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
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
References
1 Motivations and examples
2 Evolution of superoscillations
3 Classes of superoscillatory functions
4 The harmonic oscillator
5 References
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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)
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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?
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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
.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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!.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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).
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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?
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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 .
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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
].
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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).
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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)
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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)
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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).
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations
Motivations and examplesEvolution of superoscillations
Classes of superoscillatory functionsThe harmonic oscillator
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.
Fabrizio Colombo An introduction to Aharonov-Berry superoscillations