Resonances and background scattering in gedanken experiment with varying projectile flux

1. Resonances and background scattering in gedanken experiment with varying projectile flux Petra Zdanska, IOCB June 2004 – Feb 2006

2. Personal acknowledgement • Milan Sindelka and Nimrod Moiseyev • Vlada Sychrovsky and people attending my unfinished Summer course of resonances 2004 • Nimrod’s group and conferences

3. Direct density of states changes evenly  smooth spectrum Resonance metastable states density of states includes peaks Resonance and direct scattering as two mechanisms

4. Simultaneous occurrence of direct and resonance scattering mechanisms? background modulated by direct scattering resonances

5. Question: • Are direct and resonance scattering mechanisms separable at near resonance energy ? • Mathematical answer: yes by complex scaling transformation. • Physical answer: ?

6. Complex scaling method (CS) • useful non-hermitian states – “resonance poles” • purely outgoing condition is a cause to exponential divergence and complex energy eigenvalue • complex scaling transformation of Hamiltonian • non-unitary similarity transformation for taming diverging states

7. Ougoing condition for resonances and CS • Problem: • Solution:

8. Outgoing condition for resonances and CS

9. Separation of direct and resonance scattering by CS transformation Re E bound states resonance rotated continuum Im E

10. States obtained by CS as scattering states for varying projectile flux

11. Connection between gamma and theta:

12. Proofs by semiclassical and quantum simulations • Why semiclassical and not just quantum mechanics • only way to prove a correspondence between the classical notion of flux of particles and quantum wavefunctions • Cases I and II: • I. analytical proof for free-particle scattering • II. numerical evidence for direct scattering problem • Case III: • a quantum simulation of resonance scattering for varying projectile flux displaying the new effects

13. Case I: Free-particle Hamiltonian • non-hermitian solutions of CS Hamiltonian: Re E Im E

14. decays in time grows in x x t Wavefunctions of rotated continuum • exponentially modulated plane waves:

15. time-dependence:

16. Semiclassical solution to the expected physical process behind these non-hermitian states: • step I: construction of a corresponding density probability in classical phase space • 1st order emission in an asymptotic distance xe with the rate :

17. density of particles in a close neighborhood of the emitter: • analytical integration of the classical Liouville equation with the above boundary condition:

18. Classical density for free particles:

19. Step II: transformation of classical phase space density to a quantum wavefunction • non-approximate, in the case of free-Hamiltonian

20. 

21. Exact comparison with non-hermitian wavefunction as a proof • the non-hermitian and scattering wavefunctions have the same form and are equivalent supposed that, • which was to be proven.

22. Case II: Rotated complex continuum of Morse oscillator • potential: • semiclassical simulation of scattering experiment with parameters: • particles arrive with classical energy: • decay rate of the emitter:

23. Construction of classical phase space density • classical orbit [x(t),p(t)] is evaluated • phase space density:

24. Construction of semiclassical wavefunction • dividing to incoming and outgoing parts: • transformation of density to wf:

27. The expected quantum counterpart • Non-hermitian solution of CS Hamiltonian with the energy:

28. Solution of CS Hamiltonian in finite box: • box: • N=200 basis functions • solution of CS Hamiltonian: • back scaled solution:

29. Comparison of scattering wavefunction and rotated continuum state:

30. Case III: near resonance scattering • Potential: • Examined scattering energies: • resonance hit • very slightly off-resonance

31. in complex energy plane: V(x) 0.7126 0.716 Re E -0.002 -0.0034 -0.004 Im E x

32. Quantum dynamical simulations of scattering experiments • “particles” added as Gaussian wavepackets in an asymptotic distance, 40 a.u. • beginning of simulation: scattering experiment does not start abruptly but the intensity I(t) is modulated as follows:

33. slow change of gamma 0.7126 0.716 Re E -0.002 -0.0034 -0.004 Im E

34. Resonance hit:

35. Off-resonance:

36. Off-resonance

37. What is going on: • We reach stationary-like scattering states, which are characterized by a constant scattering matrix and by a constant (and complex) expectation energy value. • Are these states the non-hermitian solutions to Hamiltonian obtained by CS method?

38. Calculations of scattering matrix: • comparison of dynamical simulations with stationary solutions of complex scaled Hamiltonian • gamma<Gamma_res : • rotated continuum • gamma>Gamma_res : • resonance hit  resonance pole • slightly off-resonance  rotated continuum

39. Scattering matrix from simulations:

40. Inverted control over dynamics for gamma>Gamma_res • incoming flux decays faster than the wavefunction trapped in resonance • natural control: incoming flux disappears faster than outgoing flux – this occurs for discrete resonance energies • inverted control: outgoing flux decays according to gamma and not Gamma_res. Reason: destructive quantum interference removes the trapped particle.

41. empirical rule in CS: rotated continuum for θ> θc (γ>Γres) is not responsible for resonance cross-sections.

42. Conclusions: • resonance phenomenon studied in a new context of scattering dynamics • new light shed into complex scaling method, interference effect behind the long accepted empirical rule • first physical realization of complex scaling eventually interesting for experiment