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Physical Review A

Complementarity in a quantum nondemolition measurement

Barry C. Sanders, Gerard J Milburn

Published Jan 1, 1989• 76 citations
Publisher

Abstract

We present an analysis of a quantum nondemolition (QND) measurement, which determines which of the two possible paths a photon follows through a Mach-Zehnder interferometer. The measurement is effected by the nonlinear Kerr effect. If a photon traverses the arm of the interferometer containing the Kerr medium, it shifts the phase of a probe beam and then continues through to the interferometer output. The measurement does not destroy the photon but does disturb its phase. Using various probe states, we show how such phase disturbances reduce fringe visibility and enforce a principle of complementarity; the more accurately the QND measurement determines the path of the photon, the lower is the fringe visibility. We determine the appropriate parameters to optimize the observation of the two complementary quantities.

DOI
10.1103/physreva.39.694

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We present an analysis of a quantum nondemolition (QND) measurement, which determines which of the two possible paths a photon follows through a Mach-Zehnder interferometer. The measurement is effected by the nonlinear Kerr effect. If a photon traverses the arm of the interferometer containing the Kerr medium, it shifts the phase of a probe beam and then continues through to the interferometer output. The measurement does not destroy the photon but does disturb its phase. Using various probe states, we show how such phase disturbances reduce fringe visibility and enforce a principle of complementarity; the more accurately the QND measurement determines the path of the photon, the lower is the fringe visibility. We determine the appropriate parameters to optimize the observation of the two complementary quantities.
The role of decoherence of an electromagnetic field in the process of quantum nondemolition measurement of the number of photons using a nonlinear Mach-Zehnder interferometer is investigated. This decoherence is caused by distributed losses in a Kerr medium. A long interaction time between the field and the medium is required to achieve a high accuracy in the photon number measurement. The losses in the Kerr medium accompanying the resonance four-wave interaction between the measured and the probe fields lead to a measurement error close to unity at γτ ≫1 (γ is the rate of losses in the medium, and τ is the time of interaction between the field and the medium); consequently, nondemolition quantum measurement turns out to be impossible in the scheme considered. Under these conditions, an increase in the intensity of the probe field does not result in achievement of the required measurement accuracy.
A scheme is proposed for experimentally achieving the famous two-slit gedanken experiment using photons. A particle’s quantum pathways can interfere to produce fringes in the probability for the particle to be found at some location. If the path taken by the particle is experimentally determined, the complementarity principle says that the fringes must disappear. To carry out this experiment with photons is difficult because normally the act of determining a photon’s location destroys it. We propose to overcome this by putting a type-II optical parametric amplifier (OPA) in each arm of a Mach-Zehnder interferometer and observing fringes at the output. An OPA responds to an input photon by increasing its probability to produce a pair of photons, one having polarization orthogonal to the input photon.
A photon-number quantum nondemolition (QND) measurement theory that takes dissipative losses into account is presented. A QND measurement criterion is developed to determine the feasibility of QND measurements in a lossy optical medium. Using an analysis based on loss-error characteristics, we examine a lossy Kerr medium. We obtain restrictions on the losses and requirements for the nonlinearity and signal and probe powers necessary to observe the QND effect. We calculate the expected results for an experimental system meeting such requirements. We conclude that QND measurements are possible in existing media.
This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.
Complementarity of wave-like and particle-like properties in the Mach–Zehnder and Michelson interferometers is investigated by means of the exactly solvable model for a quantum measurement process. When the error-minimum measurement and unambiguous measurement are used for extracting the which-path information, the fringe visibility and which-path information are derived and their complementary relation is examined. It is shown that there is a possibility of the recovery of the quantum coherence in the Michelson interferometer since the which-path information can be erased by the propagating particle itself.
The complementarity for wave-like and particle-like properties of a quantum mechanical particle in the Mach–Zehnder interferometer is investigated by means of the dynamical approach. Measurement devices are placed on the paths of the interferometer to obtain the information of a propagating particle. The which-path information of the particle and the fringe visibility of the interferometer are derived by considering the dynamics of the particle and devices. When each device consists of a single qubit, the maximum amount of the which-path information is obtained and the complementary relation is examined.