Transcript Document

Some recent experiments
on weak measurements
and quantum state generation
Aephraim Steinberg
Univ. Toronto
(presently @ Institut
d'Optique, Orsay)
OUTLINE
• The 3-box problem
• Overture: an alternative introduction to
retrodiction, the 3-box problem, and weak measurements
• Experimental results
• Nonlocality?
Let's Make a Quantum Deal!
• 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...)
Let's Make a Quantum Deal!
• And now for something completely different
• Non-deterministic generation of |0,3> + |3,0>
"maximally path-entangled states"
• Phase super-resolution (Heisenberg limit?)
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
Ana Jofre (NIST?)
Samansa Maneshi
Chris Ellenor
Stefan Myrskog
(Thywissen)
Mirco Siercke
Salvatore Maone ( real world)
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Some of our theory collaborators:
Daniel Lidar, János Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman
Recall principle of weak
measurements...
Hint=gApx
System-pointer
coupling
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").
By the same token, no single event
provides much information...
Initial State of Pointer
Final Pointer Readout
But after many trials, the centre can be determined
to arbitrarily good precision...
x
x
x
x
Predicting the past...
A+B
B+C
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.
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?
A+B
= X+B+Y
X
Y
B+C=
X+B-Y
A redefinition of the redefinition...
So: the very same logic leads us to conclude the
particle was definitely in box X.
X + B'
= X+B+Y
X
Y
X + C' =
X+B-Y
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?
A Gedankenexperiment...
ee-
e-
e-
The 3-box problem: weak msmts
Prepare a particle in a symmetric superposition of
three boxes: A+B+C.
Look to find it in this other superposition:
A+B-C.
Ask: between preparation and detection, what was
the probability that it was in A? B? C?
Aw 
f Ai
f i
PA = < |A><A| >wk = (1/3) / (1/3) = 1
PB = < |B><B| >wk = (1/3) / (1/3) = 1
PC = < |C><C|>wk = (-1/3) / (1/3) = -1.
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)]
An "application": N shutters
Aharonov et al., PRA 67, 42107 ('03)
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
l/2
GP A
BS1, PBS
l/2
MS, fA
GP B
BS2, PBS
GP C
BS4,
50/50
MS, fC l/2
BS3, 50/50
PD
CCD
Camera
Screen
The pointer...
• Use transverse position of each photon as pointer
• Weak measurements can be performed by tilting a
glass optical flat, where effective
Hint  g A A px
Mode A
q
Flat
gt
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.
A negative weak value
Intensity (arbitrary units)
Perform weak msmt
on rail C.
Post-select either A,
B, C, or A+B–C.
Compare "pointer
states" (vertical
profiles).
1.4
1.2
A+B–C
(neg. shift!)
Rail C
(pos. shift)
1
0.8
0.6
Rails A and B (no shift)
0.4
220
200
180
[There exists a natural optical explanation for
this classical effect – this is left as an exercise!]
160
140
Pixel Number
120
100
Post-selected state displacement
(Units of RMS Width)
Data for PA, PB, and PC...
2
Rails A
and B
1
0
Rail C
-1
WEAK
STRONG
STRONG
-2
-3
-2
-1
0
1
2
Displacement of Individual Rail
(Units of RMS Width)
3
Is the particle "really" in
2 places at once?
• If PA and PB are both 1, what is PAB?
• For AAV’s approach, one would need an
interaction of the form
Hint  g A A B B px
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.
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.
Hint  g A A A p x  g B B B p y
We have shown that the real part of PABW
can be extracted from such correlation
measurements:
Re PABW  
2 xy
g Ag Bt
2
- Re(P
*
AW
BBW )
Non-repeatable data which happen
to look the way we want them to...
anticorrelated
particle model
exact calculation
no correlations
(PAB = 1)
And a final note...
The result should have been obvious...
|A><A| |B><B|
= |A><A|B><B|
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.
" 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:
D
C
Consider a collection of bombs so sensitive that
a collision with any single particle (photon, electron, etc.)
Bomb absent:
is guarranteed to trigger it.
Only detector C fires
BS2 that certain of
Suppose
the bombs are defective,
but differ in their behaviour in no way other than that
Bomb present:
they will not blow up when triggered.
"boom!"
1/2 bombs (or
Is there any way to identify
the working
C up? 1/4
some of them)
without blowing them
BS1
D
1/4
The bomb must be there... yet
my photon never interacted with it.
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.
Hardy's Paradox
C+
D+
D-
BS2+
C-
BS2I+
I-
O-
O+
W
BS1+
e+
BS1e-
Outcome Prob
D+
e- was
D+ and
C- in
1/16
D- e+ was in
D- and C+ 1/16
C+ and ?C- 9/16
D+DD+ and D- 1/16
But
… if they4/16
were
Explosion
both in, they should
have annihilated!
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?)
How to make the experiment
possible: The "Switch"
LO

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, 123603 (2001).

PUMP
2


LO
Coinc.
Counts
PUMP - 2 x LO
2 x LO
PUMP
+
2LO- PUMP = 
=
Experimental Setup
Det. V (D+) Det. H (D-)
50-50
BS2
CC
PBS
PBS
GaN
Diode Laser
DC BS
50-50
BS1
(W)
CC
V
H
Switch
DC BS
H Pol DC
V Pol DC
407 nm Pump
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
0
1
1
e+ out
1
-1
0
1
0
Upcoming experiment: demonstrate that "weak
measurements" (à la Aharonov + Vaidman) will
bear out these predictions.
• An experimental implementation of Hardy’s
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
Hardy’s Paradox.
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?
Which-path measurements destroy
interference (modify p-distrib!)
How is complementarity enforced?
The fringe pattern (momentum distribution) is clearly changed –
yet every moment of the momentum distribution remains the same.
The debate since then...
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...
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
<y> for each position x; this
determines <P(pi)>wk for each
final momentum pf.
Calibration of the weak
measurement
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...
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.
• 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 whichpath 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).
Highly number-entangled states
("low-noon" experiment) .
Morgan W. Mitchell et al., to appear
The single-photon superposition state |1,0> + |0,1>, 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 |2,0> + |0,2>.
States such as |n,0> + |0,n> ("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 n>2.... until now!
(But cf. related work in Vienna)
Practical schemes?
[See for example
H. Lee et al., Phys. Rev. A 65, 030101 (2002);
J. Fiurásek,
Phys. Rev. A 65, 053818 (2002)]
˘
Important factorisation:
+
=
A "noon" state
A really odd beast: one 0o photon,
one 120o photon, and one 240o photon...
but of course, you can't tell them apart,
let alone combine them into one mode!
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...
SPDC
|0> + e |2> + O(e2)
laser
|0> +  |1> + O(2)
e |3> + O(2) + O(e 2)
+ terms with <3 photons
Trick #2
How to combine three non-orthogonal photons into one spatial mode?
"mode-mashing"
Yes, it's that easy! If you see three photons
out one port, then they all went out that port.
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...
(or nothing)
(or nothing)
(or <2 photons)
The basic optical scheme
+ e i3f
Dark ports
PBS
DC
photons
HWP
to
analyzer
PP
Phas e
s hifte r
QWP
Ti:s a
It works!
7
1¥ 10
Singles:
8¥ 106
6¥ 106
4¥ 106
6
2¥ 10
Coincidences:
Triple
coincidences:
10
20
10
20
30
40
50
25000
22500
20000
17500
15000
12500
10000
30
40
50
Too good to publish?
80
60
40
20
-20
10
20
30
40
50
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
" "
"
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)
Which-path debate:
Aharonov et al, PRL 60, 1351 (1988)
Scully et al, Nature 351, 111(1991)
Ritchie, Story, & Hulet, PRL 66, 1107 (1991)
Storey et al, Nature 367 (1994) etc
Wiseman, PRA 65, 032111
Wiseman & Harrison, N 377,584 (1995) Brunner et al., quant-ph/0306108
Wiseman, PLA 311, 285 (2003)
Resch and Steinberg, quant-ph/0310113
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