March 10, @KEK
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
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
for r>>d.
ξ =kxd/2r
By the Born rule, the probability to find
a prticle at r is
P(r) = |Ψ(r)|2
= 1 + 2λρ cos2ξ
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 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 α the 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)
=α Re[λe−iξ/(λe−iξ + ρeiξ]
=α Re[ <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 y and its
eigen function ϕ(y) as a new degree of
freedom.
For the system and probe, introduce a
Hamiltonian
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 the
weak 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 interpretation
that <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 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)