Investigating Local Realism in the context of Quantum Mechanics

Investigating Local Realism in the context of Quantum Mechanics

Meredith Wieber and Kieran Balls-Barker

History

Just after quantum mechanics was introduced to explain the areas classical mechanics could not, many were skeptical of it's validity. In particular, there is a famous question proposed by Einstein, Rosen, and Podolosky, on whether quantum mechanics can be considered to be complete [1]. The issue was based on quantum mechanics violating local realism, which is the idea that any system will always have the properties of locality and realism. Locality is the idea that a measurement in one system cannot be affected by a measurement in another system unless a direct causal relationship is present between the two systems. In the case of two beams of polarization-entangled photons used in this experiment, the two beams are set up such that there is no physical connection between the two beams. The reality aspect of local realism is the idea that objects have defined properties prior to their actual measurement; something that is not necessarily true in quantum mechanics.

In this experiment, we use Hardy's Test to observe the violations of local realism that quantum mechanics imply, providing further confirmation of the validity and necessity of quantum mechanics [2]. Hardy's test is a two-particle binary test of Bell's inequalities, and has the benefit of being more accessible to wider audiences, including the undergraduate laboratory [3].

Why is this Important?

The concept of these entangled photons and the violation of local realism is a hot topic in contemporary science as it is being used to theorize on quantum computing techniques, particularly quantum encryption. This is the idea that an individual could be given a specific quantum state as their ``key"; if anyone tried to read their encryption, they would inherently change it and the owner would be alerted to the attempt. The attempt at reading another person's data would fail. Thus far, this is not a widely-distributable technique (neither are quantum computers), but this idea may end up being invaluable in the future of computing [4].

Hardy's Test

The main portion of this experiment involves the examination of a phenomenon that runs counter to classical physics and the principles of local realism. What follows is a somewhat heuristic explanation of this behavior. For the more complete description upon which this is based, see [3].

As discussed below in our apparatus section, our setup includes a nonlinear crystal known as a beta-barium-borate (BBO) crystal, which should produce pairs of polarization entangled photons. From each pair, one photon goes to "station A" and another to "station B". At each station there is equipment to measure whether the polarization is at a specific angle. In particular station A can look for polarization at θ_A1 or θ_A2 and station B can look for either θ_B1 or θ_B2. The specific angles are selected so as to make the following behavior most exaggerated (again, see beck for a more detailed description).

If we assume that local realism holds, then the moment the photons leave the BBO crystal each photon should have whatever properties it has, and from this point the photons should not affect each other. This is a very intuitive assumption, and it allows us to interpret the outcome of all four measurements as though they occur within a single probability space, which in turn allows us to use a Venn diagram to interpret the results. To stress, this assumption of local realism is flawed and is what leads to the contradiction we're looking for.

The above Venn diagram represents the aforementioned probability space, with the box representing the space of all things that can happen, and each of the circles θ_A(B)i representing the set of configurations of the photons that lead to getting a positive result at station A(B) with angle θ_A(B)i (the result of a measurement for a given photon at a given station is either 0 or 1). We represent the probability of this happening as P(θ_A(B)i). The Venn diagram is not to scale.

If we reach the specific setup of our incoming beam that we're trying to achieve, quantum mechanics gives us that we should see the following:

Observation 1: When measuring angle θ_A1 at station A and θ_B1 at station B, we should see events at both stations about 9% of the time. That is, P(θ_A1,θ_B1) ≅ 0.09. In the context of our Venn diagram this means that circles θ_A1 and θ_B1 should overlap slightly, as represented by the blue region in the Venn diagram.

Observation 2: When measuring angle θ_A1 at station A and θ_B2 at station B, we should see an event at station B whenever we see an event at station A. That is, P(θ_B2|θ_A1) = 1. In the context of our venn diagram this means that the circle θ_B2 should contain the circle θ_A1 as shown above, so that whenever a configuration lies in θ_A1 it also lies in θ_B2.

Observation 3: When measuring angle θ_A2 at station A and θ_B1 at station B, we should see an event at station A whenever we see an event at station B. That is, P(θ_A2|θ_B1) = 1. Much like for the previous observation this means that the circle θ_A2 should contain the circle θ_B1.

Looking at the Venn diagram, it seems like we should have that the section outlined in red, P(θ_A2,θ_B2), is more probable than P(θ_A1,θ_B1), with

P(θ_A2,θ_B2) ≥ P(θ_A1,θ_B1) ≅ 0.09.$

In fact there is no way to express the previous 3 observations in a Venn diagram such that we do not have this relation. However, quantum mechanics instead has that we should see:

Observation 4: When measuring angle θ_A2 at station A and θ_B2 at station B, we should never see events at both stations for the same photon pair. That is, P(θ_A2,θ_B2) = 0.

If we see this contradiction it should show that our assumption that the photons had set properties and didn't interact after leaving the BBO crystal was incorrect, and hence that our system must not be consistent with local realism.

One possible quantum mechanical system that could produce these results is the wave function

with H and V referring to horizontal and vertical polarizations for a photons, with measurement angles

θ_A1 = 71°,

θ_A2 = -55°,

θ_B1 = -71°,

θ_B2 = 55°.

The derivation of this can be seen in [3].

While the above four observations are relatively simple to explain, they rely on seeing certain outcomes 100% of the time, which is not practical experimentally. The following modification is equivalent and avoids this issue.

If under a certain circumstance a photon is observed to be polarized at a certain angle 100% of the time, it must be polarized at that angle and will never be observed at a polarization 90 degrees offset from that angle under those same circumstances. This can be used to modify observation 2, with

P(θ_B2|θ_A1) = 1 ⇔ P(θ_B2^⊥|θ_A1) = 0 ⇔ P(θ_A1,θ_B2^⊥) = 0.

From here we can modify the probabilities of observations 2 and 3 to:

2': P(θ_A1,θ_B2^⊥) = 0.

3': P(θ_A2^⊥,θ_B1) = 0.

The convenient thing about this formulation is that if observations 2' and 3' are not observed perfectly we can modify the local realistic expectation of our earlier inequality to become

P(θ_A2,θ_B2) ≥ P(θ_A1,θ_B1) - P(θ_A1,θ_B2^⊥) - P(θ_A2^⊥,θ_B1).

The above is a form of the Bell-Clauser-Horne inequality, which is a restriction on any local realistic system.

From here we define a parameter H as

H = P(θ_A1,θ_B1) - P(θ_A1,θ_B2^⊥) - P(θ_A2^⊥,θ_B1) - P(θ_A2,θ_B2).

This lets us simplify our experimental criterion to saying that if H ≤ 0 then our results are consistent with local realism, while results of H > 0 are inconsistent with local realism, though allowed in some cases by quantum mechanics.

Setting up Hardy's Test in the Lab

Figure 2: Displayed is a diagram of the apparatus used for Hardy's experiment. Additions from Figures 1 and 2 of the single and correlated photons tests

include a quarter-wave plate (λ/4 ) mounted on a rotational stage, as well as two 810nm half-wave plates (λ/2 ) and two polarizing

beam-splitters in each of the down-converted (red) beams.

Figure 2 shows a diagram of the setup for Hardy's test. A 405nm beam high-powered (80mW) UV laser was sent through a half-wave plate (HWP) to adjust the polarization, and an angled quarter-wave plate (QWP) to adjust the relative phase of the horizontal and vertical components of the polarization. Both the half- and quarter-wave plates were adjusted post-setup for fine tuning. The laser then passed through a pair of BBO crystals that are stacked such that their crystal axes are orthogonal. Due to this orientation, the first crystal converted vertically-polarized photons from the laser into horizontally-polarized entangled pairs and vice versa for the second. The two down-converted beams, A and B (810nm), went to stations A and B, respectively. It is important establish prior to conducting Hardy's Test that the BBO is producing correlated pairs of photons (as expected), and that there are single, quantized, photons passing through the down-converted beams. For a description of our correlated and single photons test, see the linked page. For Hardy's test, the components of station A and station B are the same, so we just describe the configuration of station A. As beam A reaches Station A, the beam went through a second HWP to orient a specific incident angle with the polarizing angle of the polarizing beam-splitter (PBS) that follows the half-wave plate. The orientation of the the HWP was the primary control of the experiment, as it allows the PBS to select for different angles without rotating the beam-splitter. After the HWP, the beam hits the PBS, and the photons that pass through or reflect were collected by collimators A and A', respectively, which were then fed into their corresponding SPCMs. The data from these, as well as from the corresponding SPCMs in Station B, was fed into Labview for processing.

It was very important that the 405nm beam be in a specific polarization configuration as it hits the BBO, so that the entangled photons have the state described in the quantum state equation from the previous section. This was done by adjusting the half- and quarter- wave plates along with the BBO while using our detectors for calibration. A more complete description of this process can be seen in [3].

The raw data collected via the Labview program was in the form of count rates, or numbers of events in a given time interval. The types of events that we were looking for were coincidence counts between one SPCM from Station A and one SPCM from Station B, for example the number of events detected in both SPCM A and SPCM B' in a 10-second interval. This count was referred to as R_AB.

These counts were converted to probabilities through the formula

where P(θ_Ai,θ_Bi) is the probability of observing an event when Station A is set to measure θ_Ai and Station B is set to measure θ_Bi. The principle behind this equation is that the denominator is how often a photon is detected at stations A and B simultaneously, and the numerator is how often that event is the one we're looking for.

From here we can find these probabilities for all combinations of angles, and from there find a value for the H parameter described in the previous section.

Our Results

Unfortunately, we failed to acquire a Hardy state, and thus could not complete Hardy's test.

We didn't calculate H for many configurations, as it takes some time and it's usually pretty obvious from the first measurement that we won't see an H value above 0 (which would indicate the violation of local realism we're looking for). The values we did calculate are in the following table.

Recall that P(θ_A1,θ_B1) is meant to be large, and the other three probabilities are meant to be small.

The table has two categories of results, half angle and full angle, with two subcategories, state 1 and state 2. For the half angle measurements the half-wave plates in front of the beamsplitters were set to half of the relevant measurement angle, so that the correct angle of polarization would be mapped to horizontal and sent through the BBO. The full angle measurements had the half wave plates directly set to the relevant angles. This was an extra step we did as part of our troubleshooting when we started having trouble getting the results we were expecting, but based on the way the angles are calculated the half angle approach should be correct. The full angle measurements are included here for completeness and so that future groups know not to try this particular variation again. The reason that there is only one result for half angles with four for full angles is due to the fact that it's much easier to get one of the probabilities low with full angles, and so we had a lot more configurations in which we were already measuring at least two angles.

The states refer to the wave functions we were trying to match in the measurement. State 1 is the state given in the theory section, and state 2 is the opposite wave function, with

We tried many methods to get the experiment to work properly. You can see a description of some of the more noteworthy of those methods in the troubleshooting section, linked below:

Troubleshooting

We're not entirely sure what the issue was with our setup, having not isolated the problem, but some of the more likely issues are the following:

• We could have been setting the angle of the quarter wave plate incorrectly (that is, the orientation of the fast axis). None of the literature directly relating to this experiment addresses the issue of the quarter wave plate angle directly, and so we were left with a lot of guesswork.

• There could have been minor problems with the alignment of our detectors. We had to make some big adjustments to our alignment pretty late in the project after having issues with the BBO crystals, and weren't able to spend the same amount of time that we spent with the initial alignment.

  • while conducting the single and correlated photons tests (our initial alignment) we had just one BBO in the apparatus

  • We first replaced the single BBO with the fused double BBO (two crystals mounted together in one unit by the manufacturer) but this crystal was attenuating the AB coincidence counts for some reason we couldn't explain. For time's sake we decided to replace this double BBO with two single BBOs, manually placed orthogonal to each other.

• Quantum mechanics could be wrong (though we think it's unlikely based on previous experiments ...[3], for example)

In Summary

We were able to show the presence of single pairs of correlated photons produced by two orthogonal BBO crystals. This setup was meant to be used to demonstrate Hardy's Test, and therefore quantum mechanic's violation of local realism, however this attempt was unsuccessful. Our failure to attain a positive value for our H parameter (which would show a violation of local realism) was likely due to the failure to achieve the specific Hardy State that induces the expected behavior. This failure was due to complications in alignment process as well as ambiguities in alignment instruction which should be investigated further in future iterations. In the future, we recommend students allot greater amounts of time to conducting Hardy's Test as finding the correct state proved more difficult than we anticipated. It may also be useful to further investigate the properties of the quarter wave plate in order to re-evaluate the alignment procedure originally proposed.

Works Cited

[1] Podolosky, B, Rosen, N, Einstein, A. "Can quantum-mechanical description of physical reality be considered complete?", 1935.

[2] Lucien Hardy. Nonlocality for two particles without inequalities for almost all entangled states. Physical Review Letters, 71(11):1665–1668, 1993.

[3] J. A. Carlson, M. D. Olmstead, and M. Beck. Quantum mysteries tested: An experiment implementing Hardy’s test of local realism. American Journal of Physics, 74(3):180–186, 2006.

[4] A. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett. 67, 661 (5 August 1991).