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Tokyo Tech. Akio HOSOYA

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March 10, @KEK

A Pedagogical Introduction to Weak Value and Weak MeasurementA rephrase of three box model Resch, Lundeen, andSteinberg, Phys. Lett. A324, 125 (2000)

Double Slit Experiment

1. Standard description

|Ψ〉 = λ|L〉 + ρ|R〉, λ, ρ ∈ R

positions of slits

rL= (d/2, 0)

rR= (−d/2, 0)

The probability amplitude to find a particle at r:

Ψ(r):=<r|Ψ>= λ〈r|L>+ ρ〈r|R〉

= λexp[ik|r−rL|] + ρexp[ik|r−rR|]

≈ λexp[ikr−iξ] + ρexp[ikr+iξ]

.where ξ =kxd/2r

forr>>d.

P(r) = |Ψ(r)|2

= 1 + 2λρ cos2ξ

By the Born rule, the probability to find

a prticle at r is

ξ =kxd/2r

２. Slit with width

Suppose the slits have length ℓ.

Let the (x,y) coordinates of the slits be

rL = (d/2, η, 0)

rR = (−d/2, η, 0)

( -ℓ/2<η<ℓ/2)

≈ λexp[ikr−ikyη/r-iξ] + ρexp[ikr-ikyη/r+iξ]

fromrLandrRis superposed for

−ℓ/2 ≤ η ≤ ℓ/2 to give

Ψ(r)=exp(ikr)[λ exp[-iξ]+ρexp[+iξ]]ϕ(y)、

where ϕ(y)=sin(yℓk/2r)/(yℓk/2r )

The probability distribution is

P(r) = |Ψ(r)|2

is product of the previous

x-distribution times

the y-distribution of diffraction

|ϕ(y)|2=|sin(yℓk/2r)/(yℓk/2r ) |2

y

3. Weak Measurement

Suppose the left slit is slightly tilted by a small

angle so that the optical axis is shifted by α ,

while the right slit remains as before.

LEFT

RIGHT

α

Then the probability amplitude to find a particle

at (x,y) becomes

Ψ(r) ≈ eikr[λe−iξφ(y − α) + ρeiξφ(y)]

The probability is

P(x, y) = |Ψ(r)|2

≈ λ2φ2(y − α) + ρ2φ2(y)

+2λρφ(y − α)φ(y) cos 2ξ

The tilt of the left slide slightly changes

the interference pattern in the x-y plane schematically as

y

α

x

Bby ZEBRA

For weak interaction i.e., small αthe intertference

pattern is only slightly modified.

Since the initial superposition shows up soley

through the interference pattern in

the x-direction, we can say that our weak

measurement changes the initial state only

slightly.

The average of the y-coordinate for a fixed

x is gives by

< y >=∫dyyP(x,y)/∫dyP(x,y)

≈ α (λ2+λρ cos 2ξ)/(λ2+ρ2+2λρ cos 2ξ)

ξ =kxd/2r

4. Weak Value

We have chosen |Ψ〉 = λ|L〉 + ρ|R〉

as the pre-selected state. The eigen state

of the position x <x| is post-selected.

The weak value of an observable A is defined

in general by

<A>w:= <x|A|Ψ〉/<x|Ψ〉

In particular, the weak value of the projection

operator to the left slit

PL := |L><L|

is

<PL >w= <x|L><L|Ψ>/<x|Ψ〉

=λe−iξ/(λe−iξ+ ρeiξ)

We can see that the shift of the interference pattern

< y >

≈ α (λ2+λρ cos 2ξ)/(λ2+ρ2++2λρ cos 2ξ)

=αRe[λe−iξ/(λe−iξ+ ρeiξ]

=αRe[<PL >w]

We can extract the real part of weak value

Re[<PL >w] by the average of the shift

< y > in the y-direction in the weak measurement.

This gives an information of

the initial state |Ψ〉only slightly changing the interference pattern in the x-direction,

i.e., characteristic feature of the initial state

|Ψ〉.

5. Aharonov’s original version

Prepare the initial state |Ψ〉and post-select the state <x| for the observed system.

To get the weak value:

<A>w:= <x|A|Ψ〉/<x|Ψ>

of an observable A in the system,

introduce the probe observable yand its eigen function ϕ(y) as a new degree of freedom.

For the system and probe, introduce a

Hamiltonian

H= gδ(t) A⊗Py

Pyshifts the y-coordinate.

In the previous DS model, g=α, A=|L><L|since only the left slit is tilted.

Aharonov, Albert and Vaidman, PRL 60,1351

Aharonov and Rohlich “Quantum Paradox”.

Ψ(r) =<r|exp[-i∫Hdt] |Ψ〉⊗|φ>

=<r|exp[-igA⊗Py] |Ψ〉⊗|φ>

=<r|exp[-i|L><L|⊗Py] |Ψ〉⊗|φ>

=eikr[λe−iξφ(y − α) + ρeiξφ(y)]

6. Interpretation(controversial)

How can we interpret the weak value?

Consider the DW model in which the

weak value

<PL >w= <x|L><L|Ψ>/<x|Ψ〉

can be extracted from the interference

-diffraction pattern.

The positively (negatively )larger the <PL >wis, the more upwards (downwards) shifted. This suggests the more likely the particle come from the left (right) slit.

<PL >wis a measure of tendency coming from the left slit L which we retrospectively infer when the particle is found at x for a given initial state |Ψ〉.

The probabilistic interpretation

that <PL >wis the conditional probability

for the weak value is debatable.

However, it is consistent with the Kolomgorov

measure theoretical approach dropping the

positivity from the axioms but keeping the

Probabilいty conservation

<PL >w+<PR>w=1

Note that this interpretation is equivalent

to that

<Ψ|(|x><x|L><L|)|Ψ>

is the joint probability for a particle to pass

through the left slit and then arrive at x by

the Beysian rule.

However, consistency does not imply

that it is compulsory.

7. Summary

In the double slit model ,we show the essential

feature of the weak measurement and how the

information of the initial state as the weak

value is extracted only slightly disturbing the

interference pattern.

Point: introduction of new degree of freedom

(y-coordinate) and its interaction with the

system (tilted glass)

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