90
For example, consider poor cat whose life and death depends on a quantum process such that it happens to be in a superposition state |ψcat i = α|ψdead i + β|ψalive i . On which level the process R takes place? Or: who kills the cat? I Quantum logic J 1936: introduced by Birkhoff & von Neumann Which-path & interference setups of the two-slit experiment indicate that (A ∨ B) ∧ S 6= (A ∧ S) ∨ (B ∧ S) (see Introduction) I Quantum Zeno paradox
J 1977: discovered by Sudarshan & Misra
Repeated measurements slow down or even stop (in the limiting case of infinite frequency ) the decay process. Survival probability of a decaying system for evolution without measurement: 2 P0 (t) = |hψ(0)|ψ(t)i|2 ≈ 1 − τt (see Sec. 1.5) ˆ ≡ |ψ(0)ihψ(0)| with interval ∆t = t → 0: Periodic measurement of A n h in n→∞ t n t 2 t n t n 0 P0 (t) = P0 n ≈ 1 − nτ = 1 − nτ 1+ −−−−−−→ 1 | {z } | {znτ } Note: for exponential decay no
→e−t/τ effect: P00 (t)
I “Bomb-testing” paradox
→e+t/τ −λ nt n
= [e
] = e−λt = P0 (t)
J 1993: discovered by Elitzur & Vaidman
General name “interaction-free measurement”: Measurement at one of the paths in a double-slit-type experiment destroys the interference behavior. Detection of the particle in a forbidden direction indicates that the measurement was done—it verifies functionality of the measuring device without necessarily locating the particle on the path where the device is placed. Example: photon in Mach-Zehnder interferometer with arms I, II. Symbolic expression of the photon state evolution before the 2nd beam splitter (b.s.): 1st b.s.
|1i −−−→
mirrors 1 √1 (|Ii+i|IIi) − −−−→ √2 (i|Ii−|IIi)≡|ψi 2 2nd b.s.
Since
|Ii−−−→ √12 (|2i+i|1i) 2nd b.s.
|IIi−−−→ √12 (|1i+i|2i)
) with
|1i |2i
o two exit , ≡ directions
2nd b.s.
interference is observed: |ψi −−−→ −|1i ≡ |1i. A bomb with single-photon sensitive trigger, placed n e.g. in arm II, acts as a R 50 % 1 which-path measurement device: √2 (|Ii+i|IIi) −→ |Ii |IIi 50 % . In both cases, the Ukázka elektronické knihy, UID: KOS207024