Tokyo Tech. Akio HOSOYA

1. Tokyo Tech. Akio HOSOYA 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)

3. r

4. 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)

5. 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.

6. P(r) = |Ψ(r)|2 = 1 + 2λρ cos2ξ By the Born rule, the probability to find a prticle at r is ξ =kxd/2r

7. x

8. 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)

9. Then the wave ≈ 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 )

10. 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

11. 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

12. 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

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

14. Bby ZEBRA

15. 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.

16. 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

17. 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〉

18. 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)

19. 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]

20. 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〉.

21. 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.

22. 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”.

23. 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)]

24. 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.

25. A general tendency is; 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〉.

26. 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

27. 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.

28. 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)