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Discussions: Giuseppe Mussardo , Rosario Fazio (SISSA)

Statistics of the Work done in a Quantum Quench. Alessandro Silva ICTP Trieste. Discussions: Giuseppe Mussardo , Rosario Fazio (SISSA) Natan Andrei (Rutgers) Vadim Oganesyan (Yale), Anatoli Polknovnikov (Boston).

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Discussions: Giuseppe Mussardo , Rosario Fazio (SISSA)

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  1. Statistics of the Work done in a Quantum Quench Alessandro Silva ICTP Trieste Discussions: Giuseppe Mussardo, Rosario Fazio (SISSA) Natan Andrei (Rutgers) Vadim Oganesyan (Yale), Anatoli Polknovnikov (Boston) arXiv:0806.4301to be published in Phys. Rev. Lett arXiv:0806.4301 to be published in Phys. Rev. Lett

  2. Nonequilibrium Nonequilibrium = Last unexplored frontier Partition function Mean field theory Renormalization group Equilibrium tools

  3. Non equilibrium physics in many body systems Prototype example: Kondo effect in Quantum Dots From: L. Kouwenhoven and L. Glazman, Phys. World 14(1), 33 (2001) D. Goldhaber-Gordon, et al., Nature 391, 156 (1998)

  4. Non equilibrium physics in many body systems Nonequilibrium splitting of the Kondo resonance From: De Franceschi, et al, PRL 89, 156801 (2002) Abrupt quench inside the Kondo valley From: Nordlander, et al PRL 83, 808 (1999).

  5. Non equilibrium physics in many body systems The nonequilibrium lab: cold atomic gases Superfluid Mott Superfluid From: Fisher et al, Phys Rev B 40, 546 (1989). See also Jaksch et al, PRL 81, 3108 (1998). From: Greiner et al, Nature 419, 51 (2002)

  6. Non equilibrium physics in many body systems From: Kinoshita et al., Nature 440, 900 (2006) 40 periods without thermalization:integrability ??

  7. A paradigm: the quantum quench Example: Can be quenched globally or locally

  8. Quantum quenches Early works Baruch, McCoy, Dresden, Mazur, Girardeau (’70) Universality ? Time dependence of correlators Igloi, Riegel (’01) Altman, Auerbach (’02) Sengupta, Powell, Sachdev (’04) Calabrese and Cardy (’07) Generation of excitations (defects) Zurek, Dorner, Zoller (’05) Polkovnikov (’05) Dziarmaga (’05) Cherng and Levitov (’06) Gritsev, Polkovnikov (’07) D. Patane’, A Silva, et al. (’08) Thermalization and integrability ? Rigol et al, (’06) Kollath, et al. (’07) Manmana et al. (’07) Cazalilla (’07) Gangart and Pustilnik (’08) Cramer et al (’08) Barthel and Schollwock (’08)

  9. A fundamental characterization Think thermodynamics !!!! A,B =points in parameters space A.Silva, arxiv:0806.4301 g = path g1 B Thermodynamic transformation g Work Entropy Heat g2 A g3 Closed systems

  10. Nonequilibrium=Statistics Quasistatic transformation g1 B g g2 g Out of equilibrium A g3 Statistics depends on path, time dependence, etc… Classical systems: Jarzynski (’97), Crooks (’99)

  11. Outline Statistics of the work done in a quantum quench 1- Work probability distribution P(W) Loschmidt echo (dephasing !) 2- In Quantum Critical Systems (Quantum Ising Model) Criticality Singularities in moments of P(W) Local quenches Edge singularities

  12. Work statistics and Loschmidt echo

  13. Work and Loschmidt Abrupt quench Initial energy To measure work: Final energy Initial state probability

  14. Work and Loschmidt Take a Fourier Transform: Characteristic function Loschmidt echo, Core hole correlator, etc… appears in X-ray edge problems, quantum chaos, DEPHASING Z. P. Karkuszewski, C. Jarzynski, and W. Zurek, Phys. Rev. Lett. 89, 170405 (2002) H. T. Quan, et al. Phys. Rev. Lett. 96, 140604 (2006). D. Rossini, et al. Phys. Rev. A 75, 032333 (2007). Initial state

  15. Work and Loschmidt At T=0 Loschmidt echo = Partition function (in real time)

  16. Jarzynski equalities Arbitrary quench Abrupt quench Nonequilibrium Equilibrium Jarzynski equality C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). P. Talkner, E. Lutz, and P. Haanggi, Phys. Rev. E 75, 050102 (2007)

  17. Homework Given Prove Tasaki-Crooks fluctuation theorem G. E. Crooks, Phys. Rev.E 60 2721 (1999) P. Talkner, P. Haanggi, J. Phys. A 40, F569-F571 (2007)

  18. Using Jarzynsky-Loschmidt connection I: Global quench in the Quantum Ising Model

  19. Global quantum quench Global Quench Small Fluctuations Work X unit volume Fluctuations

  20. Ising model and Landau Zener dynamics Jordan Wigner Bogoliubov rotation

  21. Loschmidt echo for global quench eigenmodes of eigenmodes of Determinant formula (full counting statistics) Klich (’02), Abanin and Levitov (’03) Or direct expansion + re-exponentiation A. LeClair, G. Mussardo, H. Saleur, S. Skorik, Nucl.Phys. B453, 581 (1995) Integrable boundary state

  22. Loschmidt echo for global quench System size Expand and get all cumulants Difference in ground state energies Excess work Thermodynamics dixit It’s Ok !!!

  23. Loschmidt echo for global quench Asymptotics for large t (low W) Measurable by dephasing Critical Casimir effect on a Cylinder t = it

  24. Using Jarzynsky-Loschmidt connection I: LOCAL quench in the Quantum Ising Model

  25. The setting Expand in cumulants Decay of Loschmidt echo Fluctuations, etc… Long “time”asymptotics Vanishing at criticality Orthogonality Catastrophe !!!

  26. Edge Singularity Start at Criticality Edge Singularity Let us get P(W) in the scaling limit !!

  27. Scaling Limit Quench=local mass term 1- Double your Majoranas

  28. Scaling Limit 1-Form Dirac fermions Quench= Local Backscattering 2- Perform nonlocal rotation (at criticality m=0) d Two chiral modes Quench = Phase shift

  29. Scaling Limit Use bosonization This is the characteristic function of the GAMMA distribution

  30. Conclusions Statistics of the work done in a quantum quench 1- Work probability distribution P(W) Loschmidt echo (dephasing !) 2- In Quantum Critical Systems (Quantum Ising Model) Criticality Singularities in moments of P(W) Local quenches Edge singularities

  31. Outlook Work, entropy, etc… as fluctuating variables. NONEQUILIBRIUM =STATISTICS 1- Other exactly solvable models (zero dimensions) [with F. Paraan] 2- General time dependence (Ising) ?? 3- More complex integrable models ?? 4- Impurity models ?? 5- Statistics of entropy ??

  32. Non equilibrium physics in many body systems From: MacKay et al., Nature 453, 76 (2008) Saturation of damping rate at low T: quantum phase slip !

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