Quantum Chaos in Quantum Graphslkaplan/graphs_cuernavaca.pdf · Wave function statistics in chaotic...
Transcript of Quantum Chaos in Quantum Graphslkaplan/graphs_cuernavaca.pdf · Wave function statistics in chaotic...
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Sep. 17, 2007 CIC, Cuernavaca 1
Quantum Chaos in Quantum Graphs
Lev KaplanLev Kaplan
Tulane UniversityTulane University
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Talk outline:� What are quantum graphs and why are they
interesting?
� Basic formulation
� Applications: what can we say about stationary quantum properties using known short-time (semiclassical) dynamics?
�Wave function statistics in chaotic graphs
�Vacuum energy and Casimir forces (with S. Fulling and J. Wilson)
� Relevance to more general chaotic systems
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What is a quantum graph?
� Physics: quantum mechanics of a particle on a set of line segments joined at vertices
� Mathematics: singular one-dimensional variety equipped with self-adjoint differential operator
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Reasons for studying quantum graphs:� Approximation for realistic physical wave systems
� Chemistry: free electron theory of conjugated molecules
� Nanotechnology: quantum wire circuits
� Optics: photonic crystals
� Laboratory for investigating general questions about
� Scattering theory and resonances
� Quantum chaos: trace formulas, localization
� Spectral theory
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Basic formulation:�� B bonds of length Lj (j = 1 … B)
�Wave function Ψj(x) on each bond 0 < x < Lj� [(-i d/dx – Aj(x))
2 + Vj(x)] Ψj(x) = k2 Ψj(x)�Often take Vj = 0, Aj = 0
� V vertices, each connecting vα bonds (α = 1 … V)
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Basic formulation:� Need boundary conditions at each vertex α: Kirchhoff
� Continuity for all bonds j starting at vertex α
� Current conservation where sum is over all bonds j starting at vertex α, and derivative is in outward direction
� λα is vertex-dependent constant
� vα=1: λα= 0 Neumann λα=∞ Dirichlet
� vα=2: delta-potential V(x) = λαδ(x)
ααΨ=Ψ∂∑ λ)0(j
j
αΨ=Ψ )0(j
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Basic formulation:� Scattering-matrix approach:
where i and j are any two bonds meeting at α, and
� Sij can be replaced with more general unitary matrix
� By adjusting graph connectivity, bond lengths Lj , and vertex S-matrices , we can construct examples of chaotic, disordered, or regular quantum systems
jii
ji evS δαωα −+= −− )1(1
)/arctan(2 kvααα λω =
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Motivation:� Transcend two extreme approaches to quantum chaos
�Brute force calculation for each specific system:
�Exact, but not insightful and must to be repeated anew for every change in Hamiltonian
�Random matrix theory (RMT):
�Universal, but no system-specific information
�� Investigate what stationary properties of general quantum systems can be reliably obtained using readily available short-time (classical) information
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Application I: Wave function statistics� Wave function on jth bond at energy k2 (assume time
reversal invariance)
� In semiclassical limit kL → ∞, statistics of intensities |Ψ(x)|2 over whole graph can be completely described by statistics of coefficients |aj(k)|2
� Normalization: 〈 |aj(k)|2 〉 = 1� RMT predicts: aj(k) is Gaussian random variable for
B → ∞ (under variation of j, k, or system parameters)� One-dimensional version of random-wave model
� Questions: Is this true? Can we do better?
ikxkj
ikxkj
kj eaeax
−+=Ψ )*()()( )(
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Application I: Wave function statistics
� Consider general quantum system, initial state | φ 〉� Autocorrelation function Aφ(t) = 〈 φ | e-iHt | φ 〉� Local density of states (weighted spectrum)
Sφ(E) = Σ |〈 φ | n 〉|2 δ(E - En) = FT [ Aφ(t) ] �Eigenstate information |〈 φ | n 〉|2 encoded in
autocorrelation function Aφ(t)
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Application I: Wave function statistics� Sφ(E) = FT [ Aφ(t) ]
� Suppose we only know dynamics for short times,Aφshort(t) = Aφ(t) exp(-t2/2T2)
� Sφsmooth(E) = FT [Aφshort(t) ]~ ∫ dE’ Sφ(E’) exp(-T2 (E-E’)2/2)
which is Sφ(E) smoothed on scale 1/T
� Knowledge of Aφshort(t) imposes constraint on Sφ(E)
� For chaotic system in B → ∞ limit, choose T � Greater than mixing time ~ log B
� Shorter than Heisenberg time ~ B
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Application I: Wave function statistics
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Application I: Wave function statistics
� Conjecture: Long-time returns given by convolutionof known short-time returns with random signal:For t àT,
Aφ(t) = ∫ dt’ A φshort(t-t’) A rnd(t’)where Arnd(t’) obeys RMT statistics
� Then full spectrum Sφ(E) obtained by multiplying Sφsmooth(E) with random (RMT-like) sum of δ-functions
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Application I: Wave function statistics
� |〈 φ | n 〉|2 = Sφsmooth(En) |rn|2where rn is drawn from random distribution
� Combining analytically known short-time dynamical information with random behavior at long times
� RMT: Sφsmooth(En) =1 ⇔ Aφshort(t) ~ δ(t)
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Application I: Wave function statistics� Moments:
� 〈 |〈 φ | n 〉|2n 〉 = [ 〈 |〈 φ | n 〉|2n 〉rnd ] [ ∫ dE (Sφsmooth(E))n ]� E.g. inverse participation ratio or mean squared intensity
IPRφ = 〈 |〈 φ | n 〉|4 〉 = Frnd ∫ dE (Sφsmooth(E))2~ Frnd ∫ dt |Aφshort(t)|2
� Properly normalizedIPRφ = Frnd ∫ dt |Aφshort(t)|2 / ∫ dt |Arndshort(t)|2
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Application I: Wave function statistics� Ring graph: periodic lattice of vertices α = 1 … V, each
vertex connected to α – v/2 , …α + v/2 (valency vαα = v)� Number of
bondsB = Vv/2
� E.g. V=12,v=6, B=36
� Set all λα=0� Randomize
L i∈[1, 1+ε]
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Application I: Wave function statistics� Want to predict distribution of eigenstate amplitudes aj(k)
� Use short-time dynamical information
� Shortest returns come from orbits that travel back and forth along single bond between two vertices
� Return probability after two bounces:Prefl = 1 – 4 (v-1)/v2
� ∫ dt |Aφ(t)|2 ~ � Similarly can include longer orbits in systematic
expansion
)P1(/)P1()(P 2refl2refl
||2refl −+=∑∞
−∞=n
n
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Application I: Wave function statistics
� Predict IPR = )(FV
b1
41-
rnd vOv +
−
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Application I: Wave function statistics� Another example: cubic lattice with disorder
� V = 37 x 37 x 37 vertices
� Valency v=6
� Fraction 1/D of all vertices randomly chosen to be occupied by scatterer, with λα drawn from random power-law distribution P(λ) ∼ λ−r (λ0 < λ < ∞)
� Free propagation otherwise
� Then power-law tail of wave function intensities
P(|a|2) ~ (λ0)2(r−1) (|a|2)-(r+1)
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Application I: Wave function statistics
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Application I: Wave function statistics� Related methods applied successfully to study
� Wave function statistics in billiards and other higher-dimensional systems
� Statistics of many-body wave functions in nuclei or quantum dots
� Statistics of extreme ocean waves
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Application II: Vacuum energy� Scalar field quantized on a graph
� Energy =
� To regularize, introduce ultraviolet cutoff t:
� Vacuum energy = cutoff-independent part of
as t → 0
mm
mn ω)2
1(
1
+∑∞=
)(dt
d
2
1
dt
d
2
1
2
1)(
11
tTeetEm
tt
mm
mm −=−== ∑∑ ∞=
−−∞
=
ωωω
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Application II: Vacuum energy� Direct approach using spectrum:
� For simple cases, e.g. line segment of length L with Dirichlet boundaries, ωm obtainable analytically:
ωm = πm/L ⇒
� Divergent term comes from Weyl density of states
� proportional to volume
� independent of geometry
� unphysical (no Casimir forces on pistons)
L482
L)(
2
π
π+=
ttE
L
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Application II: Vacuum energy� Direct approach using spectrum (numerical)
� Find ωm by solving characteristic equationdet h(ω) = 0, where h(ω) is a V by V matrix
� Vacuumenergy =
� Can speed up convergence using Richardson extrapolation if we know E(t) = E0 + O(tα)
−−∞
=→∑ 2total
10 2
L
2
1lim
te t
mm
t
m
πω ω
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Application II: Vacuum energy� Alternative approach using periodic orbits:
� Use Kirchhoff boundary conditions with λα=0 at each vertex (energy-independent scattering matrices)
� Every periodic orbit of length Lp makes contribution
to cylinder kernel T(x,x,t) if Lp goes through x
� Trace over initial/final point x gives additional factor Lp / r (r=repetition factor to avoid overcounting)
� Lp = 0: divergent (Weyl) part
( )factors scattering ofproduct L
122×
+ t
t
pπ
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Application II: Vacuum energy� For non-zero Lp, contribution to vacuum energy E0 is
� Scattering factors are
� (2/v) for transmission through Kirchhoff vertex
� (2/v – 1) for reflection from Kirchhoff vertex
� (-1) for reflection from Dirichlet reflector
� (+1) for reflection from Neumann reflector
� (eiφ) for reflection from arbitrary-phase reflector
( )factors scattering ofproduct L2
1×−
prπ
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Application II: Vacuum energy� Apply method to star graphs
� B bonds meeting at singleKirchhoff vertex at center
� Each bond j has Dirichlet,Neumann, or other reflector at distance Lj from center
� Each reflector is movable piston
� Calculate approximation to vacuum energy E0 (or Casimir force on jth piston) by summing over all orbits of length Lp ≤ Lmax
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Application II: Vacuum energy� Contribution to vacuum energy from shortest orbits only
(orbits that bounce back and forth once in one bond):
� +1 for Neumann pistons, -1 for Dirichlet
� Gives correct sign for Casimir forces at least for B>3
� repulsive for Neumann
� attractive for Dirichlet
( ) ∑=
−±− Bj jB 1 L
11
21
4
1
π
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Application II: Vacuum energy� Add up all repetitions of shortest orbits (Neumann)
� Compare with analytic result for B equal bond lengths with Neumann pistons:
� Correct only to leading order in 1/B
� Need more orbits to get good answer for finite B
L
B
B
− 3148
π
∑∑∑==
∞
=
+−=
−− Bj j
B
j jr
r
BBr 12
112 L
1...
2ln241
48L
11
21
4
1
π
π
π
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Application II: Vacuum energy
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Application II: Vacuum energy� B=4 star graph with unequal bonds and Neumann pistons
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Application II: Vacuum energy� B=4 star graph with unequal bonds and arbitrary phase
pistons
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Application II: Vacuum energy� Can determine rate of convergence with Lmax
� all Neumann pistons: Error ~ (Lmax)-1
� generic case: Error ~ (Lmax)-3/2
� Future work:
� more general graphs (not star graphs)
� including closed (non-periodic) orbits, e.g. for complex scattering matrices or vacuum energy density
� extension to higher-dimensional chaotic systems (e.g. chaotic billiards)
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Summary� Quantum graphs provide useful testing ground for techniques that
have relevance to more general quantum systems
� Major problem in quantum chaos is to predict long-time or stationary quantum behavior (where classical mechanics is not valid) using classical information
� Accurate predictions for wave function statistics in chaotic quantum graphs by combining knowledge of short periodic orbits with randomness assumption at long times
� These predictions are robust (insensitive to small changes in the Hamiltonian that dramatically affect long orbits and individual high-lying eigenstates)
� Similarly, vacuum energy and Casimir forces can be estimated using short orbit information, without detailed knowledge of thehigh-lying spectrum