politecnico di milano - mathe.tu-freiberg.debernstei/web5/fab_hannover_2016.pdf · politecnico di...

Motivations and examplesEvolution of superoscillations

Classes of superoscillatory functionsThe harmonic oscillator

References

An introduction to Aharonov-Berrysuperoscillations

Fabrizio Colombo

Politecnico di Milano

MOIMA - Symposium on Mathematical Optics, Image Modellingand Algorithms, Hannover, 2016

Fabrizio Colombo An introduction to Aharonov-Berry superoscillations



References

1 Motivations and examples

2 Evolution of superoscillations

3 Classes of superoscillatory functions

4 The harmonic oscillator

5 References




References

The Aharonov weak measuremets

A weak measurement of a quantum observable A, involving a pre-selectedstate |ψ0 > and a post-selected state |ψ1 > lead to the weak value

Aweak :=< ψ1|A|ψ0 >

< ψ1|ψ0 >= A + iA′.

The real A and the imaginary part A′, as it is now well understood can beinterpreted as the shift A and the momentum A′ of the pointer recordingthe measurement.An important feature of the weak measurement is the, in contrast to themore familiar von Neumann measurements (strong measurements), givenby the expectation value

Astrong :=< ψ|A|ψ >

the real part A of Aweak can be very large with respect to Astrong , because< ψ1|ψ0 > can be very small when the states are almost orthogonal.




References

An example from the weak measurement

Let a > 1 be a real number. From the weak measurement we have

Fn(x , a) :=(

cos(x

n

)+ ia sin

(x

n

))n=

n∑k=0

Ck(n, a)e i(1−2k/n)x ,

Ck(n, a) =

(n

k

)(1 + a

2

)n−k (1− a

2

)k

.

Fix x ∈ R, and we let n go to infinity, we immediately obtain that

limn→∞

Fn(x , a) = F (x , a) = e iax .

But the convergence is uniform just on the compact sets of R. Observethe superoscillation:

e i(1−2k/n)x , with |1− 2k/n| ≤ 1, but e iax , with a > 1.




References

Y. Aharonov and D. Rohrlich,Quantum Paradoxes: Quantum Theory for the Perplexed(Weinheim: Wiley- VCH) (2005).

Aharonov Y., Popescu S. and Tollaksen J.,A time-symmetric formulation of quantum mechanics, Phys. Today,63 (2010), 27–33.

M. V. Berry, M. R. Dennis, B. Mc Roberts, P. Shukla,Weak value distributions for spin 1/2, J. Phys. A: Math. Theor., 44(2011), 205301 (8pp)




References

M. Berry: superoscillations in optics

The Bessel wave appears for example in optics

ψ`(r) = J`(r)e i`ϕ, ` > 0 (1)

where J`(r) is the Bessel function and r = (r cosϕ, r sinϕ)) andrepresents an optical vortex of strength ` at r = 0. The vortex is thephase singularity. The function (1) is an exact solution of the free-spaceHelmholtz equation in the plane r, with wave number k = 1 andwavelength λ = 2π.

M. Berry, A note on suproscillations associated with Bessel beams,J. Opt. (2013).

There are many papers of sir. M. Berry related to optics, see hisweb page: 500 papers.




References

Problem (Aharonov’s problem)

Do superoscillations persists in time when we consider the evolutionunder Schrodinger equation?

Problem (Mathematical problem)

a) How large is the class of superoscillatory functions and how tocompute the limit explicitly?

b) How to extend to the case of several variables?




References

Evolution in the case of free particle

Cauchy problem

i∂ψ(x , t)

∂t= −∂

2ψ(x , t)

∂x2, ψ(x , 0) = Fn(x).

Theorem

The time evolution of the spatial superoscillating function Fn(x), is givenby

ψn(x , t) =n∑

k=0

Ck(n, a)e i(1−2k/n)xe−it(1−2k/n)2

.




References

Problem

To see if superoscillations persist in time we have to compute the limit

limn→∞

ψn(x , t) = ψ(x , t)

and see if ψ(x , t) is a superoscillatory function.




References

The problem to compute the limit

Theorem

The function

ψn(x , t) =n∑

k=0

Ck(n, a)e ix(1−2k/n)e−it(1−2k/n)2

. (2)

can be written as

ψn(x , t) =∞∑

m=0

(it)m

m!

d2m

dx2mFn(x)

for every x ∈ R and t ∈ R.

Idea of the proof. We consider the expansion

e−it(1−2k/n)2

=∞∑

m=0

[−it(1− 2k/n)2]m

m!.




References

We get

ψn(x , t) =∞∑

m=0

(−it)m

m!

n∑k=0

Ck(n, a)(1− 2k/n)2me ix(1−2k/n)

which can be written as

ψn(x , t) =∞∑

m=0

(it)m

m!

n∑k=0

Ck(n, a)d2m

dx2me ix(1−2k/n)

=∞∑

m=0

(it)m

m!

d2m

dx2m

n∑k=0

Ck(n, a)e ix(1−2k/n)

=∞∑

m=0

(it)m

m!

d2m

dx2mFn(x).




References

Problem

limn→∞

ψn(x , t) = limn→∞U(t,Dx)Fn(x)

= U(t,Dx) limn→∞

Fn(x)

= U(t,Dx)F (x)

where

U(t,Dx) =∞∑

m=0

(it)m

m!

d2m

dx2m.

What is the class of functions on which the operators U(t,Dx) actscontinuously?




References

We compute the limit formally

Recall thatFn(x)→ e iax

so we obtain

ψ(x , t) =∞∑

m=0

(it)m

m!

d2m

dx2me iax

=∞∑

m=0

(it)m

m!(ia)2me iax

=∞∑

m=0

(−ia2t)m

m!e iax

= e iax−ia2t .




References

Important remark

The formal solution of the above problem shows us the way to enlargethe class of superoscillatory functions:

limn→∞

ψn(x , t) = limn→∞

n∑k=0

Ck(n, a)e ix(1−2k/n)e−it(1−2k/n)2

= e iax−ia2t

If we set x = 0 we have

limn→∞

ψn(0, t) = limn→∞

n∑k=0

Ck(n, a)e−it(1−2k/n)2

= e−ia2t

so

limn→∞

n∑k=0

Ck(n, a)e−it(1−2k/n)2

= e−ia2t

This observation opens the way to construct a larger class ofsuperoscillatory functions using PDE.




References

Analytically Uniform Spaces of Leon Ehrenpreis

The action of the infinite order differential operators is well defined onspaces of holomorphic functions with growth conditions. The ingredientsare

We replace x by z ∈ C.

The operator becomes U(t,Dz) :=∑∞

m=0

(it)m

m!

d2m

dz2m

Its symbol (via the Fourier-Borel transform) is

U(t, ξ) :=∞∑

m=0

(it)m

m!ξ2m, ξ ∈ C.

Entire functions are of the form

F (z) =∞∑n=0

anzn, z ∈ C.




References

Theorem

For any value of t, the operator U(t,Dz) acts continuously on the space

A2,0 :={

f ∈ O(C) | ∀ε > 0 ∃aε | |f (z)| ≤ aεeε|z|2}

of entire functions of order less or equal 2 and of minimal type.

The sequence of functions

Fn(x) =n∑

k=0

Ck(n, a)e i(1−2k/n)x

extend to entire functions

Fn(z) =n∑

k=0

Ck(n, a)e i(1−2k/n)z ∈ A2,0.




References

Definition (Generalized Fourier sequence (or function))

We call generalized Fourier sequence a sequence of the form

Yn(x , a) :=n∑

j=0

Cj(n, a)e ikj (n)x (3)

where a ∈ R, Cj(n, a) and kj(n) are real valued functions of the variablesn, a and n, respectively.

The sequence of partial sums of a Fourier expansion is a particular caseof this notion with Cj(n, a) = Cj ∈ R and kj(n) = kj ∈ R.




References

Definition (Superoscillating sequence)

Let a ∈ R. A generalized Fourier sequence

Yn(x , a) =n∑

j=0

Cj(n, a)e ikj (n)x

is said to be a superoscillating sequence if:

|kj(n)| ≤ 1;

there exists a compact subset of R, which will be called asuperoscillation set, on which Yn converges uniformly to e ig(a)x

where g is a continuous real value function such that |g(a)| > 1.

The classical Fourier expansion is obviously not a superoscillatingsequence since its frequencies are not, in general, bounded by one.




References

Problem

Let Ck(n, a) be as for the Fn, and let a ∈ R, t ∈ [−T ,T ] where T is anyreal positive number. Show that the sequence

Yn(t, a, p) =n∑

k=0

Ck(n, a)e it(1−2k/n)p

is superoscillating for p ∈N and for a suitable function f .

A direct method for Fn(x) and e iax is based on the identity

|Fn(x)− e iax |2 = 1 +(

cos2(x

n

)+ a2 sin2

(x

n

))n−2(

cos2(x

n

)+ a2 sin2

(x

n

))n/2cos[n arctan

(a tan

(x

n

))− ax

].




References

Theorem

Consider, for p even, the Cauchy problem for the modified Schrodingerequation

i∂ψ(x , t)

∂t= −∂

pψ(x , t)

∂xp, ψ(x , 0) = Fn(x). (4)

Then the solution ψn(x , t; p), is given by

ψn(x , t; p) =n∑

k=0

Ck(n, a)e ix(1−2k/n)e it(−i(1−2k/n))p .

Moreover, for all t ∈ [−T ,T ], where T is any real positive number, wehave

limn→∞

ψn(x , t; p) = e it(−ia)pe iax ,

for x ∈ K , where K is any compact sets in R.




References

Corollary

limn→∞

ψn(0, t; p) = limn→∞

n∑k=0

Ck(n, a)e it(−i(1−2k/n))p = e it(−ia)p ,

The operators to consider

∞∑m=0

(it)m

m!

dmp

dzmp.




References

More general superoscillatory functions

We replace∂pψ(x , t)

∂xp

in

i∂ψ(x , t)

∂t= −∂

pψ(x , t)

∂xp

by a series of derivatives

G (d

dx) =

∞∑p=0

apdp

dxp, ap ∈ R.




References

Theorem

Under conditions, the convolution equation

i∂ψ(z , t)

∂t= −G (

d

dz)ψ(z , t), ψ(z , 0) = Fn(z), (5)

has the solution ψn(z , t), is given by

ψn(z , t) =n∑

k=0

Ck(n, a)e−iz(1−2k/n)e itG(−i(1−2k/n))

Moreover, for all fixed t we have

limn→∞

ψn(z , t) = e itG(ia)e iaz ,

and for z on the compact sets of C.




References

Fundamental solution harmonic oscillator

As it is well known, the solution of the Cauchy problem for the quantumharmonic oscillator can be written explicitly using its Green functionG (t, x , x ′). Thus the solution of the Cauchy problem

i∂ψ(x , t)

∂t=

1

2

(− ∂2

∂x2+ x2

)ψ(x , t), ψ(0, x) = ψ0(x) (6)

is

ψ(x , t) =

∫R

G (t, x , x ′)ψ0(x ′)dx ′, (7)

where

G (t, x , x ′) := (2πi sin t)−1/2e(2xx′−(x2+x′2) cos t)/(2i sin t).




References

Theorem

Let a ∈ R. Then the solution of the Cauchy problem

i∂ψa(x , t)

∂t=

1

2

(− ∂2

∂x2+ x2

)ψa(x , t), ψ(0, x) = e iax (8)

is

ψa(x , t) = (cos t)−1/2 exp(−(i/2)(x2 + a2) tan t + iax/ cos t). (9)




References

Theorem

Let Fn(x) be the superoscillating sequence defined above. Then thesolution of the Cauchy problem

i∂ψ(x , t)

∂t=

1

2

(− ∂2

∂x2+ x2

)ψ(x , t), ψ(x , 0) = Fn(x) (10)

is

ψn(x , t) = (cos t)−1/2 exp(−(i/2)x2 tan t)

×n∑

k=0

Ck(n, a)e−(i/2)(1−2k/n)2 tan t+ix(1−2k/n)/ cos t .

(11)

Moreover, if we set ψ(x , t) = limn→∞ ψn(x , t), then

ψ(x , t) = (cos t)−1/2e−(i/2)(x2+a2) tan t+iax/ cos t . (12)




References

Idea of the strategy

With some computations, we obtain

ψn(x , t) =e−(i/2)x

2 tan t

(cos t)1/2

∞∑m=0

1

m!

( i

2sin t cos t

)m ∂2m

∂x2mFn(x/ cos t).

Define

U(t) = U(t, ∂x) :=∞∑

m=0

1

m!

( i

2sin t cos t

)m ∂2m

∂x2m.

If U is continuous on a function space that contains function Fn, then

ψ(x , t) := limn→∞

ψn(x , t) = (cos t)−1/2e−(i/2)x2 tan t U(t) lim

n→∞Fn(x/ cos t)

= (cos t)−1/2e−(i/2)x2 tan t U(t)F (x/ cos t)

since Fn(x/ cos t) converges uniformly to F (x/ cos t) on the compact set|x | ≤ M, where M is any positive real number, for every fixed t in[0, π/2).




References

Mathematical theory and Quantum Mechanics

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,Some mathematical properties of superoscillations, J. Phys. A, 44(2011), 365304 (16pp).

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,On the Cauchy problem for the Schrodinger equation withsuperoscillatory initial data, J. Math. Pures Appl., 99 (2013),165–173.

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,Superoscillating sequences as solutions of generalized Schrodingerequations, J. Math. Pures Appl., 103 (2015), 522–534.

R. Buniy, F. Colombo, I. Sabadini, D.C. Struppa, QuantumHarmonic Oscillator with superoscillating initial datum, J. Math.Phys. 55, 113511 (2014).

Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen,The mathematics of superoscillations, to appear in Memoirs of theAmerican Mathematical Society.




References

Mathematical theory for superoscillations

B.A. Taylor, Some locally convex spaces of entire functions, 1968Entire Functions and Related Parts of Analysis (Proc. Sympos. PureMath., La Jolla, Calif., 1966) pp. 431-467 Amer. Math. Soc.,Providence, R.I.

L. Ehrenpreis, Fourier Analysis in Several Complex Variables, WileyInterscience, New York 1970.




References

Quantum Mechanics and Optics

Y. Aharonov, D. Albert, L. Vaidman, How the result of ameasurement of a component of the spin of a spin-1/2 particle canturn out to be 100, Phys. Rev. Lett., 60 (1988), 1351-1354.

M. V. Berry, Faster than Fourier, 1994, in Quantum Coherence andReality; in celebration of the 60th Birthday of Yakir Aharonov ed.J.S.Anandan and J. L. Safko, World Scientific, Singapore, pp 55-65.

M. Berry, M.R. Dennis, Natural superoscillations in monochromaticwaves in D dimension, J. Phys. A, 42 (2009), 022003.

M. V. Berry, S. Popescu, Evolution of quantum superoscillations,and optical superresolution without evanescent waves, J. Phys. A,39 (2006), 6965–6977.

J. Lindberg, Mathematical concepts of optical superresolution,Journal of Optics 14 (2012) 083001.


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