arXiv:1105.5359 · Physical Review A

Measurement of noncommuting spin components using spin-orbit interaction

D. Sokolovski, Eugene Ya. Sherman

Published Sep 22, 2011• 11 citations

Abstract

We propose a possible experiment aimed at a joint measurement of two noncommuting spin-1/2 components and analyze its physical meaning. We demonstrate that switching of a strong spin-orbit interaction, e.g., in a solid-state or a cold-atom system, for a short time interval simulates a simultaneous von Neumann measurement of the operators ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{y}$. With the spin dynamics mapped onto the quantum coordinate-space motion, such an experiment determines averages of ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{y}$ over the duration of the measurement, however short the latter may be. These time averages, unlike the instantaneous values of ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{y}$, may be evaluated simultaneously to an arbitrary accuracy.

DOI: 10.1103/physreva.84.030101

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arXiv:1105.5359 · Physical Review A

Measurement of noncommuting spin components using spin-orbit interaction

D. Sokolovski, Eugene Ya. Sherman

Published Sep 22, 2011• 11 citations

arXiv HTML PDF

Abstract

{\ensuremath{\sigma}}_{x}

and

{\ensuremath{\sigma}}_{y}

. With the spin dynamics mapped onto the quantum coordinate-space motion, such an experiment determines averages of

{\ensuremath{\sigma}}_{x}

and

{\ensuremath{\sigma}}_{y}

over the duration of the measurement, however short the latter may be. These time averages, unlike the instantaneous values of

{\ensuremath{\sigma}}_{x}

and

{\ensuremath{\sigma}}_{y}

, may be evaluated simultaneously to an arbitrary accuracy.

DOI: 10.1103/physreva.84.030101

Related papers

von Neumann spin measurements with Rashba fields

Eugene Ya. Sherman, D. Sokolovski

We show that dynamics in the spin–orbit coupling field simulate the von Neumann measurement of a particle spin. We demonstrate how the measurement influences the spin and coordinate evolution of a particle by comparing two examples of such a procedure. The first example is a simultaneous measurement of spin components, σ x and σ y , corresponding to non-commuting operators, which cannot be accurately obtained together at a given time instant due to the Heisenberg uncertainty ratio. By mapping spin dynamics onto a spatial walk, such a procedure determines measurement-time averages of σ x and σ y , which can already be precisely evaluated in a single short-time measurement. The other, qualitatively different, example is the spin of a one-dimensional particle in a magnetic field. Here, the measurement outcome depends on the angle between the spin–orbit coupling and magnetic fields. These results can be applied to studies of spin–orbit coupled cold atoms and electrons in solids.We show that dynamics in the spin–orbit coupling field simulate the von Neumann measurement of a particle spin. We demonstrate how the measurement influences the spin and coordinate evolution of a particle by comparing two examples of such a procedure. The first example is a simultaneous measurement of spin components, σ x and σ y , corresponding to non-commuting operators, which cannot be accurately obtained together at a given time instant due to the Heisenberg uncertainty ratio. By mapping spin dynamics onto a spatial walk, such a procedure determines measurement-time averages of σ x and σ y , which can already be precisely evaluated in a single short-time measurement. The other, qualitatively different, example is the spin of a one-dimensional particle in a magnetic field. Here, the measurement outcome depends on the angle between the spin–orbit coupling and magnetic fields. These results can be applied to studies of spin–orbit coupled cold atoms and electrons in solids.

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Simultaneous measurement of two noncommuting quantum variables: Solution of a dynamical model

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{\ensuremath{\sigma}}_{x}

and

{\ensuremath{\sigma}}_{y}

. With the spin dynamics mapped onto the quantum coordinate-space motion, such an experiment determines averages of

{\ensuremath{\sigma}}_{x}

and

{\ensuremath{\sigma}}_{y}

over the duration of the measurement, however short the latter may be. These time averages, unlike the instantaneous values of

{\ensuremath{\sigma}}_{x}

and

{\ensuremath{\sigma}}_{y}

, may be evaluated simultaneously to an arbitrary accuracy.

The possibility of performing simultaneous measurements in quantum mechanics is investigated in the context of the Curie-Weiss model for a projective measurement. Concretely, we consider a spin--

\frac{1}{2}

system simultaneously interacting with two magnets, which act as measuring apparatuses of two different spin components. We work out the dynamics of this process and determine the final state of the measuring apparatuses, from which we can find the probabilities of the four possible outcomes of the measurements. The measurement is found to be nonideal, as (i) the joint statistics do not coincide with the one obtained by separately measuring each spin component, and (ii) the density matrix of the spin does not collapse in either of the measured observables. However, we give an operational interpretation of the process as a generalized quantum measurement, and show that it is fully informative: The expected value of the measured spin components can be found with arbitrary precision for sufficiently many runs of the experiment.

1∕2

particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.