Bell tests with Photons
Henry Clausen
Photon Bell Tests
Outline:
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
EPR-Paradoxon
 |Ψ =
1
(|HH
2
Alice
Bob
Collapse
+ |VV )
+
 Entanglement contradicts GR:
+
Emmission
Measurement I is influenced by an event
(measurement II) outside its backwards lightcone
 Einstein also did not like the random character of QM
• “Every complete theory must assign a value to
every element at every physical reality”
Image source: http://rqgravity.net/images/paradox/Paradox-7N.gif (18.05.2013)
Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
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Local hidden variable theory
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Considerations:
•
Quantum mechanics might be incomplete
•
Unknown variables determine outcome of
measurements prior to measurement
 No randomness
 Locality
For entangled photons:
• Whether + or - is measured is determined before the
measurement
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Bell‘s theorem
 “Quantum mechanics cannot arise from a
theory of local pre-existing hidden
variables”
Necessary conditions on theory for Bell’s
theorem:
 Locality
 Counterfactual definiteness
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Locality
 No influence of events outside the
backwards light cone
Counterfactual definiteness
 A property is assigned to a system at all
times independently of whether the
measurement is carried out
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Correlation
 Hidden parameter 𝜆
Observed variables 𝑋 𝜆 , 𝑌 𝜆 ∈ −1,1
• Like polarization of photons
Expectation value: 𝐸 𝑋 : = ∫ 𝑋 𝜆 𝜎 𝜆 𝑑𝜆
𝜎 𝜆 is probability measure
Correlation:
𝜌 𝑋, 𝑌 : = ∫ 𝑋 𝜆 𝑌 𝜆 𝜎(𝜆)𝑑𝜆
Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
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Correlation
𝑅
+𝑅
−𝑅
−𝑅
 Correlation: 𝜌 𝑋, 𝑌 = ++ −− +− −+
𝑅++ +𝑅−− +𝑅+− −𝑅−+
𝑅∎∎ =measurement rate
…
Antiparallel
Pair 1
Pair 2
Pair 3
Pair 4
+
Alice
+
-
+
+
+
+
Bob
-
+
-
-
+1
+1
Correlation
-1
-1
-1
-1
Parallel
Pair 1
Pair 2
Pair 3
Pair 4
Alice
+
-
+
Bob
+
-
Correlation
+1
+1
+1
Image source: http://en.wikipedia.org/wiki/File:Bell%27s_theorem.svg (18.05.2013)
…
-1
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Correlation
 𝐴, 𝐵 measured with different detector settings
 A 𝑎, 𝜆 , 𝐵(𝑏, 𝜆)
Orthogonal
Pair 1
Pair 2
Pair 3
Pair 4
Pair 5
Alice
+
-
-
+
+
Bob
+
+
-
+
-
Correlation
+1
-1
+1
+1
-1
Image source: http://en.wikipedia.org/wiki/File:Bell%27s_theorem.svg (18.05.2013)
…
0
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
CHSH inequality
 𝐴, 𝐵 measured with different detector settings
𝑎, 𝑎′ , 𝑏, 𝑏′
𝑆: = 𝜌 𝑎, 𝑏 + 𝜌 𝑎′ , 𝑏 + 𝜌 𝑎, 𝑏 ′ − 𝜌 𝑎′ , 𝑏 ′
Image source: http://en.wikipedia.org/wiki/File:Bell%27s_theorem.svg (18.05.2013)
≤2
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
CHSH inequality
 𝐴, 𝐵 measured with different detector settings
𝑎, 𝑎′ , 𝑏, 𝑏′
𝑆: = 𝜌 𝑎, 𝑏 + 𝜌 𝑎′ , 𝑏 + 𝜌 𝑎, 𝑏 ′ − 𝜌 𝑎′ , 𝑏 ′
≤2
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
But…
 For entangled photons:
𝜌 𝑎, 𝑏 = ±cos[2 𝑎 − 𝑏 ]
𝜌 0°, 22,5° + 𝜌 45°, 67,5° + 𝜌 45°, 22,5° − 𝜌 0,67,5°

𝛼, 𝛼 ′ : 0°, 45°

𝛽, 𝛽′ : 22,5°, 67,5°
=
4
2
>2
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Bell Test by Alain Aspect, 1981
 Paper: Experimental Realization of
Einstein-Podolsky-Rosen-Bohm
Gedankenexperiment: A New Violation of
Bell's Inequalities
 A. Aspect, P. Grangier,
G. Roger
 Phys. Rev. Lett. 49, 2
(1982)
Image source: http://phototheque.institutoptique.fr/picture.php?/7212 (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Bell Test by Alain Aspect, 1981
 Done in Paris
 First experiment to measure the Bell
inequality directly
 Prior only single-channelanalyzer experiments
Image source: http://phototheque.institutoptique.fr/picture.php?/7212 (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Setup
WL-filter
WL-filter
Single rate
monitor
Single rate
monitor
Image source: http://en.wikipedia.org/wiki/File:Bell-test-photon-analyer.png (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Photon Source
 Calcium-40 cascade is exited by two
photon absorption
 Pairs of photons are emitted
 Wavelengths 𝜆1 = 551.3 nm, 𝜆2 = 422.7 nm
 |Ψ =
1
(|H 1 |H 2
2
+ |V 1 |V 2 )
 Emission rate: 5 × 107 s −1
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Polarizers
 Two prisms separated by thin
film
• Parallel polarized light
transmitted
• Orthogonal polarized
light reflected
 Rotatable
Image source: http://www.meadowlark.com/images/products_large/laserline_beamsplitting_polarizer_fig1_17_2_13895.gif (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Polarizers
 𝑇1∥ = 𝑅1⊥ = 0.950
for 𝜆1
 𝑇1⊥ = 𝑅1∥ = 0.007
 𝑇2∥ = 𝑅2⊥ = 0.930
for 𝜆2
 𝑇2⊥ = 𝑅2∥ = 0.007
Image source: http://www.meadowlark.com/images/products_large/laserline_beamsplitting_polarizer_fig1_17_2_13895.gif (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Advantage of two-channel polarizers
 A one-channel polarizer
always blocks one
polarization (-)
 Coincidence rates 𝑅± (𝑎, 𝑏) and 𝑅−− (𝑎, 𝑏)
cannot be measured directly
𝑅++ +𝑅−− −𝑅+− −𝑅−+
𝑅++ +𝑅−− +𝑅+− −𝑅−+
 Measurement of 𝜌 𝑎, 𝑏 =
not direct  No sufficient violation
Image source: http://en.wikipedia.org/wiki/File:Single-channel_Bell_test.svg (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Detection
 One Photomultiplier for each channel (4 in
total)
 Photon-coincidence and single-photondetection are both measured and stored
 Single-photon detection rate ~104 s −1
 Dark rate ~102 s −1
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Setup
WL-filter
WL-filter
Single
monitor
Single
monitor
Image source: http://en.wikipedia.org/wiki/File:Bell-test-photon-analyer.png (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Detection
 Coincidence window = 20 ns
 Lifetime of exited state ~5 ns
All photon pairs included
ρ(45°,22,5°)
70
60
50
40
30
20
10
0
R++
Coincidence
 From single-photon rate the accidental
coincidences are estimated to be ~10 s −1
 Substracted from total coincidences
 True coincidence rate 𝑅±,± 𝑎, 𝑏 ≈ 0 − 40 𝑠 −1
R-+
R+-
R--
Accidential Coincidence
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Result
 Five runs at 100 s
 𝑆𝑚𝑒𝑎𝑠 = 2.697 ± 0.015
 Correction:
𝜌 𝑎, 𝑏 =
=0.984
𝐹
𝑇1∥ − 𝑇1⊥ 𝑇2∥ − 𝑇2⊥
𝑇1∥
+
𝑇1⊥
 𝑆𝑒𝑥𝑝 = 2.70 ± 0.05
Image source: [2]
𝑇2∥
+
𝑇2⊥
cos 2 𝑎𝑏
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Result
 𝑆𝑚𝑒𝑎𝑠 = 2.697 ± 0.015
 𝑆𝑒𝑥𝑝 = 2.70 ± 0.05
 Overall corrections and
errors:
• Accidental coincidences
• Poisson deviation
• Visibility (Not perfect transmission & reflection)
• Asymetry of detectors
Image source: [2]
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Detection Loophole
 Not every photon pair was measured
•
•
Emission rate: ~5 × 107 𝑠 −1
Coincidence rate: ~80 𝑠 −1
4
𝜂
 Corrected Bell ineq: 𝑆𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 ≤ − 2
 Possible violation in a test:
• 𝑆𝑞𝑚 > 𝑆𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 ⇒ 𝜂min = 0.82
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Locality Loophole
 No communication between measurements
assumed
 Even if separation spacelike:
 Communication after changing the angle
possible
 Measurement basis (angle) has to be
changed after photon emission
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Bell Test by Gregor Weihs, 1998
 Paper: Violation of Bell’s
Inequality under Strict Einstein
Locality Conditions
 G. Weihs, T. Jennewein, C. Simon,
H. Weinfurter, and A. Zeilinger
 Phys. Rev. Lett. 81, 23
(1998)
Image source: [3]
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Bell Test by Gregor Weihs, 1998
 Spacelike separation of
measurements
 Random number generator to
determine measurement basis
 Done during photon flight
 First time Locality Loophole was closed
Image source: [3]
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Setup Aspect
WL-filter
WL-filter
Image source: http://en.wikipedia.org/wiki/File:Bell-test-photon-analyer.png (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Setup Weihs
Laser
Modulator
Modulator
400 m
Random-number-generators
Image source: http://en.wikipedia.org/wiki/File:Bell-test-photon-analyer.png (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Fibers
 Single mode optical fibers
 Depolarization ~1%
 Each cable 500 m long (600 m coiled up)
 Difference less than 1 m (~5 ns)
Image source: S. Johnson, Quantum Electronics FS2013 Summary of lecture notes, (30.07.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Rotation
 Rotation of the polarized light, not the polarizers
 Done by electro-optic modulator
Voltage controlled optic axis
 If voltage applied, polarization rotates by 45°
 Voltage controlled by RNG
Image source: S. Johnson, Quantum Electronics FS2013 Summary of lecture notes, (30.07.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Rotation
 Toggle frequency 𝜔 ≈ 10 𝑀𝐻𝑧
(𝑇 = 100 ns)
 Each setting is stored with time-tag
 Atomic clock at each polarizer for
synchronization
• Accuracy: ~20 ns
Image source: S. Johnson, Quantum Electronics FS2013 Summary of lecture notes, (30.07.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Locality of measurement
 Measurement delays:
•
•
•
•
•
RNG + Amplifier
Electro-optic modulator
Photomultiplier
Sample storing
fiber length difference
 Total 𝟏𝟎𝟎 𝐧𝐬
 Separation of measurement stations:
400 m
400 m ⇔ 𝑡 =
= 𝟏. 𝟑 𝛍𝐬
𝑐
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Locality of measurement
 Measurement delays:
•
•
•
•
•
RNG + Amplifier
Electro-optic modulator
Photomultiplier
Sample storing
fiber length difference
 Total 𝟏𝟎𝟎 𝐧𝐬
 Time between entanglement and measurement:
500 m
= 𝟏. 𝟔𝟔𝟖 𝛍𝐬
𝑐
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Result
 Visibility = 0.97 ⇒ 𝑆exp = 2.74
 Measured value: 𝑆meas = 2.73 ± 0.02
 Measurement 10 s long, 14700 coincidences
collected
 Efficiency ~5%  Still detection loophole
Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
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Outlook
 Detection loophole for photons closed 2013 by Zeilinger
group
 Used Eberhard inequality, not CHSH
• Only 𝜂 = 0.66 required
 Not maximally entangled photons
•
𝛹 =
1
1+𝑟 2
( 𝐻𝑉 + 𝑟 𝑉𝐻 )
 Superconducting TES-calorimeter
• Arm efficiency: 𝜂 ≈ 0.75
Image source: http://phonon.gsfc.nasa.gov/qcal/qcal_f4.html (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Outlook
Weihs‘ Bell test
Freedman &
Clauser
Locality
loophole closed
First Bell test
1970
010
1975
1980
Aspect’s
Bell Test
1985
1990
1995
2000
Wineland et
al.
First time
detection
loophole
closed with
ions
Zeilinger’s
test
Matsukevich
& Moehring
Detection
loophole
closed for
photons
Bell test with
two remote
atomic qubits
2005
2010
Ansmann et. al.
Detection
loophole closed
with Josephson
Qubits
Sources
[1] L. Maccone, A simple proof of Bell’s inequality, arXiv:1212.5214v2 (2013) [quant-ph]
[2] A. Aspect, P. Grangier, G. Roger, Experimental Realization of Einstein-Podolsky-Rosen-Bohm
Gedankenexperiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 2 (1982)
[3] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Violation of Bell’s Inequality
under Strict Einstein Locality Conditions, Phys. Rev. Lett. 81, 23 (1998)
[4] D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, C. Monroe, Bell Inequality Violation
with Two Remote Atomic Qubits, Phys. Rev. Lett. 100, 150404
[5] M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. Kofler, J. Beyer, A. Lita, B. Calkins, T. Gerrits,
S.W. Nam, R. Ursin. A. Zeilinger, Bell violation with entangled photons, free of fair-sampling
assumption, arXiv:1212.0533 [quant-ph] (2013)
[6] Wikipedia Foundation, Bell’s theorem, http://en.wikipedia.org/wiki/Bell_inequalities
(18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Source
 Argon ion laser 351 nm
 Parametric down conversion
with BBO-crystal
 Entangled 702 nm photon pairs
 |Ψ =
1
(|H 1 |V 2
2
− |V 1 |H 2 )
Image source: http://en.wikipedia.org/wiki/File:Scheme_of_spontaneous_parametric_down-conversion.pdf (18.05.2013)
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
RNG
Diode
 Diode emits light onto beam splitter
 One photomultiplier corresponds
to “0”, one to “1”
1
BS
PM
 Toggle frequency 𝜔 ≈ 10 𝑀𝐻𝑧
(𝑇 = 100 ns)
 “0” and “1” not necessarily equally often
 Stored in coincidence monitor
PM
0
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Bell‘s theorem
Photon Bell Test by Aspect
Loopholes
Photon Bell Test by Weihs
Outlook
Detection
 Rotation switched many time between two coincidences
 Very high resolved time tags (75 ps)
and synchronization required
 Atomic clock at each station
Synchronized before measurement ~20 ns
 Coincidences + polarization setting can be identified
without confusion