Single and Correlated Photons

These are two properties we needed to establish in our setup before performing Hardy's Test. The setup and results of each of these two tests during Fall 2020 is described below. The calculations are adapted from Pearson and Jackson [1]

Correlated Photons

To begin our experiment, we sent a 405nm, 80mW laser beam through a beta-Barium-Borate (BBO) crystal, creating two subsequent 810nm down-converted beams (let's refer to them as A and B). An SPCM (single photon counting module) detector was connected via fiber-optic cables to collimating detectors placed in the path of each of the two 810nm beams at stations A and B, as shown in Figure 1, below. These detectors sent the count rates observed to a desktop computer running a Labview program which recorded the data. The Labview program recorded how often detectors A and B registered coincident photons, where coincident photons are defined as simultaneous counts between detectors A and B within an interval of about 10ns (a window determined by the limitations of the electronics used to digitize the data).

Figure 1: Shown is a diagram of the apparatus used for the correlated photons test. HWP refers to the 405nm half-wave plate which adjusts the linear polarization of the 405nm beam before it passes through the down-converting crystals (BBO)

After performing this test we expected to observe a number of ``coincidence counts" greater than what could be explained by uncorrelated events at detectors A and B. Exploiting the Poissonian nature of the pump laser, the rate we expected assuming the events at A and B were completely uncorrelated is the following, defined here as the rate of ``accidental" coincidence counts.

The pulse width, τ is the duration of the digital pulse produced by the SPCM detectors, and R_AB represents the count rates detected at A (B). Twice the pulse width is used in this expression because the longest period of time acceptable to consider an event ``coincident" between the two detectors occurs when the detected events are within one pulse width (τ) of each other.

With knowledge of the predicted amount of coincident counts from the nature of our Poissonian light source, we can now introduce the ``anticorrelation parameter", α_2D, which gives a measure of how correlated the two down-converted beams are.

We expected to find α_2D ≥ 1 if our photons were indeed correlated, because the amount of detected coincidences was greater than the amount of predicted coincidences. With our apparatus, this difference was explained by the process of spontaneous parametric down-coversion in the BBO crystals, or the splitting of photons which creates pairs of entangled photons [2]. A sampling of the data we collected for this test is displayed below in Table 1.

Table 1: Observed α_2D at various stages of our experiment with corresponding average counts rates at detectors A and B (R_A and R_B). It is clear that changes in attenuation

make a difference in the value, while measurements with and without the PBS's in place are near each other. Small differences could be due to

differences in detector alignment and fine-attenuation.

Single Photons

After we established the two 810nm beams as correlated sources, it is possible to verify that single photons are passing through the BBO. This is due to their correlated nature; we know that if there is a photon detected at Station A, there should also be one at Station B. To show this experimentally, it is necessary to introduce a non-polarizing beam-splitter and third detector (B'). These new components will be set up as in Figure 2. Beam B will pass through the beam-splitter, forcing the single photon to be transmitted (toward detector B) or reflected (toward detector B'). The observations from all three detectors are again passed to the Labview program for processing, where the following calculations will be performed.

If single photons were present in our apparatus, we expected to see an α_3D << 1. This is due to the fact that a single photon would be forced to either transmit or reflect at the beam-splitter, hitting only detector B OR B', but never both. Contrastingly, the classical notion of light as a wave would tell us we should measure an α of 1 because a light wave would be able to split at the beam-splitter and register counts at both detectors B and B'.

It is important to note why the third detector was necessary to prove the existence of single photons with respect to this equation. As described in the α_3D equation, we are conditioning observations of triple coincidences on the observation of a photon at A. If we had just measured the coincidence counts between detectors B and B', without conditioning on A, the best α we could have found was 1 because we'd always be observing a count at B or B'. With the inclusion of the A detector and knowledge that we had two correlated beams of down-converted photons (from the previous test), we knew exactly when to look for a correlated pair at B and B', allowing us to ignore spurious or background counts. The data we collected for this test is displayed in Table 2, below.

Table 2: Observed α_3D values at various stages and intensities during the experiment. Overall it seems clear that we had single photons passing

through our apparatus at any point. The asterisk on the second value of α_3D denotes that only one triple-coincidence count was recorded

over a period of at least 500s. This explains the large uncertainty assigned to the value.

Works Citied

[1] Brett J. Pearson and David P. Jackson. A hands-on introduction to single photons and quantum mechanics for undergraduates. American Journal of Physics, 78(5):471-484, 2010.

[2] Paul G. Kwiat, Edo Waks, Andrew G. White, Ian Appelbaum, and Philippe H. Eber- hard. Ultrabright source of polarization-entangled photons. Physical Review A - Atomic, Molecular, and Optical Physics, 60(2):R773–R776, 1999.