Some recent experiments on weak measurements and quantum state generation

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Some recent experimentson weak measurementsand
quantum state generation
Aephraim Steinberg Univ. Toronto
(presently _at_ Institut d'Optique, Orsay)
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OUTLINE

• The 3-box problem
• Overture an alternative introduction to
• retrodiction, the 3-box problem, and weak
  measurements
• Experimental results
• Nonlocality?
• Hardy's Paradox and retrodiction
• Retrodiction is claimed to lead to a paradox in
  QM
• "Weak probabilities" seem to "resolve" the
  "paradox"?
• Experiment now possible, thanks to 2-photon
  "switch"
• Which-path experiments (collab. w/ Howard
  Wiseman)
• Old debate (Scully vs. Walls, e.g.)
• When which-path measurements destroy
  interference, must
• momentum necessarily be disturbed?
• Weak values allow one to discuss this momentum
  shift, and
• reconcile some claims of both sides
• (Negative values essential, once more...)

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U of T quantum optics laser cooling
group PDFs Morgan Mitchell Marcelo Martinelli
(back ??Brazil) Optics Kevin
Resch(?Zeilinger) Jeff Lundeen Krister Shalm
Masoud Mohseni (?Lidar) Reza Mir?real
world(?) Rob Adamson Karen Saucke (back
?????????) Atom Traps Jalani Fox Stefan
Myrskog (?Thywissen) Ana Jofre (????NIST?)
Mirco Siercke Samansa Maneshi Salvatore Maone
(? real world) Chris Ellenor Some of our
theory collaborators Daniel Lidar, János
Bergou, Mark Hillery, John Sipe, Paul Brumer,
Howard Wiseman
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Recall principle of weak measurements...
By using a pointer with a big uncertainty
(relative to the strength of the measurement
interaction), one can obtain information, without
creating entanglement between system and
apparatus (effective "collapse").
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By the same token, no single event provides much
information...
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Predicting the past...
What are the odds that the particle was in a
given box (e.g., box B)?
It had to be in B, with 100 certainty.
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Consider some redefinitions...
In QM, there's no difference between a box and
any other state (e.g., a superposition of
boxes). What if A is really X Y and C is
really X - Y?
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A redefinition of the redefinition...
So the very same logic leads us to conclude
the particle was definitely in box X.
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What does this mean?

• Then we conclude that if you prepare in (X Y)
  B and postselect in (X - Y) B,
  you know the particle was in B.
• But this is the same as preparing (B Y) X and
  postselecting (B - Y)
  X, which means you also know the particle was
  in X.
• If P(B) 1 and P(X) 1, where was the particle
  really?

But back up is there any physical sense in which
this is true? What if you try to observe where
the particle is?
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A Gedankenexperiment...
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The 3-box problem weak msmts
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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An "application" N shutters
Aharonov et al., PRA 67, 42107 ('03)
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The implementation A 3-path interferometer
(Resch et al., Phys Lett
A 324, 125('04))
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...
cf. Ritchie et al., PRL 68, 1107 ('91).
The position of each photon is uncertain to
within the beam waist... a small shift does not
provide any photon with distinguishing info. But
after many photons arrive, the shift of the beam
may be measured.
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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).
There exists a natural optical explanation for
this classical effect this is left as an
exercise!
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Data for PA, PB, and PC...
Rails A and B
Rail C
WEAK
STRONG
STRONG
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Is the particle "really" in 2 places at once?

• 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
Resch Steinberg, PRL 92,130402 ('04)
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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Non-repeatable data which happen to look the way
we want them to...
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The joint probabilities
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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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" Quantum seeing in the dark "

• (AKA The Elitzur-Vaidman bomb experiment)
• A. Elitzur, and L. Vaidman, Found. Phys. 23, 987
  (1993)
• P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci.
  Am. (Nov., 1996)

Problem Consider a collection of bombs so
sensitive that a collision with any single
particle (photon, electron, etc.) is guarranteed
to trigger it. Suppose that certain of the bombs
are defective, but differ in their behaviour in
no way other than that they will not blow up when
triggered. Is there any way to identify the
working bombs (or some of them) without blowing
them up?
Bomb absent Only detector C fires
Bomb present "boom!" 1/2 C 1/4
D 1/4
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What do you mean, interaction-free?
Measurement, by definition, makes some quantity
certain. This may change the state, and (as we
know so well), disturb conjugate variables. How
can we measure where the bomb is without
disturbing its momentum (for example)? But
if we disturbed its momentum, where did the
momentum go? What exactly did the bomb interact
with, if not our particle? It destroyed the
relative phase between two parts of the
particle's wave function.
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Hardy's Paradox
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
D- e was in DD- ? But if they
were both in, they should have annihilated!
D e- was in
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What does this mean?
Common conclusion We've got to be careful
about how we interpret these
"interaction-free measurements." You're not
always free to reason classically about what
would have happened if you had measured
something other than what you actually did.
(Do we really have to buy this?)
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How to make the experiment possible The "Switch"
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But what can we say about where the particles
were or weren't, once D D fire?
Y. Aharanov, A. Botero, S. Popescu, B. Reznik,
J. Tollaksen, quant-ph/0104062
Probabilities e- in e- out
e in 1
e out 0
1 0
0
1
1
-1
Upcoming experiment demonstrate that
"weak measurements" (à la Aharonov Vaidman)
will bear out these predictions.
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Measurement Pointer Position Uncertainty
Ideal Dirac Delta
Real Width ltlt Change in Position
Weak Width gtgt Change in Position
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• An experimental implementation of Hardys
  Paradox is now possible.
• A single-photon level switch allows photons to
  interact with a high efficiency.
• A polarization based system is now running.
• Once some stability problems solved, we will
  look at the results of weak measurements in
  Hardys Paradox.

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PROBLEM SOLVED!(?)
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Which-path controversy(Scully, Englert, Walther
vs the world?)
Suppose we perform a which-path measurement using
a microscopic pointer, e.g., a single photon
deposited into a cavity. Is this really
irreversible, as Bohr would have
all measurements? Is it sufficient to destroy
interference? Can the information be erased,
restoring interference?
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Which-path measurements destroy interference
(modify p-distrib!)
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How is complementarity enforced?
The fringe pattern (momentum distribution) is
clearly changed yet every moment of the
momentum distribution remains the same.
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The debate since then...
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Why the ambiguity?
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Weak measurements to the rescue!
To find the probability of a given momentum
transfer, measure the weak probability of each
possible initial momentum, conditioned on the
final momentum observed at the screen...
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Convoluted implementation...
Glass plate in focal plane measures P(pi) weakly
(shifting photons along y)
Half-half-waveplate in image plane measures path
strongly
CCD in Fourier plane measures ltygt for each
position x this determines ltP(pi)gtwk for
each final momentum pf.
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Calibration of the weak measurement
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A few distributions P(pi pf)
EXPERIMENT
THEORY
(finite width due to finite width of measuring
plate)
Note not delta-functions i.e., momentum may
have changed. Of course, these "probabilities"
aren't always positive, etc etc...
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The distribution of the integrated
momentum-transfer
EXPERIMENT
THEORY
Note the distribution extends well beyond
h/d. On the other hand, all its moments are (at
least in theory, so far) 0.
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• Weak-measurement theory can predict the output
  of meas-urements without specific reference to
  the measurement technique.
• They are consistent with the surprising but
  seemingly airtight conclusions classical logic
  yields for the 3-box problem and for Hardy's
  Paradox.
• They also shed light on tunneling times, on the
  debate over which-path measurements, and so
  forth.
• Of course, they are merely a new way of
  describing predictions already implicit in QM
  anyway.
• And the price to pay is accepting very strange
  (negative, complex, too big, too small) weak
  values for observables (inc. probabilities).

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Highly number-entangled states("low-noon"
experiment) .
Morgan W. Mitchell et al., to appear
The single-photon superposition state 1,0gt
0,1gt, which may be regarded as an entangled
state of two fields, is the workhorse of
classical interferometry. The output of a
Hong-Ou-Mandel interferometer is 2,0gt 0,2gt.
States such as n,0gt 0,ngt ("high-noon"
states, for n large) have been proposed for
high-resolution interferometry related to
"spin-squeezed" states. A number of proposals
for producing these states have been made, but
so far none has been observed for ngt2.... until
now! (But cf. related work in Vienna)
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Practical schemes?
Important factorisation
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Trick 1
Okay, we don't even have single-photon
sources. But we can produce pairs of photons in
down-conversion, and very weak coherent states
from a laser, such that if we detect three
photons, we can be pretty sure we got only one
from the laser and only two from the
down-conversion...
0gt e 2gt O(e2)
?? 3gt O(?2) O(? 2) terms with lt3 photons
0gt ? 1gt O(?2)
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Trick 2
How to combine three non-orthogonal photons into
one spatial mode?
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Trick 3
But how do you get the two down-converted photons
to be at 120o to each other? More post-selected
(non-unitary) operations if a 45o photon gets
through a polarizer, it's no longer at 45o. If
it gets through a partial polarizer, it could be
anywhere...
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The basic optical scheme
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More detailed schematic of experiment
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It works!
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A 3f fringe in - coincidences
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Too good to publish?
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SUMMARY

• Three-box paradox implemented
• Some more work possible on nonlocal observables
• Hardy's paradox implemented
• Setting up to perform the joint weak
  measurements
• Wiseman's proposal re which-path measurements
  carried out
• Paper in preparation
• What to do next? (Suggestions welcome!)
• 3-photon entangled state produced.
• What next? (Probably new sources required.)
• Other things I didn't have time to tell you
  about
• Process tomography working in both photonic and
  atomic systems.
• Next steps adaptive error correction
  (bang-bang, DFS,...)
• Optimal (POVM) discrimination of non-orthogonal
  states
• Using decoherence-free-subspaces for optical
  implementations of q. algorithms
• BEC project .... plans to probe tunneling atoms
  in the forbidden region
• Coherent control of quantum chaos in optical
  lattices
• Tunneling-induced coherence " "
  "

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Some references
Tunneling times et cetera Hauge and Støvneng,
Rev. Mod. Phys. 61, 917 (1989) Büttiker and
Landauer, PRL 49, 1739 (1982) Büttiker, Phys.
Rev. B 27, 6178 (1983) Steinberg, Kwiat, Chiao,
PRL 71, 708 (1993) Steinberg, PRL 74, 2405 (1995)
Weak measurements Aharonov Vaidman, PRA 41,
11 (1991) Aharonov et al, PRL 60, 1351
(1988) Ritchie, Story, Hulet, PRL 66, 1107
(1991) Wiseman, PRA 65, 032111 Brunner et al.,
quant-ph/0306108 Resch and Steinberg,
quant-ph/0310113
Which-path debate Scully et al, Nature 351,
111(1991) Storey et al, Nature 367
(1994) etc Wiseman Harrison, N 377,584
(1995) Wiseman, PLA 311, 285 (2003)
Hardy's Paradox Hardy, PRL 68, 2981
(1992) Aharonov et al, PLA 301, 130 (2001).
The 3-box problem Aharonov et al, J Phys A 24,
2315 ('91) PRA 67, 42107 ('03) Resch, Lundeen,
Steinberg, quant-ph/0310091