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Annual Meeting Optical Society of America

Measuring amplitude probability distribution for photon number-operator eigenstates

Bernard Yurke, David Stoler

Published Jan 1, 1987• 0 citations
Publisher

Abstract

In parametric downconversion, pump photons are converted into pairs of signal and idler photons. A photodetector counting photons in the idler beam can be used to gate a homodyne detector in the signal beam so that it is on only when an n-photon wave packet enters its input. The probability distribution for the charge delivered by the homodyne detector has the same form as the position probability distribution of a harmonic oscillator in its nth energy eigenstate. In particular, the probability distribution exhibits fringes. We have calculated the visibility of these fringes for the case when both the photodetector and the homodyne detector have less than unit quantum efficiency. By reducing the photon flux in the signal and idler beam sufficiently, the effect of idler photodetector loss on the fringe visibility can be made negligible. The degree to which the fringes have been degraded then depends only on the inefficiency of the homodyne detector. For the case n = 1, the fringe is still visible when the homodyne detector efficiency is 0.5. Hence the proposed experiment should be feasible with existing technology.

DOI
10.1364/oam.1987.tui6

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In parametric downconversion, pump photons are converted into pairs of signal and idler photons. A photodetector counting photons in the idler beam can be used to gate a homodyne detector in the signal beam so that it is on only when an n-photon wave packet enters its input. The probability distribution for the charge delivered by the homodyne detector has the same form as the position probability distribution of a harmonic oscillator in its nth energy eigenstate. In particular, the probability distribution exhibits fringes. We have calculated the visibility of these fringes for the case when both the photodetector and the homodyne detector have less than unit quantum efficiency. By reducing the photon flux in the signal and idler beam sufficiently, the effect of idler photodetector loss on the fringe visibility can be made negligible. The degree to which the fringes have been degraded then depends only on the inefficiency of the homodyne detector. For the case n = 1, the fringe is still visible when the homodyne detector efficiency is 0.5. Hence the proposed experiment should be feasible with existing technology.
A homodyne detector measures a field-amplitude component of the incoming signal. If the incoming signal is in an nnn-photon eigenstate the homodyne detector's output probability distribution exhibits nnn fringes. Here the degree to which the visibility of these fringes is degraded by a homodyne detector with less than unit quantum efficiency is evaluated. An experiment employing conjugate pairs of photons generated via a parametric down conversion or four-wave mixing is proposed by which these fringes could be observed.
A theoretical study is made of the process in which incident pump photons that interact with a nonlinear medium (such as a crystal lacking inversion symmetry) are spontaneously split into lower-frequency signal and idler photons. The down-converted fields are quantized and described by a continuum of modes, a subset of which interacts with each photodetector. It is shown that when two ideal photodetectors are appropriately located so that they receive the conjugate signal and idler photons, then the joint probability of two-photon detection by the two detectors can equal the single-photon detection probability. The time correlation between the two detected photons is shown to be limited either by the resolving time of the detectors, or by the bandwidth of the down-converted light, and to be independent of the coherence time of the pump field or of the length of the nonlinear medium. These conclusions are compared with the results of recent experiments.
Taking advantage of our previous experimental results and calculations that establish a new approach to the calibration of photo detectors using correlated photon pairs, we developed an experimental setup with a fast electronic control unit for the direct measurement of photon statistics. The most convenient source for generating such correlated photon pairs is a nonlinear optical process of spontaneous parametric down conversion (SPDC). When a photon of the pump beam is incident to a nonlinear crystal with phase matching condition, a pair of photons (signal and idler) is created at the same time. The photons are detected in the signal beam, and a controllable shutter is placed in the idler beam. When a detection event is made in the signal beam the fast optical shutter opens for a certain (pre-programmed) time. While the opening time of the shutter is adjustable with good time resolution, the time interval between the photons in the idler beam can be scanned, so the photon statistics can be determined.
We report the first measurement of the joint photon-number probability distribution for a two-mode quantum state created by a nondegenerate optical parametric amplifier. The measured distribution exhibits inherently quantum correlations between the signal and idler photon numbers, whereas the marginal distributions are thermal as expected for parametric fluorescence. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
For measurements describable by a discrete set of possibilities, numerous demonstrations of the violation of Bell's inequalities have been reported in optical physics based on observations of polarization correlations for photon pairs. By contrast for observables with a continuous spectrum, there appears to be no general inequalities corresponding to those of Bell and no experimental realization of the original Einstein-Podolsky-Rosen paradox. Following Reid and Drummond,1 we are attempting to demonstrate the EPR paradox in the context of the measurement of continuous quantum variables with a subthreshold optical parametric oscillator operated in a frequency degenerate but polarization nondegenerate mode. In this case, the EPR paradox is formulated in terms of two sets of quadrature phase amplitudes for the spatially separated signal and idler fields. As the parametric gain is increased toward threshold, the quadrature amplitudes of the signal and idler beams become increasingly faithful quantum copies of one another. Our measurements of quadrature amplitudes are carried out using independent balanced homodyne detectors at two sites. When operated for the detection of squeezing at one site, noise reductions of 4 dB relative to the vacuum state limit are observed, which is most encouraging since noise reductions >3 dB in the difference photocurrent from the two sites are sufficient for inference at a distance of a signal observable from a measurement of the corresponding idler variable.
Measurement of photon-number statistics of fields composed of photon pairs generated in spontaneous parametric downcoversion pumped by strong ultrashort pulses is described. Final detection quantum efficiencies, noises as well as possible loss of one or both photons from a pair are taken into account. Measured data provided by an intensified single-photon CCD camera are analyzed along the developed model. The joint signal-idler photon-number distribution is obtained using the expectation maximization algorithm. Covariance of the signal and idler photon-numbers equals 80 %. Statistics of the generated photon pairs are identified to be Poissonian in our case. Distribution of the integrated intensities of the signal and idler fields shows strong correlations between the fields. Negative values of this distribution occurring in some regions clearly demonstrate a nonclassical character of the light composed of photon pairs.