Abstract
We present a generalization of continuous position measurements that accounts for a spatially inhomogeneous measurement strength. This describes many real measurement scenarios, in which the rate at which information is extracted about position has itself a spatial profile, and includes measurements that detect whether a particle has crossed from one region into another. We show that such measurements can be described, in their averaged behavior, as stochastically fluctuating potentials of vanishing time average. Reasonable constraints restrict the form of the measurement to have degenerate outcomes, which tend to drive the system to spatial superposition states. We present the results of quantum-trajectory simulations for measurements with a step-function profile (a ‘which-way’ measurement) and a Gaussian profile. We find that the particle can coherently reflect from the measurement region in both cases, despite the stochastic nature of the measurement back-action. In addition, we explore the connection to the quantum Zeno effect, where we find that the reflection probability tends to unity as the measurement strength increases. Finally, we discuss two physical realizations of a spatially varying position measurement using atoms.
Highlights
Continuous measurements have an important role in quantum control and in the transition to the classical limit of quantum mechanics
Continuous position measurements in particular can be employed in feedback control loops for cooling quantum systems [11]
It has been shown previously by a number of authors that a measurement that determines whether a particle is on one side of a dividing line or the other can exclude the wave function from the region in which the particle is initially absent [21]–[29]. This causes the particle to reflect from the dividing line, and is due to the quantum Zeno effect. In our case this corresponds to a measurement with the following two properties: (i) the measurement strength takes one of just two values, and makes a jump from one value to the other at the dividing line; (ii) the measurement strength is much greater than the kinetic energy of the incoming particle, suitably scaled
Summary
We will derive the equation of motion describing a spatially varying position measurement. If we mix in a ‘local oscillator’ with this signal, we pass over to a master equation similar in form to the usual position-measurement master equation [8]. This corresponds to the transformation μ μ. (8b) where α = |α|eiφ is the complex amplitude of the local oscillator, which leaves the unconditioned evolution (i.e. the evolution averaged over all the possible measurement results) unchanged. For large |α|, we can pass over to the white-noise limit. In the large |α| limit, the resulting white-noise master equation is dρ. Where we have taken an ensemble average over all possible noise realizations
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