probed to two additional systems, which serve as measurement pointers. In this contribution we investigate an extended scheme to measure position and momentum of a massive particle. By considering an explictly time-dependent coupling we can quantify the timing of the measurement. We investigate how the noise resulting from such a measurement process depends on the interaction strength and the size of the pointers. In particular, we focus on the question of which measurement timing minimizes the corresponding uncertainty product.
We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.
The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty \nrelations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. \nSpecifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances \nin terms of the commutators. \nFurthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality \nis found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive \nvarious types of inequalities. \nFinally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of \nthree canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.
We propose an optical scheme for the simultaneous measurement of the position and momentum of a single atom. The scheme involves the coupling of the atom of two light fields with different spatial and polarization characteristics. The proposed technique is closely related to the Arthurs-Kelly measurement scheme; the principal difference is that in the present case the values of the position and momentum are inferred from phase shifts in electromagnetic fields rather than from shifts in the position of a pointer.
Abstract A modern state-of-the-art “quantum measurement” [The term “quantum measurement” as used here implies that parameters of atomic particles are measured that emerge from a single scattering process of quantum particles.] of momentum and position of a single electron at a given time [“at a given time” means directly after the scattering process. (It should be noticed that the duration of the reaction process is typically extremely short => attoseconds).] and the precision limits for their experimental determination are discussed from an experimentalists point of view. We show—by giving examples of actually performed experiments—that in a single reaction between quantum particles at a given time only the momenta of the emitted particles but not their positions can be measured with sub-atomic resolution. This fundamental disparity between the conjugate variables of momentum and position is due to the fact that during a single-event measurement only the total momentum but not position is conserved as function of time. We highlight, that (other than prevalently perceived) Heisenberg’s “Uncertainty Relation” UR [1] does not limit the achievable resolution of momentum in a single-event measurement. Thus, Heisenberg’s statement that in a single-event measurement only either the position or the momentum (velocity) of a quantum particle can be measured with high precision contradicts a real experiment. The UR states only a correlation between the mean statistical fluctuations of a large number of repeated single-event measurements of two conjugate variables. A detailed discussion of the real measurement process and its precision with respect to momentum and position is presented.
We combine traditional pointer-based simultaneous measurements of conjugate observables with the concept of quantum Brownian motion of multipartite systems to phenomenologically model simultaneous measurements of conjugate observables in a thermal environment. This approach provides us with a formal solution of the complete measurement dynamics for quadratic Hamiltonians and we can therefore discuss the measurement uncertainty and optimal measurement times. As a main result, we obtain a lower bound for the uncertainty of a noisy measurement, which is an extension of a previously known uncertainty relation and in which the squeezing of the system state to be measured plays an important role. This also allows us to classify minimal uncertainty states in more detail.
The von Neumann interaction between a particle and an apparatus has been considered in the measurement of the position of a particle when the interaction lasts for a finite amount of time. When the measurement has finite duration, both the motion of the pointer and the particle influences the result of the measurement. Provided that the particle is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle. Furthermore, the probability that the pointer will indicate a given average value is equal to the transition probability for the undisturbed particle to experience the change in position. If the initial state of the pointer is a narrow wave packet, then for any initial state of the particle, the measurement yields, approximately, the undisturbed probability distribution for the position of the particle at the end of the measurement.