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An Experimental Implementationof Hardy’s Paradox
Jeff Lundeen, Kevin ReschAephraim SteinbergUniversity of TorontoJune 2003
Funding by:NSERC, PRO, Sumner.
Zeno’s Paradox : Achilles and The Tortoise
BS1
BS2
IO
DC
Bang½NeitherPresent¼DNone¼C
ResultProb.Detector
Bomb Present:
Bomb Absent:Only detector C fires
Interaction-Free MeasurementA. C. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993)
IFM On A Quantum Object
BS1
BS2
IO
DC
A click at D collapses the bomb’s superposition
BS1-
BS2-
O-
C-
BS1+
BS2+
I+
e+ e-
I-O+
D+C+ D-
W
9/16C+ and C-
4/16Explosion
1/16D+ and D-
1/16D- and C+
1/16D+ and C-
ProbOutcome
Hardy’s ParadoxL. Hardy, Phys. Rev. Lett. 68, 2981 (1992)
The Switch
ϕPUMP
ω
ω
Coinc.Counts2ω
ϕPUMP - 2 x ϕLO
ϕLO
ω
ωϕLO
+
ϕPUMP 2 x ϕLO
=
2ϕLO- ϕPUMP = π
K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, 123603 (2001).
GaNDiode Laser
PBS
Det. H (D-)Det. V (D+)
DC BS DC BS
50-50BS1
50-50BS2
Switch H
V
CC
CC
PBS
BS1-
BS2-
O-
C-
BS1+
BS2+
I+
e+ e-
I-O+
D+C+ D-
WBS1-
BS2-
O-
C-
BS1+
BS2+
I+
e+ e-
I-O+
D+C+ D-
W
Experimental Setup
(W)
What do we need to measure?Ideally
If D+ clicks ⇒ - photon is in I-
IFM+
If D- clicks ⇒ + photon is in I+
IFM-
There is never a photon in I- and I+
Annihilation at W
Sometimes both D+ and D- click
The paradox
For a Real Apparatus
VisHor Int
VisVert Int
VisSwitch
VisSwitch + VisInt2 ≥ 7/4
Vs + Vmz2 ≥ 7/4
Required Interferometer Visibilities To Show Hardy’s Paradox
0
50
100
150
200
250
300
350
400
-1 -0.5 0 0.5 1 1.5 2 2.5 3
Switch Visibility
Delay (microns)
Vis = 85.7 +/- 0.1 %C
oin
cid
ence
Co
un
ts /2
0s
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6
Vertical Polarization Mach-Zehnder
Delay (microns)
Vis = 90 +/- 1%
Co
inci
den
ce C
ou
nts
/s
Width >> Change in Position
Weak
Width << Change in Position
Real
Dirac DeltaIdeal
Pointer Position Uncertainty
Measurement
Weak Measurements
∆X∆P ≥ h/2π⇒ small disturbance⇒ little system –pointer entanglement
No system disturbance⇒ simultaneous measure of different weak values.
Y. Aharonov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, e-print quant-ph/0104062 (2001).
Pointer(X) = exp[-(X - gAW)2 / ∆X]
<φ|A|ψ>AW= <φ|ψ>
<φ|A|ψ>AW= <φ|ψ>
Weak Measurements in Hardy’s ParadoxY. Aharanov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, e-print quant-ph/0104062 (2001).
Det. H (D-)Det. V (D+)
-11N(O+)
10N(I+)
N(O−)N(I-)# In Arm
1
1
00-11N(O+)
10N(I+)
N(O−)N(I-)# In Arm
1
1
00
λ/2
λ/2
N(I-)
N(O−)
N(I+)
N(O+)Pol.
BS1-
BS2-
O-
C-
BS1+
BS2+
I+
e+ e-
I-O+
D+C+ D-
WBS1-
BS2-
O-
C-
BS1+
BS2+
I+
e+ e-
I-O+
D+C+ D-
W
BS2+
BS2-
Conclusions
• An experimental implementation of Hardy’s Paradox is now possible.
• A single-photon level switch allows photons to interact with a high efficiency.
• The first version of the experiment will be a polarization based system.
• A later extension will look at the results of weak measurements in Hardy’s Paradox.