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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:0806.4301 to be published in Phys. Rev. Lett
arXiv:0806.4301 to be published in Phys. Rev. Lett
Nonequilibrium
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, McCoy, Dresden, Mazur, Girardeau (’70)
A fundamental characterization
Think thermodynamics !!!!
A.Silva, arxiv:0806.4301
A,B = points in parameters space
g = path
g1
B
Thermodynamic transformation
g
g2
g3
A
Work
Entropy
Heat
Closed systems
Nonequilibrium=Statistics
Quasistatic transformation
g1
B
g
g2
g3
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, F569-F571 (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
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
Loschmidt echo for global quench
System size
Expand and 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: MacKay et al., Nature 453, 76 (2008)
Saturation of damping rate at low T: quantum phase slip !