Positive Hamiltonians can give purely exponential decay

Transcription

1 Positive Hamiltonians can give purely exponential decay Paolo Facchi Università di Bari & INFN Italy Joint work with Daniel Burgarth, Aberystwyth University Policeta - San Rufo 26 June 2018

2 Exponential decay law N(t) = N 0 e γt O s (t) t

3 Exponential decay l 1 1 r -dl 10" N(t) = N 0 e γt t 1/2 = hours CD 10" 6 E 56 Mn radioactive decay Norman, et al., Phys. Rev. Lett. 60, 2246 (1988) 1^ S. o a) N I ^10" Time ( 56 Mn half-lives)

4 Exponential decay ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 566 (2006) Testing the exponential decay law of gold 198 Au D. Novkovic, L. Nadderd, A. Kandic, I. Vukanac, M. Durasˇ evic, D. Jordanov Institute of Nuclear Sciences Vinča, Belgrade, PB 522, Serbia and Montenegro Received 31 January 2006; received in revised form 14 July 2006; accepted 20 July 2006 Available online 11 August 2006 Abstract A long time scale experimental testing of the exponential decay law of gold, 198 Au, is presented. The experiment has been processed by using a contemporary digital spectrum analyzer. Within the limits of the experimental errors no deviation from the exponential decay law of 198 Au has been observed. r 2006 Elsevier B.V. All rights reserved. PACS: Mr; e; Kv Keywords: Non-exponential decay; Zeno effect

5 Quantum theory Gamow Weisskopf - Wigner Friedrichs - Lee Fermi Fock - Krylov

6 Quantum systems Survival amplitude Survival probability a(t) = ψ e ith ψ p(t) = a(t) 2 a(t) = ψ e ith ψ = e iet d μ ψ (E) Fourier transform E spectrum μ ψ (Ω) = Prob ( E Ω ) μ ψ = μ pp + μ ac + μ sc spectral measure Lebesgue decomposition theorem

7 Pure point spectrum Survival amplitude a(t) = ψ e ith ψ a(t) = ψ e ith ψ = e iet d μ ψ (E) Fourier transform μ ψ = μ pp = c j j 2 δ(e E j ) c pure point j = E j ψ E a(t) = ψ e ith ψ = j c j 2 e ie j t almost periodic motion p(t) j c j 4 = const > 0, as t

8 Continuous spectrum Survival amplitude a(t) = ψ e ith ψ a(t) = ψ e ith ψ = e iet d μ ψ (E) Fourier transform μ ψ = μ ac + μ sc continuous spectrum E p(t) 0, as t RAGE theorem

9 Quantum unstable systems Survival amplitude Survival probability a(t) = ψ e ith ψ p(t) = a(t) 2 ψ e ith ψ = e iet d μ ψ (E) μ ψ = μ ac p ψ (E) = d μ ψ (E) d E absolutely continuous p ψ (E) = E ψ 2 0 lim t a(t) = lim t e iet p ψ (E)d E = 0 Riemann-Lebesgue lemma unstable system

10 Short times Survival amplitude a(t) = ψ e ith ψ p(t) = a (t) 2 = ψ exp( ith)ψ 2 ψ D(H) Hψ < finite variance p(t) = 1 t 2 ( Hψ Hψ ψ Hψ 2 ) + o (t 2 ) as t 0

11 Short times

12 Quantum Zeno effect 1 pn (t) N 0 t t t 9 3

13 Large times Paley-Wiener theorem p ψ (E) = E ψ 2 = 0 for E < 0 Ground energy a(t) = 0 e iet p ψ (E) de 0 E R ln a(t) 1 + t 2 dt < a(t) C t α

14 Quantum decay survival probability 1 Z quadratic (Zeno) 1 exponential t 2 2 Z Ze t power t time t

15 Exponential decay? a(t) = R e iet p ψ (E) de p ψ (E) = E ψ 2 a(t) = exp ( γ 2 t i ω 0 t ) Fourier transform p ψ (E) = 1 2π R e iet a(t) dt = γ 2π 1 (E ω 0 ) 2 + γ2 4 E ψ = ϕ C (E) = p ψ (E) e iα(e) ϕ C (E) = γ 2π 1 E ω 0 i γ 2

16 Exponential decay ξ C (x) 2 Cauchy ϕ C (E) = γ 2π 1 E ω 0 i γ x 0 H = q position operator spec H = R E ϕ C e itq ϕ C = exp ( γ 2 t i ω 0 t ) exponential

17 Subsystem evolution H = C 2 L 2 (R) spin-1/2 particle ϕ 0 ψ = 0 ϕ ϕ 1 ( ϕ ) 1 Hamiltonian q + H = 0 0 q q ( q ) x x + q + ϕ(x) = x + ϕ(x) q ϕ(x) = x ϕ(x) ramp functions

18 Subsystem evolution H = ( q + q ) ψ = ( ϕ 0 ϕ 1 ) ψ Hψ = ϕ 0 q + ϕ 0 + ϕ 1 q ϕ 1 = + 0 x( ϕ 0 (x) 2 + ϕ 1 ( x) 2 )dx 0, positive Hamiltonian x x + q = q + q

19 Subsystem evolution H = 0 0 q q U(t) = 0 0 e itq e itq unitary group ρ ϕ C ϕ C factorized initial state ρ(t) = tr 2 (U(t)(ρ ϕ C ϕ C )U(t) ) reduced density matrix ρ(t) = ( ρ 00 f(t) ρ 01 f(t) ρ 10 ρ 11 ) subsystem evolution ξ C(x) 2 exponential overlap function f(t) = ϕ C e itq ϕ C = e γ 2 t i ω 0 t x

20 Subsystem evolution ρ(t) = ( ρ 00 e γ 2 t i ω 0t ρ 01 e γ 2 t+i ω 0t ρ 10 ρ 11 ) dephasing channel GKLS master equation Markovian! ρ(t) = e tl ρ Lρ(t) = i ω 0 [σ z, ρ(t)] γ 2 [σ z, [σ z, ρ(t)]] σ z =

21 Exponential decay! ρ(t) = e tl ρ Lρ(t) = i ω 0 [σ z, ρ(t)] γ 2 [σ z, [σ z, ρ(t)]] σ z = ω 0 = 0 σ x = polarization σ x (t) = tr(σ x ρ(t)) = σ x (0) e γ 2 t exponential

22 Paley-Wiener? ρ tot (t) = e ith ρ tot e +ith = e itg (ρ tot ) Gρ tot = [H, ρ tot ] Liouvillian spec G = spec H spec H 0 0 A = A generic observable Hamiltonian spectrum Liouvillian spectrum A(t) = tr(ae ith ρ tot e +ith ) = tr(a e itg (ρ tot ))

23 Paley-Wiener? A(t) = tr(ae ith ρ tot e +ith ) = tr(a e itg (ρ tot )) Gρ tot = [H, ρ tot ] Survival probability A = ρ tot = ψ ψ A(t) = tr( ψ ψ e ith ψ ψ e +ith ) = ψ e ith ψ 2 = p(t) Very special! It depends on absolute values of energy

24 Non-analitic potential? ramp function x x + H = 0 0 V(q) V( q) W(x) = V(x) V( x) increasing function V( x) V(x)

25 Non-analitic potential? H = 0 0 V(q) V( q) W(x) = V(x) V( x) increasing function ρ ϕ ϕ ρ(t) = ( ρ 00 f(t) ρ 01 f(t) ρ 10 ρ 11 ) f(t) = ϕ e itw(q) ϕ subsystem evolution V( x) V(x)

26 H = 0 0 V(q) V( q) W(x) = V(x) V( x) ρ(t) = ( ρ 00 f(t) ρ 01 f(t) ρ 10 ρ 11 ) ρ ϕ ϕ f(t) = ϕ e itw(q) ϕ ϕ(x) = W (W 1 (x)) 1/2 ϕ C (W 1 (x)) initial state Example V(x) = exp(x) W(x) = 2 sinh x ϕ(x) = γ 4π cosh y 1 y ω 0 i γ 2 y = arcsinh (x/2)

27 Thank you!