Dispersive Quantization the Talbot Effect Peter J. Olver - PowerPoint PPT Presentation

Dispersive Quantization — the Talbot Effect Peter J. Olver University of Minnesota http://www.math.umn.edu/ ∼ olver Varna, June, 2012

Peter J. Olver Introduction to Partial Differential Equations Pearson Publ., to appear (2012?) Amer. Math. Monthly 117 (2010) 599–610

Dispersion Definition. A linear partial differential equation is called dispersive if the different Fourier modes travel unaltered but at different speeds. Substituting u ( t, x ) = e i ( k x − ω t ) produces the dispersion relation ω = ω ( k ) relating frequency ω and wave number k . c p = ω ( k ) Phase velocity: k c g = dω Group velocity: (stationary phase) dk

A Simple Linear Dispersive Wave Equation: ∂t = ∂ 3 u ∂u ∂x 3 = ⇒ linearized Korteweg–deVries equation ω = k 3 Dispersion relation: c p = ω k = k 2 Phase velocity: c g = dω dk = 3 k 2 Group velocity: Thus, wave packets (and energy) move faster (to the right) than the individual waves.

Linear Dispersion on the Line ∂t = ∂ 3 u ∂u u (0 , x ) = f ( x ) ∂x 3 Fourier transform solution: � ∞ 1 f ( k ) e i ( k x + k 3 t ) dk � u ( t, x ) = √ 2 π −∞ Fundamental solution u (0 , x ) = δ ( x ) � � � ∞ u ( t, x ) = 1 1 x −∞ e i ( k x + k 3 t ) dk = √ 3 t Ai √ 2 π 3 3 3 t

Linear Dispersion on the Line ∂t = ∂ 3 u ∂u u (0 , x ) = f ( x ) ∂x 3 Fourier transform solution: � ∞ 1 f ( k ) e i ( k x + k 3 t ) dk � u ( t, x ) = √ 2 π −∞ Fundamental solution u (0 , x ) = δ ( x ) � � � ∞ u ( t, x ) = 1 1 x −∞ e i ( k x + k 3 t ) dk = √ 3 t Ai √ 2 π 3 3 3 t

Linear Dispersion on the Line ∂t = ∂ 3 u ∂u u (0 , x ) = f ( x ) ∂x 3 Fourier transform solution: � ∞ 1 f ( k ) e i ( k x + k 3 t ) dk � u ( t, x ) = √ 2 π −∞ Fundamental solution u (0 , x ) = δ ( x ) � � � ∞ u ( t, x ) = 1 1 x −∞ e i ( k x + k 3 t ) dk = √ 3 t Ai √ 2 π 3 3 3 t

t = . 03 t = . 1 t = 1 / 3 t = 1 t = 5 t = 20

Linear Dispersion on the Line ∂t = ∂ 3 u ∂u u (0 , x ) = f ( x ) ∂x 3 � x − ξ � � ∞ 1 u ( t, x ) = √ −∞ f ( ξ ) Ai √ dξ 3 3 3 t 3 t � 0 , x < 0 , Step function initial data: u (0 , x ) = σ ( x ) = 1 , x > 0 . � � u ( t, x ) = 1 x 3 − H √ − 3 3 t � 2 � � 1 9 z 3 � � 2 � � 2 9 z 3 � z 2 Γ 3 ; 2 3 , 4 3 ; 1 3 ; 4 3 , 5 3 ; 1 z Γ 1 F 2 1 F 2 3 3 H ( z ) = − � 2 � � 4 � � 4 � � 5 � 3 5 / 3 Γ 3 7 / 3 Γ Γ Γ 3 3 3 3

Linear Dispersion on the Line ∂t = ∂ 3 u ∂u u (0 , x ) = f ( x ) ∂x 3 � x − ξ � � ∞ 1 u ( t, x ) = √ −∞ f ( ξ ) Ai √ dξ 3 3 3 t 3 t � 0 , x < 0 , Step function initial data: u (0 , x ) = σ ( x ) = 1 , x > 0 . � � u ( t, x ) = 1 x 3 − H √ − 3 3 t � 2 � � 1 9 z 3 � � 2 � � 2 9 z 3 � z 2 Γ 3 ; 2 3 , 4 3 ; 1 3 ; 4 3 , 5 3 ; 1 z Γ 1 F 2 1 F 2 3 3 H ( z ) = − � 2 � � 4 � � 4 � � 5 � 3 5 / 3 Γ 3 7 / 3 Γ Γ Γ 3 3 3 3

Linear Dispersion on the Line ∂t = ∂ 3 u ∂u u (0 , x ) = f ( x ) ∂x 3 � x − ξ � � ∞ 1 u ( t, x ) = √ −∞ f ( ξ ) Ai √ dξ 3 3 3 t 3 t � 0 , x < 0 , Step function initial data: u (0 , x ) = σ ( x ) = 1 , x > 0 . � � u ( t, x ) = 1 x 3 − H √ − 3 3 t � 2 � � 1 9 z 3 � � 2 � � 2 9 z 3 � z 2 Γ 3 ; 2 3 , 4 3 ; 1 3 ; 4 3 , 5 3 ; 1 z Γ 1 F 2 1 F 2 3 3 H ( z ) = − � 2 � � 4 � � 4 � � 5 � 3 5 / 3 Γ 3 7 / 3 Γ Γ Γ 3 3 3 3

t = . 005 t = . 01 t = . 05 t = . 1 t = . 5 t = 1 .

Periodic Linear Dispersion ∂t = ∂ 3 u ∂u ∂x 3 ∂ 2 u ∂x 2 ( t, − π ) = ∂ 2 u ∂u ∂x ( t, − π ) = ∂u u ( t, − π ) = u ( t, π ) ∂x ( t, π ) ∂x 2 ( t, π ) Step function initial data: � 0 , x < 0 , u (0 , x ) = σ ( x ) = 1 , x > 0 . sin( (2 j + 1) x − (2 j + 1) 3 t ) ∞ u � ( t, x ) ∼ 1 2 + 2 � . π 2 j + 1 j =0

Periodic Linear Dispersion ∂t = ∂ 3 u ∂u ∂x 3 ∂ 2 u ∂x 2 ( t, − π ) = ∂ 2 u ∂u ∂x ( t, − π ) = ∂u u ( t, − π ) = u ( t, π ) ∂x ( t, π ) ∂x 2 ( t, π ) Step function initial data: � 0 , x < 0 , u (0 , x ) = σ ( x ) = 1 , x > 0 . sin( (2 j + 1) x − (2 j + 1) 3 t ) ∞ u � ( t, x ) ∼ 1 2 + 2 � . π 2 j + 1 j =0

Periodic Linear Dispersion ∂t = ∂ 3 u ∂u ∂x 3 ∂ 2 u ∂x 2 ( t, − π ) = ∂ 2 u ∂u ∂x ( t, − π ) = ∂u u ( t, − π ) = u ( t, π ) ∂x ( t, π ) ∂x 2 ( t, π ) Step function initial data: � 0 , x < 0 , u (0 , x ) = σ ( x ) = 1 , x > 0 . sin( (2 j + 1) x − (2 j + 1) 3 t ) ∞ u � ( t, x ) ∼ 1 2 + 2 � . π 2 j + 1 j =0

t = 0 . t = . 1 t = . 2 t = . 3 t = . 4 t = . 5

t = 1 t = 1 t = 1 30 π 15 π 10 π t = 2 t = 1 t = 1 15 π 6 π 5 π

t = 1 t = 1 t = π 2 π 3 π t = 1 t = 1 t = 1 4 π 5 π 6 π t = 1 t = 1 t = 1 7 π 8 π 9 π

At rational time t = 2 πp/q , the solution u � ( t, x ) is Theorem. constant on every subinterval 2 πj/q < x < 2 π ( j + 1) /q . At irrational time u � ( t, x ) is a non-differentiable continuous function.

Lemma. ∞ � c k e i k x f ( x ) ∼ k = −∞ is piecewise constant on intervals 2 πj/q < x < 2 π ( j + 1) /q if and only if c k = � c l , k ≡ l �≡ 0 mod q, c k = 0 , 0 � = k ≡ 0 mod q. � � where 2 πk c k c k = k �≡ 0 mod q. � i q ( e − 2 i π k/q − 1) = ⇒ DFT

The Fourier coefficients of the solution u � ( t, x ) at rational time t = 2 πp/q are � � 2 π p = b k (0) e i ( k x − 2 π k 3 p/q ) , c k = b k q where, for the step function initial data,  − i / ( πk ) , k odd,    b k (0) = 1 / 2 , k = 0 ,    0 , 0 � = k even. Crucial observation: then k 3 ≡ l 3 mod q if k ≡ l mod q, and so e i ( k x − 2 π k 3 p/q ) = e i ( lx − 2 π l 3 p/q )

The Fundamental Solution F (0 , x ) = δ ( x ) Theorem. At rational time t = 2 πp/q , the fundamental solution to the initial-boundary value problem is a linear combination of finitely many periodically extended delta functions, based at 2 πj/q for integers − 1 2 q < j ≤ 1 2 q . Corollary. At rational time, any solution profile u (2 πp/q, x ) to the periodic initial-boundary value problem depends on only finitely many values of the initial data, namely u (0 , x j ) = f ( x j ) where x j = x + 2 πj/q for integers − 1 2 q < j ≤ 1 2 q .

The Fundamental Solution F (0 , x ) = δ ( x ) Theorem. At rational time t = 2 πp/q , the fundamental solution to the initial-boundary value problem is a linear combination of finitely many periodically extended delta functions, based at 2 πj/q for integers − 1 2 q < j ≤ 1 2 q . Corollary. At rational time, any solution profile u (2 πp/q, x ) to the periodic initial-boundary value problem depends on only finitely many values of the initial data, namely u (0 , x j ) = f ( x j ) where x j = x + 2 πj/q for integers − 1 2 q < j ≤ 1 2 q .

⋆ ⋆ The same quantization/fractalization phenomenon appears in any linearly dispersive equation with “integral polynomial” dispersion relation: n � c m k m ω ( k ) = m =0 where c m = α n m n m ∈ Z

Linear Free-Space Schr¨ odinger Equation ∂t = ∂ 2 u i ∂u ∂x 2 ω = k 2 Dispersion relation: c p = ω Phase velocity: k = − k c g = dω Group velocity: dk = − 2 k

Periodic Linear Schr¨ odinger Equation ∂t = ∂ 2 u i ∂u ∂x 2 ∂u ∂x ( t, − π ) = ∂u u ( t, − π ) = u ( t, π ) ∂x ( t, π ) • Michael Berry, et. al. • Bernd Thaller, Visual Quantum Mechanics • Oskolkov • Kapitanski, Rodnianski “Does a quantum particle know the time?” • Michael Taylor • Fulling, G¨ unt¨ urk

William Henry Fox Talbot (1800–1877)

⋆ Talbot’s 1835 image of a latticed window in Lacock Abbey = ⇒ oldest photographic negative in existence.

The Talbot Effect Fresnel diffraction by periodic gratings (1836) “It was very curious to observe that though the grating was greatly out of the focus of the lens . . . the appearance of the bands was perfectly distinct and well defined . . . the experiments are communicated in the hope that they may prove interesting to the cultivators of optical science.” — Fox Talbot = Lord Rayleigh calculates the Talbot distance (1881) ⇒

The Quantized/Fractal Talbot Effect • Optical experiments — Berry & Klein • Diffraction of matter waves (helium atoms) — Nowak et. al.

Quantum Revival • Electrons in potassium ions — Yeazell & Stroud • Vibrations of bromine molecules — Vrakking, Villeneuve, Stolow

Periodic Linear Schr¨ odinger Equation ∂t = ∂ 2 u i ∂u ∂x 2 ∂u ∂x ( t, − π ) = ∂u u ( t, − π ) = u ( t, π ) ∂x ( t, π ) Integrated fundamental solution: e i ( k x + k 2 t ) ∞ u ( t, x ) = 1 � . 2 π k 0 � = k = −∞ ⋆ For x/t ∈ Q , this is known as a Gauss (or, more generally, Weyl) sum, of importance in number theory = ⇒ Hardy, Littlewood, Weil, I. Vinogradov, etc.