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Fortschritte der Physik

Determination of Multimode Moments of Creation and Annihilation Operators via Multimode Balanced Homodyning with a Single Homodyne Detector

Jaromı́r Fiurášek

Published Oct 1, 2001• 1 citation
Publisher

Abstract

We investigate reconstruction of moments of creation and annihilation operators of multimode traveling light field from data recorded in multimode balanced homodyne detection with a single homodyne detector. We derive sampling functions whose average over the measured quadrature statistics provides the estimates of the moments. We show that detection losses may be compensated by a simple rescaling of the sampling functions. We also show that all moments of given order can be reconstructed from the statistics of a finite number of quadratures.

DOI
10.1002/1521-3978(200110)49:10/11<955::aid-prop955>3.0.co;2-b

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We investigate reconstruction of moments of creation and annihilation operators of multimode traveling light field from data recorded in multimode balanced homodyne detection with a single homodyne detector. We derive sampling functions whose average over the measured quadrature statistics provides the estimates of the moments. We show that detection losses may be compensated by a simple rescaling of the sampling functions. We also show that all moments of given order can be reconstructed from the statistics of a finite number of quadratures.
We consider multimode homodyne tomography with a single local oscillator and derive sampling functions for multimode s-parametrized quasidistributions and s-ordered moments of creation and annihilation operators. In both cases, loss-compensating sampling functions, applicable to realistic measurement with imperfect detection, are provided. Results of Monte Carlo simulations for a squeezed three-mode state are reported and the feasibility of reconstruction of the three-mode Q function from 107{10}^{7}107 sampled data points is demonstrated.
We consider the moment operators of the observable (i.e. a semispectral measure or POM) associated with the balanced homodyne detection statistics, whilst paying attention to the correct domains of these unbounded operators. We show that the high amplitude limit, when performed on the moment operators, actually determines uniquely the entire statistics of a rotated quadrature amplitude of the signal field, thereby verifying the usual assumption that the homodyne detection achieves a measurement of that observable. We also consider, in a general setting, the possibility of constructing a measurement of a single quantum observable from a sequence of observables by taking the limit on the level of moment operators of these observables. In this context, we show that under some natural conditions (each of which is satisfied by the homodyne detector example), the existence of the moment limits ensures that the underlying probability measures converge weakly to the probability measures of the limiting observable. The moment approach naturally requires that the observables be determined by their moment operator sequences (which does not automatically happen), and it turns out, in particular, that this is the case for the balanced homodyne detector.
We consider double balanced homodyne detection schemes with and without (local oscillator) phase-randomized detection. We discuss the reconstruction of the photon statistics from phase-randomized measurements. We show how the Wigner function of a photon-number state can be synthesized from phase-randomized double homodyne measurements of two properly prepared field states. Moreover we propose a new procedure to determine the whole density matrix from the Q function. Finally we express the Q function in terms of normally ordered moments and discuss their reconstruction.
Balanced homodyne correlation allows the measurement of moments of creation and annihilation operators with unequal or equal powers with high signal to noise ratios, thus giving access to high order moments. In this work we implement this method for the first time. Our realisation with eight detectors gives access to field moments up to fourth order. We use a laser direct written waveguide structures to implement the beam splitter cascade, and we use a photodiode array for multi-beam detection. The set-up is successfully verified with squeezed- and coherent states.