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: to be published in Phys. Rev. Lett
arXiv: 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, (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 Universality ? Thermalization and integrability ?
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:
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, (2002)
H. T. Quan, et al. Phys. Rev. Lett. 96, (2006).
D. Rossini, et al. Phys. Rev. A 75, (2007).
Initial state

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

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

17. Homework Given Prove Tasaki-Crooks fluctuation theorem
G. E. Crooks, Phys. Rev. E (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 !