Optical measurements in quantum communication and quantum radiometry are quite often based on two-photon correlated channels and the coincidence between them. To quantify the noise level in these measurements, the authors focus on the statistics of photon coincidence in a basic experimental setup, accounting for all experimental effects contributing to the noise increment. By the proposed theory the measurement performance is evaluated in terms of noise fluctuations. Particularly, an ultimate limit in noise reduction is established by the optimal estimation model herein presented.
Optical measurements in quantum communication and quantum radiometry are quite often based on two-photon correlated channels and the coincidence between them. To quantify the noise level in these measurements, the authors focus on the statistics of photon coincidence in a basic experimental setup, accounting for all experimental effects contributing to the noise increment. By the proposed theory the measurement performance is evaluated in terms of noise fluctuations. Particularly, an ultimate limit in noise reduction is established by the optimal estimation model herein presented.
Photon correlated beams, generated by parametric down-conversion in nonlinear crystals (PDC), have been proved successfully for the measurement of quantum efficiency, of single photon detectors [1, 2, 3, 4] and can be used for measuring transmittance, as well. The main feature relies on the realization of two correlated quantum channels yielding coincident events. To reduce uncertainty [4, 5, 6] and determine its ultimate limit, contributions to statistical noise in a real experimental set-up are here investigated, and a general model is proposed to calculate the maximum likelihood best estimators and evaluate the probability of coincidence. In typical measurement scheme a channel is set as trigger (idler, i) and attention is paid to catch at least the same amount of correlated photons in the other channel (signal, s) [4]. Unwanted uncorrelated photons due to PDC itself result in a decreased degree of correlation, noise fluctuations in coincidence measurements and asymmetry between the two channels, whose mean photon rate can be quite different. We indicate by with the mean rate of correlated photons in s and i channels We address the total mean photon rates in both channels as where terms take into account also additive background photon rates due to straylight and detector noise. Actually statistical noise is also due to losses of correlated photons by optical elements, non ideal detectors and electronic devices in both channels. We distinguish, therefore, between optical losses ( transmittance of the correlated photon path), detection losses detectorquantum efficiency), electronic losses associated to the dead times time during which the detection system is unable to detect photons). It is usual to identify dead times as non-extending and extending: in the first case, all photons following a revealed photon within the fixed time interval are ignored, while in the second case any incoming photon produces or prolongs Eventually the finite duration of the time coincidence window, modifies the distribution probability of coincident events. Let us consider the distribution probability of single channel
The accuracy of optical measurements at low light levels is limited by the quantum noise of the source and by the random nature of the interaction with the measured object. The source noise may be reduced by use of nonclassical photon-number squeezed light. We consider the use of two photon-correlated beams (generated, for example, by spontaneous parametric downconversion) to measure the optical transmittance of an object. The photons of each beam obey a random Poisson process but are synchronized in time. One beam is used to probe the object, and the other is used as a reference providing information on the realization of the random arrival of photons at the object. The additional information available by such measurement may be exploited to improve the accuracy of the measurement. Various estimators, including the maximum-likelihood estimator, are considered, and their performance is evaluated and compared with the measurement based on a single-beam conventional (Poissonian) source and a maximally squeezed (fixed-photon-number) source. The performance advantage that is established depends on parameters such as the intensity of the source, the transmittance of the object, the quantum efficiency of the detectors, the background noise, and the degree of correlation of the photon numbers in the two beams.
The accuracy of optical measurements at low light levels is limited by the quantum noise of the source and by the random nature of the interaction with the measured object. The source noise may be reduced by use of nonclassical photon-number squeezed light. This paper considers the use of two photon-correlated beams (generated, for example, by spontaneous parametric downconversion) to measure the optical transmittance of an object. The photons of each beam obey a random Poisson process, but are synchronized in time. One beam is used to probe the object while the other is used as a reference providing information on the realization of the random arrival of photons at the object. The additional information available by such measurement may be exploited to improve the accuracy of the measurement. Various estimators, including the maximum likelihood estimator, are considered and their performance is evaluated and compared with the measurement based on single-beam conventional (Poisson) source and maximally squeezed (fixed photon number) source. The performance advantage established in this paper depends on parameters such as the intensity of the source, the transmittance of the object, the quantum efficiency of the detectors, the background noise, and the degree of correlation of the photon numbers in the two beams.
Quantum-intensity-correlated twin beams of light can be used to measure absorption with precision beyond the classical shot-noise limit. The degree to which this can be achieved with a given estimator is defined by the quality of the twin-beam intensity correlations, which is quantified by the noise reduction factor. We derive an analytical model of twin-beam experiments, incorporating experimental parameters such as the relative detection efficiency of the beams, uncorrelated optical noise, and uncorrelated detector noise. We show that for twin beams without excessive noise, measured correlations can be improved by increasing the detection efficiency of each beam; notwithstanding, this may unbalance detection efficiency. However, for beams with excess intensity or other experimental noise, one should balance detection efficiency, even at the cost of reducing detection efficiency—we specifically define these noise conditions and verify our results with statistical simulation. This has application in design and optimization of absorption spectroscopy and imaging experiments.
We investigated statistical fluctuations in the numbers of photons contained in the correlated twin photon beams generated by an optical parametric process. The coincidence between the signal- and idler-photoelectron pulses from a pair of avalanche photo-diodes was measured. The measured fluctuation in the number of signal-photons coinciding with idler-photons was equal to the standard deviation of a binomial distribution and was smaller than that of a Poisson distribution. We determined that even when the quantum efficiency of the photon detectors was much lower than unity, the uncertainty in the number of photons obtained with the twin photon beams was smaller, by a factor of 1/√2, than that with a classical light beam.
Imaging is the estimation of the spatial distribution of a physical object by measuring the optical radiation it emits, or by making use of an optical wave that interacts with the object, via reflection or transmission, for example, before being measured by a detector. The resolution of an imaging system is limited by the inability to localize the optical field at points of the object. Under otherwise ideal conditions, resolution is limited by diffraction. The sensitivity of an imaging system is limited by the uncertainty in the measurement. Under ideal conditions, this is determined by photon noise, which depends on the statistical fluctuations of the light. In conventional imaging systems, an extended detector, such as a CCD camera or an array detector, measures the spatial distribution of the optical intensity, which is proportional to the photon flux density. In interferometric systems, the spatial distribution of the optical field is inferred from measurements of the optical intensity. With the emergence of coherence theory, imaging systems based on measurements of the second-order coherence function at pairs of points in the detection plane were developed. An example is the imaging of an incoherent object based on the van Cittert-Zernike theorem. Imaging systems based on measurement of intensity correlation, or the photon coincidence rate, at pairs of points, were developed in the 1960s. A classic example of the photon-correlation imaging of an object emitting thermal light is stellar imaging using a Hanbury-Brown-Twiss intensity-correlation interferometer. More recently, two-photon light, which may be generated via spontaneous parametric downconversion in a second-order nonlinear optical crystal, has been used for imaging. This type of two-photon (or biphoton) imaging, which has come to be called quantum imaging, is also based on the measurement of photon coincidence by the use of photon-counting array detectors or by scanning two photon-counting detectors at pairs of points. To compare the resolution and sensitivity of imaging systems based on the aforementioned types of measurements, it is necessary to derive expressions for the measured quantities in terms of the object distribution. The point-spread functions based on such expressions can be used to assess the resolution. One measure of the sensitivity of the imaging process is the signal-to-noise ratio (SNR) of the measured variables. The statistical nature of the light source must be known in order to determine the SNR. The purpose of this chapter is to compare the resolution and sensitivity of photon-correlation imaging systems that make use of thermal light and two-photon light. We will henceforth refer to these two imaging modalities as classical and quantum photon-correlation imaging, or simply classical and quantum imaging, respectively. Clearly, light in other quantum states can also be used for imaging.