Open quantum systems

Transcription

1 Open quantum systems

2 Wikipedia: An open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment. The open quantum system can be viewed as a distinguished part of a larger closed quantum system, the other part being the environment. Example: EM radiation INTERDISCIPLINARY Small quantum systems, whose properties are profoundly affected by environment, i.e., continuum of scattering and decay channels, are intensely studied in various fields of physics: nuclear physics, atomic and molecular physics, nanoscience, quantum optics, etc.

3 Quasistationary States (alpha decay) For the description of a decay, we demand that far from the force center there be only the outgoing wave. The macroscopic equation of decay is dn dt = wn; N = N 0e wt N is a number of radioactive nuclei, i.e., number of particles inside of sphere r=r: N d 3 r

4 We should thus seek a solution of the form = (r)e ie 0t/ wt/ = (r)e iet/ E = E 0 i ; = w J.J. Thompson, 1884 G. Gamow, 198 The time dependent equation i t = m relation between decay width and decay probability + V (r) can be reduced by the above substitution to the stationary equation E + The boundary condition m V (r) (r) =0 (r) e ikr for r!1 takes care of the discrete complex values of E lim r E = ~ k m d ln (r) = i dr me

5 In principle, resonances and decaying particles are different entities. Usually, resonance refers to the energy distribution of the outgoing particles in a scattering process, and it is characterized by its energy and width. A decaying state is described in a time dependent setting by its energy and lifetime. Both concepts are related by: T 0 = This relation has been checked in numerous precision experiments. See more discussion in R. de la Madrid, Nucl. Phys. A81, 13 (008) U. Volz et al., Phys. Rev. Lett 76, 86(1996)

6 A comment on the time scale i! ψ t = ˆ H ψ Time Dependent Schrödinger Equation T 1/ = ln! Γ,! = MeV sec Can one calculate Γ with sufficient accuracy using TDSE? T s. p sec = 3babysec 38 U: T 1/ =10 16 years 56 Fm: T 1/ =3 hours For very narrow resonances, explicit time propagation impossible!

7 How do resonances appear? A square well example square-well potential spherical symmetry l=0 (s-wave) Radial Schrödinger equation χ''+ M (E V )χ = 0 ( χ = ϕr)! Region I: Region II: χ I = Asin pr, p = ME! χ II = c + e q(r a ) + c e q(r a ), q = M(V b E) Region III: χ III = c 1 e ip(r b ) + c e ip(r b )!

8 In almost all cases χ III is much larger than χ I. We are now interested in those situations where χ III is as small as possible. The condition c + = 0 tan( pa) = p q defines virtual levels in region I: particle is well localized; very small penetrability through the barrier

9 When c + =0, the penetrability becomes proportional to χ III χ exp # I! M(V & b E)(b a) $ % ' ( This is the semi-classical WKB result: P = χ III χ I $ r ' exp& k(r)dr) %& () r 1

10 open closed Width of a narrow resonance H(t) = H 0 + V (t) (V << H 0 ) time-dependent Hamiltonian i! ψ t = H ˆ ψ expansion of ψ in the basis of H o H ˆ 0 φ n = E n φ n ψ = c n (t) φ n e ie n t /! n i! dc k dt = c n (t) φ k V φ n e iω knt, ω kn = E k E n n ( ) /! As initial conditions, let us assume that at t=0 the system is in the state φ 0 c n (0) = 1 for n = 0 0 for n 0 If the perturbation is weak, in the first order, we obtain: i! dc k dt = φ k V φ 0 e iω k 0t

11 Furthermore, if the time variation of V is slow compared with exp(iω ko t), we may treat the matrix element of V as a constant. In this approximation: c k (t) = φ k V φ 0 E k E 0 ( 1 e iω k 0t ) The probability for finding the system in state k at time t if it started from state 0 at time t=0 is: c k (t) = φ V φ k 0 E k E 0 ( 1 cosω ( ) k 0 t) The total probability to decay to a group of states within some interval labeled by f equals: c k (t) =! φ V φ k 0 1 cosω k 0 t k f ω k 0 ( ) ρ ( E k )de k

12 The transition probability per unit time is W = d dt c k (t) = sinω φ! k V φ k 0 t 0 ρ ( E k )de k k f ω k 0 Since the function sin(x)/x oscillates very quickly except for x~0, only small region around E 0 can contribute to this integral. In this small energy region we may regard the matrix element and the state density to be constant. This finally gives: W 0 f = π! φ f V φ 0 ρ ( E f ) Fermi s golden rule Although named after Fermi, most of the work leading to the Golden Rule was done by Dirac, who formulated an almost identical equation 197. It is given its name because Fermi called it "Golden Rule No.." in sin x x dx = π