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arXiv:0706.4436 · Journal of Modern Optics

On the moment limit of quantum observables, with an application to the balanced homodyne detection

Jukka Kiukas, P. Lahti†

Published Apr 10, 2008• 15 citations
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Abstract

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.

DOI
10.1080/09500340701624658

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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.
The questions studied in this thesis are centered around the moment operators of a quantum observable, the latter being represented by a normalized positive operator measure. The moment operators of an observable are physically relevant, in the sense that these operators give, as averages, the moments of the outcome statistics for the measurement of the observable. The main questions under consideration in this work arise from the fact that, unlike a projection valued observable of the von Neumann formulation, a general positive operator measure cannot be characterized by its first moment operator. The possibility of characterizing certain observables by also involving higher moment operators is investigated and utilized in three different cases: a characterization of projection valued measures among all the observables is given, a quantization scheme for unbounded classical variables using translation covariant phase space operator measures is presented, and, finally, a mathematically rigorous description is obtained for the measurements of rotated quadratures and phase space observables via the high amplitude limit in the balanced homodyne and eight-port homodyne detectors, respectively. In addition, the structure of the covariant phase space operator measures, which is essential for the above quantization, is analyzed in detail in the context of a (not necessarily unimodular) locally compact group as the phase space.
It is usually assumed that a balanced homodyne detector measures a variable called the quadrature phase amplitude of a signal field. We examine this assumption by obtaining analytic expressions for the probability statistics for the output of such a detector for a single-mode signal with both signal and local oscillator treated quantum mechanically. We investigate the conditions under which the statistics of the quadrature phase are reproduced in the actual output of the detector. We show that the most obvious condition---that the number of quanta in the local oscillator be much larger than the number in the signal---is not sufficient to ensure that a balanced homodyne detector acts like an ideal detector of quadrature phases. Furthermore, we obtain the explicit conditions that are necessary and sufficient to reproduce the interference features in the homodyne statistics of a superposition of coherent states.
We discuss a balanced homodyne detection scheme with imperfect detectors in the framework of the operational approach to quantum measurement. We show that a realistic homodyne measurement is described by a family of operational observables that depends on the experimental setup, rather than a single field quadrature operator. We find an explicit form of this family, which fully characterizes the experimental device and is independent of a specific state of the measured system. We also derive operational homodyne observables for the setup with a random phase, which has been recently applied in an ultrafast measurement of the photon statistics of a pulsed diode laser. The operational formulation directly gives the relation between the detected noise and the intrinsic quantum fluctuations of the measured field. We demonstrate this with two examples: the operational uncertainty relation for the field quadratures, and the homodyne detection of suppressed fluctuations in photon statistics.
It is well known that the Husimi Q-function of the signal field can actually be measured by the eight-port homodyne detection technique, provided that the reference beam (used for homodyne detection) is a very strong coherent field so that it can be treated classically. Using recent rigorous results on the quantum theory of homodyne detection observables, we show that any phase space observable, and not only the Q-function, can be obtained as a high amplitude limit of the signal observable actually measured by an eight-port homodyne detector. The proof of this fact does not involve any classicality assumption.