Statistics of the Work done in a Quantum

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) ar. Xiv: 0806. 4301 to be published in Phys. Rev. Lett

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

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)

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).

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)

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

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

Quantum quenches 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) Early works Baruch, Mc. Coy, Dresden, Mazur, Girardeau (’ 70)

A fundamental characterization Think thermodynamics !!!! A. Silva, arxiv: 0806. 4301 A, B = points in parameters space g = path g 1 B Thermodynamic transformation g g 2 g 3 A Work Entropy Heat Closed systems

Nonequilibrium=Statistics Quasistatic transformation g 1 B g g 2 g 3 A g Out of equilibrium Statistics depends on path, time dependence, etc… Classical systems: Jarzynski (’ 97), Crooks (’ 99)

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

Work statistics and Loschmidt echo

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

Work and Loschmidt Take a Fourier Transform: Characteristic function 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 Loschmidt echo, Core hole correlator, etc… appears in X-ray edge problems, quantum chaos, DEPHASING

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

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)

Homework Given Prove Tasaki-Crooks fluctuation theorem G. E. Crooks, Phys. Rev. E 60 2721 (1999) P. Talkner, P. Haanggi, J. Phys. A 40, F 569 -F 571 (2007)

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

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

Ising model and Landau Zener dynamics Jordan Wigner Bogoliubov rotation

Loschmidt echo for global quench eigenmodes of Determinant formula (full counting statistics) Klich (’ 02), Abanin and Levitov (’ 03) Or direct expansion + re-exponentiation A. Le. Clair, G. Mussardo, H. Saleur, S. Skorik, Nucl. Phys. B 453, 581 (1995) Integrable boundary state

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

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

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

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

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

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

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

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

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

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 ? ?

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