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.