1 / 45

Chaos and Irreversibility: An introduction to the Loschmidt echo

Chaos and Irreversibility: An introduction to the Loschmidt echo. Diego A. Wisniacki. UBA. Overview. Introduction Loschmidt echo Loschmidt echo and chaos Regimes of Loschmidt echo Decoherence and Loschmidt echo Experiments Final Remarks. Colaboradores-Referencias. Colaborators.

loman
Télécharger la présentation

Chaos and Irreversibility: An introduction to the Loschmidt echo

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chaos and Irreversibility: An introduction to the Loschmidt echo Diego A. Wisniacki UBA

  2. Overview • Introduction • Loschmidt echo • Loschmidt echo and chaos • Regimes of Loschmidt echo • Decoherence and Loschmidt echo • Experiments • Final Remarks

  3. Colaboradores-Referencias Colaborators • Horacio Pastawski (UNC) • Fernando Cuccietti (Los Alamos) • Eduardo Vergini (TANDAR, Buenos Aires) • Doron Cohen (BGU) • Florentino Borondo (UAM, Madrid) • Rosa Benito (UPM, Madrid)

  4. Colaboradores-Referencias Introduction What is chaos in classical mechanics?

  5. Colaboradores-Referencias Introduction

  6. Colaboradores-Referencias Introduction and

  7. Colaboradores-Referencias Introduction

  8. Colaboradores-Referencias Introduction

  9. Colaboradores-Referencias Introduction Sensitivity to initial conditions How it can be measure?

  10. Colaboradores-Referencias Introduction How it can be measure? Liapunov Exponents

  11. Colaboradores-Referencias Introduction Lets make the same program in quantum mechanics: and So

  12. Colaboradores-Referencias Loschmidt Echo In 1984 A. Peres proposed: Perturbed evolution Josef Loschmidt (1821-1895)

  13. Colaboradores-Referencias Loschmidt Echo Sensitivity to perturbations

  14. Colaboradores-Referencias Loschmidt Echo Irreversibility

  15. Colaboradores-Referencias Loschmidt Echo Peres, 1984 PRA Coupled rotator model:

  16. Loschmidt Echo and Chaos Colaboradores-Referencias Jalabert-Pastawski PRL 2001 • Analytical semiclassical study of the LE • Initial state: localized state • Semiclassical aproximation for propagator K

  17. Loschmidt Echo and Chaos Colaboradores-Referencias Jalabert-Pastawski PRL 2001 • Perturbation: static disordered potential • The LE results • The Loschmidt echo has two contributions: • For strong perturbation: l is the Lyapunov exponent of the unperturbed Hamiltonian!!!!

  18. Loschmidt Echo and Chaos Colaboradores-Referencias What is the behavior of LE if H0 is integrable? Jacquod et al 2003 • Semiclassical aproximation for K idem Jalabert-Pastawski • The Loschmidt echo has two contributions: • For strong perturbation: Power law decay

  19. Loschmidt Echo and Chaos Colaboradores-Referencias Jacquod et al 2003 • Numerical check: kicked top Increase of the perturbation

  20. Loschmidt Echo and Chaos Colaboradores-Referencias Jacquod et al 2003

  21. Colaboradores-Referencias Regimes of the LE The LE depends on • The perturbation S • The initial state Y • The time t

  22. Colaboradores-Referencias Regimes of the LE Regimes of the LE with perturbation S Jacquod Silvestrov Beenakker PRE 2001 S

  23. Colaboradores-Referencias Regimes of the LE Regimes of the LE with perturbation S If the perturbation matrix Perturbation theory element is much smaller than D Gaussian decay Variance of level velocities

  24. Colaboradores-Referencias Regimes of the LE Regimes of the LE with perturbation S If S Relates old and new eigenstates LDOS

  25. Colaboradores-Referencias Regimes of the LE Regimes of the LE with perturbation S If LDOS Relates old and new eigenstates Width of LDOS FGR decay

  26. Colaboradores-Referencias Regimes of the LE Regimes of the LE with perturbation S If Liapunov exponent Liapunov Regime !!!!!!!!!

  27. Colaboradores-Referencias Regimes of the LE Regimes of the LE in the stadium billiard S

  28. Colaboradores-Referencias Regimes of the LE Regimes of the LE in the stadium billiard g exp(-g t) l G(S)/2 S Non-universal

  29. Colaboradores-Referencias Regimes of the LE Regimes of the LE in the Lorentz gas M(t)=0.09

  30. Colaboradores-Referencias Regimes of the LE Regimes of the LE in the Lorentz gas

  31. Colaboradores-Referencias Regimes of the LE Dependence of the LE with the initial state Wisniacki-Cohen 2002 Is universal the Lyapunov regime? Initial state: eigenstate But Then Physics of the LE = LDOS??

  32. Colaboradores-Referencias Regimes of the LE Dependence of the LE with the initial state Wisniacki-Cohen 2002 New V_ij=random(-1)*V_ij No lyapunov regime!!!!

  33. Colaboradores-Referencias Regimes of the LE Dependence of the LE with the initial state Wisniacki-Cohen 2002 No lyapunov regime!!!!

  34. Colaboradores-Referencias Regimes of the LE Short time decay of the LE Wisniacki 2003 Why? Experimental relevant regime?? Perturbed Hamiltonian Width of LDOS We show depends on Y and V

  35. Colaboradores-Referencias Regimes of the LE Short time decay of the LE Wisniacki 2003

  36. Colaboradores-Referencias Regimes of the LE Short time decay of the LE Initial state: eigenfunction of Ho

  37. Colaboradores-Referencias Regimes of the LE Short time decay of the LE Initial state: gaussian wave packet Initial state: evolved gaussian wave packet

  38. Colaboradores-Referencias Decoherence and the LE Decoherence -> lost of quantum coherence -> quantum-classical transition Zurek-Paz (1994) S Environment Chaotic System t Lyapunov exponent independent of the coupling with the environment Perturbation independent regime As Loschmidt echo but with non-unitary evolution

  39. Colaboradores-Referencias Decoherence and the LE Direct connection between decoherence and the LE Cucchietti et al (2003) Density matrix evolved by unperturbed U Unitary evo. Non Unitary evo. Lyapunov regime FGR They showed

  40. Colaboradores-Referencias Experiments • MNR polarization echo Physica A 00 Pastawski • Microwave cavity PRL 05 Stockmann • NMR Information processor PRL 05 Laflamme

  41. Colaboradores-Referencias Experiments MNR polarization echo Physica A 00 Pastawski • Single crystal of ferrocene • Many-body system • Gaussian decay • Perturbation independent • regime

  42. Colaboradores-Referencias Experiments Microwave cavity PRL 05 Stockmann • Electromagnetic cavity: equivalence of • Helmholtz and Schrodinger eq. • Measure the stationary scattering matrix element • RMT theoretical result 200 mm 438 mm

  43. Colaboradores-Referencias Experiments NMR Information processor PRL 05 Laflamme et al • Measure of the LE in an Nuclear Magnetic Resonance experiment. • Idea: Characterization of Complex Quantum Dynamics with a • Scalable NMR Information Processor ---> understanding the • performance and improvement of the device • It is implementing in an scalable circuit in which the measure is • done in one q-bit • U unitary map, P perturbation • U chaotic o regular • 5 q-bits

  44. Colaboradores-Referencias Experiments NMR Information processor PRL 05 Laflamme et al Regular U different perturbations FGR decay Chaotic U different perturbations FGR decay

  45. Colaboradores-Referencias Final Remarks • Is the LE a good measure of 'quantum chaos'? • Regimes of the LE ---> complex behaviour • Irreversibility and LE • Experiments: -nobody see the Lyapunov regime • -microwave billiards and NMR processor • FGR regime • -PID in the many body system • Other works: LE in a many body system, LE freeze,...

More Related