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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.

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Chaos and Irreversibility: An introduction to the Loschmidt echo

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  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,...

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