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Subtracting accidental coincidences is a common practice quantum optics experiments. For zero mean Gaussian states, such as squeezed vacuum, we show that if one removes accidental coincidences the measurement results are quantitatively the same, both for photon coincidences at very low flux and for intensity covariances. Consequently, pure quantum effects at the photon level, like interference of photon wave functions or photon bunching, are reproduced in the correlation of fluctuations of macroscopic beams issued from spontaneous down conversion. This is true both in experiment if the detection resolution is smaller than the coherence cell (size of the mode), and in stochastic simulations based on sampling the Wigner function. We discuss the limitations of this correspondence, such as Bell inequalities (for which one cannot substract accidental coincidences), highly multimode situations such as quantum imaging, and higher order correlations., Comment: 10 pages, 2 figures

Experimental demonstrations of entangled quantum images produced through parametric downconversion have so far been confined to studying two photon correlations. Here we show that multiphoton correlations between quantum images are accessible experimentally and exhibit many new features including being sensitive to the phase of the bi-photon wavefunction. As a concrete example, we consider a modification of existing quantum imaging experiments in which the CCD cameras are moved out of focus, provide detailed analytical predictions for the resulting 4 photon intereferences, and support these by numerical simulations. The proposed experiment can also be interpreted as entanglement swapping: Bob's photons are initially unentangled, but the joint detection of Alice's photons projects Bob's photons onto an entangled state. The general approach proposed here can be extended to other quantum optics experiments involving high dimensional entanglement., Comment: 13 pages, 3 figures

We report experimental results where a momentum entangled biphoton state with a giant dimensionality of 8000 is retrieved and manipulated when only one photon of the pair is transmitted through a thin scattering medium. For this purpose, the transmission matrix of the complex medium is first measured with a phase-shifting interferometry measurement method using a spatial light modulator (SLM) illuminated with a laser source. From this matrix, different phase masks are calculated and addressed on the SLM to spatially control the focusing of the laser through the complex medium. These same masks are used to manipulate the phase of the biphoton wave function transmitted by the thin diffuser in order to restore and control in the same way the momentum correlations between the far-field images of twin beams issued from strongly spatial-multi-mode spontaneous parametric down conversion., Comment: 5 pages, 4 figures

This article addresses the measurement of the power spectrum of red noise processes at the lowest frequencies, where the minimum acquisition time is so long that it is impossible to average on a sequence of data record. Therefore, averaging is possible only on simultaneous observation of multiple instruments. This is the case of radio astronomy, which we take as the paradigm, but examples may be found in other fields such as climatology and geodesy. We compare the Bayesian confidence interval of the red-noise parameter using two estimators, the spectrum average and the cross-spectrum. While the spectrum average is widely used, the cross-spectrum using multiple instruments is rather uncommon. With two instruments, the cross-spectrum estimator leads to the Variance-Gamma distribution. A generalization to $q$ devices is provided, with the example of the observation of millisecond pulsars with 5 radio telescopes., Comment: 10 pages, 7 figures and 3 tables

At first sight, the use of an everywhere positive Wigner function as a probability density to perform stochastic simulations in quantum optics seems equivalent to the introduction of local hidden variables, thus preventing any violation of Bell inequalities. However, because of the difference between symmetrically and normally ordered operators, some trajectories in stochastic simulations can imply negative intensities, despite a positive mean. Hence, Bell inequalities do not apply. Here, we retrieve for a weakly squeezed Gaussian state the maximum violation on polarization states allowed by quantum mechanics, for the Clauser-Horn-Shimony-Holt (CHSH), as well as for the Clauser-Horn Bell inequalities. For the case of the Clauser-Horn Bell inequality, the influence of the quantum efficiency of the detectors is studied, and for both inequalities, the influence of the degree of squeezing is assessed, as well as the uncertainty range versus the number of trajectories used in the simulations., Comment: 10 pages, 5 figures

We first extend our recent experiments of correlation imaging through scattering media to the case of a thick medium, composed of two phase scatterers placed respectively in the image and the Fourier planes of the crystal. The spatial correlations between twin photons are still detected but no more in the form of a speckle. Second, a numerical simulation of the biphoton wave function is developed and applied to our experimental situation, with a good agreement.

The cross-spectrum method consists in measuring a signal $c(t)$ simultaneously with two independent instruments. Each of these instruments contributes to the global noise by its intrinsec (white) noise, whereas the signal $c(t)$ that we want to characterize could be a (red) noise. We first define the real part of the cross-spectrum as a relevant estimator. Then, we characterize the probability density function (PDF) of this estimator knowing the noise level (direct problem) as a Variance-Gamma (V$\Gamma$) distribution. Next, we solve the "inverse problem" thanks to Bayes' theorem to obtain an upper limit of the noise level knowing the estimate. Checked by massive Monte Carlo simulations, V$\Gamma$ proves to be perfectly reliable to any number of degrees of freedom (dof). Finally we compare this method with an other method using the Karhunen-Lo\`{e}ve transfrom (KLT). We find an upper limit of the signal level slightly different as the one of V$\Gamma$ since KLT better takes into account the available informations., Comment: 10 pages (2 columns), 5 figures, 3 tables, 34 references

We report the experimental observation of a spatio-temporal Hong-Ou-Mandel (HOM) interference of bi-photon states of extremely high Schmidt number. Two-photon interference of 1500 spatial modes and a total of more than 3x10^6 spatio-temporal modes is evidenced by measuring momentum spatial coincidences between the pixels of the far-field images of two strongly multimode spontaneous parametric down conversion (SPDC) beams propagating through a HOM interferometer. The outgoing SPDC beams are recorded onto two separate detectors arrays operating in the photon-counting regime. The properties of HOM interference are investigated both in the time and space domains. We show that the two-photon interferences exhibit temporal and two-dimensional spatial HOM dips with visibilities of 30% and widths in good agreement with the spatio-temporal coherence properties of the bi-photon state. Moreover, we demonstrate that a peak of momentum spatial coincidences is evidenced inside each image, in correspondence with this dip., Comment: 10 pages, 8 figures

The concepts of Fourier optics were established in France in the 1940s by Pierre-Michel Duffieux, and laid the foundations of an extensive series of activities in the French research community that have touched on nearly every aspect of contemporary optics and photonics. In this paper, we review a selection of results where applications of the Fourier transform and transfer functions in optics have been applied to yield significant advances in unexpected areas of optics, including the spatial shaping of complex laser beams in amplitude and in phase, real-time ultrafast measurements, novel ghost imaging techniques, and the development of parallel processing methodologies for photonic artificial intelligence.

We image with cameras entangled photon light transmitted through a random medium. Near-field and far-field spatial quantum correlations show that entangled photon pairs (bi-photons) generated by spontaneous optical parametric down-conversion exhibit speckle pattern. In contrast, no information from the random medium can be extracted using incoherent light issued from one photon of the pair. Although these measurements require several hours to record thousands of image pairs, our method is instantaneous for the recording of one pair of twin images and involve all the photons of the images.

We develop a fully spatio-temporal numerical model, based on stochastic simulations, simulating the generation of spatio-temporal multimode spontaneous parametric down conversion, the propagation of the signal and idler beams through a Hong-Ou-Mandel interferometer and the detection of the outgoing beams with two separate detectors arrays. Spatial and temporal properties of the two-photon interference are investigated by measuring the spatial distribution of the momenta coincidence counts between the time integrated outgoing far-field images and a full spatio-temporal HOM dip is exhibited. Numerical results also demonstrate that the spatio-temporal coherence properties of bi-photon wave packets can be fully characterized with detectors arrays with no temporal resolution.

The three-cornered hat / Groslambert Covariance methods are widely used to estimate the stability of each individual clock in a set of three, but no method gives reliable confidence intervals for large integration times. We propose a new KLTS (Karhunen-Lo\`eve Tansform using Sufficient statistics) method which uses these estimators to take into account the statistics of all the measurements between the pairs of clocks in a Bayesian way. The resulting Cumulative Density Function (CDF) yields confidence intervals for each clock AVAR. This CDF provides also a stability estimator which is always positive. Checked by massive Monte-Carlo simulations, KLTS proves to be perfectly reliable even for one degree of freedom. An example of experimental measurement is given., Comment: 7 pages, 5 figures

The dynamical Casimir effect is the generation of pairs of real particles or photons from the vacuum as a result of a non-adiabatic change of a system parameter or boundary condition. As opposed to standard parametric amplification where the modulation occurs both in space and in time, this fundamental process requires a pure modulation in time, which makes its detection particularly challenging at optical frequencies. In this paper we experimentally demonstrate a realisation of the optical analogue of the mechanical dynamical Casimir effect in the near-infrared optical regime in a dispersion-oscillating photonic crystal fibre. The experiments are based on the equivalence of the spatial modulation of the fibre core diameter to a pure temporal modulation when this is considered in the co-moving frame of the travelling pump pulse. We provide evidence of optical dynamical Casimir effect by measuring quantum correlations between the spectrally resolved photon pairs. The non-classical nature of the measured light is supported by evidence of anti-bunching photon statistics.

The three-cornered hat method and the Groslambert Covariance are very often used to estimate the frequency stability of each individual oscillator in a set of three oscillators by comparing them in pairs. However, no rigorous method to assess the uncertainties over their estimates has yet been formulated. In order to overcome this lack, this paper will first study the direct problem, i.e. the calculation of the statistics of the clock stability estimates by assuming known values of the true clock stabilities and then will propose a first attempt to solve the inverse problem, i.e. the assessment of a confidence interval over the true clock stabilities by assuming known values of the clock stability estimates. We show that this method is reliable from 5 Equivalent Degrees of Freedom (EDF) and beyond., Comment: 27 pages, 8 figures, 2 tables

We report results of two-photon quantum holography where spatial information stored in phase holograms is retrieved by measuring quantum spatial correlations between two images formed by spatially entangled twin photons with a dimensionality of 1790 in the two-dimensional transverse space. In our experiments, the entire flux of spontaneous down converted photons illuminates the phase holograms and the photons of pairs signal-idler transmitted by the holograms are detected separately in far field on two electron-multiplying charge coupled device cameras., Comment: 6 pages, 4 figures

We use twin photons generated by spontaneous parametric down conversion (SPDC) to perform temporal ghost imaging of a single time signal. The retrieval of a binary signal containing eight bits is performed with an error rate below 1%.

We report ghost imaging of a single non-reproducible temporal signal in the range of tens kHz by using pseudo-thermal speckle light patterns and a single detector array with a million of pixels working without any temporal resolution. A set of speckle patterns is generated deterministically at radio-frequency rate, multiplied by the temporal signal and time integrated in a single shot by the camera. The temporal information is retrieved by computing the spatial intensity correlations between this time integrated image and each speckle pattern of the set., Comment: 6 pages, 4 figures

Ghost imaging is a fascinating process, where light interacting with an object is recorded without resolution, but the shape of the object is nevertheless retrieved, thanks to quantum or classical correlations of this interacting light with either a computed or detected random signal. Recently, ghost imaging has been extended to a time object, by using several thousands copies of this periodic object. Here, we present a very simple device, inspired by computational ghost imaging, that allows the retrieval of a single non-reproducible, periodic or non-periodic, temporal signal. The reconstruction is performed by a single shot, spatially multiplexed, measurement of the spatial intensity correlations between computer-generated random images and the images, modulated by a temporal signal, recorded and summed on a chip CMOS camera used with no temporal resolution. Our device allows the reconstruction of either a single temporal signal with monochrome images or wavelength-multiplexed signals with color images.

Spatially entangled twin photons provide a test of the Einstein-Podolsky-Rosen (EPR) paradox in its original form of position (image plane) versus impulsion (Fourier plane). We show that recording a single pair of images in each plane is sufficient to safely demonstrate an EPR paradox. On each pair of images, we have retrieved the fluctuations by subtracting the fitted deterministic intensity shape and then have obtained an intercorrelation peak with a sufficient signal to noise ratio to safely distinguish this peak from random fluctuations. A 95% confidence interval has been determined, confirming a high degree of paradox whatever the considered single pairs. Last, we have verified that the value of the variance of the difference between twin images is always below the quantum (poissonian) limit, in order to ensure the particle character of the demonstration. Our demonstration shows that a single image pattern can reveal the quantum and non-local behavior of light, without any need of averaging after repeating the experiment.

1/f noise is very common but is difficult to handle in a metrological way. After having recalled the main characteristics of stongly correlated noise, this paper will determine relationships giving confidence intervals over the arithmetic mean and the linear drift parameters. A complete example of processing of an actual measurement sequence affected by 1/f noise will be given.

To evidence multimode spatial entanglement of spontaneous down-conversion, detector arrays allow a full field measurement, without any a priori selection of the paired photons. We show by comparing results of the recent literature that electron-multiplying CCD (EMCCD) cameras allow, in the present state of technology, the detection of quantum correlations with the best signal-to-noise ratio (SNR), while intensified CCD (ICCD) cameras allow at best to identify pairs. The SNR appears to be proportional to the square root of the number of coherence cells in each image, or Schmidt number. Then, corrected estimates are derived for extended coherence cells and not very low and not space-stationary photon fluxes. Finally, experimental measurements of the SNR confirm our model.

Spatially entangled twin photons provide both promising resources for modern quantum information protocols, because of the high dimensionality of transverse entanglement, and a test of the Einstein-Podolsky-Rosen(EPR) paradox in its original form of position versus impulsion. Usually, photons in temporal coincidence are selected and their positions recorded, resulting in a priori assumptions on their spatio-temporal behavior. Here, we record on two separate electron-multiplying charge coupled devices (EMCCD) cameras twin images of the entire flux of spontaneous down-conversion. This ensures a strict equivalence between the subsystems corresponding to the detection of either position (image or near-field plane) or momentum (Fourier or far-field plane). We report then highest degree of paradox ever reported and show that this degree corresponds to the number of independent degrees of freedom or resolution cells, of the images.

With a small number of measurements, the three most well known estimators of a single parameter give very different results when estimating a scale parameter. A transformation of this scale parameter to a location parameter by using logarithms of the data rends the three estimators equivalent in the absence of any a priori information., Comment: 8 pages, 1 figure

Searching the complete space of possible Bayesian networks is intractable for problems of interesting size, so Bayesian network structure learning algorithms, such as the commonly used Sparse Candidate algorithm, employ heuristics. However, these heuristics also restrict the types of relationships that can be learned exclusively from data. They are unable to learn relationships that exhibit "correlation-immunity", such as parity. To learn Bayesian networks in the presence of correlation-immune relationships, we extend the Sparse Candidate algorithm with a technique called "skewing". This technique uses the observation that relationships that are correlation-immune under a specific input distribution may not be correlation-immune under another, sufficiently different distribution. We show that by extending Sparse Candidate with this technique we are able to discover relationships between random variables that are approximately correlation-immune, with a significantly lower computational cost than the alternative of considering multiple parents of a node at a time., Comment: Appears in Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007)

We demonstrate Einstein-Podolsky-Rosen (EPR) entanglement by detecting purely spatial quantum correlations in the near and far fields of spontaneous parametric down-conversion generated in a type-2 beta barium borate crystal. Full-field imaging is performed in the photon-counting regime with an electron-multiplying CCD camera. The data are used without any postselection, and we obtain a violation of Heisenberg inequalities with inferred quantities taking into account all the biphoton pairs in both the near and far fields by integration on the entire two-dimensional transverse planes. This ensures a rigorous demonstration of the EPR paradox in its original position momentum form.