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Weak Measurements
Weak Measurements
Dan Elton
Stony Brook University
Graduate Physics AMO Seminar
11/10/10
Overview
Overview
I.Historical background of the m
easurement problem
II.
The Measurement Hamiltonian & “Two State Vector
Form
alism”
III.
Weak Measurement
IV.
Experimental Realization
The problem started before the cat, with a powder keg.
August 8, 1935:Einstein to Schrödinger: Imagine an unstable powder keg.
“Afte
r a year…
the p
si-fu
nctio
n th
en desc
ribes
a sort
of blen
d of not-y
et an
d alrea
dy-
expl
oded sy
stems
. Thr
ough
no art o
f int
erpret
ation ca
n th
is ps
i-fun
ction
be t
urne
d into an
adequa
te descr
iptio
n of a re
al st
ate o
f affairs;
in re
ality
there
is ju
st no
inter
mediary
betw
een ex
plod
ed and
not-ex
plod
ed.”
September 19, 1935: Schrödinger to Einstein:
“I hav
e con
struc
ted an exam
ple v
ery si
milar t
o you
r exp
loding
pow
der k
eg..”
September, 1935:Einstein to Schrödinger:
“You
r cat sh
ows t
hat w
e are
in co
mplet
e agreem
ent c
oncer
ning
our
asse
ssmen
t of t
he
charac
ter of t
he cu
rrent
theory
.... A
psi-
func
tion th
at co
ntains
the l
iving as w
ell as t
he
dead
cat c
anno
t be t
aken
as a
desc
riptio
n of th
e real s
tate
of affairs.
”
Indeed, both Einstein andSchrödinger were m
etaphysical realists
and could not accept this description as fundamental.
From Einstein: his life and universe, by Walter Isaacson
The Measurement Problem
The Measurement Problem --History
History
-1926 –
Compton-Simons experiment
-The m
easurement of a single observable can be m
ade
arbitrarily precise.
-Two consecutive m
easurements will yield the same
result.
-Conclusion: A measurement causes a collapse of the
wavefunction.
The Measurement Problem
The Measurement Problem --History
History
Can State Reduction be m
ade consistent with Unitary Evolution?
If not, and they are independent phenomena,
What are the necessary conditions for State Reduction to occur?
Uni
tary
Evo
lutio
n:
Det
erm
inis
ticC
ontin
uous
Tim
e-re
vers
ible
The
rmod
ynam
ical
ly r
ever
sibl
e
Sta
te R
educ
tion:
(th
e “Q
uant
um L
eap”
)N
on-d
eter
min
istic
Dis
cont
inuo
usN
ot ti
me-
reve
rsib
le*
Not
ther
mod
ynam
ical
ly r
ever
sibl
e, in
gen
eral
.
Unitary Evolution vs. State Reduction
Unitary Evolution vs. State Reduction
The
The ““Measurement problem
Measurement problem””
Von Neumann, in his famous work The Mathem
atical Fou
ndations of Quantum
Mechanics(1932), struggled to form
alize collapse m
athematically but was forced to
conclude that consciousness causes collapse.
Today there are m
any competing interpretations of quantum m
echanics.
However, Von Neumann came up with a description of the interaction between I and II.
I
II
III
Von
Neu
man
n's
Mea
sure
men
t Sch
eme
Von
Neu
man
n's
Mea
sure
men
t Sch
eme
I Microscopic system
IIMacroscopic m
easuring device
III
The observer
Ideal Measurement Hamiltonian
Ideal Measurement Hamiltonian
q(t) = Coupling function (compactly supported, norm
alized to 1)
P = Conjugate m
omentum operator of pointer variable Q
A= operator for what is being m
easured
PA
tq
HM
)(
=
PA
tq
HH
HII
I)
(+
+=
In time window of measurement, t1-t 2, dynamics from H
I& H
IIare ignored.
For simplicity, assume the system is in a pure state with eigenvalue a
III
ϕϕ
ϕ⊗
=
)(
)ex
p()
(1
2t
gPa
it
ϕϕ
η−
=
This is simply a translation operator –translation is proportional to
quantity m
easured. The system is not disturbed by the m
easurement –-ie.
It collapses perfectly without further disruption.
Ideal measurements are ideal!
)(
)(
exp(
)(
12
2 1
tdt
PA
tg
it
t t
ϕϕ
∫−
=η
Two State Vector Form
alism
Two State Vector Form
alism
Aharonov, Bergmann, Lebowitz(1964)
Measurements become time-sym
metric
Pre
Measurement
Post
Measurement
⟩Ψ
Φ⟨=
ii
TSV
||<Φ| is the backwards traveling state vector. It is not a bra vector!
|ψII>
is the forw
ards traveling state vector.
(weak
measurement
will occur
here)
-Yields the same results as conventional QM
-Can describe certain things better (Hardy’s paradox, three-box
paradox …)
-Is controversial because it often references counterfactuals. (if a
measurement had been perform
ed, it would have yielded ___ )
-Is the form
alism in which weak measurement is usually defined
and understood. According to SanduPopescu, weak measurement
can be explained with conventional quantum m
echanics, but “the
explanation is cumbersome and involves very intricate interference
effects in the m
easuring device.”
Two State Vector Form
alism
Two State Vector Form
alism
++
=
++
≅
++
≅
)2(1
)2(
)2(1
OA
iqp
Oiq
PA
e
Oiq
PA
eiq
PA
iqP
A
ψφ
ψφ
ψφ
ψφ
ψφ
ψφ
Whe
n q(
t) b
ecom
es v
ery
smal
l or
P b
ecom
es v
ery
smal
l, w
e m
ove
into
the
wea
k m
easu
rem
ent r
egim
e.
(not
e, Q
bec
omes
larg
e)
Sch
emat
ical
ly:P
At
qH
M)
(=
TSV
Weak Value
Weak Measurement
Weak Measurement
Weak Measurement Properties
Weak Measurement Properties
⟩Ψ
Φ⟨⟩
ΨΦ⟨
≡|
||A
Aw
-The weak measurement of a purely pre-selected system becomes regular
expectation value.
-The weak value is in general complex. real part = position of the pointer
and imaginary = m
omentum ofpointer.
-In a weak measurement, the change in the position of the m
easuring
device can be less than it's own quantum uncertainty.
-Awbecomes very large when Φ
and ψare nearly orthogonal. This is called
“weak value amplification”and attracted a lot of attention. It is the subject
of current ongoing research.
-According to Hulet, et. Al (1997), all real measurements m
ust “lie on a
spectrum between weak and ideal.”Thus, understanding weak
measurements is important to understanding m
easurements in general.
⟩Ψ
Ψ⟨|
|A
Weak Measurement: First example
Weak Measurement: First example
SGx Apparatus
Strong B-field
SGz Apparatus:
Weak B-field
Pre-selection:
Spins in ξdirection
Post-selection:
Spins in X+ direction
Weak
Value
From Aharonov, Albert & Vaidman: How the Result of a Measurement of
a Component of the Spin of a Spin-1/2 Particle Can Turn Out to be 100
(1987)
Weak Measurement : a simple thought
Weak Measurement : a simple thought
experiment
experiment
)si
n()
cos(
θθ
θz
xS
SS
+=
)si
n()2/
1()
cos(
)2/1(
θθ
θ+
=SPost select a particle from the blue beam. What if we “went
back in time”and m
easured in between along an angle theta?
SGz
SGx
One description of weak measurement is it is due to the intricate
interference effects leading to a large m
easurement error. The other,
argued by Aharonov, Albert and Vaidman,is that it is fully explained by
TSVF.
2/2
4=
πS
Outcomes with different measurement strength
Outcomes with different measurement strength
N1∝
σ
From Ahronov& Vaidman, TSVF: An Updated Review
Weak measurement
∆=
1P
Strong measurement
Weak Measurement
Weak Measurement --History
History
Theorized by Aharonov, Albert, & Vaidmanin 1987.
Theory revised by M. Duck, P. M. Stevenson, and E. C. G. Sudarshan
(1989)
(and other articles)
Hulet, Ritchie, Story, (1991) –First experimental realization.
Hosten& Kwiat(2008) -used weak measurements to m
easure the Spin Hall
Effect for photons. Splitting of light beam ~ 1 Angstrom.
Dixon et.al(2009) measured angular deflection of a light beam with the precision
of a hairs breadth at the distance of the m
oon.
However, although impressive, many remained skeptical as to whether weak-
measurements can actually be m
ore precise than traditional measuring schemes.
Reducing SNR in beam-defection would benefit
-Spectroscopy (phototherm
al, etc)
-interferometry
-ultra-precise position m
easurements
-atomic force m
icroscopy
Optimizing the signal
Optimizing the signal-- toto-- noise ration of a beam
noise ration of a beam-- deflection
deflection
measurement with
measurement with interferometric
interferometricweak values.
weak values.
David J. Starling, P. Ben Dixon, Andrew N. Jordan, and John C. Howell (2009)
Piezoelectric
actuator causes a
small beam
deflection here.
Quadrant-cell detector to measure beam position.
Normally, destructive
interference occurs
here.
CCD is used to
analyze the mode
quality (an
element of
technical noise) in
laser beam
Where is the weak
measurement??
The weak measurement: qualitative
The weak measurement: qualitative
The thing being measured is the
deflection. It corresponds to a tiny
shift in the transverse momentum of
the beam.
There are clockwise (CW) and
counterclockwise (CCW) beams,
they receive opposite shifts.
The tiny shifts in momentum is
coupled to how much light emerges
on either side of the beam splitter.
The “postselection”is to only look at
light emerging on the dark side
(darkport) or bottom of the BS.
Incidentally, the same result obtained from the weak value theory also
can be derived from classical Fourier beam optics. The effect here really
isn’t quantum, but it’s quite amazing there is a correspondence. John C.
Howell, David J. Starling, PHYSICAL REVIEW A 81, 033813 (2010)
The m
ain results
The m
ain results
Optimizing the signal
Optimizing the signal-- toto-- noise ration of a beam
noise ration of a beam-- deflection m
easurement with
deflection m
easurement with interferometric
interferometricweak values.
weak values.
David J. Starling, P. Ben Dixon, Andrew N. Jordan, and John C. H
David J. Starling, P. Ben Dixon, Andrew N. Jordan, and John C. Howell (2009)
owell (2009)
Future prospects
Future prospects
Laboratory applications:
Laboratory applications:
----SNR improvements (in particular case of large beam diameters)
SNR improvements (in particular case of large beam diameters)
----Possible large increase in precision (controversial)
Possible large increase in precision (controversial)
----Quantum eavesdropping (?)
Quantum eavesdropping (?)
Implications to foundations of QM:
Implications to foundations of QM:
----If weak measurements become better understood / realized, they
If weak measurements become better understood / realized, they could
could
help answ
er a lot of previously unansw
erable questions.
help answ
er a lot of previously unansw
erable questions.
----Wiseman, et. al. argues that one can determ
ine which path the e
Wiseman, et. al. argues that one can determ
ine which path the electron
lectron
goes through in the double slit using weak measurement.
goes through in the double slit using weak measurement.
----Work by
Work by Vaidman
Vaidmanand others suggest that weak measurements could test
and others suggest that weak measurements could test
to see if
to see if Bohmian
Bohmianquantum m
echanics is correct.
quantum m
echanics is correct.
----Aharonov
Aharonovsays that when Feynman pronounced that we can never truly
says that when Feynman pronounced that we can never truly
comprehend quantum m
echanics, he was "too hasty". "I think peopl
comprehend quantum m
echanics, he was "too hasty". "I think peopl e will
e will
remove the m
ystery that Feynman said could never be removed,
remove the m
ystery that Feynman said could never be removed, ……
you should
you should
never say never."
never say never."
Selected references
Selected references
Y. Aharonov, P. G. Bergmann, and J. Lebowitz, Phys. Rev 134, B1410 (1964).
Aharonov, Albert & Vaidman: How
the Result of a Measurement of a Com
pon
ent of the
Spin of a Spin-1/2 Particle Can
Turn Out to be 100(1987)
Aharonov, Vaidman. The Two-State Vector Form
alism: an Updated Review.
M. Duck, P. M. Stevenson, and E. C. G. Sudarshan,. Phys. Rev. D. 40, 2112 (1989)
Popescu, Sandu. Weak measurements just got stron
ger
APS Physics 2, 32 (2009)
G. Hulet, N. W M
. Ritchie, and 1.G. Story. Measurement of a W
eak Value(1997)
R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Steinberg, J. L. Garretson, H. M. Wiseman
A dou
ble-slit `which-w
ay' experiment on
the complementarity--uncertainty debate
Vaidman, L. The Reality in BohmianQuantum Mechanics or Can You
Kill with an Empty
Wave Bullet?
Shalm, L.K.;
Kocsis, S.;Ravets, S.;Braverm
an, B.;Stevens, M. J.;Mirin, R. P.;
Steinberg,
A. M.;
Observation of Boh
miantrajectories of a single photon
using weak measurements.
IEEE Con
ference proceedings, M
ay 2010.
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