Optical implementation of the Quantum Box Problem

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Optical implementation of the Quantum Box Problem

• Kevin Resch
• Jeff Lundeen
• Aephraim Steinberg

Department of Physics, University of Toronto
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Outline

• Motivation subensembles, postselection,
  and "Let's Make a Deal"
• Weak measurements
• The Quantum 3-Box Problem
• An optical implementation
• Can a particle be in 2 places at 1 time?
• Summary

Funding (sources not scared to admit association
with this research)
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Motivation
In QM, one can make predictions
(probabilities) about future observables. Can
one "retrodict" anything about past
observables? What can one say about
"subensembles" defined both by preparation
post-selection?
N.B. Questions about postselected
subensembles becoming more and more widespread
in quantum optics, quantum info, etc.
Cf. Wiseman, PRA 65, 032111 ('02) and
quant-ph/0303139
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Pick a box, any box...
What are the odds that the particle was in a
given box?
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Weak Measurements
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Conditional measurements(Aharonov, Albert, and
Vaidman)
AAV, PRL 60, 1351 ('88)
Prepare a particle in igt try to "measure" some
observable A postselect the particle to be in fgt
Does ltAgt depend more on i or f, or equally on
both? Clever answer both, as Schrödinger
time-reversible. Conventional answer i, because
of collapse.
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A (von Neumann) Quantum Measurement of A
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A Weak Measurement of A
Poor resolution on each shot.
Negligible back-action (system
pointer separable) Mean pointer shift is given
by ltAgtwk.
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The Quantum 3-Box Problem
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The 3-box problem
Prepare a particle in a symmetric superposition
of three boxes ABC. Look to find it in this
other superposition AB-C. Ask between
preparation and detection, what was the
probability that it was in A? B? C?
Questions were these postselected particles
really all in A and all in B? can this negative
"weak probability" be observed?
Aharonov Vaidman, J. Phys. A 24, 2315 ('91)
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Aharonov's N shutters
PRA 67, 42107 ('03)
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An Optical Implementation
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The implementation A 3-path interferometer
Diode Laser
Spatial Filter 25um PH, a 5cm and a 1 lens
GP A
l/2
BS1, PBS
l/2
MS, fA
GP B
BS2, PBS
BS3, 50/50
CCD Camera
BS4, 50/50
GP C
MS, fC
l/2
Screen
PD
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The pointer...

• Use transverse position of each photon as pointer
• Weak measurements can be performed by tilting a
  glass optical flat, where effective

q
gt
Flat
Mode A
cf. Ritchie et al., PRL 68, 1107 ('91).
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A negative weak value
Perform weak msmt on rail C. Post-select
either A, B, C, or ABC. Compare "pointer
states" (vertical profiles).
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Data for PA, PB, and PC...
Rails A and B
Rail C
WEAK
STRONG
STRONG
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So can one really "detect" that a particle is in
box A and that it is in box B ????
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Measuring joint probabilities

• If PA and PB are both 1, what is PAB?
• For AAVs approach, one would need an interaction
  of the form

OR STUDY CORRELATIONS OF PA PB... - if PA
and PB always move together, then the
uncertainty in their difference never
changes. - if PA and PB both move, but never
together, then D(PA - PB) must increase.
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Practical Measurement of PAB
Use two pointers (the two transverse
directions) and couple to both A and B then use
their correlations to draw conclusions about PAB.
We have shown that the real part of PABW can be
extracted from such correlation measurements
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Extracting the joint probability...
anticorrelated particle model
exact calculation
no correlations (PAB 1)
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And a final note...
The result should have been obvious... AgtltA
BgtltB AgtltABgtltB is identically zero
because A and B are orthogonal. Even in a
weak-measurement sense, a particle can never be
found in two orthogonal states at the same time.
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Summary

• You have won the fabulous vacation!
• We have implemented the quantum box
  problem and confirmed the predictions,
  including the strange "negative probability."
• New method for joint weak measurements
• Each photon appears to be definitely in each of
  two places but never both (cf. Aharonov et al.,
  Phys. Lett. A 301, 130 (2002) on Hardy's Paradox)
• Much more to explore in the strange magical
  land of weak measurements!