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Einstein’s Recoiling-Slit Experiment:Uncertainty and Complementarity
Radhika Vathsan
BITS Pilani K K Birla Goa Campus
QIPA 2013, 4th Dec...
Collaborator: Tabish QureshiCenter for Theoretical PhysicsJamia Millia Islamia, New Delhi
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 1 / 43
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Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 2 / 43
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Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 3 / 43
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The Two-Slit Experiment with Quantum particles
Setup
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 4 / 43
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The Two-Slit Experiment with Quantum particles
Slit 1 open
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 5 / 43
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The Two-Slit Experiment with Quantum particles
Slit 2 open
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 6 / 43
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The Two-Slit Experiment with Quantum particles
Both slits open
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 7 / 43
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Two-slit experiment with electronsTonomura, Endo, Matsuda, Kawasaki, Ezawa, Am. J. Phys. 57(2) (1989).
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 8 / 43
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Two-slit experiment with electronsTonomura, Endo, Matsuda, Kawasaki, Ezawa, Am. J. Phys. 57(2) (1989).
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 8 / 43
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Two-slit experiment with electronsTonomura, Endo, Matsuda, Kawasaki, Ezawa, Am. J. Phys. 57(2) (1989).
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 8 / 43
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Two-slit experiment with electronsTonomura, Endo, Matsuda, Kawasaki, Ezawa, Am. J. Phys. 57(2) (1989).
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 8 / 43
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Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 9 / 43
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Which slit did the electron pass through?Getting the “Welcher-Weg" (which-way) information
No Interference!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 10 / 43
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Which slit did the electron pass through?Getting the “Welcher-Weg" (which-way) information
Which-way Detector
No Interference!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 10 / 43
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Bohr’s Complementarity Principle
Niels Bohr in 1928
In describing the results ofquantum mechanical experiments,certain physical concepts arecomplementary. If two conceptsare complementary, an experimentthat clearly illustrates one conceptwill obscure the othercomplementary one.. . .("The Quantum Postulate and the Recent Development ofAtomic Theory," Supplement to Nature, April 14, 1928, p.580)
In the two-slit experiment: the “which-way" information vsexistence of interference pattern.
They can NEVER be observed at the same time, in the sameexperiment.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 11 / 43
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Bohr’s Complementarity Principle
Niels Bohr in 1928
In describing the results ofquantum mechanical experiments,certain physical concepts arecomplementary. If two conceptsare complementary, an experimentthat clearly illustrates one conceptwill obscure the othercomplementary one.. . .("The Quantum Postulate and the Recent Development ofAtomic Theory," Supplement to Nature, April 14, 1928, p.580)
In the two-slit experiment: the “which-way" information vsexistence of interference pattern.
They can NEVER be observed at the same time, in the sameexperiment.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 11 / 43
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Bohr’s Complementarity Principle
Niels Bohr in 1928
In describing the results ofquantum mechanical experiments,certain physical concepts arecomplementary. If two conceptsare complementary, an experimentthat clearly illustrates one conceptwill obscure the othercomplementary one.. . .("The Quantum Postulate and the Recent Development ofAtomic Theory," Supplement to Nature, April 14, 1928, p.580)
In the two-slit experiment: the “which-way" information vsexistence of interference pattern.
They can NEVER be observed at the same time, in the sameexperiment.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 11 / 43
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Bohr’s Complementarity Principle
Niels Bohr in 1928
In describing the results ofquantum mechanical experiments,certain physical concepts arecomplementary. If two conceptsare complementary, an experimentthat clearly illustrates one conceptwill obscure the othercomplementary one.. . .("The Quantum Postulate and the Recent Development ofAtomic Theory," Supplement to Nature, April 14, 1928, p.580)
In the two-slit experiment: the “which-way" information vsexistence of interference pattern.
They can NEVER be observed at the same time, in the sameexperiment.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 11 / 43
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5th Solvay Conference (1927)
A Piccard, E Henriot, P Ehrenfest, E Herzen, T de Donder, E Schrodinger, J-E Verschaffelt,W Pauli, W Heisenberg, R H Fowler, LBrillouin,P Debye, M Knudsen, W L Bragg, H A Kramers, P Dirac, A Compton, L de Broglie, M Born, N Bohr,
I Langmuir, M Planck, M Sklodowska Curie, H Lorentz,A Einstein, P Langevin, C Guye, C T R Wilson, O W Richardson
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5th Solvay Conference (1927)
A Piccard, E Henriot, P Ehrenfest, E Herzen, T de Donder, E Schrodinger, J-E Verschaffelt, W Pauli, W Heisenberg, R H Fowler,L Brillouin,P Debye, M Knudsen, W L Bragg, H A Kramers, P Dirac, A Compton, L de Broglie, M Born, N Bohr,
I Langmuir, M Planck, M Sklodowska Curie, H Lorentz, A Einstein, P Langevin, C Guye, C T R Wilson, O W Richardson
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Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 13 / 43
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Einstein’s Recoiling-Slit Gedanken Experiment
... Einstein thought he had found a counterexample tothe uncertainty principle. "It was quite a shock forBohr .... he did not see the solution at once. Duringthe whole evening he was extremely unhappy, goingfrom one to the other and trying to persuade them thatit couldn’t be true, that it would be the end of physics ifEinstein were right; but he couldn’t produce anyrefutation. I shall never forget the vision of the twoantagonists leaving the club [of the FondationUniversitaire]: Einstein a tall majestic figure, walkingquietly, with a somewhat ironical smile, and Bohrtrotting near him, very excited ....
The next morningcame Bohr’s triumph."
ROSENFELD (1968)Fundamental Problems in Elementary Particle Physics
Proceedings of the Fourteenth Solvay Conference, Interscience, New York, p. 232.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 14 / 43
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Einstein’s Recoiling-Slit Gedanken Experiment
... Einstein thought he had found a counterexample tothe uncertainty principle. "It was quite a shock forBohr .... he did not see the solution at once. Duringthe whole evening he was extremely unhappy, goingfrom one to the other and trying to persuade them thatit couldn’t be true, that it would be the end of physics ifEinstein were right; but he couldn’t produce anyrefutation. I shall never forget the vision of the twoantagonists leaving the club [of the FondationUniversitaire]: Einstein a tall majestic figure, walkingquietly, with a somewhat ironical smile, and Bohrtrotting near him, very excited ....
The next morningcame Bohr’s triumph."
ROSENFELD (1968)Fundamental Problems in Elementary Particle Physics
Proceedings of the Fourteenth Solvay Conference, Interscience, New York, p. 232.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 14 / 43
Page 24
Einstein’s Recoiling-Slit Gedanken Experiment
... Einstein thought he had found a counterexample tothe uncertainty principle. "It was quite a shock forBohr .... he did not see the solution at once. Duringthe whole evening he was extremely unhappy, goingfrom one to the other and trying to persuade them thatit couldn’t be true, that it would be the end of physics ifEinstein were right; but he couldn’t produce anyrefutation. I shall never forget the vision of the twoantagonists leaving the club [of the FondationUniversitaire]: Einstein a tall majestic figure, walkingquietly, with a somewhat ironical smile, and Bohrtrotting near him, very excited .... The next morningcame Bohr’s triumph."
ROSENFELD (1968)Fundamental Problems in Elementary Particle Physics
Proceedings of the Fourteenth Solvay Conference, Interscience, New York, p. 232.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 14 / 43
Page 25
Einstein’s Recoiling-Slit Gedanken Experiment
Replace the static source slit
by a movable slitto obtain which-way information without disturbing the particle
Figures after Bohr
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 15 / 43
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Einstein’s Recoiling-Slit Gedanken Experiment
Replace the static source slit
by a movable slit
to obtain which-way information without disturbing the particle
Figures after Bohr
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 15 / 43
Page 27
Einstein’s Recoiling-Slit Gedanken Experiment
Replace the static source slit
by a movable slitto obtain which-way information without disturbing the particle
Figures after Bohr
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 15 / 43
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Einstein’s Recoiling-Slit Gedanken Experiment
Particle going through upper/lower slit has momentum ±p0
Momentum conservation =⇒ recoil ∓p0 of slitMomentum of slit→ which-way information
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 16 / 43
Page 29
Einstein’s Recoiling-Slit Gedanken Experiment
Particle going through upper/lower slit has momentum ±p0
Momentum conservation =⇒ recoil ∓p0 of slitMomentum of slit→ which-way information
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 16 / 43
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Einstein’s Recoiling-Slit Gedanken Experiment
Particle going through upper/lower slit has momentum ±p0
Momentum conservation =⇒ recoil ∓p0 of slit
Momentum of slit→ which-way information
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 16 / 43
Page 31
Einstein’s Recoiling-Slit Gedanken Experiment
Particle going through upper/lower slit has momentum ±p0
Momentum conservation =⇒ recoil ∓p0 of slitMomentum of slit→ which-way information
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 16 / 43
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Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 17 / 43
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Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2)
≈ pθ =hλθ =
hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.Fringe separation = λL
d .Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
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Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2) ≈ pθ =hλθ
=hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.Fringe separation = λL
d .Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
Page 35
Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2) ≈ pθ =hλθ =
hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.Fringe separation = λL
d .Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
Page 36
Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2) ≈ pθ =hλθ =
hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.Fringe separation = λL
d .Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
Page 37
Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2) ≈ pθ =hλθ =
hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.Fringe separation = λL
d .Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
Page 38
Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2) ≈ pθ =hλθ =
hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.
Fringe separation = λLd .
Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
Page 39
Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2) ≈ pθ =hλθ =
hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.Fringe separation = λL
d .
Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
Page 40
Bohr’s reply
For particles passing through Slit Aand those through slit B:
∆px = 2p sin(θ/2) ≈ pθ =hλθ =
hλ
dL
This is the limit on accuracy ofmeasuring recoil momentum.
Min uncertainty in position of source slit: ∆x =~
2∆px=
λL4πd
.
This is the uncertainty in position of a fringe.Fringe separation = λL
d .Interference pattern is lost!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 18 / 43
Page 41
Implication of Bohr’s resolution
Complementarity enforced by Uncertainty Principle?
Getting which-way information will necessarily disturb the state ofthe particle.Disturbance will be enough to wash out interference.This viewed as a restatement of Uncertainty Principle
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 19 / 43
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Implication of Bohr’s resolution
Complementarity enforced by Uncertainty Principle?Getting which-way information will necessarily disturb the state ofthe particle.
Disturbance will be enough to wash out interference.This viewed as a restatement of Uncertainty Principle
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 19 / 43
Page 43
Implication of Bohr’s resolution
Complementarity enforced by Uncertainty Principle?Getting which-way information will necessarily disturb the state ofthe particle.Disturbance will be enough to wash out interference.
This viewed as a restatement of Uncertainty Principle
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 19 / 43
Page 44
Implication of Bohr’s resolution
Complementarity enforced by Uncertainty Principle?Getting which-way information will necessarily disturb the state ofthe particle.Disturbance will be enough to wash out interference.This viewed as a restatement of Uncertainty Principle
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 19 / 43
Page 45
Realization of Recoiling-Slit Experiment
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 20 / 43
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Realization of Recoiling-Slit Experiment
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 21 / 43
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Is uncertainty a requirement for Complementarity?
Now it turns out that the concept of Uncertainty is not necessary forexplaining complementarity!
Obtaining information about a quantum system is throughMeasurement, which yields classical result.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 22 / 43
Page 48
Is uncertainty a requirement for Complementarity?
Now it turns out that the concept of Uncertainty is not necessary forexplaining complementarity!
Obtaining information about a quantum system is throughMeasurement, which yields classical result.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 22 / 43
Page 49
Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 23 / 43
Page 50
Quantum measurementAccording to von Neumann
A quantum measurement consists of two processes:
1 Process 1: Unitary operation establishes correlation betweensystem & detector.
Initial states: System:∑n
i=1 ci|ψi〉; Detector:|d0〉
|d0〉n∑
i=1
ci|ψi〉Unitary evolution−−−−−−−−−→
Process 1
n∑i=1
ci|di〉|ψi〉
2 Process 2: Non-unitary selection of a single state |ψk〉 withprobability |ck|2:
n∑i=1
ci|di〉|ψi〉 −−−−−→Process 2
|dk〉|ψk〉
"The Measurement Problem".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 24 / 43
Page 51
Quantum measurementAccording to von Neumann
A quantum measurement consists of two processes:1 Process 1: Unitary operation establishes correlation between
system & detector.
Initial states: System:∑n
i=1 ci|ψi〉; Detector:|d0〉
|d0〉n∑
i=1
ci|ψi〉Unitary evolution−−−−−−−−−→
Process 1
n∑i=1
ci|di〉|ψi〉
2 Process 2: Non-unitary selection of a single state |ψk〉 withprobability |ck|2:
n∑i=1
ci|di〉|ψi〉 −−−−−→Process 2
|dk〉|ψk〉
"The Measurement Problem".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 24 / 43
Page 52
Quantum measurementAccording to von Neumann
A quantum measurement consists of two processes:1 Process 1: Unitary operation establishes correlation between
system & detector.Initial states: System:
∑ni=1 ci|ψi〉; Detector:|d0〉
|d0〉n∑
i=1
ci|ψi〉Unitary evolution−−−−−−−−−→
Process 1
n∑i=1
ci|di〉|ψi〉
2 Process 2: Non-unitary selection of a single state |ψk〉 withprobability |ck|2:
n∑i=1
ci|di〉|ψi〉 −−−−−→Process 2
|dk〉|ψk〉
"The Measurement Problem".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 24 / 43
Page 53
Quantum measurementAccording to von Neumann
A quantum measurement consists of two processes:1 Process 1: Unitary operation establishes correlation between
system & detector.Initial states: System:
∑ni=1 ci|ψi〉; Detector:|d0〉
|d0〉n∑
i=1
ci|ψi〉Unitary evolution−−−−−−−−−→
Process 1
n∑i=1
ci|di〉|ψi〉
2 Process 2: Non-unitary selection of a single state |ψk〉 withprobability |ck|2:
n∑i=1
ci|di〉|ψi〉 −−−−−→Process 2
|dk〉|ψk〉
"The Measurement Problem".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 24 / 43
Page 54
Quantum measurementAccording to von Neumann
A quantum measurement consists of two processes:1 Process 1: Unitary operation establishes correlation between
system & detector.Initial states: System:
∑ni=1 ci|ψi〉; Detector:|d0〉
|d0〉n∑
i=1
ci|ψi〉Unitary evolution−−−−−−−−−→
Process 1
n∑i=1
ci|di〉|ψi〉
2 Process 2: Non-unitary selection of a single state |ψk〉 withprobability |ck|2:
n∑i=1
ci|di〉|ψi〉 −−−−−→Process 2
|dk〉|ψk〉
"The Measurement Problem".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 24 / 43
Page 55
Quantum measurementAccording to von Neumann
A quantum measurement consists of two processes:1 Process 1: Unitary operation establishes correlation between
system & detector.Initial states: System:
∑ni=1 ci|ψi〉; Detector:|d0〉
|d0〉n∑
i=1
ci|ψi〉Unitary evolution−−−−−−−−−→
Process 1
n∑i=1
ci|di〉|ψi〉
2 Process 2: Non-unitary selection of a single state |ψk〉 withprobability |ck|2:
n∑i=1
ci|di〉|ψi〉 −−−−−→Process 2
|dk〉|ψk〉
"The Measurement Problem".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 24 / 43
Page 56
Which-way Detection in Einstein’s experimentUsing von Neumann’s process 1
Two orthogonal states of the particle depending on the path:slit 1: |ψ1〉 slit 2: |ψ2〉
Two momentum states of the recoiling slit: |p1〉 and |p2〉.
(a) Final state of particle+slit: necessary entanglement :
|Ψ〉 = |ψ1〉|p1〉+ |ψ2〉|p2〉
(b) Reading out of which-way information: correlation of “readout"states with detector states without affecting the states of theparticle
Point (a) was not part of Bohr’s reply.
and is enough to rule out interference!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 25 / 43
Page 57
Which-way Detection in Einstein’s experimentUsing von Neumann’s process 1
Two orthogonal states of the particle depending on the path:slit 1: |ψ1〉 slit 2: |ψ2〉
Two momentum states of the recoiling slit: |p1〉 and |p2〉.
(a) Final state of particle+slit: necessary entanglement :
|Ψ〉 = |ψ1〉|p1〉+ |ψ2〉|p2〉
(b) Reading out of which-way information: correlation of “readout"states with detector states without affecting the states of theparticle
Point (a) was not part of Bohr’s reply.
and is enough to rule out interference!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 25 / 43
Page 58
Which-way Detection in Einstein’s experimentUsing von Neumann’s process 1
Two orthogonal states of the particle depending on the path:slit 1: |ψ1〉 slit 2: |ψ2〉
Two momentum states of the recoiling slit: |p1〉 and |p2〉.
(a) Final state of particle+slit: necessary entanglement :
|Ψ〉 = |ψ1〉|p1〉+ |ψ2〉|p2〉
(b) Reading out of which-way information: correlation of “readout"states with detector states without affecting the states of theparticle
Point (a) was not part of Bohr’s reply.
and is enough to rule out interference!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 25 / 43
Page 59
Which-way Detection in Einstein’s experimentUsing von Neumann’s process 1
Two orthogonal states of the particle depending on the path:slit 1: |ψ1〉 slit 2: |ψ2〉
Two momentum states of the recoiling slit: |p1〉 and |p2〉.
(a) Final state of particle+slit: necessary entanglement :
|Ψ〉 = |ψ1〉|p1〉+ |ψ2〉|p2〉
(b) Reading out of which-way information: correlation of “readout"states with detector states without affecting the states of theparticle
Point (a) was not part of Bohr’s reply.
and is enough to rule out interference!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 25 / 43
Page 60
Which-way Detection in Einstein’s experimentUsing von Neumann’s process 1
Two orthogonal states of the particle depending on the path:slit 1: |ψ1〉 slit 2: |ψ2〉
Two momentum states of the recoiling slit: |p1〉 and |p2〉.
(a) Final state of particle+slit: necessary entanglement :
|Ψ〉 = |ψ1〉|p1〉+ |ψ2〉|p2〉
(b) Reading out of which-way information: correlation of “readout"states with detector states without affecting the states of theparticle
Point (a) was not part of Bohr’s reply.
and is enough to rule out interference!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 25 / 43
Page 61
Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 26 / 43
Page 62
Which-way Information and Interference
Without which-way informationAmplitude for finding the particle at point x on the screen is
Ψ(x) = ψ1(x) + ψ2(x).
Probability (intensity):
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x) + ψ∗
2(x)ψ1(x)︸ ︷︷ ︸.interference
WITH which-way information
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x)〈p1|p2〉+ ψ∗
2(x)ψ1(x)〈p2|p1〉
= |ψ1(x)|2 + |ψ2(x)|2, since 〈p1|p2〉 = 0
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 27 / 43
Page 63
Which-way Information and Interference
Without which-way informationAmplitude for finding the particle at point x on the screen is
Ψ(x) = ψ1(x) + ψ2(x).
Probability (intensity):
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x) + ψ∗
2(x)ψ1(x)︸ ︷︷ ︸.
interference
WITH which-way information
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x)〈p1|p2〉+ ψ∗
2(x)ψ1(x)〈p2|p1〉
= |ψ1(x)|2 + |ψ2(x)|2, since 〈p1|p2〉 = 0
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 27 / 43
Page 64
Which-way Information and Interference
Without which-way informationAmplitude for finding the particle at point x on the screen is
Ψ(x) = ψ1(x) + ψ2(x).
Probability (intensity):
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x) + ψ∗
2(x)ψ1(x)︸ ︷︷ ︸.interference
WITH which-way information
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x)〈p1|p2〉+ ψ∗
2(x)ψ1(x)〈p2|p1〉
= |ψ1(x)|2 + |ψ2(x)|2, since 〈p1|p2〉 = 0
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 27 / 43
Page 65
Which-way Information and Interference
Without which-way informationAmplitude for finding the particle at point x on the screen is
Ψ(x) = ψ1(x) + ψ2(x).
Probability (intensity):
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x) + ψ∗
2(x)ψ1(x)︸ ︷︷ ︸.interference
WITH which-way information
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x)〈p1|p2〉+ ψ∗
2(x)ψ1(x)〈p2|p1〉
= |ψ1(x)|2 + |ψ2(x)|2, since 〈p1|p2〉 = 0
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 27 / 43
Page 66
Which-way Information and Interference
Without which-way informationAmplitude for finding the particle at point x on the screen is
Ψ(x) = ψ1(x) + ψ2(x).
Probability (intensity):
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x) + ψ∗
2(x)ψ1(x)︸ ︷︷ ︸.interference
WITH which-way information
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x)〈p1|p2〉+ ψ∗
2(x)ψ1(x)〈p2|p1〉
= |ψ1(x)|2 + |ψ2(x)|2, since 〈p1|p2〉 = 0
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 27 / 43
Page 67
Which-way Information and Interference
Without which-way informationAmplitude for finding the particle at point x on the screen is
Ψ(x) = ψ1(x) + ψ2(x).
Probability (intensity):
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x) + ψ∗
2(x)ψ1(x)︸ ︷︷ ︸.interference
WITH which-way information
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x)〈p1|p2〉+ ψ∗
2(x)ψ1(x)〈p2|p1〉= |ψ1(x)|2 + |ψ2(x)|2,
since 〈p1|p2〉 = 0
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 27 / 43
Page 68
Which-way Information and Interference
Without which-way informationAmplitude for finding the particle at point x on the screen is
Ψ(x) = ψ1(x) + ψ2(x).
Probability (intensity):
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x) + ψ∗
2(x)ψ1(x)︸ ︷︷ ︸.interference
WITH which-way information
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉
|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2 + ψ∗1(x)ψ2(x)〈p1|p2〉+ ψ∗
2(x)ψ1(x)〈p2|p1〉= |ψ1(x)|2 + |ψ2(x)|2, since 〈p1|p2〉 = 0
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 27 / 43
Page 69
Which-way Information and Interference
Interference vanishes if which-way information is obtained.
Another interpretation: the recoil of the slit stores which-wayinformation.No need to invoke uncertainty!
If this entanglement between the particle and the recoiling-slit hadbeen recognized and its implications understood
Bohr could have provided a simpler rebuttal to Einstein!
Can this argument be made more quantitative?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 28 / 43
Page 70
Which-way Information and Interference
Interference vanishes if which-way information is obtained.Another interpretation: the recoil of the slit stores which-wayinformation.
No need to invoke uncertainty!
If this entanglement between the particle and the recoiling-slit hadbeen recognized and its implications understood
Bohr could have provided a simpler rebuttal to Einstein!
Can this argument be made more quantitative?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 28 / 43
Page 71
Which-way Information and Interference
Interference vanishes if which-way information is obtained.Another interpretation: the recoil of the slit stores which-wayinformation.No need to invoke uncertainty!
If this entanglement between the particle and the recoiling-slit hadbeen recognized and its implications understood
Bohr could have provided a simpler rebuttal to Einstein!
Can this argument be made more quantitative?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 28 / 43
Page 72
Which-way Information and Interference
Interference vanishes if which-way information is obtained.Another interpretation: the recoil of the slit stores which-wayinformation.No need to invoke uncertainty!
If this entanglement between the particle and the recoiling-slit hadbeen recognized and its implications understood
Bohr could have provided a simpler rebuttal to Einstein!
Can this argument be made more quantitative?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 28 / 43
Page 73
Which-way Information and Interference
Interference vanishes if which-way information is obtained.Another interpretation: the recoil of the slit stores which-wayinformation.No need to invoke uncertainty!
If this entanglement between the particle and the recoiling-slit hadbeen recognized and its implications understood
Bohr could have provided a simpler rebuttal to Einstein!
Can this argument be made more quantitative?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 28 / 43
Page 74
Which-way Information and Interference
Interference vanishes if which-way information is obtained.Another interpretation: the recoil of the slit stores which-wayinformation.No need to invoke uncertainty!
If this entanglement between the particle and the recoiling-slit hadbeen recognized and its implications understood
Bohr could have provided a simpler rebuttal to Einstein!
Can this argument be made more quantitative?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 28 / 43
Page 75
Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 29 / 43
Page 76
Path-distinguishability and Interference
Suppose our detector distinguishes the two paths inaccurately.
This means “which-way" states 〈d1|d2〉 6= 0.
Define Distinguishability:
D =√
1− |〈d1|d2〉|2,
Amplitude that the paths are perfectly distinguishedDefine Visibility:
V ≡ Imax − Imin
Imax + Imin,
measure of the interference observed.
Is there a relationship between them to capture complementarity?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 30 / 43
Page 77
Path-distinguishability and Interference
Suppose our detector distinguishes the two paths inaccurately.
This means “which-way" states 〈d1|d2〉 6= 0.
Define Distinguishability:
D =√
1− |〈d1|d2〉|2,
Amplitude that the paths are perfectly distinguishedDefine Visibility:
V ≡ Imax − Imin
Imax + Imin,
measure of the interference observed.
Is there a relationship between them to capture complementarity?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 30 / 43
Page 78
Path-distinguishability and Interference
Suppose our detector distinguishes the two paths inaccurately.
This means “which-way" states 〈d1|d2〉 6= 0.
Define Distinguishability:
D =√
1− |〈d1|d2〉|2,
Amplitude that the paths are perfectly distinguished
Define Visibility:
V ≡ Imax − Imin
Imax + Imin,
measure of the interference observed.
Is there a relationship between them to capture complementarity?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 30 / 43
Page 79
Path-distinguishability and Interference
Suppose our detector distinguishes the two paths inaccurately.
This means “which-way" states 〈d1|d2〉 6= 0.
Define Distinguishability:
D =√
1− |〈d1|d2〉|2,
Amplitude that the paths are perfectly distinguishedDefine Visibility:
V ≡ Imax − Imin
Imax + Imin,
measure of the interference observed.
Is there a relationship between them to capture complementarity?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 30 / 43
Page 80
Path-distinguishability and Interference
Suppose our detector distinguishes the two paths inaccurately.
This means “which-way" states 〈d1|d2〉 6= 0.
Define Distinguishability:
D =√
1− |〈d1|d2〉|2,
Amplitude that the paths are perfectly distinguishedDefine Visibility:
V ≡ Imax − Imin
Imax + Imin,
measure of the interference observed.
Is there a relationship between them to capture complementarity?
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 30 / 43
Page 81
Path-distinguishability and InterferenceGaussian Wave-packet Model
t = 0: particle emerges from the double-slit with amplitude
Ψ(x, 0) =
A(|d1〉e−
(x−d/2)2
4ε2 + |d2〉e−(x+d/2)2
4ε2
), A =
14√
8πε2
After time t, traveling a distance L, amplitude for particle to arrive at xon screen:
Ψ(x, t) = At
(|d1〉e
− (x−d/2)2
4ε2+2i~t/m + |d2〉e− (x+d/2)2
4ε2+2i~t/m
),
where At =1√2
[√
2π(ε+ i~t/2mε)]−1/2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 31 / 43
Page 82
Path-distinguishability and InterferenceGaussian Wave-packet Model
t = 0: particle emerges from the double-slit with amplitude
Ψ(x, 0) = A(|d1〉e−
(x−d/2)2
4ε2 + |d2〉e−(x+d/2)2
4ε2
),
A =1
4√
8πε2
After time t, traveling a distance L, amplitude for particle to arrive at xon screen:
Ψ(x, t) = At
(|d1〉e
− (x−d/2)2
4ε2+2i~t/m + |d2〉e− (x+d/2)2
4ε2+2i~t/m
),
where At =1√2
[√
2π(ε+ i~t/2mε)]−1/2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 31 / 43
Page 83
Path-distinguishability and InterferenceGaussian Wave-packet Model
t = 0: particle emerges from the double-slit with amplitude
Ψ(x, 0) = A(|d1〉e−
(x−d/2)2
4ε2 + |d2〉e−(x+d/2)2
4ε2
), A =
14√
8πε2
After time t, traveling a distance L, amplitude for particle to arrive at xon screen:
Ψ(x, t) = At
(|d1〉e
− (x−d/2)2
4ε2+2i~t/m + |d2〉e− (x+d/2)2
4ε2+2i~t/m
),
where At =1√2
[√
2π(ε+ i~t/2mε)]−1/2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 31 / 43
Page 84
Path-distinguishability and InterferenceGaussian Wave-packet Model
t = 0: particle emerges from the double-slit with amplitude
Ψ(x, 0) = A(|d1〉e−
(x−d/2)2
4ε2 + |d2〉e−(x+d/2)2
4ε2
), A =
14√
8πε2
After time t, traveling a distance L, amplitude for particle to arrive at xon screen:
Ψ(x, t) = At
(|d1〉e
− (x−d/2)2
4ε2+2i~t/m + |d2〉e− (x+d/2)2
4ε2+2i~t/m
),
where At =1√2
[√
2π(ε+ i~t/2mε)]−1/2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 31 / 43
Page 85
Path-distinguishability and InterferenceGaussian Wave-packet Model
t = 0: particle emerges from the double-slit with amplitude
Ψ(x, 0) = A(|d1〉e−
(x−d/2)2
4ε2 + |d2〉e−(x+d/2)2
4ε2
), A =
14√
8πε2
After time t, traveling a distance L, amplitude for particle to arrive at xon screen:
Ψ(x, t) = At
(|d1〉e
− (x−d/2)2
4ε2+2i~t/m + |d2〉e− (x+d/2)2
4ε2+2i~t/m
),
where At =1√2
[√
2π(ε+ i~t/2mε)]−1/2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 31 / 43
Page 86
Path-distinguishability and InterferenceGaussian Wave-packet Model
Probability of finding particle at point x on the screen
|Ψ(x, t)|2 = 2|At|2e− x2+d2/4
2σ2t cosh(xd/2σ2
t )
×
1 + |〈d1|d2〉|cos(
xdλL/2π4ε4+(λL/2π)2 + θ
)cosh(xd/2σ2
t )
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 32 / 43
Page 87
Path-distinguishability and InterferenceGaussian Wave-packet Model
Probability of finding particle at point x on the screen
|Ψ(x, t)|2 = 2|At|2e− x2+d2/4
2σ2t cosh(xd/2σ2
t )
×
1 + |〈d1|d2〉|cos(
xdλL/2π4ε4+(λL/2π)2 + θ
)cosh(xd/2σ2
t )
〈d1|d2〉 = |〈d1|d2〉|eiθ
p0 = h/λ =⇒ ~t/m = λL/2π,
σ2t = ε2 +
(~t
2mε
)2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 32 / 43
Page 88
Path-distinguishability and InterferenceGaussian Wave-packet Model
Probability of finding particle at point x on the screen
|Ψ(x, t)|2 = 2|At|2e− x2+d2/4
2σ2t cosh(xd/2σ2
t )
×
1 + |〈d1|d2〉|cos(
xdλL/2π4ε4+(λL/2π)2 + θ
)cosh(xd/2σ2
t )
Fringe width =
λLd
+16π2ε4
λdL.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 32 / 43
Page 89
Visibility of Interference
Visibility V ≡ Imax − Imin
Imax + Imin
=|〈d1|d2〉|
cosh(xd/2σ2t )
cosh(y) ≥ 1 =⇒ V ≤ |〈d1|d2〉|.
UsingD2 = 1− |〈d1|d2〉|2,
we get
V2 +D2 ≤ 1.
Englert-Greenberger-Yasin duality relation
A quantitative statement of complementarity
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 33 / 43
Page 90
Visibility of Interference
Visibility V ≡ Imax − Imin
Imax + Imin=
|〈d1|d2〉|cosh(xd/2σ2
t )
cosh(y) ≥ 1 =⇒ V ≤ |〈d1|d2〉|.
UsingD2 = 1− |〈d1|d2〉|2,
we get
V2 +D2 ≤ 1.
Englert-Greenberger-Yasin duality relation
A quantitative statement of complementarity
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 33 / 43
Page 91
Visibility of Interference
Visibility V ≡ Imax − Imin
Imax + Imin=
|〈d1|d2〉|cosh(xd/2σ2
t )
cosh(y) ≥ 1 =⇒ V ≤ |〈d1|d2〉|.
UsingD2 = 1− |〈d1|d2〉|2,
we get
V2 +D2 ≤ 1.
Englert-Greenberger-Yasin duality relation
A quantitative statement of complementarity
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 33 / 43
Page 92
Visibility of Interference
Visibility V ≡ Imax − Imin
Imax + Imin=
|〈d1|d2〉|cosh(xd/2σ2
t )
cosh(y) ≥ 1 =⇒ V ≤ |〈d1|d2〉|.
UsingD2 = 1− |〈d1|d2〉|2,
we get
V2 +D2 ≤ 1.
Englert-Greenberger-Yasin duality relation
A quantitative statement of complementarity
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 33 / 43
Page 93
Visibility of Interference
Visibility V ≡ Imax − Imin
Imax + Imin=
|〈d1|d2〉|cosh(xd/2σ2
t )
cosh(y) ≥ 1 =⇒ V ≤ |〈d1|d2〉|.
UsingD2 = 1− |〈d1|d2〉|2,
we get
V2 +D2 ≤ 1.
Englert-Greenberger-Yasin duality relation
A quantitative statement of complementarity
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 33 / 43
Page 94
Visibility of Interference
Visibility V ≡ Imax − Imin
Imax + Imin=
|〈d1|d2〉|cosh(xd/2σ2
t )
cosh(y) ≥ 1 =⇒ V ≤ |〈d1|d2〉|.
UsingD2 = 1− |〈d1|d2〉|2,
we get
V2 +D2 ≤ 1.
Englert-Greenberger-Yasin duality relation
A quantitative statement of complementarity
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 33 / 43
Page 95
Origin of Complementarity?
Quantum correlations ?D.M. Greenberger, A. Yasin, Phys. Lett. A 128, 391 (1988),“Simultaneous wave and particle knowledge in a neutron interferometer",B-G. Englert, Phys. Rev. Lett. 77, 2154 (1996),“Fringe visibility and which-way information: an inequality"M.O. Scully, B.G. Englert, H. Walther, Nature 375, 367 (1995),
“Complementarity and uncertainty."
Uncertainty principleS.M. Tan, D.F. Walls,Phys. Rev. A 47, 4663-4676 (1993),“Loss of coherence in interferometry".E.P. Storey, S.M. Tan, M.J. Collett, D.F. Walls, Nature 367, 626 (1994).H. Wiseman, F. Harrison, Nature 377, 584 (1995),“Uncertainty over complementarity?"H. Wiseman, Phys. Lett. A 311, 285 (2003),
“Directly observing momentum transfer in twin-slit which-way experiments"
Does the particle really receive a “momentum kick"?S. Durr, T. Nonn, G. Rempe, Nature 395, 33 (1998),“Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer."C.S. Unnikrishnan, Phys. Rev. A 62, 015601 (2000),
“Origin of quantum-mechanical complementarity without momentum back action in atom-interferometry
experiments".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 34 / 43
Page 96
Origin of Complementarity?
Quantum correlations ?D.M. Greenberger, A. Yasin, Phys. Lett. A 128, 391 (1988),“Simultaneous wave and particle knowledge in a neutron interferometer",B-G. Englert, Phys. Rev. Lett. 77, 2154 (1996),“Fringe visibility and which-way information: an inequality"M.O. Scully, B.G. Englert, H. Walther, Nature 375, 367 (1995),
“Complementarity and uncertainty."
Uncertainty principleS.M. Tan, D.F. Walls,Phys. Rev. A 47, 4663-4676 (1993),“Loss of coherence in interferometry".E.P. Storey, S.M. Tan, M.J. Collett, D.F. Walls, Nature 367, 626 (1994).H. Wiseman, F. Harrison, Nature 377, 584 (1995),“Uncertainty over complementarity?"H. Wiseman, Phys. Lett. A 311, 285 (2003),
“Directly observing momentum transfer in twin-slit which-way experiments"
Does the particle really receive a “momentum kick"?S. Durr, T. Nonn, G. Rempe, Nature 395, 33 (1998),“Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer."C.S. Unnikrishnan, Phys. Rev. A 62, 015601 (2000),
“Origin of quantum-mechanical complementarity without momentum back action in atom-interferometry
experiments".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 34 / 43
Page 97
Origin of Complementarity?
Quantum correlations ?D.M. Greenberger, A. Yasin, Phys. Lett. A 128, 391 (1988),“Simultaneous wave and particle knowledge in a neutron interferometer",B-G. Englert, Phys. Rev. Lett. 77, 2154 (1996),“Fringe visibility and which-way information: an inequality"M.O. Scully, B.G. Englert, H. Walther, Nature 375, 367 (1995),
“Complementarity and uncertainty."
Uncertainty principleS.M. Tan, D.F. Walls,Phys. Rev. A 47, 4663-4676 (1993),“Loss of coherence in interferometry".E.P. Storey, S.M. Tan, M.J. Collett, D.F. Walls, Nature 367, 626 (1994).H. Wiseman, F. Harrison, Nature 377, 584 (1995),“Uncertainty over complementarity?"H. Wiseman, Phys. Lett. A 311, 285 (2003),
“Directly observing momentum transfer in twin-slit which-way experiments"
Does the particle really receive a “momentum kick"?S. Durr, T. Nonn, G. Rempe, Nature 395, 33 (1998),“Origin of quantum-mechanical complementarity probed by a which-way experiment in an atom interferometer."C.S. Unnikrishnan, Phys. Rev. A 62, 015601 (2000),
“Origin of quantum-mechanical complementarity without momentum back action in atom-interferometry
experiments".
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 34 / 43
Page 98
Uncertainty principle and complementarityOther work
G. Bjork, J. Soderholm, A. Trifonov, T. Tsegaye, A. Karlsson, Phys. Rev.A 60, 1874 (1999), “Complementarity and the uncertainty relations".
K-P Marzlin, B.C. Sanders, P.L. Knight, Phys. Rev. A 78, 062107 (2008),“Complementarity and uncertainty relations for matter-waveinterferometry",
J-H Huang, S-Y Zhu, arXiv:1011.5273 [physics.optics],“Complementarity and uncertainty in a two-way interferometer".
G.M. Bosyk, M. Portesi, F. Holik, A. Plastino, arXiv:1206.2992 [quant-ph]“On the connection between complementarity and uncertainty principlesin the Mach-Zehnder interferometric setting".
Paul Busch, Christopher R. Shilladay. arXiv:quant-ph/0609048, PhysRep 435, 1-31 (2006)
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 35 / 43
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Outline
1 Two-Slit Experiment and ComplementarityTwo Slit Experiment with Quantum ParticlesComplementarityEinstein’s Recoiling Slit Experiment...and Bohr’s Reply
2 Complementarity and Entanglementvon Neuman MeasurementsWhich-way Information and InterferencePath Distinguishability and Fringe Visibility
3 Complementarity and UncertaintyDuality and Uncertainty
4 Conclusions
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 36 / 43
Page 100
Complementarity and UncertaintyUncertainty and duality
“Which-way" states of the recoiling slit: |d1〉 and |d2〉(normalized, not necessarily orthogonal)
Orthonormal basis for recoiling slit: {|p1〉, |p2〉}Eigenstates of some observable P̂ with eigenvalues ±1.
Which-way states in the P-basis:
|d1〉 = c1|p1〉+ c2|p2〉,|d2〉 = c∗2|p1〉+ c∗1|p2〉.
|c1| = 1, c2 = 0→ full which-way information|c1| = |c2| = 1/
√2→ no which-way information
Uncertainty: ∆P2 = 〈P̂2〉 − 〈P̂〉2 = 4|c1|2|c2|2.Distinguishability:
D2 = 1− |〈d1|d2〉|2 = 1− 4|c1|2|c2|2
= 1−∆P2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 37 / 43
Page 101
Complementarity and UncertaintyUncertainty and duality
“Which-way" states of the recoiling slit: |d1〉 and |d2〉(normalized, not necessarily orthogonal)
Orthonormal basis for recoiling slit: {|p1〉, |p2〉}
Eigenstates of some observable P̂ with eigenvalues ±1.Which-way states in the P-basis:
|d1〉 = c1|p1〉+ c2|p2〉,|d2〉 = c∗2|p1〉+ c∗1|p2〉.
|c1| = 1, c2 = 0→ full which-way information|c1| = |c2| = 1/
√2→ no which-way information
Uncertainty: ∆P2 = 〈P̂2〉 − 〈P̂〉2 = 4|c1|2|c2|2.Distinguishability:
D2 = 1− |〈d1|d2〉|2 = 1− 4|c1|2|c2|2
= 1−∆P2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 37 / 43
Page 102
Complementarity and UncertaintyUncertainty and duality
“Which-way" states of the recoiling slit: |d1〉 and |d2〉(normalized, not necessarily orthogonal)
Orthonormal basis for recoiling slit: {|p1〉, |p2〉}Eigenstates of some observable P̂ with eigenvalues ±1.
Which-way states in the P-basis:
|d1〉 = c1|p1〉+ c2|p2〉,|d2〉 = c∗2|p1〉+ c∗1|p2〉.
|c1| = 1, c2 = 0→ full which-way information|c1| = |c2| = 1/
√2→ no which-way information
Uncertainty: ∆P2 = 〈P̂2〉 − 〈P̂〉2 = 4|c1|2|c2|2.Distinguishability:
D2 = 1− |〈d1|d2〉|2 = 1− 4|c1|2|c2|2
= 1−∆P2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 37 / 43
Page 103
Complementarity and UncertaintyUncertainty and duality
“Which-way" states of the recoiling slit: |d1〉 and |d2〉(normalized, not necessarily orthogonal)
Orthonormal basis for recoiling slit: {|p1〉, |p2〉}Eigenstates of some observable P̂ with eigenvalues ±1.
Which-way states in the P-basis:
|d1〉 = c1|p1〉+ c2|p2〉,|d2〉 = c∗2|p1〉+ c∗1|p2〉.
|c1| = 1, c2 = 0→ full which-way information|c1| = |c2| = 1/
√2→ no which-way information
Uncertainty: ∆P2 = 〈P̂2〉 − 〈P̂〉2 = 4|c1|2|c2|2.Distinguishability:
D2 = 1− |〈d1|d2〉|2 = 1− 4|c1|2|c2|2
= 1−∆P2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 37 / 43
Page 104
Complementarity and UncertaintyUncertainty and duality
“Which-way" states of the recoiling slit: |d1〉 and |d2〉(normalized, not necessarily orthogonal)
Orthonormal basis for recoiling slit: {|p1〉, |p2〉}Eigenstates of some observable P̂ with eigenvalues ±1.
Which-way states in the P-basis:
|d1〉 = c1|p1〉+ c2|p2〉,|d2〉 = c∗2|p1〉+ c∗1|p2〉.
|c1| = 1, c2 = 0→ full which-way information|c1| = |c2| = 1/
√2→ no which-way information
Uncertainty: ∆P2 = 〈P̂2〉 − 〈P̂〉2 = 4|c1|2|c2|2.Distinguishability:
D2 = 1− |〈d1|d2〉|2 = 1− 4|c1|2|c2|2
= 1−∆P2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 37 / 43
Page 105
Complementarity and UncertaintyUncertainty and duality
“Which-way" states of the recoiling slit: |d1〉 and |d2〉(normalized, not necessarily orthogonal)
Orthonormal basis for recoiling slit: {|p1〉, |p2〉}Eigenstates of some observable P̂ with eigenvalues ±1.
Which-way states in the P-basis:
|d1〉 = c1|p1〉+ c2|p2〉,|d2〉 = c∗2|p1〉+ c∗1|p2〉.
|c1| = 1, c2 = 0→ full which-way information|c1| = |c2| = 1/
√2→ no which-way information
Uncertainty: ∆P2 = 〈P̂2〉 − 〈P̂〉2 = 4|c1|2|c2|2.
Distinguishability:
D2 = 1− |〈d1|d2〉|2 = 1− 4|c1|2|c2|2
= 1−∆P2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 37 / 43
Page 106
Complementarity and UncertaintyUncertainty and duality
“Which-way" states of the recoiling slit: |d1〉 and |d2〉(normalized, not necessarily orthogonal)
Orthonormal basis for recoiling slit: {|p1〉, |p2〉}Eigenstates of some observable P̂ with eigenvalues ±1.
Which-way states in the P-basis:
|d1〉 = c1|p1〉+ c2|p2〉,|d2〉 = c∗2|p1〉+ c∗1|p2〉.
|c1| = 1, c2 = 0→ full which-way information|c1| = |c2| = 1/
√2→ no which-way information
Uncertainty: ∆P2 = 〈P̂2〉 − 〈P̂〉2 = 4|c1|2|c2|2.Distinguishability:
D2 = 1− |〈d1|d2〉|2 = 1− 4|c1|2|c2|2
= 1−∆P2
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 37 / 43
Page 107
Uncertainty and Duality
Correlation of detector states with particle states:
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉.
Consider a basis change:
|p1〉+ |p2〉 → ψ1(x) + ψ2(x)
|p1〉 − |p2〉 → ψ1(x)− ψ2(x)
=⇒ ∃ another observable Q̂ with eigenvalues ±1 andcorresponding eigenstates
|q1〉 = (|p1〉+ |p2〉)/√
2
|q2〉 = (|p1〉 − |p2〉)/√
2
The particle states can be correlated with these states:
Ψ(x) =c1√
2[ψ1(x) + ψ2(x)]|q1〉+
c2√2
[ψ1(x)− ψ2(x)]|q2〉
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 38 / 43
Page 108
Uncertainty and Duality
Correlation of detector states with particle states:
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉.
Consider a basis change:
|p1〉+ |p2〉 → ψ1(x) + ψ2(x)
|p1〉 − |p2〉 → ψ1(x)− ψ2(x)
=⇒ ∃ another observable Q̂ with eigenvalues ±1 andcorresponding eigenstates
|q1〉 = (|p1〉+ |p2〉)/√
2
|q2〉 = (|p1〉 − |p2〉)/√
2
The particle states can be correlated with these states:
Ψ(x) =c1√
2[ψ1(x) + ψ2(x)]|q1〉+
c2√2
[ψ1(x)− ψ2(x)]|q2〉
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 38 / 43
Page 109
Uncertainty and Duality
Correlation of detector states with particle states:
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉.
Consider a basis change:
|p1〉+ |p2〉 → ψ1(x) + ψ2(x)
|p1〉 − |p2〉 → ψ1(x)− ψ2(x)
=⇒ ∃ another observable Q̂ with eigenvalues ±1 andcorresponding eigenstates
|q1〉 = (|p1〉+ |p2〉)/√
2
|q2〉 = (|p1〉 − |p2〉)/√
2
The particle states can be correlated with these states:
Ψ(x) =c1√
2[ψ1(x) + ψ2(x)]|q1〉+
c2√2
[ψ1(x)− ψ2(x)]|q2〉
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 38 / 43
Page 110
Uncertainty and Duality
Correlation of detector states with particle states:
Ψ(x) = ψ1(x)|p1〉+ ψ2(x)|p2〉.
Consider a basis change:
|p1〉+ |p2〉 → ψ1(x) + ψ2(x)
|p1〉 − |p2〉 → ψ1(x)− ψ2(x)
=⇒ ∃ another observable Q̂ with eigenvalues ±1 andcorresponding eigenstates
|q1〉 = (|p1〉+ |p2〉)/√
2
|q2〉 = (|p1〉 − |p2〉)/√
2
The particle states can be correlated with these states:
Ψ(x) =c1√
2[ψ1(x) + ψ2(x)]|q1〉+
c2√2
[ψ1(x)− ψ2(x)]|q2〉
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 38 / 43
Page 111
Uncertainty and Duality
Correlate the detected particles on the screen with the measuredeigenstate of Q̂ (c1 = c2 case)
Two complementary interference patterns corresponding to |q1〉 and|q2〉.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 39 / 43
Page 112
Uncertainty and Duality
Correlate the detected particles on the screen with the measuredeigenstate of Q̂ (c1 = c2 case)
0-1-2-3-4-5 1 2 3 4 5
Two complementary interference patterns corresponding to |q1〉 and|q2〉.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 39 / 43
Page 113
Uncertainty and Duality
Correlate the detected particles on the screen with the measuredeigenstate of Q̂ (c1 = c2 case)
0-1-2-3-4-5 1 2 3 4 5
Two complementary interference patterns corresponding to |q1〉 and|q2〉.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 39 / 43
Page 114
Uncertainty and Duality
Correlate the detected particles on the screen with the measuredeigenstate of Q̂ (c1 = c2 case)
0-1-2-3-4-5 1 2 3 4 5
Two complementary interference patterns corresponding to |q1〉 and|q2〉.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 39 / 43
Page 115
Uncertainty and Duality
For any c1, c2,
|Ψ(x)|2 =|ψ1(x)|2 + |ψ2(x)|2
2+|c1|2 − |c2|2
2[ψ∗
1(x)ψ2(x) + ψ∗2(x)ψ1(x)] .
Fringe visibility: V2 ≤ (|c1|2 − |c2|2)2.
The uncertainty in Q̂, in this entangled state:
∆Q2 = 1− (|c1|2 − |c2|2)2.
ThusV2 ≤ 1−∆Q2.
Combining with the earlier result D2 = 1−∆P2, we get
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 40 / 43
Page 116
Uncertainty and Duality
For any c1, c2,
|Ψ(x)|2 =|ψ1(x)|2 + |ψ2(x)|2
2+|c1|2 − |c2|2
2[ψ∗
1(x)ψ2(x) + ψ∗2(x)ψ1(x)] .
Fringe visibility: V2 ≤ (|c1|2 − |c2|2)2.
The uncertainty in Q̂, in this entangled state:
∆Q2 = 1− (|c1|2 − |c2|2)2.
ThusV2 ≤ 1−∆Q2.
Combining with the earlier result D2 = 1−∆P2, we get
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 40 / 43
Page 117
Uncertainty and Duality
For any c1, c2,
|Ψ(x)|2 =|ψ1(x)|2 + |ψ2(x)|2
2+|c1|2 − |c2|2
2[ψ∗
1(x)ψ2(x) + ψ∗2(x)ψ1(x)] .
Fringe visibility: V2 ≤ (|c1|2 − |c2|2)2.
The uncertainty in Q̂, in this entangled state:
∆Q2 = 1− (|c1|2 − |c2|2)2.
ThusV2 ≤ 1−∆Q2.
Combining with the earlier result D2 = 1−∆P2, we get
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 40 / 43
Page 118
Uncertainty and Duality
For any c1, c2,
|Ψ(x)|2 =|ψ1(x)|2 + |ψ2(x)|2
2+|c1|2 − |c2|2
2[ψ∗
1(x)ψ2(x) + ψ∗2(x)ψ1(x)] .
Fringe visibility: V2 ≤ (|c1|2 − |c2|2)2.
The uncertainty in Q̂, in this entangled state:
∆Q2 = 1− (|c1|2 − |c2|2)2.
ThusV2 ≤ 1−∆Q2.
Combining with the earlier result D2 = 1−∆P2, we get
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 40 / 43
Page 119
Uncertainty and Duality
For any c1, c2,
|Ψ(x)|2 =|ψ1(x)|2 + |ψ2(x)|2
2+|c1|2 − |c2|2
2[ψ∗
1(x)ψ2(x) + ψ∗2(x)ψ1(x)] .
Fringe visibility: V2 ≤ (|c1|2 − |c2|2)2.
The uncertainty in Q̂, in this entangled state:
∆Q2 = 1− (|c1|2 − |c2|2)2.
ThusV2 ≤ 1−∆Q2.
Combining with the earlier result D2 = 1−∆P2, we get
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 40 / 43
Page 120
Uncertainty and DualityThe Sum Uncertainty Relation
Sum uncertainty relation for angular momenta 1
∆L2x + ∆L2
y + ∆L2z ≥ `
Implication for Pauli spin matrices
∆σ2x + ∆σ2
y + ∆σ2z ≥ 2, ∆σ2
x + ∆σ2y ≥ 1.
In our case, P̂ = σ̂z, Q̂ = σ̂x. So, ∆P2 + ∆Q2 ≥ 1 .Using this on
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
we getD2 + V2 ≤ 1.
The duality relation also emerges from the sum uncertainty relation.
1Hoffmann, Takeuchi, Phys. Rev. A 68, 032103 (2003).Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 41 / 43
Page 121
Uncertainty and DualityThe Sum Uncertainty Relation
Sum uncertainty relation for angular momenta 1
∆L2x + ∆L2
y + ∆L2z ≥ `
Implication for Pauli spin matrices
∆σ2x + ∆σ2
y + ∆σ2z ≥ 2, ∆σ2
x + ∆σ2y ≥ 1.
In our case, P̂ = σ̂z, Q̂ = σ̂x. So, ∆P2 + ∆Q2 ≥ 1 .Using this on
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
we getD2 + V2 ≤ 1.
The duality relation also emerges from the sum uncertainty relation.
1Hoffmann, Takeuchi, Phys. Rev. A 68, 032103 (2003).Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 41 / 43
Page 122
Uncertainty and DualityThe Sum Uncertainty Relation
Sum uncertainty relation for angular momenta 1
∆L2x + ∆L2
y + ∆L2z ≥ `
Implication for Pauli spin matrices
∆σ2x + ∆σ2
y + ∆σ2z ≥ 2, ∆σ2
x + ∆σ2y ≥ 1.
In our case, P̂ = σ̂z, Q̂ = σ̂x. So, ∆P2 + ∆Q2 ≥ 1 .
Using this onD2 + V2 ≤ 2− [∆P2 + ∆Q2].
we getD2 + V2 ≤ 1.
The duality relation also emerges from the sum uncertainty relation.
1Hoffmann, Takeuchi, Phys. Rev. A 68, 032103 (2003).Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 41 / 43
Page 123
Uncertainty and DualityThe Sum Uncertainty Relation
Sum uncertainty relation for angular momenta 1
∆L2x + ∆L2
y + ∆L2z ≥ `
Implication for Pauli spin matrices
∆σ2x + ∆σ2
y + ∆σ2z ≥ 2, ∆σ2
x + ∆σ2y ≥ 1.
In our case, P̂ = σ̂z, Q̂ = σ̂x. So, ∆P2 + ∆Q2 ≥ 1 .Using this on
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
we getD2 + V2 ≤ 1.
The duality relation also emerges from the sum uncertainty relation.
1Hoffmann, Takeuchi, Phys. Rev. A 68, 032103 (2003).Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 41 / 43
Page 124
Uncertainty and DualityThe Sum Uncertainty Relation
Sum uncertainty relation for angular momenta 1
∆L2x + ∆L2
y + ∆L2z ≥ `
Implication for Pauli spin matrices
∆σ2x + ∆σ2
y + ∆σ2z ≥ 2, ∆σ2
x + ∆σ2y ≥ 1.
In our case, P̂ = σ̂z, Q̂ = σ̂x. So, ∆P2 + ∆Q2 ≥ 1 .Using this on
D2 + V2 ≤ 2− [∆P2 + ∆Q2].
we getD2 + V2 ≤ 1.
The duality relation also emerges from the sum uncertainty relation.1Hoffmann, Takeuchi, Phys. Rev. A 68, 032103 (2003).
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 41 / 43
Page 125
Conclusions
For any two orthogonal states of the recoiling slit (say) |ξ1〉 and|ξ2〉, one can always find operators P̂ and Q̂ whose uncertaintiesenforce complementarity.
P̂ = |ξ1〉〈ξ1| − |ξ2〉〈ξ2| Q̂ = |ξ1〉〈ξ2|+ |ξ2〉〈ξ1|
Englert-Greenberger-Yasin duality relation emerges fromcorrelations and also from the sum uncertainty relation.Complementarity enforced by correlations and the uncertaintyrelations are two sides of a coin (provided the observables arecorrectly identified).Momentum back-action of the recoiling slit on the particle plays norole in complementarity.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 42 / 43
Page 126
Conclusions
For any two orthogonal states of the recoiling slit (say) |ξ1〉 and|ξ2〉, one can always find operators P̂ and Q̂ whose uncertaintiesenforce complementarity.P̂ = |ξ1〉〈ξ1| − |ξ2〉〈ξ2| Q̂ = |ξ1〉〈ξ2|+ |ξ2〉〈ξ1|
Englert-Greenberger-Yasin duality relation emerges fromcorrelations and also from the sum uncertainty relation.Complementarity enforced by correlations and the uncertaintyrelations are two sides of a coin (provided the observables arecorrectly identified).Momentum back-action of the recoiling slit on the particle plays norole in complementarity.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 42 / 43
Page 127
Conclusions
For any two orthogonal states of the recoiling slit (say) |ξ1〉 and|ξ2〉, one can always find operators P̂ and Q̂ whose uncertaintiesenforce complementarity.P̂ = |ξ1〉〈ξ1| − |ξ2〉〈ξ2| Q̂ = |ξ1〉〈ξ2|+ |ξ2〉〈ξ1|Englert-Greenberger-Yasin duality relation emerges fromcorrelations and also from the sum uncertainty relation.
Complementarity enforced by correlations and the uncertaintyrelations are two sides of a coin (provided the observables arecorrectly identified).Momentum back-action of the recoiling slit on the particle plays norole in complementarity.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 42 / 43
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Conclusions
For any two orthogonal states of the recoiling slit (say) |ξ1〉 and|ξ2〉, one can always find operators P̂ and Q̂ whose uncertaintiesenforce complementarity.P̂ = |ξ1〉〈ξ1| − |ξ2〉〈ξ2| Q̂ = |ξ1〉〈ξ2|+ |ξ2〉〈ξ1|Englert-Greenberger-Yasin duality relation emerges fromcorrelations and also from the sum uncertainty relation.Complementarity enforced by correlations and the uncertaintyrelations are two sides of a coin (provided the observables arecorrectly identified).
Momentum back-action of the recoiling slit on the particle plays norole in complementarity.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 42 / 43
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Conclusions
For any two orthogonal states of the recoiling slit (say) |ξ1〉 and|ξ2〉, one can always find operators P̂ and Q̂ whose uncertaintiesenforce complementarity.P̂ = |ξ1〉〈ξ1| − |ξ2〉〈ξ2| Q̂ = |ξ1〉〈ξ2|+ |ξ2〉〈ξ1|Englert-Greenberger-Yasin duality relation emerges fromcorrelations and also from the sum uncertainty relation.Complementarity enforced by correlations and the uncertaintyrelations are two sides of a coin (provided the observables arecorrectly identified).Momentum back-action of the recoiling slit on the particle plays norole in complementarity.
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 42 / 43
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Tabish Qureshi, Radhika VathsanEinstein’s Recoiling Slit Experiment, Complementarity andUncertaintyArxiv: 1210.4248 [quant-ph]Quanta Vol. 2 (April 2013)
THANK YOU!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 43 / 43
Page 131
Tabish Qureshi, Radhika VathsanEinstein’s Recoiling Slit Experiment, Complementarity andUncertaintyArxiv: 1210.4248 [quant-ph]Quanta Vol. 2 (April 2013)
THANK YOU!
Radhika Vathsan (BITS Goa) Einstein’s Recoiling-Slit Experiment QIPA 2013, 4th Dec, Allahabad 43 / 43