Optics 223: Entanglement and Bell’s Inequalities

Stephen Eggers and Daniel Balonek

Group 3: Friday, April 10, 2009

Abstract

By creating two different photons whose polarization states were entangled through the use of BBO crystals through a process called Spontaneous Parametric Down-Conversion, we were able to measure the coincidence counts between two single-photon detectors to calculate Bell’s Inequalities and verify that they are indeed violated, thus proving that there is no classical explanation for this phenomenon.

Theory

Entanglement is one of the fascinating results of Quantum Mechanics where the property of one particle depends on the property of the other. Two particles that are entangled have wave functions that cannot be separated. Any measurement performed on one particle would change the state of the other. The quantum mechanical state describing the particle’s momentum, spin, or polarization may be entangled. We entangled the polarization in this lab through Beta Barium Borate crystals. For each horizontally polarized photon of wavelength λ incident on the crystal, two photons of wavelength 2λ exit with vertical polarization. We used two BBP crystals to create two exiting cones overlapped with one another composed of vertical and horizontally polarized and entangled photons.

Set-Up and Procedure

As shown in figure 1 a ~100mW pump argon ion laser of wavelength λ=363.3 nm and a vertical polarization passes through a blue filter to remember parasite fluorescence inherent in the argon laser, then through abirefringent quartz material to create a phase difference. It is then redirected through two BPO Type-1 crystals mounted back-to-back and perpendicular to one another. The down-converted photons from the BPO are emitted in two overlapping cones (one horizontally-polarized and the other vertically-polarized) with a wavelength of 2λ=727.6nm. One side of the cone is passed through Polarizer B, and the other side of the cone is passed through Polarizer A, and then each is detected using avalanche photodiodes (APD) detectors. The polarizers are adjusted, and the APDs (which are equidistant from the center of the crystal) then can count the photons arriving at each detector. This data is fed to a computer which calculates a coincidence count which consists of how similar the counts are in relation to APD B and APD A. We made two sets of measurements in our lab. In the first set, we set Polarizer A to 45 degrees, and then rotated Polarizer B from 0 to 360 degrees while taking measurements every 10 degree interval. In the second set, we set Polarizer A to 135 degrees, and rotated B again from 0 to 360 degrees. We would expect for entangled photons that the coincidence count would be highest when Polarizer A and Polarizer B were at the same angle. Our measured data is presented in the table on the next page.

Figure 1[1]

Schematic of Experimental Setup

Measured Data

Set 1: Polarizer A at 45 Degrees / Set 2: Polarizer A at 135 Degrees
Polarizer B (degrees) / Coincidence Count / Single A Count / Single B Count / Polarizer B (degrees) / Coincidence Count / Single A Count / Single B Count
0 / 210 / 29738 / 27458 / 0 / 324 / 31082 / 33497
10 / 242 / 28306 / 26536 / 10 / 210 / 30761 / 33189
20 / 380 / 28786 / 27054 / 20 / 140 / 30582 / 33879
30 / 414 / 27631 / 26682 / 30 / 80 / 30406 / 34203
40 / 448 / 27645 / 26774 / 40 / 35 / 29897 / 34309
50 / 439 / 27472 / 26875 / 50 / 20 / 30042 / 34437
60 / 442 / 27587 / 26903 / 60 / 58 / 29043 / 34488
70 / 385 / 27607 / 26951 / 70 / 111 / 30824 / 34284
80 / 310 / 27967 / 26861 / 80 / 190 / 30600 / 34200
90 / 271 / 28484 / 26456 / 90 / 280 / 31000 / 34000
100 / 180 / 28410 / 26780 / 100 / 400 / 31000 / 34000
110 / 107 / 28791 / 26379 / 110 / 490 / 32143 / 33060
120 / 60 / 29019 / 26607 / 120 / 540 / 32000 / 34000
130 / 32 / 29378 / 26485 / 130 / 590 / 32700 / 33900
140 / 22 / 29156 / 26301 / 140 / 586 / 32300 / 33400
150 / 37 / 29441 / 26494 / 150 / 563 / 32500 / 33500
160 / 85 / 28991 / 26301 / 160 / 508 / 32300 / 33300
170 / 145 / 28599 / 26204 / 170 / 419 / 32000 / 33600
180 / 230 / 27824 / 26325 / 180 / 317 / 31000 / 33700
190 / 254 / 27517 / 26099 / 190 / 227 / 30800 / 33400
200 / 361 / 26960 / 25920 / 200 / 158 / 30400 / 33900
210 / 382 / 26229 / 26036 / 210 / 87 / 30300 / 33900
220 / 420 / 26345 / 25962 / 220 / 38 / 29700 / 33800
230 / 448 / 25949 / 25779 / 230 / 28 / 29800 / 34200
240 / 445 / 26451 / 25743 / 240 / 55 / 30300 / 34100
250 / 381 / 26672 / 25774 / 250 / 112 / 30500 / 34000
260 / 353 / 26589 / 25601 / 260 / 200 / 31300 / 34200
270 / 251 / 27723 / 26069 / 270 / 387 / 31200 / 33800
280 / 185 / 28296 / 25800 / 280 / 404 / 31900 / 33900
290 / 111 / 28309 / 25749 / 290 / 500 / 32700 / 34300
300 / 59 / 28882 / 25879 / 300 / 556 / 33000 / 33600
310 / 27 / 29189 / 25871 / 310 / 630 / 33100 / 34100
320 / 18 / 29181 / 25926 / 320 / 602 / 33100 / 33700
330 / 44 / 28781 / 25895 / 330 / 589 / 32600 / 33600
340 / 79 / 28496 / 25458 / 340 / 522 / 32800 / 33800
350 / 138 / 28056 / 25558 / 350 / 436 / 31900 / 33500
360 / 210 / 27271 / 25207 / 360 / 357 / 31600 / 34000

Table 1

Raw Data collected

Figure 2

Coincidences at 45 and 135 degrees with theoretical fit

Figure 3

Singles counts for 45 and 135 degrees

Results:

Data was taken with respect to coincidence count vs. polarizer B angle and single counts vs. polarizer B angle. The data was plotted in figures 2 and 3. As can be seen from figure 2 the measured data correlated very well with a theoretical fit. The theoretical fit function was cos(a-b)^2, where ‘a’ represents the angle at which polarizer A was orientated and ‘b’ represents the orientation of polarizer B. The single photon counts were plotted on figure 3. As can be seen from the graph our single photon count for each detector, A and B, were fairly consistent. The visibility of the coincidence count was also calculated with equation 1. At 45 degrees the visibility was calculated to be .923 and at 135 degrees it was calculated to be .938. The discrepancy in the height of each of the curves in figure 2 is due to the fact that the power level of the input laser was changed in between each trial, from 80mW to 98mW for 45 degrees and 135 degrees respectively.

References

Lukishova, S. G. (2008). Lab. 1. Entanglement and Bell’s Inequalities [Laboratory Manual]. University of Rochester

  1. [1] Lukishova, S. G. (2008). Lab. 1. Entanglement and Bell’s Inequalities [Laboratory Manual]. University of Rochester