1.

Principle of a Bell inequalities test

EPR “gedankenexperiment” with polarization entangled photons, first presented by Bohm, is shown on the next figure. Polarization state of the entangled photon pairs created by the source is:

On both sides, each photon is analysed by a polarization beam splitter cube. The result of this measurement is + 1 if the photon is detected with a vertical polarization in direction a (or b on the other side), or -1 if the photon is detected with a polarization perpendiculare to direction a (or b on the other side). One can measure the probabilities of simple detections or the probabilities of double simultaneaous detections with different orientations of the beam splitter cubes.

The correlations between the results on the polarization tests, for any directions a and b of the analysers can be defined by:

E(I a, II b) = P(V_{I a}, V_{I b}) + P(H_{I a}, H_{I b})–P(H_{I a}, V_{I b})–P(V_{I a}, H_{I b})

If there is not any correlation between the two measurements: ^{E(I a, II b) = 0}.

At the opposite extreme, if ^{E(I a, II b) = 1 or -1}, the correlation is perfect.

For an EPR pair of entangled photons, Quantum Mechanics predicts:

E(I a, II b) = cos[2(a – b)]

Then if you measure the Bell parameter given by:

S_Bell(a,a',b,b') = E(I a, II b)–E(I a, II b') + E(I a', II b) + E(I a' II b') with the orientations shown on the next figure, you should obtain .

Figure 3: Orientations which give the maximum conflict between Bell inequalities and Quantum Mechanics For , (Q. M.)

Figure 1:

experimental set-up

Figure 2:

Einstein-Podolsky-Rosen-Bohm « gedankenexperiment » with photons.

Figure 3:

two crystals downconversion source

Bell famous theorem states that any Local Hidden Variable Theories will give: −2 ≤ S_Bell (a,a',b,b') ≤ 2

With the simple setup experiment that we decided to build in our engeneering school, our third year students can create polarization-entangled photons pairs and perform a test of Bell inequalities in 4 or 5 hours.

2.

The source of polarization-entangled photon pairs

Entangled photons are created by type I spontaneous parametric downconversion in BBO (-Borate de Baryum) non linear crystals. The pump, a 60 mW at 405 nm blue diode, gives 810 nm entangled photons.

We use two crystals, 5 mm x 5 mm 0.5 mm thick) cut with their crystal axis at 30° from normal to the large face. For this angle the indexes are:

In a downconversion process, energy and photon momentum are conserved:

With these indexes, we expect type I phase matching, which means that an extraordinary polarized pump photon will create two ordinary polarised “daughter photons”.

We select “degenerate” daughter photons, those which have the same wavelength: λ_signal = λ_iddler = 2λ_pump.

Degenerated Photons pairs are emitted in a cone of angle about 3° around the pump direction.

To produce polarization entangled pairs of photons, we use two identical crystals, rotated by 90° from each other about the pump beam direction. In this configuration, 45° polarized pump photons can downconvert in either crystal (but it is not possible to know in which crystal the photon pairs were created!).

3.

Detection of the polarization-entangled photon pairs

Downconverted photons are collected by two lenses (focal: 75 mm, diameter: 10 mm) at about one meter from the crystals and focussed on single photon counting avalanche photodiodes.

Filters at 810 nm, 10 nm width, are placed just in front of each lens. Polarization is analysed by rotating the half wave plates in front of the polarization beam splitter cubes. Black plastic tubes prevent from stray light and protect single photon counting modules. For each detected single photon these modules give a 25 ns TTL pulse which is sent to a simple coincidence detector. The coincidence window is 30 ns width. We have 3 counters to measure both single detection rates and the coincidence rate.

In our experiment, for single detection rates of about 23000 on each side, we detect about 1600 coincidences per second.

4.

Adjustment of the EPR polarization state

For a rectilinearly polarised pump at 45°, the pump photons state is and two downconversion processes are equally probable:

Thus the photon pairs are in the superposition state:

where is a the phase which depends on many different parameters (wavelengths of pump and downconverted photons, for example).

This state leads to a detection probability of coincidence for both analysers oriented at 45°:

If , entangled photons are actually in the EPR state as defined before.

If , entangled photons are in the state:

If there is not any correlation between the polarization measurements in the diagonal base:

P(V_{I 45°}, V_{II 45°}) = 1/4

To perform a Bell test, it is very important to adjust precisely this phase between the two possible dowconversion processes.

We have decided to control this phase by a Babinet compensator on pump beam. It is possible to measure P(V_{I 45°}, V_{II 45°}) versus the position of the Babinet compensator when both analysers are oriented at 45°. Thus, we adjust precisely the Babinet to obtain a maximum number of coincidences per second in this configuration

5.

Bell’s Parameter Measurement

The Bell parameter is obtained by the measurements of 4 coefficients of correlation, E(V_a, V_b)_:

S_Bell (a,a',b,b') = E(V_a, V_b)–E(V_a, V_b')+E(V_a', V_b)+E(V_a', V_b')

Each of these coefficients needs 4 coincidence rate measurements, n(a,b).

n_total = n(a, b) + n(a + 90, b + 90) + n(a, b + 90) + n(a + 90, b)

P(V_a, V_b) = n(a, b)/n_total

P(H_a, H_b) = n(a + 90, b + 90)/n_tot

P(V_a, H_b) = n(a, b + 90)/n_tot

P(H_a, V_b) = n(a + 90, b)/n_tot

In our experiment, we measure these coincidence rates in 15 seconds each. We obtain ^{S_Bell(a,a',b,b') = 2,48} with a standard deviation . This result is in complete disagreement with any hidden variable local theory.

6.

Short Conclusion

Since the eighties, the EPR paradox is no longer a “gedankenexperiment”. Now, it has become a very exciting lab work for students. We believe it is enlightening not only for quantum mechanics knowledge but also in many different fields of modern physics.