Experiment: Hardy's Test of Local Realism

Figure 1: Complete optical set-up for Hardy's Test. For a description of the individual components see below.

Introduction to the Experiment

This experiment verifies Hardy's Test of Local Realism.

Hardy's Test is based on Bell's theorem of 1964 which is a proof that certain plausible propositions about locality, reality, and theoretical completeness are incompatible with some predictions of two-particle quantum mechanics. This contradiction is revealed by deriving an inequality, Bell’s Inequality, which is violated by certain quantum mechanical statistical predictions. [1]

In 1993 Hardy derived what Mermin has referred to as “the best version of Bell’s theorem.” Hardy conceived of a system of two particles in which local realism predicts a certain outcome never occurs, while quantum mechanics predicts that it sometimes occurs. In principle, the observation of a single occurrence of this event is enough to demonstrate that quantum mechanics violates local realism. However, in practice, a real experiment with an imperfect apparatus must measure an inequality. [2]

The experimental effort required for this test is essentially the same as that required for a test of a Bell inequality. However, this test is based on concepts that are easier to understand and more compelling than those behind the original Bell inequality. [2]

In our experiment, we found a 10 standard deviation violation of the predictions of local realism.

Theory

The theory behind Hardy’s Test is well documented and instead of repeating it, refer to the sources listed below:

• A detailed mathematical treatment of this subject is found in Greenberger’s, Horne’s, Shimony’s and Zeilinger’s paper Bell's theorem without inequalities. [1]

• For a less mathematical treatment on the subject please read Mermin’s paper Quantum mysteries refined.” [3]

• The book Quantum Mechancis by Mark Beck also is an excellent resource for this topic. [4]

• Finally, the paper by J. A. Carlson, M. D. Olmstead, and M. Beck paper: Quantum mysteries tested [2] was used as the primary resource for understanding this experiment.

  • This PDF, VennDiagramsofBeckObservations.pdf, attatched to this page (see bottom of page), gives a brief visualization (using Venn diagrams) of the arguments given by Carlson et al.

Read this section for more theory behind the optical components of this experiment. Specifically, read about the wave plates as the QWP and the HWP's are crucial for this experiment. In truth, some experiments use a Quartz Plate rather than a QWP. The physics behind the Quartz Plate is not as well understood as that of the QWP, which is why it is recommended that the QWP be used when first performing Hardy's Test

Experimental Set-Up

The set-up for this experiment closely follows the paper by J. A. Carlson, M. D. Olmstead, and M. Beck: Quantum mysteries tested [2]. Since this experiment is usually done after completing the Single Photon Quantum Interference (SPQI) experiment by Pearson and Jackson [6] many of the initial set-up steps to create single pairs of down converted photons have already been performed; these are in great detail described elsewhere on this wiki.

Nevertheless, the Hardy's Test set-up differs in two major ways from the SPQI experiment:

• The pump beam is down converted to create two entangled pairs of photons which are orthogonally aligned with each other. This is achieved by using two, back-to-back orthogonally aligned BBO crystals and a phase matching wave plate in front of them. A more detailed description on how to create and test for these orthogonally entangled pairs of photons is given in Kwiat's paper [5] which you are strongly encouraged to study..

• We use all 4 Single Photon Counting Modules (SPCM) to detect the photons from the 2 entangled pairs and we use rotatable linear polarizers in front of the detectors to select the particular quantum state that we want to observe.

Equipment Description

Here is a list of the major optical components and their function: (see also Figure 1 at the top of this page.)

1. PUMP: The pump laser (Blue Diode Laser: Power Technology, Inc. (PTI) IQ1C50(405-60B)G26, power output: 85.4mW and wavelength of 403nm) produces linearly polarized light.

2. BS: This light enters an (optional) polarizing beam splitter which establishes an absolute reference of the polarization axis and also produces a pure polarization state. (The pump laser has been rotated so that most of its light is transmitted through the polarizing beam splitter.)

3. HWP: A half-wave plate rotates the linearly polarized light so that it reaches the two BBO crystals at about 45 degrees to their orthogonal axes. This ensures that each of the BBO crystal will receive about the same amount of photons to down convert and, therefore, will produce about the same amount of horizontally and vertically polarized photon pairs.

4. QWP: The quarter-wave plate is rotated by the angle φ (see Figure 3 below) to adjust the phase between the vertical and horizontal component of the 45 degree polarized light. This “phase matching” is required to account for the spatial separation of the two BBO crystals. Adjusting this phase allows us to produce pairs of (down-converted) photons from each crystal which are coherent with each other. (Note: though we used a quarter-wave plate for this component any birefringent component or wave plate should also work.)

5. BBO: Two identical, adjacent BBO crystals are oriented with their optic axes aligned perpendicularly. Since the linear polarized light striking the crystal has been rotated 45 degrees with respect to the crystals’ optic axes, the vertical component of the incoming light will be down-converted only in the crystal whose optic axis is vertical; similarly, the horizontal component will be down-converted only in the BBO crystal with its optic axis aligned horizontally. A 45° polarized pump photon will be equally likely to down-convert in either crystal (neglecting losses from passing through the first), and these two possible down-conversion processes are coherent with one another, as long as the emitted spatial modes for a given pair of photons are indistinguishable for the two crystals. Consequently, the photons will automatically be created in the state HH+exp(i*phi)*VV [5] where phi is controlled by the QWP in step 4.

6. HWP1 and HWP2 rotate the polarization of the down converted light by angles θ1 and θ2 (see Figure 4 below); hence, they set up the “detection” angles for the polarizing beam splitter immediately following them. Adjusting these half-wave plates allows us to detect the particular states for Hardy’s Test.

7. PBS1 and PBS2 either transmit or reflect the photons, depending on their polarization, into the corresponding detectors.

Figure 2: This diagram shows the polarization of the pump beam and the down converted light during the experiment. Note that the difference between the cones is greatly exaggerated in order to help distinguish between the two cones. In reality, these cones are (nearly) perfectly aligned. Because we are using two BBO crystals to make two cones of down converted light(one horizontally polarized and the other vertically polarized), there is a difference in phase that is introduced. In order to compensate for this, the QWP is used. It introduces a phase difference(as indicated by the red color) that will cancel with that of the BBO crystals. This guarantees that the two cones are in phase with each other

Figure 3: Down Conversion Setup: polarizing beam splitter, half-wave plate, quarter wave plate and the two orthogonal aligned, back-to-back BBOs.

Figure 4: Detector Setup: filters, adjustable halfwave plates and polarizing beam splitters that will send the beams to detectors A, B, C and D.

Figure 5: Detector Setup: filters, adjustable halfwave plates and polarizing beam splitters that will send the beams to detectors A, B, C and D. (Note: the "actual" detectors, the Single Photon Counters or SPCs are inside the large black box in the background; however, for simplicity we refer to the fiber optics collimators labeled in the Figure above as "Det. X" as the "detectors." Since the collimators are are directly connected to the individual SPCs we call them "detectors.")

Figure 6: Another view of the complete Setup. (λ/2 and λ/4 indicate half- and quarter wave plates, i.e., labeled HWP and QWP in the diagram above.)

General Alignment

Follow the above link which describes the alignment procedure used in the Summer of 2014 to conduct Hardy's test.

Waveplate Alignment

The purpose of this page is to describe the behavior of the HWP and QWP and how they affect the coincidence counts of A and B.

Figure 1: This graph shows the effects of rotating the 405nm HWP while keeping the QWP fixed(in this case, θ_QWP=162). Since the Labview Program measures the coincidence counts with respect to time, the graph doesn't look like a perfect sine wave because changing θ_HWP at a constant rate is difficult to do manually.The yellow 'X' indicates which value the HWP was set to when taking measurements on 10/01/14.

Figure 2: This graph shows the effects of rotating the QWP while keeping the HWP fixed(in this case, θ_HWP=280). Notice that the amplitude changes between consecutive wavelengths. It changes between 2 value, a higher one and a lower one. The yellow 'X' indicates which value the QWP was set to when taking measurements on 10/01/14.

Here is a procedure to use when trying to set the wave plates for data collection.

1. Set all four wave plates( the 3 HWP's and the QWP) to their zero mark.

2. Adjust the 405nm HWP so that you get a maximum AB coincidence rate.

3. Rotate the QWP one full turn so that you get a graph similar to that of Figure 2. Note that, unless you get a maximum AB count rate at θ_HWP =28 0, your graph will be out of phase with Figure 2.

4. Notice on Figure 2 how the yellow 'X' is located about 1/4 or 1/3 up from the trough of one of the high amplitude cycles? Set your QWP in a similar spot.

5. You should now be able to reproduce Fig. 2b of the Kwiat paper(Figures 1 and 2 of the Results page).

Now that you know your two wave plates are in good alignment, you can conduct the Hardy's test experiment. The Summer 2014 group found Section III, Part C: "Tuning the state" of the Beck paper to be very helpful.

-- Main.booth135 - 03 Oct 2014

Results

This section provides that results that were obtained by the Summer 2014 group.

References

1. D. M. Greenberger, M. A. Horne, A. Shimony, A. Zeilinger, “Bell's theorem without inequalities,” Am. J. Phys., 58, (12), 1131-1143 (1990).

2. 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).

3. N. D. Mermin, “Quantum mysteries refined,” Am. J. Phys. 62, 880–886 (1994).

4. M Beck, ”Quantum Mechanics: Theory and Experiment.” (Oxford University Press, USA, 2012).

5. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).

6. 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).