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

2. IOCB February 20, 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. IOCB February 20, 2006 3 Resonance and direct scattering as two mechanisms Direct –density of states changes evenly  smooth spectrum Resonance – metastable states – density of states includes peaks

4. IOCB February 20, 2006 4 Simultaneous occurrence of direct and resonance scattering mechanisms?

5. IOCB February 20, 2006 5 Question: Are direct and resonance scattering mechanisms separable at near resonance energy ? Mathematical answer: yes by complex scaling transformation. Physical answer: ?

6. IOCB February 20, 2006 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. IOCB February 20, 2006 7 Ougoing condition for resonances and CS Problem: Solution:

8. IOCB February 20, 2006 8 Outgoing condition for resonances and CS

9. IOCB February 20, 2006 9 Separation of direct and resonance scattering by CS transformation Im E Re E bound states resonance rotated continuum

10. IOCB February 20, 2006 10 States obtained by CS as scattering states for varying projectile flux

11. IOCB February 20, 2006 11 Connection between gamma and theta:

12. IOCB February 20, 2006 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. IOCB February 20, 2006 13 Case I: Free-particle Hamiltonian non-hermitian solutions of CS Hamiltonian: Im E Re E

14. IOCB February 20, 2006 14 Wavefunctions of rotated continuum exponentially modulated plane waves: grows in x decays in time

15. IOCB February 20, 2006 15 time-dependence:

16. IOCB February 20, 2006 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 –1 st order emission in an asymptotic distance x e with the rate :

17. IOCB February 20, 2006 17 –density of particles in a close neighborhood of the emitter: –analytical integration of the classical Liouville equation with the above boundary condition:

18. IOCB February 20, 2006 18 Classical density for free particles:

19. IOCB February 20, 2006 19 Step II: transformation of classical phase space density to a quantum wavefunction –non-approximate, in the case of free- Hamiltonian

20. IOCB February 20, 2006 20 

21. IOCB February 20, 2006 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. IOCB February 20, 2006 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. IOCB February 20, 2006 23 Construction of classical phase space density classical orbit [x(t),p(t)] is evaluated phase space density:

24. IOCB February 20, 2006 24 Construction of semiclassical wavefunction dividing to incoming and outgoing parts: transformation of density to wf:

25. IOCB February 20, 2006 25

26. IOCB February 20, 2006 26

27. IOCB February 20, 2006 27 The expected quantum counterpart Non-hermitian solution of CS Hamiltonian with the energy:

28. IOCB February 20, 2006 28 Solution of CS Hamiltonian in finite box: box: N=200 basis functions solution of CS Hamiltonian: back scaled solution:

29. IOCB February 20, 2006 29 Comparison of scattering wavefunction and rotated continuum state:

30. IOCB February 20, 2006 30 Case III: near resonance scattering Potential: Examined scattering energies: –resonance hit –very slightly off-resonance

31. IOCB February 20, 2006 31 in complex energy plane: Im E Re E -0.0034 -0.002 V(x) x 0.71260.716 -0.004

32. IOCB February 20, 2006 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. IOCB February 20, 2006 33 slow change of gamma Im E Re E -0.0034 -0.002 0.71260.716 -0.004

34. IOCB February 20, 2006 34 Resonance hit:

35. IOCB February 20, 2006 35 Off-resonance:

36. IOCB February 20, 2006 36 Off-resonance

37. IOCB February 20, 2006 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. IOCB February 20, 2006 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. IOCB February 20, 2006 39 Scattering matrix from simulations:

40. IOCB February 20, 2006 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. IOCB February 20, 2006 41 empirical rule in CS: rotated continuum for θ> θ c (γ>Γ res ) is not responsible for resonance cross-sections.

42. IOCB February 20, 2006 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