Einstein-Podolsky-Rosen paradox in single pairs of images. E. Lantz, S. Denis, P.-A. Moreau and F. Devaux in Opt. Express 23:26472 (2015). What the paper says!?
This is an EPR paper from SPDC which relies on a single image to demonstrate the correlations. This improves on their previous paper.[1]
Interesting opening statement for our quantum-state ergodicity hypothesis:
Statistical properties of fluctuations in quantum mechanics are described by ensemble averages, which are often estimated by time averages if the signal is stationary in time, but which can also be estimated by spatial averages if the signal is stationary in space on a sufficiently large area.
This is closely the same in space as ergodicity in time: here independent data sets (independent biphotons) are averaged in space from the same image. The biphotons either come from independent modes from the same pulse, or from different pulses. The non-uniform background (Gaussian) means this is non strictly-erdogic, though, but this is a technical aside to a conceptual observation.
Equally interesting continuation:
Most of the experiments in quantum imaging record averages of temporal coincidences, i.e. characterize the spatial repartition of temporal averages, rather than spatial averages, with accent on the high dimensionality of the underlying entanglement, in order to demonstrate that an image conveys a great number of spatially correlated quantum (temporal) channels in parallel [1–3].
Here they have a sentence ambiguity, the "in order to demonstrate" means one should rely on spatial averages to do that (exploit the potential of high-dimensional entanglement).
They specifically target the notion of average over a single frame!
[...] leaving open the question of demonstrating pure spatial quantum effects in single images, without the need of either temporal averages or averages of set of images.
Comparing to [their] previous works:
all photons were recorded with an equal chance in our experiments with one [13] or two [8] EMCCD cameras, showing a high degree of paradox by using spatial correlation of the images. However, it was necessary in these works to record a set of pairs of images, at least 20 pairs in the near field, in order to, first, retrieve the fluctuations by subtracting an average image and, second, compute the intercorrelation coefficient by adding the results of each pair of images.
Here:
In the present work we show that a single pair of images in each plane (near and far field) is sufficient to safely demonstrate an EPR paradox.
So what they do is that they consider one frame only, for instance this one:
which shows the signal on the left and the idler on the right, both in momentum space (they could image real space instead; they could also image, say, momentum-signal and real-space-idler but that would be uncorrelated). They use a 300×300 window in that (the square above), so that 90000×(0.15 photon per pixel) (midpoint of their stated mean fluency of SPDC) gives about 13500 photons on the frame. They say they have a correlation C=0.23, which is the ratio of biphotons to the total number, so 13500×0.23=3100 bi-photons per frame. They get this from a 1kHz pump rate with an exposure time of 0.03s, so 30 pulses per frame. Each pulse has $\ll1$ photon per spatiotemporal mode, but since here they consider the whole spatio-temporal field, they deal with a considerable amount of modes, amounting to something like 1700 biphotons per pulse. Note that it's still mainly temporal
More details can be found in their previous paper, in particular their Eq. (1).[2] This is an interesting estimate of the SNR which is linked to "coherence cells" and Schmidt numbers.
They can thus complete their program:
To demonstrate an EPR paradox, we have to use one pair of images in each plane.
The average they consider is basically a spatial-average of correlations (products of intensities) over the camera, thus reducing the high-resolution 300×300 pixels from Fig. 1 into a ±20 pixels in Fig. 2.
Since position-momentum uncertainty products are from independent measurements, repeating the one-frame program 900 times (as they did), they can then test departure from $\hbar^2/4$ from a large dataset:
Because there is no reason to associate a particular pair to another, we have calculated the degree of violation for all the 900 × 900 possible combinations of two pairs.
Note that they excluded «the 2,5% lowest and highest product values» (is "2,5%" a typo for "2.5%"?)
They successfully demonstrate that a single image is enough:
we have shown that a two dimensional EPR paradox can be safely demonstrated with single images, without any aspect implying temporal averaging of a set of images. Hence, an image can exhibit quantum properties by itself: all quantum correlations can be demonstrated by ”repeating the experiment” over the different resolution cells, or spatial modes, of the image, and by using neither detection of temporal coincidences nor repeat of the experiment on a set of images.
Their results are a degree of violation of EPR correlations, coming in the form of numbers (their figures are illustrative only)
They have this weird and confusing notation of putting in square bracket the lower and upper ranges of their calculations (also in Table 1).
It is remarkable that, on the one hand, «This improvement of our previous results has been made possible by suppressing the fluorescence parasitic light» and that, on the other hand, «The experimental setup, shown in Fig. 1, is similar to that used in [8], except that the mount of optical components no more includes polymer, that emitted parasitic fluorescent light»