effect of nonlinearity on anderson localization of...
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Effect of nonlinearityon Anderson localization
of classical waves
D. M. Basko
Laboratoire de Physique et Modélisation des Milieux Condensés,
CNRS and Université Joseph Fourier, Grenoble, France
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Effect of Anderson localizationon the dynamics
of classical nonlinear waves
D. M. Basko
Laboratoire de Physique et Modélisation des Milieux Condensés,
CNRS and Université Joseph Fourier, Grenoble, France
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extended
state
localized
state x
x
localization length ξ
de Broglie wavelengthψ(x)
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OutlineLecture 1
• Nonlinear dynamical systems, integrability and chaos
• Statistics of chaos in disordered nonlinear chains
Lecture 2
• Statistical physics of chaotic systems
• Transport in disordered nonlinear chains
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Disordered nonlinear wave equations
disorder
+nonlinearity
continuous
discrete
lattice
In one spatial dimension
all linear normal modes are localized
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Superposition principle in linear systems
Lattice with L sites: n =‒L/2+1, ..., L/2,
linear Hermitian operator
Algorithm:
1. Find the eigenfunctions and eigenvalues
2. For any initial condition, the solution is
Linear systems: finding L independent periodic solutions
gives access to all possible solutions
Nonlinear systems: - localized periodic solutions still exist
Fröhlich, Spencer & Wayne, J. Stat. Phys. 42, 247 (1986),
- their localization length is the same as in the linear case
Iomin & Fishman, Phys. Rev. E 76, 056607 (2007),
but they are a just small minority among all possible solutions!
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Lyapunov exponentChoose an initial condition → trajectory
A nearby initial condition → trajectory
Lyapunov
exponentmind the order of limits
initial deviation
grows exponentially
Numerical simulation with double precision: ε ~ 10‒16
reliable for times t < 36/λ
λ = 0 for
linear systems:no exponential
growth
Also, λ = 0 for (quasi)periodic solutions of nonlinear equations
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Hamiltonian systemsPhase space: coordinates q = (q
1,...,q
N), momenta p = (p
1,...,p
N)
Hamiltonian function H(q,p) → equations of motion:
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Hamiltonian systemsPhase space: coordinates q = (q
1,...,q
N), momenta p = (p
1,...,p
N)
Hamiltonian function H(q,p) → equations of motion:
its complex conjugate
Example 1: DNLS chain with disorder
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Hamiltonian systemsPhase space: coordinates q = (q
1,...,q
N), momenta p = (p
1,...,p
N)
Hamiltonian function H(p,q) → equations of motion:
Example 2: continuous NLS equation with disorder
its complex conjugate
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Integrable Hamiltonian systemsIntegral of motion
Poisson brackets
Completely integrable system: N degrees of freedom, N integrals of motion
Canonical transformation : choose
as new momenta
Lyapunov exponent = 0
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Invariant toriThe choice of is not unique
From V. I. Arnold, Usp. Mat. Nauk 18, 13 (1963)
Confined motion: choose so that for all α
– action-angle variables
motion on an N-dimensional torus
in the 2N-dimensional phase space
Incommensurate Trajectory covers the whole torus
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Examples of integrable systems1. Any linear system: projections on eigenvectors are conserved
2. Any conservative system with 1 degree of freedom: energy is conserved
3. Two-site DNLS chain:
4. Clean continuous NLS eq.:
two conserved quantities: energy and norm
complete infinite set
of conserved quantitiesZakharov & Shabat,
JETP 34, 62 (1972)
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Examples of integrable systems
5. Toda lattice: M. Toda, J. Phys. Soc. Jpn. 22, 431 (1967)
6. Ablowitz-Ladik chain:
Ablowitz & Ladik,
J. Math. Phys. 16, 698 (1975)
Clean DNLS chain: non-integrable!
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Kolmogorov-Arnold-Moser theorem
integrable
small parameter
integrability-breaking perturbation
At small ε most of the tori are preserved,
measure of destroyed tori → 0 as ε → 0
The phase space splits into regular and chaotic regions
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Pendulum:
the period divergesat the separatrix
Spectrum
the separatrix motionhas a continuous spectrum
Phase space:
oscillations
rotations
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Perturbed pendulum:
oscillations
rotations
ergodic trajectorieswithinthe stochastic layer
regular motionsurvives
Stochastic layer area:
Continuous spectrumof the chaotic motion:
review: B. Chirikov (1979)
Melnikov-Arnold integral
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When and howdoes chaos appear
in disordered nonlinear chains?
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Probability of chaos
Disorder realization
Initial conditiontrajectory
regular
chaotic
Probability to be on a chaotic trajectory for a chain of length L:
or
Lyapunov
exponent
0 for regular, 1 for chaotic
Initial condition
over a length L:
disorder average
box distribution
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Locality of chaos
Basko, Ann. Phys. 326, 1577 (2011), Pikovsky & Fishman, Phys. Rev. E 83, 025201 (2011)
locality in space
Hypothesis:strong
localizationweak
nonlinearity
Switch to localized normal modes:
Nonlinearity couples
the modes:
overlap between the modes
only not too far apart
Transition to chaos should occur independently in different spatial regions
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Pikovsky & Fishman, Phys. Rev. E 83, 025201 (2011)
Direct numerics
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Pendulum in the DNLS equationtwo-site
Hamiltonian:
canonical transformation:
another canonicaltransformation:
transformedHamiltonian:
almostconstant
complete the squareconstant
conserved
very similar
to a pendulum
A third site:
perturbation of the pendulum
Three sites are sufficient
to generate chaos
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guidingresonance
smalldeviation
cutoffpoint
Geometrical view of
Chaos on three sites
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guidingresonance
smalldeviation
cutoffpoint
Geometrical view of
Need
Strong disorder, weak nonlinearity:
Look for a resonance
(guiding resonance)
Chaos on three sites
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guidingresonance
smalldeviation
cutoffpoint
Geometrical view of
Need
Strong disorder, weak nonlinearity:
Look for a resonance
The pendulum frequency
The stochastic layer widthLook for another resonance
(guiding resonance)
Need (layer resonance)
Chaos on three sites
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effective coupling 1 ↔ 3:
works when
Guiding and layer resonances can be generated
in high orders of the perturbation theory
Coupling + nonlinearity → effective couplings of the form
Competition: number of combinations ↔ power of the coupling constants
Chaos comes from rare resonant triples:
Basko, Ann. Phys. 326, 1577 (2011)
Chaos on more sites
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Rescaled Hamiltonian of a triple close to resonance:
rescaled
detunings: initial condition:
No parameters in the rescaled Hamiltonian
A numerical coefficient to be determined numerically
disorder : coupling + nonlinearity:
A one-parameter model
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Numerics for the resonant triple
Confinement in both directions:
a numerical proof that two resonances are needed for chaos
Phase-averaged
probability of chaos
(greyscale)
coefficient
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Numerics for the full chain
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Single-site excitation
Initial excitation is concentrated on a single site in the middle of the chain
probability
of chaos
The probability for the initial site
to be on a resonant triple
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Conclusions of Lecture 1
1. Anderson localization + weak nonlinearity → weak chaos
2. Such chaos is characterized by a probability
L→∞, finite energy density probability → 1
3. At weak nonlinearity chaos nucleates on rare local spots
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Bosonic atoms in disordered optical lattices
level
position tunneling
amplitude
interpatricle
repulsion
Bose-Hubbard Hamiltonian:
bosonic
operator
condensate wave functionquantum fluctuation
Roati et al., Nature 453, 895 (2008)
Billy et al., Nature 453, 891 (2008)
The classical limit of a bosonic field:
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Light in evanescently coupled 1D waveguides
Wave equation for the electric field :
inhomogeneous
dielectric
structure
Kerr
nonlinearity
Lahini et al., PRL 100, 013906 (2008)
transverse eigenmodes
discrete
nonlinear
Schrödinger
equation
paraxial approximation
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Anderson localization and nonlinearityanharmonicoscillators
nearest-neighborcoupling
disorder → Anderson localization
nonlinearity
classical
Hamiltonian
extended
state
localized
state
de Broglie wavelength
localization length
In one dimension
all eigenstates are localized
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WEAK coupling
↓
STRONG localization
WEAK
nonlinearity
Anderson localization and nonlinearity
What is the effect of a WEAK nonlinearity
on STRONG Anderson localization?
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The problem of wave packet spreadingLinear system: Anderson localization
normal modes
initial condition
The wave packet remains exponentially localized forever
Nonlinear system: interaction between the normal modessmall correction?
chaotic behavior?
Numerical integration: subdiffusion
Shepelyansky (1993); Molina (1998); Kopidakis et al. (2008); Pikovsky & Shepelyansky (2008);
Skokos et al. (2009); Skokos & Flach (2010); Laptyeva et al. (2010); Bodyfelt et al. (2011)
KAM theorem, perturbation theory:Bourgain & Wang (2008); Wang & Zhang (2009); Fishman et al. (2009); Johansson et al. (2010)
Indications for slowing down: Mulansky et al. (2011); Michaely & Fishman (2012)
Experiment: subdiffusion (non-universal exponent) Lucioni et al. (2011)
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Other disordered nonlinear chains
Fröhlich, Spencer & Wayne, J. Stat. Phys. 42, 247 (1986)
existence of localized solutions
corresponding to invariant tori
(one conserved quantity;
two different terms responsible for anharmonicity and coupling, like in NLS)
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Other disordered nonlinear chains
Fröhlich, Spencer & Wayne, J. Stat. Phys. 42, 247 (1986)
existence of localized solutions
corresponding to invariant tori
Classical spins, :
Oganesyan, Pal & Huse, Phys. Rev. B 80, 115104 (2009)
Energy transport → fast decay of time correlations on local spots of the chain
(one conserved quantity;
two different terms responsible for anharmonicity and coupling, like in NLS)
(one conserved quantity; nonlinearity and coupling are governed by the same J)