an experimental implementation of hardy’s paradox jeff lundeen, kevin resch aephraim steinberg...
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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
Bang½ Neither
Present¼ D
None¼ C
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
Outcome Prob
D+ and C- 1/16
D- and C+ 1/16
C+ and C- 9/16
D+ and D- 1/16
Explosion 4/16
Hardy’s ParadoxL. Hardy, Phys. Rev. Lett. 68, 2981 (1992)
The Switch
PUMP
Coinc.Counts2
PUMP - 2 x LO
LO
LO
+
PUMP 2 x LO
=
2LO- PUMP =
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)
H Pol DC
V Pol DC
407 nm Pump
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 %
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6
Vertical Polarization Mach-Zehnder
Delay (microns)
Vis = 90 +/- 1%
Measurement
Pointer Position Uncertainty
Ideal Dirac Delta
Real Width << Change in Position
Weak Width >> Change in Position
Weak Measurements
XP h/2
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]
<|A|>AW= <|>
<|A|>AW= <|>
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
0
0-11N(O+)
10N(I+)
N(O )N(I-)# In Arm
1
1
0
0
/2
/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.