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Quantum Tomography As a Tool for the Characterization of Optical Devices

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Quantum Tomography As a Tool for the Characterization of Optical Devices INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS J. Opt. B: Quantum Semiclass. Opt. 3 (2001) S1–S6 PII: S1464-4266(01)30946-1 Quantum tomography as a tool for the characterization of optical devices G Mauro D'Ariano1, Martina De Laurentis2, Matteo G A Paris1, Alberto Porzio2;3 and Salvatore Solimeno2;3 1 Quantum Optics and Information Group, Unita` INFM and Dipartimento di Fisica ‘Alessandro Volta’, Universita` di Pavia, via Bassi 6, I-27100 Pavia, Italy 2 Dipartimento di Scienze Fisiche, Universita` ‘Federico II’, Complesso Universitario di Monte Sant’Angelo, via Cintia, 80126 Napoli, Italy 3 INFM Unita´ di Napoli, Italy Received 19 November 2001 Published Online at stacks.iop.org/JOptB/3 Abstract Processing/JOB/job130946/SPE We describe a novel tool for the quantum characterization of optical devices. Printed 8/12/2001 The experimental setup involves a stable reference state that undergoes an Issue no unknown quantum transformation and is then revealed by balanced Total pages homodyne detection. Through tomographic analysis of the homodyne data First page Last page we are able to characterize the signal and to estimate parameters of the File name OB .TEX interaction, such as the loss of an optical component or the gain of an Date req amplifier. We present experimental results for coherent signals, with (BJ) application to the estimation of losses introduced by simple optical components, and show how these results can be extended to the characterization of more general optical devices. Keywords: 1. Introduction In this paper we address QHT as a tool for the quantum characterization of optical devices, as in the estimation of Q.1 Quantum homodyne tomography (QHT) is certainly the the coupling constant of an active medium or the quantum most successful technique for measuring the quantum state efficiency of a photodetector. The goal is to link the of radiation. It is based on homodyne detection, where estimation of such parameters with the results from feasible the signal mode is amplified by the local oscillator. This measurement schemes, such as homodyne detection, and to means that there is no need for single-photon resolving make the estimation procedure as efficient as possible. We photodetectors, whence it is possible to achieve quantum present our experimental results for the reconstruction of the efficiency η approaching the ideal unit value by using linear photodiodes [1]. Moreover, QHT is efficient and statistically quantum state of coherent signals, together with application reliable, such that it can be used on-line with the experiment. to the estimation of the losses introduced by simple optical Indeed, among other proposed state reconstruction methods, components. Moreover, we show how these preliminary QHT is the only one which has been implemented in quantum results can be extended to the characterization of more general optical experiments [2,3]. optical devices. Possible applications of QHT range from the measurement In the next section we review some basic elements of of photon correlations on a subpicosecond time-scale [2] to quantum tomography, and in section 3 we describe the basic the characterization of squeezing properties [3, 4], photon requirements needed to implement a quantum characterization statistics in parametric fluorescence [5], quantum correlations in down-conversion [1] and non-classicality of states [6]. In tool based on tomographic measurements. In section 4 the general, the key point is that QHT provides information about experimental apparatus is described with some details, and the quantum state within a chosen sideband, thus allowing in section 5 the experimental data are analysed and discussed. for precise spectral characterization of the light beam under Section 6 closes the paper by discussing the possible extensions investigation. of the present work. 1464-4266/01/000001+06$30.00 © 2001 IOP Publishing Ltd Printed in the UK S1 G Mauro D’Ariano et al 2. Quantum homodyne tomography tomographic samples should be devised, in order to minimize the number of measurements. Quantum tomography of a single-mode radiation field consists The first point can be satisfied by considering Gaussian of a set of repeated measurements of the field-quadrature signals, like coherent or squeezed states. Indeed, quantum 1 iφ † iφ xφ .ae + a e / at different values of the reference signals that are most likely to be reliably generated in a D 2 − phase φ. The expectation value of a generic operator can be laboratory are Gaussian states. The most general Gaussian obtained by averaging a suitable kernel function R[O].x; φ/ state can be written as % D.µ) S.r/ ν S†.r/ D†(µ), where D 1 a†a as follows [7] ν denotes a thermal state ν .nth + 1/− [nth=.nth + 1/] , 2 †2 D π S.r/ exp[r.a a /=2] the squeezing operator and D(µ) dφ 1 D † − D Tr %O dx p.x; φ/ R[O].x; φ/ (1) exp(µa µ∗a/ the displacement operator. However, thermal f g D 0 π − Z Z−1 excitations can be neglected at optical frequencies, such that we where p.x; φ/ denotes the probability distribution of the may generally consider ν as the vacuum state. The homodyne distribution of the state % at phase φ with respect to the outcomes x for the quadrature xφ, and R[O].x; φ/ is given by local oscillator is Gaussian and, remarkably, such Gaussian character is not altered by many transformations induced by optical devices, such as the loss of a component, the gain of an R[O].x; φ/ 1 1dr Tr O cos[pr.x x /] : (2) 4 φ amplifier or the quantum efficiency of a detector. In this paper D 0 f − g Z we consider the reference signal excited in a coherent state. Actually, the tomographic kernel R[O].x; φ/ for a given More general signals will be considered elsewhere. operator O is not unique, since a large class of null The need for effective data processing leads us to consider functions [9, 10] F.x; φ/ exists that have zero tomographic the use of either adaptive or maximum-likelihood (ML) average for an arbitrary state. This degree of freedom can procedures with the tomographic data. In section 5 we apply be exploited to adapt the kernel to data and achieve an adaptive tomography for the estimation of losses induced by optimized determination of the quantity of interest. For optical filters. Here we illustrate the use of an ML procedure example, quantities like the photon number, the field amplitude for the characterization of a general (active or passive) optical and any normally ordered moment can be the tomographically medium, which we plan to perform experimentally in the near estimated by averaging the following kernel future. K [O].x; φ/ R[O].x; φ/ Let us start by reviewing the ML approach. Let p.x λ) D j M 1 M 1 be the probability density of a random variable x, conditioned − − to the value of the parameter λ. The analytical form of p + µk Fk .x; φ/ + µk∗ Fk∗.x; φ/ (3) k 0 k 0 is known, but the true value of the parameter λ is unknown, XD XD where the kernels R[O] for the moments are given by [11] and should be estimated from the result of a measurement of x. Let x1; x2; : : : ; xN be a random sample of size N. The joint probability density of the independent random variable †n m i.m n)φ Hn+m .p2 x/ R[a a ].x φ/ e − (4) x ; x ; : : : ; x (the global probability of the sample) is given I D p2n+m n+m 1 2 N n by N Hn.x/ being the Hermite polynomial of order n, and the null k i.k+2/φ L(λ) p.xk λ) (5) functions Fk are expressed as Fk .x; φ/ x e ; k D j D D k 1 0; 1; : : : . The coefficients µk are obtained by minimizing YD the rms error for the given kernel on the given homodyne and is called the likelihood function of the given data sample. sample. A similar approach can be applied to optimize the The ML estimator of the parameter λ is defined as the quantity λML that maximizes L(λ) for variations of λ. Since the reconstruction of the matrix elements %mn m % n , thus achieving an effective quantum state characterizationD h j j. i likelihood is positive this is equivalent to maximizing N 3. QHT as a tool for estimating parameters L(λ) log L(λ) log p.xk λ) (6) D D k 1 j XD The state reconstruction method provided by QHT is effective which is the so-called log-likelihood function. and reliable, such that it can be exploited to build a tool Let us consider a generic optical medium: the propagation to characterize quantum devices. The general scheme for of a signal is governed, for negligible saturation effects, by the such a tool should be as follows. First, we need a stable master equation source of quantum states, i.e. a source able to provide † repeated preparations of a reference signal. The signal can %˙ G1 L[a] % + G2 L[a ] % (7) be characterized by QHT, and then employed as the input to a D given device which we want to characterize by the estimation where % is the density matrix describing the quantum state of some relevant parameters. At the output, the transformed of the signal mode a and L[O] denotes the Lindblad † 1 † 1 † state can be analysed by QHT, so as to characterize the input– superoperator L[O]A O A O 2 O O A 2 A O O.
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