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Applications of Quantum Mechanics

Analogías clásico-cuánticas en

Arreglos de guías de onda

Héctor Manuel Moya-Cessa

[email protected] Nacional de Astrofísica,

Optica y Electrónica

Tonantzintla, Pue.

Mexico


Grupo:

Dr. Arturo Zuniga Segundo (IPN)

Estudiantes de maestría y doctorado


Resumen:

•Motivación

•Notación de Dirac

•Préstamo de estructuras de cuántica a clásica

(Gerd Leuchs: Classical entanglement: Oxymoron or resource)

•Ejemplos


Other lectures:

2. Special functions

3. Quantum mechanical invariants

4. Optical realization of a quantum Kerr medium

5. Wigner function

Application

s of

Quantum

Mechanics

From Physics &

Probability,

Grandy Jr. &

Milonni, eds.

Application

s of

Quantum

Mechanics

p.87


01 0 1

2 2

, 1

0 0

0 1 0 0

0 0 0 0ˆ ˆ| 0 1|

0 0 0 0

0 0 0 0

0 1 0 0

0 0 1 0| 1|

0 0 0 1

0 0 0 0

T

j j

j j

V r r

V j j

Bessel states as nonlinearcoherent states


Infinite waveguides array

1 1 0j

z j j

dEik E E

dz

A beam injected

into one of the

waveguides in the

array spreads to the

rest of them by

wave coupling.

Discrete Diffraction


This phenomenon has been referred to as DISCRETE

DIFFRACTION

( ) ( ) 2j

j j zE z i J k z


†

,

| 1|,

| 1 |,

|

n

n

n m

V n n

V n n

n m

†|( ) | ,

| |

z

j

j

d Ei k V V E

dZ

E E j

Schrödinger-like equation

1 1 0j

z j j

dEik E E

dz

01 0 1

2 2

, 1

0 0

0 1 0 0

0 0 0 0ˆ ˆ| 0 1|

0 0 0 0

0 0 0 0

0 1 0 0

0 0 1 0| 1|

0 0 0 1

0 0 0 0

T

j j

j j

V r r

V j j


†

†

( )

1( )

|( ) | ,

| | (0)

|

(2 ) |

| | (2 ) |

z

z

z

ik V V z

k iV ziV

n n

n z

n

n

n z

n

d Ei k V V E

dz

E e E

e m

i J k V m

j E j i J k m n

†

,

1 †

| 1|,

| 1 |,

|

n

n

n m

V n n

V n n

n m

V V

( ) ( ) 2j

j j zE z i J k z


Caminatas aleatorias cuánticas


010 E

Zd

Edi 01 11 nn

n EnEndZ

Edi

†|( ) | 0

d Ei a a E

dZ

Arreglo tipo Glauber/Fock


)0(exp)( aaiZz )(ZmEm

kaZiaZimZEm )exp()(exp)2/exp( 2

1],[ aa

1( ) ( )

| | 1

n na x n x

a n n n

1( ) 1 ( ),

| 1 | 1

n na x n x

a n n n

Ver Arfken: Capítulo Hermite


)()!(

!)()2/(exp)( 22 ZL

sk

kZiZZE s

k

s

sk

)(

!

)!()()2/exp()( 22 ZL

k

skZiZZE s

sk

s

sk


f(n)

A. Perez-Leija, H. MOYA-CESSA, F. Soto-Eguibar, O. Aguilar-Loreto, A.

Szameit, and D.N. Christodoulides, Opt. Commun. 284, No. 7, 1833-1836

(2011). “CLASSICAL ANALOGS TO QUANTUM NONLINEAR

COHERENT STATES IN PHOTONIC LATTICES.”

† † †1 1ˆ ˆ ˆ( 1) , ( 1),A f n a A a f n n a a

su(1,1)


†02 lncoshtan tan| ( ) | (0)

A Zi ZA i ZAz e e e


A. Zúñiga-Segundo, B.M. Rodríguez-Lara, D.J.

Fernández and H.M. MOYA-CESSA,

Optics Express (2013)

Jacobi photonic lattices and their SUSY partners

http://arxiv.org/abs/1310.5176











Optical realization of light matterinteraction

†0

† 0

†0

ˆ ( )2

ˆ(2

(

)

ˆ )( )2

z

n g a a

g a

g a

a

n

n

H a


Application

s of

Quantum

Mechanics

Atom-field interaction detuning=0 Ion-laser interaction

Application

s of

Quantum

Mechanics

Eigenestados

Application

s of

Quantum

Mechanics

ˆ ˆ

ˆ ˆ

( 1) 1 ( 1) 11

2 ( 1) 1 1 ( 1)

n n

n nT

†0

† 0

†0

ˆ ( )2

ˆ(2

(

)

ˆ )( )2

z

n g a a

g a

g a

a

n

n

H a

Application

s of

Quantum

Mechanics

ˆ †0

†

ˆ †0

ˆ ( 1) ( ) 02

ˆ0 ( 1) ( )2

n

T

n

n g a a

H THT

n g a a

( , ) ( , )| 2 0 0 | 2

| (0) , | (0) ,0 | 2 | 2 0

e e e g

T T

n nT T

n n

( , ) ( , )| 2 1 | 2 1 0 0

| (0) , | (0) .0 0 | 2 1 | 2 1

o e o g

T T

n nT T

n n

Application

s of

Quantum

Mechanics

ˆ †0ˆ ( 1) ( )2

n

eH n g a a

.

0 0

0 1

0

1 1

0,2

( 1) 1 0, 12

nn

n n n

dEi E gE

dZ

dEi n E g n E g nE n

dZ


Paraxial wave equation

Suponemos ahora dos medios GRIN pegados

GRaded INdex referring to an optical material

with refractive index in the form of a parabolic curve,

decreasing from the center towards the cladding.


Optical realization of a quantum beam splitter


Splitting in a 50:50 beam splitter

R.A. Campos, B.E.A. Saleh, and M. C. Teich, Phys. Rev. A 40, 1371 (1989).


Paraxial wave equation

Consider the copropagation of two beams, probe and signal, in a

Kerr medium. The probe beam produces the index of refraction

If the probe beam has a Gaussian profile,

Is astigmatic and slightly tilted,a term xy is produced


1 1 2 0 0 2( ) ( ) ( ) ( ) ( ) ( )u x u y u x u y u x u y

† †, , , ,2 2

q q

q q q q q

q ip q ipa a n a a q x y


Tomado de: http://www.idquantique.com

Quantis uses Quantum Physics to create truly-random numbers

Existing randomness sources can be grouped in two classes: software

solutions, which can only generate pseudo-random bit streams, and

physical sources. In the latter, most random generators rely on classical

physics to produce what looks like a random stream of bits. In reality,

determinism is hidden behind complexity.

Contrary to classical physics, quantum physics is fundamentally

random. It is the only theory within the fabric of modern physics

that integrates randomness. Quantis uses this property to generate

random numbers from quantum origin


Conclusiones:


References:

1. H.S Eisenberg et al, PRL 81, 3383 (1998).

2. A. L. Jones, JOSA 55, 261-271 (1965).

3. HMC, F. Soto-Eguibar, Differential Equations: An Operational

Approach, RINTON PRESS, New Jersey, 2011. ISBN 1-58949-060-

4

4. R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, HMC, S. Nolte, D.

N. Christodoulides, and A. Szameit, Phys. Rev. Lett. (2011)

“DISPLACED FOCK STATES AND QUANTUM

CORRELATIONS IN GLAUBER-FOCK PHOTONIC

LATTICES”

5. HMC, F. Soto-Eguibar, J.M. Vargas Martínez, R Juárez-Amaro and A.

Zúñiga-Segundo Physics Reports 513, 229 (2012).

6. R. Mar-Sarao and HMC, Optics Letters 33, No. 17, 1966-1968 (2008),

“OPTICAL REALIZATION OF A QUANTUM BEAM

SPLITTER.”

7. A. Perez-Leija, HMC and D.N. Christodoulides, Physica Sripta T 147,

014023 (2012).

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