Demonstrating Light Quantization & Testing Local Realism

Johnny Greavu & Kyle Schlicht, Spring 2016

To demonstrate the quantum nature of light, a blue diode pump laser with a nonlinear crystal is used to create an entangled photon source. By placing a polarizing beam splitter in one of the entangled beam arms, a three-detector anticorrelation parameter α—the ratio of the coincidence rate to the accidental rate—is determined, where α < 1 (anticorrelated) implies the existence of photons. We measure an α = 0.009(3) proving photons by 330σ. By adding a fourth detector and additional polarizing beam splitter to the other beam arm, a simpler version of Bell’s inequality is tested (Hardy’s test). By measuring coincidence probabilities between these four detectors, quantum mechanics is shown to violate our classical assumptions of local realism by 165σ.

I. INTRODUCTION

In the era immediately preceding the development of quantum mechanics, light was thought to have only classical, wave-like properties. It was discovered that as the intensity of a light source diminishes, light is observed to travel in a more quantized manner. The existence of photons can be shown by measuring the anticorrelation parameter α for multiple-detector coincidence experiments.

Generally, the anticorrelation parameter is the ratio of the coincidence rate to the expected accidental rate. An α > 1 indicates correlation, α < 1 indicates anticorrelation, and α = 1 indicates no correlation. In two-detector, beam splitter experiments (Figure 1), classical and quantum mechanics are equally valid in describing coincidences from a typical laser [1]. As waves are continuous and split equally at the beam splitter, only quantized light can be anticorrelated.

In 1935, Einstein, Podolsky, and Rosen suggested that quantum theory is incomplete, based on assumptions of “local hidden variables” or “local realism” [2]. Local realism is the combination of locality and realism. Locality, or local relativistic causality, is the notion that a system is only directly influenced by its immediate surroundings. It takes time for change to propagate and for light to travel, i.e., there are no superluminal effects. Realism is the notion that measurements are defined before they are measured. In 1964, John Stewart Bell proposed a test to show that assumptions of local realism can never reproduce all the predictions of quantum mechanics [3] [4]. Here, we experimentally perform Hardy’s test of local realism, which is a simpler adaption of Bell’s work [5]. We simultaneously measure the polarizations of entangled photons when there is no possibility of communication between detectors due to a sufficient spatial separation between them. The probabilities of observed polarizations for the photon pair divert from the expected probabilities calculated while assuming local realism, proving that quantum mechanics does not follow classical notions about reality.

II. THEORY

A. Light through a beam splitter

Suppose a HeNe laser passes through a "50/50" polarizing beam splitter to two single-photon detectors (Figure 1).

FIG. 1. HeNe laser light through a 50/50 beam splitter. Figure reprinted from [1].

If photons exist, it is extremely unlikely that B and B’ will simultaneously detect photons, within a coincidence window τ_c on the order of several nanoseconds. A photon can be observed to transmit or reflect through the beam splitter, but not both. The only way B and B’ can detect a coincidence is either through false coincidences (from the expected accidental rate), or two photons being bunched tightly enough in the laser beam upon entering the beam splitter so that one transmits right after the other reflects, or vice versa.

1. The anticorrelation parameter

It is more useful to talk of the anticorrelation parameter α, rather than of rates. For two detectors, this parameter is defined as,

where the P's are the probability of registering a two-detector coincidence or a single count in each detector, the N's are the number of counts (where N_p is the possible number of hits, how long we take data for divided by τ_c), and the R's are the coincidence and accidental rates.

As shown, α is the ratio of the rate of measured coincidences to the rate of calculated accidentals. If there are the same number of coincidences as accidentals, α = 1. The wave/classical model of light predicts α >= 1 while the photon/quantum model predicts α >= 0. Therefore, only a measurement of α < 1 can definitively prove the existence of photons [6]. With just two detectors and a beam splitter, α = 1, within the statistical uncertainty (see Table 1). This leads to two equally valid interpretations. Classically, the laser is emitting waves that divide equally at the splitter, simultaneously hitting both detectors continuously. Quantum mechanically, the laser is emitting a random stream of photons, which can only be observed to go one way through the beam splitter.

B. Down-converted light & entanglement

Through spontaneous parametric down conversion, we can correlate the light striking the detectors. The HeNe laser is swapped for a more powerful pump laser, and the beam splitter is replaced by a nonlinear crystal. A nonlinear crystal takes in a photon and outputs a pair of entangled photons simultaneously, each at half the frequency of the parent photon. The photons are in an energy superposition or an “entangled” state, and due to the conservation of energy and momentum, a measurement of a property of one photon will immediately tell you the same property of the other. Every time a down-converted photon is observed at one detector, there must be its entangled twin at the other. This leads to α >> 1 (see Tables 2, 3, and 4).

1. Entangled source with three-detector correlations

The existence of photons can finally be demonstrated by adding a third detector and a beam splitter to one of the entangled arms as shown in Figure 2.

FIG. 2. A third detector and nonlinear crystal (BBO) added to the setup. The HeNe laser has been exchanged with a pump laser. Figure reprinted from [1].

As the detection of a photon at A implies a photon at B or B’, it is sufficient to check for coincidences between all three detectors only when A triggers. This significantly decreases the maximum number of possible events (now N_p = N_A) and increases the probability of two-or-more detector coincidences. The anticorrelation parameter for this three-detector setup is then

At last, this leads to an α < 1, demonstrating the existence of photons.

C. Hardy’s test of local realism

Assuming local realism is true, suppose we send two entangled, linearly polarized streams of light to Alice and Bob. Each is equipped with a polarizer and single-photon detector. They are separated so that light takes longer to travel between them than it does for a photon to reach one of them individually (they cannot communicate during the experiment, i.e. locality). They take a series of measurements with various polarizer orientations and compare data afterward. They notice:

1) If Alice and Bob set their polarizers at angles A1 and B1 respectively, there is a small, but non-zero chance that they both detected photons or P(A1, B1) > 0.

2) If Alice set her polarizer to A1, and Bob set his to angle B2, Bob always detected a photon whenever Alice did, or P(B2|A1) = 1.

3) If Alice set her polarizer to an angle A2, and Bob set his to B1, Alice always detected a photon whenever Bob did, or P(A2|B1) = 1.

Say Alice and Bob initially set their polarizers to A1 and B1 and detect photons, which is allowed by 1). However, unbeknownst to Alice, Bob changes his mind just before his photon reaches his polarizer and moves his polarizer to B2. Assuming that the photons’ polarizations were already determined when they left their source (realism), since Alice still observes her photon then by 2), Bob always detects a photon polarized along his new angle. Likewise, if Bob stays and still detects and Alice switches, then by 3), they also will both detect. By the same reasoning we can deduce that if they both switch, to A2 and B2, they should still both detect. This implies that they will simultaneously measure photons polarized along A2 and B2 at least as often as they measure photons polarized along A1 and B1. In other words, our reasoning based on the assumption of local realism predicts

However, when comparing their data, the final thing they noticed was:

4) If they both orient their polarizers along the angles A2 and B2, they never detect photons simultaneously, or P(A2, B2) = 0.

This is a contradiction. We can first experimentally verify 2), 3), and 4), and through classical reasoning arrive at the implication that they will simultaneously measure photons along A2 and B2 at least as often as along A1 and B1. Since 4) says photons are never measured along A2 and B2 simultaneously, if photons are ever measured along A1 and B1 simultaneously, the above inequality fails and our initial assumptions and classical reasoning are invalid, showing quantum mechanics violates local realism.

It is much more difficult to prove P = 1 than it is to prove P = 0. A single unexpected event can show that P is non-zero while proving P = 1 would require infinite trials and time. Because of this, we state observations 2) and 3) equivalently as follows:

2) If Alice set her polarizer to A1, and Bob set his to B2 +/- 90 degrees, Alice and Bob never both detect photons or P(A1, B2 +/- 90) = 0.

3) If Alice set her polarizer to A2 +/- 90 degrees, and Bob set his to B1, Alice and Bob never both detect photons or P(A2 +/- 90, B1) = 0.

It is still impossible to verify a measured probability is zero. However, local realism can still be experimentally tested even if 2) and 3) don’t happen all the time. Suppose they only happen some arbitrary percentage of the time. Then the inequality becomes

We rewrite this inequality as the parameter H for convenience

where H > 0 shows quantum mechanics violates local realism.

III. EXPERIMENTAL METHODS

A. Validation of the existence of single photons

Light from an 85.4mW, 403nm polarized blue diode pump laser toward a beta-barium borate (BBO) crystal, where it undergoes spontaneous parametric down-conversion and produces two entangled photons of identical energy, and momentum, and polarization. Since the energy is evenly distributed in the entangled photons, their wavelengths are twice that of the original blue photon and are now infrared. Before the BBO crystal is a 405nm half-wave plate (HWP) which allows us to control the light’s plane of polarization upon entering the BBO crystal. Preceding the HWP is placed a linearly polarizing cube to ensure all light is polarized in the same direction. After passing through the BBO, one entangled photon travels to detector A while the other travels to a beam splitter that then sends the photon to either detector B or B’ (Figure 2). Photon events at detectors B and B’ are only considered when detector A is triggered. From these rates, α is determined.

The “detectors”, as referenced previously for ease, are actually collimators focusing and transmitting the light through optical fiber cables. In front of the collimators are 810nm bandpass filters. Light is sent through the collimators and cables to single-photon counting modules (SPCM’s). SPCM pulses and coincidences are counted by an FPGA and output to a PC where the events are read out in a pre-built LabVIEW program which determines rates, coincidences, and α. Alignment is made precise by making slight adjustments to the collimators, wave-plates, and BBO while monitoring the rates and coincidences in the LabVIEW program.

B. Hardy’s test

FIG. 3. Experimental setup for Hardy’s test. Note that photo is upside-down compared to diagram.Figure reprinted from [6].

Hardy’s test is performed using the experimental setup depicted above in Figure 3. The setup produces two entangled photons through two orthogonal BBO crystals (DC) in tandem with a half-wave plate (λ/2) and quarter-wave plate (QWP). The produced photons are either both vertically or horizontally aligned so that they are sent to the appropriate detectors. The photons then each pass through 810nm HWP’s and polarizing beam splitters (PBS) which reflect or transmit the photons depending on whether they are vertically or horizontally aligned, routing them to detectors A, A’, B, and B’. The 810nm HWP’s act as linear polarizers so that the desired quantum state may be selected. The photons are then counted by four SPCM’s. A quartz plate, cheaper and more easily available than a quarter-wave plate, is also tested in the QWP’s place to see if it produces the same results.

IV. RESULTS

A. Data collection process

To measure the anticorrelation parameter, the room is darkened while photons are counted over multiple runs and then averaged for coincidence windows of 10, 20, 40, and 60 ns.

In Hardy’s test, we test two states each with their own set of angles. A1 = -B1 = β and -A2 = B2 = α (not to be confused with the anticorrelation parameter), where α and β determine the angles of the polarizers placed in front of the detectors. H is measured in two configurations. In the first state, the ratio of the probability of horizontal polarization to vertical polarization is 4:1, α = 55 degrees, and β = 71 degrees. In the second state, the ratio of the probability of horizontal polarization to vertical polarization is 1:4, α = 35 degrees, and β = 19 degrees, where these angles have been selected to give the maximum H value [5]. We ensured that these angles were optimized by making slight adjustments to the 810nm HWP’s. We take ten 20s measurements for every probability in each state.

B. Uncorrelated two-detector data (B, B’)

Data for two detectors with a beam splitter in the arm (Figure 1) is shown in Table 1. We averaged 25 five-second runs for each coincidence window.

C. Correlated two-detector data (A, B)

Data for two detectors with an entangled source is shown in Tables 2, 3, and 4 for different laser intensities. We averaged 25 five-second runs for each coincidence window.

D. Anticorrelated three-detector data (A, B, B’)

Data for the three-detector setup (Figure 2) is shown in Table 5. We adjusted the laser intensity so that the double coincidence rates match those from [1]. We averaged 25 120s runs for each coincidence window.

E. Hardy’s test

Data for Hardy’s test for averages of 10 20s runs are shown in Tables 6, 7, 8, 9, 10, and 11.

When trying to maximize our H value, we took samples variating over different angles of the QWP/quartz plate and the 405nm HWP while minimizing/maximizing the four probabilities. We noticed that for both states, there is a strong anticorrelation between H and probability P(-α', -β), as shown below in Figure 4.

FIG 4. The four probabilities vs. H.

V. CONCLUSIONS

Our best value of the anticorrelation parameter gave α = 0.009(3), 330σ below unity, conclusively demonstrating the existence of photons. For Hardy’s test, our best H value (which was with the quarter-wave plate and state 2) was H = 0.329(2), 165σ above zero, showing that quantum mechanics is inconsistent with classical assumptions of local realism. Finally, our state 2 H value with the quartz plate was H = 0.309(4), 77σ above zero, while the state 1 values were almost identical. This shows that the quartz plate is an acceptable substitute for the quarter-wave plate.

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[1] B. J. Pearson and D. P. Jackson, "A hands-on introduction to single photons and quantum mechanics for undergraduates," Am. J. Phys., 78, 471 (2010).

[2] A. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description of physical reality be considered complete?" Physical Review 41, 777 (1935).

[3] J.S. Bell: "On the Einstein Podolsky Rosen paradox" Physics 1, 3, 195 (1964).

[4] There are many loopholes to Bell’s test, notably detector inefficiencies, and superdeterminism. Superdeterminism, which Bell acknowledged, is the absence of free will in the strictest sense. That is, there is no need for superluminal signals to tell detector A what happened at detector B, because the whole Universe (and therefore, detector A) already knows what will happen. Such loopholes were not explored further.

[5] J. A. Carlson, M. D. Olmstead, and M. Beck, “Quantum mysteries tested: An experiment implementing Hardy’s test of local realism,” Am. J. Phys., 74, (3), 180-186 (2006).

[6] Light quantization could be proven with just two-detectors if we use a confirmed single-photon source.