Sep. 17, 2007 CIC, Cuernavaca 1

Quantum Chaos in Quantum Graphs

Lev KaplanLev Kaplan

Tulane UniversityTulane University

Sep. 17, 2007 CIC, Cuernavaca 2

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

Sep. 17, 2007 CIC, Cuernavaca 3

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

Sep. 17, 2007 CIC, Cuernavaca 26

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

Sep. 17, 2007 CIC, Cuernavaca 28

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)

Sep. 17, 2007 CIC, Cuernavaca 34

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