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.