Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set

Quantum dynamics for a fractal spectrum

Ariane SORET

Supervised by Eric AKKERMAN ENS Cachan - Technion Institute of Technology

6 August 2014

1 Mathematical prerequisites

2 Experimental example : Fibonacci cavity

3 Anaytical study in the case of a Cantor set

Measure Correlation function

Table of contents

1 Mathematical prerequisites

2 Experimental example : Fibonacci cavity

3 Anaytical study in the case of a Cantor set

Table of contents

1 Mathematical prerequisites

2 Experimental example : Fibonacci cavity

3 Anaytical study in the case of a Cantor set

Mandelbrot’s definition of a fractal : COMPLETER In our case : objects with a scaling symmetry. Ex. : Sierpinsky gasket, triadic Cantor set. Cantor set : "zoom in" by a factor 3 and get the same object rescaled by factor \frac{1}{2} Scaling property and log-periodic oscillations :

f (x) = \frac{1}{b}f (ax)

f (x) = x^{ln(b)/ln(a)}g (\frac{ln(x)}{ln(a)})

g periodic of period 1.

Table of contents

1 Mathematical prerequisites

2 Experimental example : Fibonacci cavity

3 Anaytical study in the case of a Cantor set

Table of contents

1 Mathematical prerequisites

2 Experimental example : Fibonacci cavity

3 Anaytical study in the case of a Cantor set

Idea : spacially varying potential yields modification of the density of states of propagating waves ; quasiperiodic potential -> ? ? ? Longitudinal cavity, quasiperiodically varying width

Figure: Source : [1]

Figure: Source : [1]

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Table of contents

1 Mathematical prerequisites

2 Experimental example : Fibonacci cavity

3 Anaytical study in the case of a Cantor set

Measure Correlation function

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Table of contents

1 Mathematical prerequisites

2 Experimental example : Fibonacci cavity

3 Anaytical study in the case of a Cantor set

Measure Correlation function

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Hamiltonian H, spectrum = triadic Cantor set. Which physical quantities highlight the fractal feature ? Analytical expression ? Conversely : determine the spectrum from these quantities ?

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Define an appropriate measure :

∫_{C} f (x)dμ(x) = lim_{n→+∞}(\frac{3}{2})^{n} ∑_{P_{n}}∫_{a_{j}}^{a_{j}+3^{−n}} f (x)dx

Verifies :

∫_{C} dμ(x) = 1 ∫_{C} f (x)dμ(x) = \frac{1}{2} ∫_{C} f (\frac{x}{3})dμ(x) + \frac{1}{2} ∫_{C} f (\frac{x+2}{3})dμ(x) "dμ(ϵ) = ρ(ϵ)dϵ"

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Integrated density of states (IDOS) :

N(ϵ) = ∫_{spectre} θ(ϵ − ϵ′)dμ(ϵ′)

= \frac{1}{2} ∫_{spectre} θ (ϵ − \frac{ϵ′}{3}) dμ(ϵ′) + \frac{1}{2} ∫_{spectre} θ (ϵ − \frac{ϵ′ + 2}{3}) dμ(ϵ′)

= \frac{1}{2}N(3ϵ) + g(ϵ)

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Correlation function or return probability function. Results for the Fibonacci cavity : log-periodic oscillations. Likely to be the same for Cantor.

|ψ(t) > wave function. Definition of the time-averaged correlation function :

C(t) = \frac{1}{t} ∫_{0}^{t} | < ψ(0)|ψ(t′) > |^{2}dt′

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Decomposition over the eigenfunction (orthonormal basis)

|ψ(t) >= ∫_{C} g(ϵ)e^{−iϵt}|ϕ(ϵ) > dμ(ϵ)

Normalization :

< ψ(t)|ψ(t) >= ∫_{C} |g(ϵ)|^{2}dμ(ϵ)

= 1

Define : dμ̃(ϵ) = |g(ϵ)|^{2}dμ(ϵ)

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

C(t) = \frac{1}{t} ∫_{0}^{t} | < ψ(0)|ψ(t′) > |^{2}dt′

= \frac{1}{t} ∫_{0}^{t} | ∫_{C} |g(ϵ|^{2}e^{−iϵt′}dμ(ϵ)|^{2}dt′
= \frac{1}{t} ∫_{0}^{t} ∫∫_{C×C} e^{i(ϵ′−ϵ)t′}dμ̃(ϵ)dμ̃(ϵ′)
= \frac{1}{t} ∫∫_{C×C} \frac{e^{i(ϵ′−ϵ)t} − 1}{i(ϵ′ − ϵ)}dμ̃(ϵ)dμ̃(ϵ′)
= ∫∫_{C×C} sinc((ϵ′ − ϵ)t)dμ̃(ϵ)dμ̃(ϵ′) + i ∫∫_{C×C} \frac{sinc^{2}((ϵ′ − ϵ)t/2)}{(ϵ′ − ϵ)t}

Ariane SORET  Quantum dynamics for a fractal spectrum

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

C(t) = ∫∫_{C×C} sinc((ϵ′ − ϵ)t)dμ̃(ϵ)dμ̃(ϵ′)

Change of variable : l = |ϵ′ − ϵ| dS(l, μ) =area for measure μ of the Cantor carpet in the strips.

C(t) = ∫_{0}^{1} sinc(lt)dS(l, μ)

Figure:

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Scaling property + Mellin to calculate dS(l, μ). Mellin transform :

M_{f} (s) = ∫_{0}^{∞} f (x)x^{s−1}dx

M_{ϕ}^{−1}(x) = \frac{1}{2iπ} ∫_{γ−i∞}^{γ+i∞}ϕ(s)x^{−s}ds

ζ(α) = ∫_{0}^{1} l^{−α}dS(l, μ)
= ∫∫_{C×C} \frac{dμ(x)dμ(y)}{|x − y|^{α}}
= \frac{1}{2} ∫∫_{C×C} \frac{dμ(x)dμ(y)}{3^{−α}|x − y|^{α}}+ ∫∫_{C×C} \frac{dμ(x)dμ(y)}{3^{−α}|x − y + 2|^{α}}
= \frac{3^{α}}{2}ζ(α) + ϵ(α)

Ariane SORET  Quantum dynamics for a fractal spectrum

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

Therefore : ζ(α) = \frac{ϵ(α)}{1− \frac{3^{α}}{2}} and

dS(l, μ) = \frac{1}{2iπ} ∫_{γ−i∞}^{γ+i∞}ζ(1 − α)l^{−α}dα

= ∑_{n}[\frac{α − (1 − α_{n})}{1 − \frac{3^{1−α}}{2}}l^{−α}ϵ(α)]_{α=α_{n}}

1 − α_{n} = \frac{ln(2)}{ln(3)} , d = \frac{ln(2)}{ln(3)}

dS(l, μ) = \frac{l^{d−1}}{ln(3)} ∑_{n} l^{\frac{2iπn}{ln(3)}} ϵ (d + \frac{2iπn}{ln(3)}) = \frac{l^{d−1}}{ln(3)} ∑_{n} l^{\frac{2iπn}{ln(3)}} ϵ_{n}

Mathematical prerequisites

Experimental example : Fibonacci cavity Anaytical study in the case of a Cantor set Measure Correlation function

d_{n} = d + \frac{2iπn}{ln(3)}