Analogías clásico-cuánticas en Arreglos de guías de onda. Héctor Manuel Moya-Cessa

Transcription

1 Analogías clásico-cuánticas en Arreglos de guías de onda Héctor Manuel Moya-Cessa Instituto Nacional de Astrofísica, Optica y Electrónica Tonantzintla, Pue. Mexico

2 Grupo: Dr. Arturo Zuniga Segundo (IPN) Estudiantes de maestría y doctorado

3 Resumen: Motivación Notación de Dirac Préstamo de estructuras de cuántica a clásica (Gerd Leuchs: Classical entanglement: Oxymoron or resource) Ejemplos

4 Other lectures: 2. Special functions 3. Quantum mechanical invariants 4. Optical realization of a quantum Kerr medium 5. Wigner function

5 From Physics & Probability, Grandy Jr. & Milonni, eds. Application s of Quantum Mechanics

6 p.87 Application s of Quantum Mechanics

7 Notación de Dirac rˆ 1 rˆ 2 rˆ 3 rˆ , 1, 2, 3, ˆ j 0 j 0 r j T j k rˆ rˆ j k jk 3 3 x c rˆ x c n n n n n0 n0 T

8 V rˆˆ r T V j j j, j1 j0 j0

9 Bessel states as nonlinear coherent states Infinite waveguides array de dz j ik E E z j1 j1 0

10 Discrete Diffraction A beam injected into one of the waveguides in the array spreads to the rest of them by wave coupling. This phenomenon has been referred to as DISCRETE DIFFRACTION E ( z) ( i) j J 2k z j j z

11 V n n 1, n V n n n n m 1, nm, V rˆˆ r V j j T j, j1 j0 j0 Schrödinger-like equation d E i k V V E dz E E j j j z ( ), de dz j ik E E z j1 j1 0

12 d E i k V V E dz z ( ), ikz ( V V ) z E e E(0) e 1 kz ( iv ) z iv n n m n j E j i J (2 k ) m n n z n n n i J (2 k ) V m z V n n 1, n V n n n n m V V 1 1, nm, E ( z) ( i) j J 2k z j j z

13 Caminatas aleatorias cuánticas

14 Arreglo tipo Glauber/Fock d E d E 0 n i E1 0 i n 1 En 1 n En 1 0 d Z dz d E i a a E dz ( ) 0

15 iz a a (0) ( z) exp E m m (Z) E m exp( Z / 2) m exp ( iza 2 ) exp( iza) k Ver Arfken: Capítulo Hermite a ( x) n ( x) n [ a, a ] 1 n 1 a n n n 1 a ( x) n 1 ( x), n n1 a n n 1 n 1

16 ) ( )! (! ) 2)( / exp ( ) ( 2 2 Z L s k k iz Z Z E s k s s k ) (! )! ( ) 2)( / exp( ) ( 2 2 Z L k s k iz Z Z E s s k s s k

17 A. Perez-Leija, H. MOYA-CESSA, F. Soto-Eguibar, O. Aguilar-Loreto, A. Szameit, and D.N. Christodoulides, Opt. Commun. 284, No. 7, (2011). CLASSICAL ANALOGS TO QUANTUM NONLINEAR COHERENT STATES IN PHOTONIC LATTICES. f(n) 1 1 ˆ ˆ ˆ A f ( n 1) a, A a f ( n 1), n a a

18 su(1,1) i tan ZA 2 A0 ln cosh Z i tan ZA ( z) e e e (0)

19 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

20 Optical realization of light matter interaction ˆ ( ) 2 ˆ ( 2 ( ) ˆ )( ) 2 z n g a a ga ga a n n H a

21 Atom-field interaction detuning=0 Ion-laser interaction Application s of Quantum Mechanics

22 Eigenestados Application s of Quantum Mechanics

23 H 0 nˆ z ga ( a )( ) 2 n g a a 2 0 ga ( a ) nˆ 2 0 ˆ ( ) T nˆ nˆ 1 ( 1) 1 ( 1) 1 nˆ nˆ 2 ( 1) 1 1 ( 1) Application s of Quantum Mechanics

24 H T 0 nˆ nˆ ( 1) g( a a ) 0 2 THT 0 nˆ 0 nˆ ( 1) g( a a ) 2 2n 0 0 2n 0 2 n 2n 0 ( e, e) ( e, g ) T (0) T, T (0) T, 2n1 2n n1 2n1 ( o, e) ( o, g ) T (0) T, T (0) T. Application s of Quantum Mechanics

25 . de 0 0 i E0 ge1 dz de He n g a a 2 0 nˆ ˆ ( 1) ( ) 0, 2 ( 1) 1 0, 1 2 n 0 n i n En g n En 1 g nen 1 n dz Application s of Quantum Mechanics

26 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.

27 Optical realization of a quantum beam splitter

28 R.A. Campos, B.E.A. Saleh, and M. C. Teich, Phys. Rev. A 40, 1371 (1989). Splitting in a 50:50 beam splitter

29

30 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

31 q ipq q ip q a q, aq, nq aqaq, q x, y 2 2 u ( x) u ( y) u ( x) u ( y) u ( x) u ( y)

32

33 Tomado de: 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

34 Conclusiones:

35 References: 1. H.S Eisenberg et al, PRL 81, 3383 (1998). 2. A. L. Jones, JOSA 55, (1965). 3. HMC, F. Soto-Eguibar, Differential Equations: An Operational Approach, RINTON PRESS, New Jersey, ISBN 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, (2008), OPTICAL REALIZATION OF A QUANTUM BEAM SPLITTER. 7. A. Perez-Leija, HMC and D.N. Christodoulides, Physica Sripta T 147, (2012).