A Pedagogical Introduction to Weak Value

and Weak Measurement

A rephrase of three box model Resch, Lundeen, and Steinberg,

Phys. Lett. A 324, 125 (2000)

Tokyo Tech. Akio HOSOYA

March 10, @KEK

r

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⟨ ⟨ ⟩ = λ exp[ik|r−rL|] + ρ exp[ik|r−rR|]≈ λ exp[ikr −iξ] + ρ exp[ikr+iξ]

.where ξ =kxd/2r

for r>>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

x

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)

Then the wave ≈ λ exp[ikr −ikyη/r-iξ] + ρ exp[ikr-ikyη/r+iξ]

from rL and rR is superposed for

−ℓ/2 ≤ η ≤ ℓ/2 to give

Ψ(r)= exp(ikr)[λ exp[-iξ]+ρ exp[+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 smallangle so that the optical axis is shifted by α ,while the right slit remains as before.

LEFT RIGHT

α

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

Then the probability amplitude to find a particle

at (x,y) becomes

The tilt of the left slide slightly changes the interference pattern in the x-y plane schematically as

x

y

α

Bby ZEBRA

For weak interaction i.e., small α the intertferencepattern is only slightly modified.

Since the initial superposition shows up soleythrough the interference pattern inthe x-direction, we can say that our weak measurement changes the initial state only slightly.

The average of the y-coordinate for a fixedx 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 stateof the position x <x| is post-selected.

The weak value of an observable A is definedin general by

<A>w := <x|A|Ψ /⟩ <x|Ψ⟩

In particular, the weak value of the projectionoperator to the left slit

PL := |L><L|is

<PL >w = <x|L><L|Ψ>/<x|Ψ⟩ =λe−iξ/(λe−iξ + ρeiξ)

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

=α Re[λe−iξ/(λe−iξ + ρeiξ]

=α Re[ <PL >w]

We can see that the shift of the interference pattern

We can extract the real part of weak valueRe[ <PL >w] by the average of the shift < y > in the y-direction in the weak measurement. This gives an information ofthe 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 versionPrepare 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 y and its eigen function ϕ(y) as a new degree of freedom.

For the system and probe, introduce aHamiltonian

H= g δ(t) A⊗Py

Py shifts 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[-i g A⊗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 theweak value <PL >w = <x|L><L|Ψ>/<x|Ψ⟩

can be extracted from the interference-diffraction pattern.

A general tendency is;

The positively (negatively )larger the <PL >w

is, the more upwards (downwards) shifted. This suggests the more likely the particle come from the left (right) slit.

<PL >w is 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 interpretationthat <PL >w is 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 equivalentto that

<Ψ|(|x><x|L><L|)|Ψ>

is the joint probability for a particle to passthrough the left slit and then arrive at x bythe Beysian rule.

However, consistency does not implythat it is compulsory.

7. SummaryIn the double slit model ,we show the essentialfeature of the weak measurement and how the information of the initial state as the weak value is extracted only slightly disturbing theinterference pattern.

Point: introduction of new degree of freedom (y-coordinate) and its interaction with the system (tilted glass)