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Physical Review A

Using feedback to eliminate back-action in quantum measurements

Howard M. Wiseman

Published Mar 1, 1995• 72 citations
Publisher

Abstract

A quantum-nondemolition (QND) measurement is one which does not disturb the quantity it measures. Any measurement can be regarded as a QND measurement followed by an additional back-action which may disturb that quantity. By feeding back the measurement result to control the system dynamics, this back-action can always be eliminated in principle. A practical device based on homodyne measurement of the x quadrature of a cavity mode is investigated. First, a ${\mathrm{\ensuremath{\chi}}}^{(2)}$ crystal is inserted in the cavity so that it becomes a degenerate parametric oscillator at threshold. Then the photocurrent is used to control coherent driving of the cavity, so as to give positive feedback with unit gain. This feedback, combined with the nonlinearity, turns the homodyne measurement into a QND measurement of x. The device can also be used to measure the x quadrature of a traveling wave, and gives near-perfect QND correlations over a large bandwidth. However, the quality of the measurement is badly degraded by even slightly inefficient photodetectors.

DOI
10.1103/physreva.51.2459

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A quantum-nondemolition (QND) measurement is one which does not disturb the quantity it measures. Any measurement can be regarded as a QND measurement followed by an additional back-action which may disturb that quantity. By feeding back the measurement result to control the system dynamics, this back-action can always be eliminated in principle. A practical device based on homodyne measurement of the x quadrature of a cavity mode is investigated. First, a \ensuremathχ(2){\mathrm{\ensuremath{\chi}}}^{(2)}\ensuremathχ(2) crystal is inserted in the cavity so that it becomes a degenerate parametric oscillator at threshold. Then the photocurrent is used to control coherent driving of the cavity, so as to give positive feedback with unit gain. This feedback, combined with the nonlinearity, turns the homodyne measurement into a QND measurement of x. The device can also be used to measure the x quadrature of a traveling wave, and gives near-perfect QND correlations over a large bandwidth. However, the quality of the measurement is badly degraded by even slightly inefficient photodetectors.
Conventional devices for measuring the amplitude of a light wave absorb the light, thus destroying it. Recently, quantum nondemolition (QND) methods based on the optical Kerr effect have been shown capable of measuring optical amplitude fluctuations with near quantum-limited sensitivity.1 In these experiments, amplitude fluctuations of a signal beam are transferred to the phase fluctuation quadrature of a probe beam which co-propagates with the signal beam, by the intensity-dependent refractive index, without absorption of the signal beam. A pure Kerr nonlinearity can cause only phase modulation of the light wave, and the amplitude of the signal wave is preserved. Such methods should more properly be called back-action evading measurement schemes since the quantity being measured is not perturbed by the back-action that quantum mechanics requires for any measurement. The signal-to-noise ratio achieved in these experiments was limited by the small magnitude of the Kerr nonlinearity relative to the excess phase noise imposed on the probe beam by thermal light scattering processes in the glass fibers that were employed.
Measuring a quantum system can randomly perturb its state. The strength and nature of this back-action depend on the quantity that is measured. In a partial measurement performed by an ideal apparatus, quantum physics predicts that the system remains in a pure state whose evolution can be tracked perfectly from the measurement record. We demonstrated this property using a superconducting qubit dispersively coupled to a cavity traversed by a microwave signal. The back-action on the qubit state of a single measurement of both signal quadratures was observed and shown to produce a stochastic operation whose action is determined by the measurement result. This accurate monitoring of a qubit state is an essential prerequisite for measurement-based feedback control of quantum systems.
Quantum nonlinear feedback control is developed for traveling optical fields. We first describe the discretization of the traveling optical fields. The discrete-time formulation is used to describe the stochastic master equation subject to homodyne measurement. Nonlinear feedback is formulated by directly feeding the measurement outcomes back to the traveling field through a multiplicative action. Since the measurement outcomes have a correlation with the system, the multiplicative feedback control can create nonlinear effects in the traveling field. In this formulation, a time delay is naturally introduced in the feedback loop. This is essentially different from instantaneous feedback in a continuous-time setting. As an example of the feedback scheme, a quantum nondemolition sum gate is considered. Numerical results show that quantum superposition state can be created by applying the feedback to a squeezed state.
A quantum-nondemolition (QND) measurement of the vibrational energy of a trapped ion may be made by placing it at a node of a cavity optical standing wave. In certain parameter regimes there is a backaction evading coupling between the ion's vibrational energy eigenstates and the coherent states of the cavity field. This facilitates a QND measurement of the ion trap state via a homodyne measurement on the cavity output field. This scheme has the advantage that the required measurements minimally disturb the ion.
We explore experimentally the fundamental projective properties of a quantum measurement and their application in the control of a system's evolution. We perform quantum non-demolition (QND) photon counting on a microwave field trapped in a very-high- Q superconducting cavity, employing circular Rydberg atoms as non-absorbing probes of light. By repeated measurement of the cavity field we demonstrated the freeze of its initially coherent evolution, illustrating the back action of the photon number measurement on the field's phase. On the contrary, by utilizing a weak QND measurement in combination with the control injection of coherent pulses, we plan to force the field to deterministically evolve towards any target photon-number state. This quantum feedback procedure will enable us to prepare and protect photon-number states against decoherence.
We present the quantum theory of optical cavity feedback mediated by homodyne detection, with an arbitrary time delay. We apply this theory to a system with nonclassical dynamics, a sub-Poissonian pumped laser. By using the feedback to phase lock the laser it is possible to produce output light which exhibits perfect quadrature squeezing on resonance, rather than just sub-Poissonian intensity statistics. However, we also show that feedback mediated by homodyne detection (or any other extracavity measurement) cannot produce nonclassical light unless the cavity dynamics can do so without feedback. Furthermore, in systems which already exhibit squeezing, such feedback can only degrade the squeezing in the output. With feedback mediated by an intracavity measurement, these theorems do not apply. We show that an (admittedly unrealistic) intracavity quantum nondemolition quadrature measurement allows arbitrary squeezing to be produced by controlling the amplitude of a coherent driving field.