Download Single and Entangled Photon Sources

Single and Entangled Photon Sources
Renald Dore1 Jaime Cristian Vela2
1
Institute of Optics, 2Department of Physics and Astronomy, University of Rochester
December 11th, 2013
Abstract
In this experiment we demonstrate the violation of Bell’s inequalities by producing two polarization
entangled photons through spontaneous parametric down conversion. Using two avalanche photodiodes, we
measure how the orientation of two polarization angles affects the coincidence counts of entangled photon pairs
incident on the avalanche photodiodes. Bell’s inequalities state that observable S shall be no larger than 2, yet we
observe a value of 2.54, meaning we have violated Bell’s inequality and proved that we have entangled photons.
Introduction
Today’s research in quantum entanglement offers new and exciting possibilities in future technology.
These include quantum cryptography, quantum computing, and even quantum state teleportation where entangled
particles give information about each other no matter how far apart they are located, due to them sharing the same
wave function. In quantum mechanics, a particles state doesn’t exist until it is measured, but in entangled species,
by making a measurement on one entangled particle we can reliably know information about the other without
actually making any measurement on it [1]. In this experiment we test the nonlocality hypothesis of nature by
studying entangled photons through spontaneous parametric down conversion and calculating bell’s inequalities for
these photons whose polarization states are measured. By violating Bell’s inequality, a classical relationship, we
prove that the relationship between the particles is indeed quantum and cannot be explained by classical
interpretation.
Theory
Two entangled particles (A and B) reliably give information about one another due to them sharing the
same wave function, or that is, that their wave functions cannot be separated [2]. Any measurement performed on
particle A would simultaneously give information of B without and consideration of the separation between them.
This phenomenon is completely quantum mechanical and has now classical explanation. In the laboratory we can
produce entangled photons through a nonlinear process called spontaneous parametric down conversion (SPDC)
[3]. In SPDC, a photon is down converted to two lower energy photons, where the conservation of momentum and
energy is conserved. Using type 1 beta barium borate crystals (BBO), which are especially cut with regard to their
optical axis, a horizontally polarized photon is incident on the crystal with wavelength λ and consequentially, two
photons of wavelength 2λ emerge with vertical polarization with respect to the incident photon. Figure 1 depicts
this process.
Figure 1. Down conversion of photon with horizontal polarization.
For this process, if we rotate the basis, we achieve the same results but with opposite polarizations.
Figure 2. Down conversion of photon with vertical polarization.
By using a pair of BBO crystals, as shown in figure 3, we can produce the down converted quantum state shown in
equation 1.
Figure 3. A pair of type one BBO crystals aligned perpendicular with respect to their axis of polarization.
(eqs.1)
The down converted pair has the probability of detection described by
(eqs. 2)
which, after some calculations, can be written as
(eqs. 3)
Where the angle
corresponds to the angle of a half wave plate.
In the special case where
, then we have
(eqs. 4)
So finally, the coincidence counts is written as
(eqs. 5)
Where C is an offset to account for the imperfections in the polarizers and alignment of the crystals [2].
We will use the Clauser-Horne-Shimony-Holt (CHSH) inequality to measure entanglement of photons in
this experiment [2]. The inequality is shown in equation 6 and 7 below, where
(eqs. 6)
and
(eqs. 7)
and N(a,b) is the coincidence count when polarizer A is at a and polarizer B is at b.
Experimental Setup
The laboratory setup was composed by several components including: 100 mW 363.8nm argon-ion laser, filter,
mirror, Beta Barium Borate (BBO) crystals, polarizers and APD’s. It utilizes spontaneous down conversion in
conjunction with CHSH inequality to demonstrate the entanglement of polarization states of the photons.
Figure 4. Schematic of the experimental setup.
The above setup uses a high power 100mw Argon-Ion laser (363.8nm wavelength) as a light source. The laser
beam first passes through the narrowband blue filter to remove all the unwanted wavelengths and to transmit
photons with double the wavelength of the Argon-ion laser at 727.6nm. The light then passes through the quartz
plate which compensates the phase of the laser beam and then the light hits the high reflective mirror to be directed
towards the BBO Crystals. Beta Barium Borate (BBO) Crystals are the main component of the parametric down
conversion where an incident photon is converted into two photons of longer (twice) wavelength called signal and
idler photons. In this process the momentum and energy are conserved. The BBO crystals were type I meaning the
polarization of the signal and idler photons was orthogonal to the polarization of the incident photon. There will be
two cone of light with equal number of photons produced by the BBO crystals, a horizontal cone and a vertical
cone. The overlapping of this this two cones is performed thanks to the use of the quartz plate which adds phase to
the lagging cone. Using this nonlinear scheme, which produces the signal and idler photons, can create polarized
entangled states. Light after passing through the BBO crystal gets directed towards two different polarizers, where
the Bell’s Inequalities can be calculated at different polarizer orientations. The final parts of this experiment are the
pair of single-photon counting avalanche photodiodes (APD) modules. Coincidence counts can be measured using
the two polarizers in front of the APDs. When the two polarizers are at parallel angles relative to each other both
the signal and idle photons hit the APDs, while when the two polarizers are at perpendicular angles relative to each
other only one of the photons hits the APDs. In the latter case only the entangled photon pair passes and this results
in the minimum photon coincidence count. As discussed in the theory part, there should be a cos2 dependence of the
coincidence counts on the difference between each of the polarization angles.
Procedure
1- Align the system with the help of the TA and use the EM-CCD camera to overlap the horizontally
polarized cone of light with the vertically polarized cone of light.
2- Observe the fringe visibility (Visibility=
) of the photon coincidence count and see if it exceeds
the value of 0.71. If the value of the fringe visibility is greater than 0.71 then the Bell’s Inequality is
violated.
3- Check to see if the single counts of each APD are dependent or independent of its accompanying polarizer
angle. If there is dependence on the polarizer angle, then there is phase misalignment and the quartz plate
must be adjusted accordingly.
4- Take measurements of the coincidence count while changing the angle of the polarizers in order to
calculate Bell’s Inequalities. The specific angles to be measured are α={00,450,900,1350} and
β={22.50,67.50,112.50,157.50}.
5- After the system is aligned and the visibility is >0.71, repeat step for one more time. 16 measurements will
be used to calculate inequality.
6- Violate and confirm the violation of Bell’s Inequalities.
7- Possibility to try and obtain results from the usage of random angles.
Results and Analysis
After making the proper adjustments and aligning the system, the results were recorded in excel files and plotted
accordingly. The measurements were recorded using a laser power of 25mW, an acquisition time of 500ms and 20
measurements per each angle.
Average Coincidence
Count
70
60
0,0
50
45,45
40
30
90,90
20
135,13
5
10
0
-30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
18
20
Rotation Angle (Horizontal Axis)
Figure 5. Coincidence Count vs. Angle of Rotation of the Quartz plate in the horizontal axis. The optimal angle for the Quartz in the
0
horizontal axis would be at the 3 angle on the graph.
Average Coincidence Count
90
80
70
60
0,0
50
45,45
90,90
40
135,135
30
20
10
0
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
18
20
Rotation Axis (Vertical Axis)
Figure 6. Coincidence Count vs. Angle of Rotation of the Quartz plate in the vertical axis. The optimal angle for the Quartz in the
0
vertical axis would be at the -2 angle on the graph.
100
90
Coincedence Count
80
70
60
50
40
30
20
10
0
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
Polarizer Angle
Polarizer A: 90
Polarizer A: 0
Figure 7: Coincidence Count vs. Angle of Rotation of Polarizer. Fringe Visibility of 0.958435.
160
Coincedence Count
140
120
100
80
60
40
20
0
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
Polarizer Angle
Polarizer A: 135
Polarizer A:45
Figure 8: Coincidence Count vs. Angle of Rotation of Polarizer. Fringe Visibility of 0.9588565.
320
340
The graphs above represent the effect that the orientation of the polarizers has on the Fringe Visibility. The
Fringe Visibility was calculated to be 0.958435 when the polarizers positions were 1350 and 450. The Fringe
Visibility was calculated to be 0.9588565 when the polarizers positions were 900 and 00. Both these values are
above 0.71 suggesting that the Bell’s Inequality was violated. The measurements were recorded using a laser power
of 25mW, an acquisition time window of 500ms, total acquisition time of 10s, coincidence time window of 2.6*108
s and 20 measurements per each angle.
The below measurements were recorded using a laser power of 25mW, acquisition time window of
500mW, total acquisition time of 15s, coincidence time window of 2.6*10-8s and 30 measurements taken for each
angle.
Polarizer
B
Polarizer A
-45
-22.5
-45
22.5
-45
67.5
-45
112.5
0
-22.5
0
22.5
0
67.5
0
112.5
45
-22.5
45
22.5
45
67.5
45
112.5
90
-22.5
90
22.5
90
67.5
90
112.5
Net
coincidence
Std. Dev. Net
Coincidence
118.0197
9.77147
29.71924
5.98667
9.149151
3.19644
92.47203
8.79237
85.00017
8.94074
71.19628
8.50213
8.52593
3.07306
23.55562
4.52883
4.814389
2.59287
42.98622
5.83184
50.05641
8.3308
13.69465
2.88496
40.65945
5.73846
3.950726
2.52709
48.09043
5.42493
82.69109
9.53734
E
Std. Dev. of E
0.846588
0.043130257
0.696722
0.053005732
0.46032
0.060650716
-0.51458
0.063136022
S
2.518212
Std. Dev. of S
0.111060764
Violated Bell's Inequality
4.666019
Table 1: The angles above are chosen specifically to maximize the difference between quantum and classical
results. The coincidence counts correspond to the combinations of these angles and are used to calculate
Bell’s Inequality. The data show that Bell’s Inequality is violated and that also CHSH inequality is violated,
because S=2.51821 >2.0.
Polarizer
A
Polarizer
B
0
22.5
0
67.5
0
112.5
0
157.5
45
22.5
45
67.5
45
112.5
45
157.5
90
22.5
90
67.5
90
112.5
90
157.5
135
22.5
135
67.5
135
112.5
135
157.5
Net
coincidence
Std. Dev. Net
Coincidence
69.01223
7.58219
8.711241
2.61209
21.30755
5.3687
86.73302
8.20149
43.135
5.78782
49.9647
6.01808
14.19915
4.44235
5.375227
2.79758
4.029181
2.96822
49.12256
7.41263
83.52402
9.54746
39.37713
6.31218
26.82713
5.00172
9.88968
3.30864
88.87463
8.21829
112.9117
11.9397
S
Std.
Dev. of
S
2.546682
0.119412
Violated Bell's Inequality by # of Standard Deviations
4.578123
Table 2: The angles above are chosen specifically to maximize the difference between quantum and classical
results. The coincidence counts correspond to the combinations of these angles and are used to calculate
Bell’s Inequality. The data show that Bell’s Inequality is violated and that also CHSH inequality is violated,
because S=2.546682 >2.0.
Polarizer
A
Polarizer
B
-10
20
-10
40
-10
60
-10
80
35
20
35
40
35
60
35
80
80
20
80
40
80
60
80
80
125
20
125
40
125
60
125
80
Net
coincidence
Std. Dev. Net
Coincidence
S
0.708909
Std.
Dev. of
S
66.27565
8.03326
0.15709
33.80179
6.14162
8.010204
2.93297
Violated Bell's Inequality by # of Standard Deviations
2.113004
2.05918
-8.21879
54.80029
7.2715
56.85992
5.82346
45.29528
8.54441
27.52615
5.12869
7.044696
2.60878
23.99263
4.79331
45.12073
6.41487
62.77326
6.94279
19.1446
5.89964
1.423591
1.21343
7.988232
3.45147
38.54907
5.94428
Table 3: The angles above are chosen arbitrarily. The coincidence counts correspond to the combinations of
these angles and are used to calculate Bell’s Inequality. The data show that Bell’s Inequality is not violated
and that also CHSH inequality is not violated, because S=0.708909 < 2.0.
Conclusion
In this experiment we were able to produce entangled photons with and Argon Ion laser and BBO crystals. We
showed the cos2 dependence of the coincidence counts on the difference between each of the polarization angles.
We also violated the Bell’s and CHSH Inequalities. This experiment verified the phenomenon of quantum
entanglement in the polarization state of photon pairs.
Contributions
Jaime Cristian Vela worked on: Abstract, Introduction, and Theory.
Renald Dore worked on: Experimental Setup, Procedure, Results and Analysis, Conclusion.
References
[1] J. Gribbin. In Search of Schrodinger’s Cat” Bantam Books. New York. 1984.
[2] S. Lukishova. Lab 1 Entanglement and Bell’s Inequalites. University of Rochester, Institute of Optics. 2008.
[3] J. Eberly. Bell’s Inequalities and Quantum Mechanics. Amer. J. Phys. 70 (3), 286, March, 2002.