Nonadiabatic dynamics in closed Hamiltonian systems.

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Nonadiabatic dynamics in closed Hamiltonian
systems.
Anatoli Polkovnikov, Boston University
R. Barankov BU Claudia De Grandi
BU Vladimir Gritsev U. of Fribourg
University of Utah, Condensed Matter Seminar,
10/27/2009
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Cold atoms (controlled and tunable Hamiltonians,
isolation from environment)
Experimental examples
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Optical Lattices
I. Bloch, Nature Physics 1, 23 - 30 (2005)
OL are tunable (in real time) from weak
modulations to tight binding regime. Can change
dimensionality and study 1D, 2D, and 3D
physics. Both fermions and bosons.
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Superfluid insulator phase transition in an
optical lattice Greiner et. al. 2003
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Repulsive Bose gas.
Lieb-Liniger, complete solution 1963.
M. Olshanii, 1998
interaction parameter
Experiments
T. Kinoshita, T. Wenger, D. S. Weiss ., Science
305, 1125, 2004
Also, B. Paredes1, A. Widera, V. Murg, O. Mandel,
S. Fölling, I. Cirac, G. V. Shlyapnikov, T.W.
Hänsch and I. Bloch, Nature 277 , 429 (2004)
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Local pair correlations.
Kinoshita et. Al., Science 305, 1125, 2004
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Cold atoms (controlled and tunable Hamiltonians,
isolation from environment)
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In the continuum this system is equivalent to an
integrable KdV equation. The solution splits into
non-thermalizing solitons Kruskal and Zabusky
(1965 ).
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Qauntum Newton Craddle.(collisions in 1D
interecating Bose gas Lieb-Liniger model)
T. Kinoshita, T. R. Wenger and D. S. Weiss,
Nature 440, 900 903 (2006)
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Cold atoms (controlled and tunable Hamiltonians,
isolation from environment)
3. 12 Nonequilibrium thermodynamics?
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Thermalization in classical systems (origin of
ergodicity) chaotic many-body dynamics implies
exponential in time sensitivity to initial
fluctuations.
Eignestate thermalization hypothesis (Srednicki
1994 M. Rigol, V. Dunjko M. Olshanii, Nature
452, 854 , 2008.) An,n const (n) so there is no
dependence on ?nn.
Information about equilibrium is fully contained
in diagonal elements of the density matrix.
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Information about equilibrium is fully contained
in diagonal elements of the density matrix.
This is true for all thermodynamic observables
energy, pressure, magnetization, . (pick your
favorite). They all are linear in ?.
The usual way around coarse-grain density matrix
(remove by hand fast oscillating off-diagonal
elements of ?. Problem not a unique procedure,
explicitly violates time reversibility and
Hamiltonian dynamics.
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Von Neumann entropy always conserved in time (in
isolated systems). More generally it is invariant
under arbitrary unitary transfomations
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Connection between two adiabatic theorems allows
us to define heat (A.P., Phys. Rev. Lett. 101,
220402, 2008 ).
Consider an arbitrary dynamical process and work
in the instantaneous energy basis (adiabatic
basis).

• Adiabatic energy is the function of the state.
• Heat is the function of the process.
• Heat vanishes in the adiabatic limit. Now this is
  not the postulate, this is a consequence of the
  Hamiltonian dynamics!

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Isolated systems. Initial stationary state.
Unitarity of the evolution
In general there is no detailed balance even for
cyclic processes (but within the Fremi-Golden
rule there is).
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yields
The statement is also true without the detailed
balance but the proof is more complicated
(Thirring, Quantum Mathematical Physics, Springer
1999).
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Properties of d-entropy (A. Polkovnikov,
arXiv0806.2862. ).
Jensens inequality
Therefore if the initial density matrix is
stationary (diagonal) then
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Classic example freely expanding gas
Suddenly remove the wall
by Liouville theorem
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Thermodynamic adiabatic theorem.
In a cyclic adiabatic process the energy of the
system does not change no work done on the
system, no heating, and no entropy is generated .
General expectation
- is the rate of change of external parameter.
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Adiabatic theorem in quantum mechanics
Landau Zener process
In the limit ? ? 0 transitions between different
energy levels are suppressed.
This, for example, implies reversibility (no work
done) in a cyclic process.
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Adiabatic theorem in QM suggests adiabatic
theorem in thermodynamics

1. Transitions are unavoidable in large gapless
   systems.
2. Phase space available for these transitions
   decreases with the rate. Hence expect

Low dimensions high density of low energy
states, breakdown of mean-field approaches in
equilibrium
Breakdown of Taylor expansion in low dimensions,
especially near singularities (phase transitions).
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Three regimes of response to the slow ramp A.P.
and V.Gritsev, Nature Physics 4, 477 (2008)

1. Mean field (analytic) high dimensions
2. Non-analytic low dimensions, near singularities
   like QCP
3. Non-adiabatic low dimensions, bosonic
   excitations

In all three situations quantum and thermodynamic
adiabatic theorem are smoothly connected. The
adiabatic theorem in thermodynamics does follow
from the adiabatic theorem in quantum mechanics.
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Origin of quadratic scaling adiabatic
perturbation theory
Can we say anything about system response in the
limit ??0?
Assume that we start in the ground state.
Need to solve
Convenient to work in the adiabatic (co-moving,
instantaneous) basis
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Schrödinger equation in the adiabatic basis
Assume the proximity to the instantaneous ground
state (small excitation probability)
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Landau-Zener problem
For finite interval of excitation the transition
probability scales as the second power of the
rate (not exponentially).
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Gapless systems with quasi-particle excitations
Ramping in generic gapless regime (low energy
contribution)
Uniform system
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Low energy contribution
High dimensions high energies dominate
dissipation, low-dimensions low energies
dominate dissipation.
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Example crossing a QCP.
? ? ? t, ? ? 0
Gap vanishes at the transition. No true adiabatic
limit!
How does the number of excitations scale with ? ?
A.P. 2003, Zurek, Dorner, Zoller 2005
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Another Example loading 1D condensate into an
optical lattice or merging two 1D condensates (C.
De Grandi, R. Barankov, AP, PRL 2008 )
K Luttinger liquid parameter
Relevant sineGordon model
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Results
K2 corresponds to a SF-IN transition in an
infinitesimal lattice (H.P. Büchler, et.al. 2003)
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Sudden and slow quenches starting from the QCP
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Probing quasi-particle statistics in nonlinear
dynamical probes.
T
0
1
K
massive fermions (hard core bosons)
massive bosons
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Conclusions

1. Adiabatic theorems in quantum mechanics and
   thermodynamics are directly connected.
2. Diagonal entropy satisfies laws of thermodynamics
   from microscopics. Heat and entropy change result
   from the nonadiabatic transitions between
   microscopic energy levels.
3. Maximum entropy state with ?nnconst is the
   natural attractor of the Hamiltonian dynamics.
4. Universal scaling of density of excitations,
   heat, entropy for sudden and slow dynamics near
   QCP. These scaling laws are closely related to
   scaling of fidelity and other susceptibilities.

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M. Rigol, V. Dunjko M. Olshanii, Nature 452,
854 (2008)
a, Two-dimensional lattice on which five
hard-core bosons propagate in time. b, The
corresponding relaxation dynamics of the central
component n(kx 0) of the marginal momentum
distribution, compared with the predictions of
the three ensembles c, Full momentum
distribution function in the initial state, after
relaxation, and in the different ensembles.
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Non-adiabatic regime (bosons on a lattice).
Nonintegrable model in all spatial dimensions,
expect thermalization.
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T0.02
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2D, T0.2
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Classical systems.
Instead of energy levels we have orbits.