frank-condon principle and adjustment of optical waveguides with nonhomogeneous refractive index

Transcription

1 arxiv: v1 [quant-ph] 6 Mar 009 frank-condon principle and adjustment of optical waveguides with nonhomogeneous refractive index Vladimir I. Man ko 1, Leonid D. Mikheev 1, and Alexandr Sergeevich 1 P. N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Pr. 53, Moscow , Russia School of Physics, The University of Sydney, New South Wales 006, Australia s: manko@sci.lebedev.ru mikheev@sci.lebedev.ru a.sergeevich@physics.usyd.edu.au Abstract The adjustment of two different selfocs is considered using both exact formulas for mode connection coefficients expressed in terms of Hermite polynomials of several variables and qualitative approach based on Frank-Condon principle. Several examples of the refractive index dependence are studied and illustrative plots for these examples are presented. Connection with tomographic approach to quantum states of two-dimensional oscillator and Frank-Condon factors is established. Keywords: Frank-Condon principle, entanglement, optical waveguide, molecular spectra, Hermite polynomials. 1. Introduction It is known [1, ] that the radiation beams propagating in optical waveguides can be described by Schrödinger-like equations [3, 4]. The role of time t in this equation played by longitudinal coordinate z and the role of Plank constant is played by wavelength of the radiation. The refractive index profile is an analogue of the potential energy in the quantum Schrodinger equation. The modes of the electromagnetic radiation in the optical waveguides are analogs of the wave functions of a quantum system. This quantum-like picture of the field propagating in the waveguide was intensively used to study connection of the energy distributions among the modes of the successive waveguides for two (or several waveguides with different refractive index profiles provided the distribution in the initial waveguide is known. This problem is equivalent to finding in quantum mechanics of two-atom (or several-atom molecules the vibronic structure of electronic lines in the absorption or emission spectra of polyatomic molecules. In case of harmonic potential the 1

2 structure is determined in terms of Frank-Condon factors expressed through Hermite polynomials of several variables [5, 6, 7]. Recently technique of femtosecond pulses was developed and applied in different domains of ultrapower laser physics [8, 9]. In this domain the problem of waveguiding such kind of radiation also arises [10, 11]. One of the aims of this work is to consider analogs of the quantum problems of the molecular spectra intensively discussed in [1, 13, 14, 15] to transport the results of these investigations into domain of femtosecond laser pulse physics. Another aim of this work is to connect the classical problem of optical waveguide properties with quantum problem of entanglement [16, 17]. The point is that for states with two degrees of freedom the notion of entanglement corresponds to the degree of correlation of observables related to these different degrees of freedom. In waveguide picture the entanglement analog is related to the structure of the modes propagating in the waveguides. When the initial separable two-mode field in first waveguide is propagating into second waveguide with different profile of refractive index the entanglement in the arising modes in this waveguide appears. This entanglement can be considered and related to the energy distribution among the field modes in the second waveguide. In the work we discuss this relation and consider possibility to find connection with different entanglement criteria [18, 19, 0, 1, ]. The paper is organized as follows. In next Section we review the results, related to modes in planar waveguides with quadratic refractive index. In Section 3 the two-mode problem in selfocklike waveguides will be considered and analog of two-mode squeezed light wave function will be used for considering the Frank-Condon factors. The entanglement analogs of the waveguides modes will be discussed in Section 4. The conclusion and perspectives are presented in Section 5.. Planar Waveguides In this section we will consider a transition of a light beam from one planar waveguide with quadratic refractive index to another along z coordinate. The refractive index, being constant by y and z represents here a symmetric potential energy curve of a one-electron atom: n(x = kx U(x = ω x. (1 The second waveguide has different dependence of a refractive index and its axe is shifted from the axe of the first waveguide by d: n (x = k (x d U(x = ω (x d, ( which is equivalent to shifting and stretching or shrinking of the potential energy curve. Here we take m = m = 1. The most probable energy level, to which electron jumps could be found by the Frank-Condon principle. Taking into consideration that energy levels in a parabolic potential are distributed as E n = hω(n + 1, the final state could be easily found.

3 Fig. 1. The plot of P n 0 for ω ω = 3 and ωd = 9. Let us consider a one-electron atom. approximation is equal to The nuclear hamiltonian in the Born-Oppenheimer H = 1 hω{a, a+ }, (3 where ω is a frequency, corresponding to the coordinate q, a and a + are the operators of birth and annihilation. The wave function of electron in atom with hamiltonian (1 is described by ψ (x, n, ω = ( π 1 n n!l(ω ( ( 1 exp x x H l n, (4 (ω l(ω where l(ω = h, n is a vibrational quantum number and H ω n(ξ is an nth Hermite polynomial. The shift of the center of potential from x = 0 to x = d and changing ω ω, n n gives a wave function ψ (x, n, ω, d = ( π 1 n n!l(ω 1 exp ( (x d H l (ω n ( x d. (5 l(ω The overlap integral n n = ψ(x, n, ω, 0ψ(x, n, ω, ddx (6 3

4 Fig.. The plot of P n 3 for ω ω = 3 and ωd = 16. describes the amplitude of probability of the transfer ψ(x, n, ω, 0 ψ(x, n, ω, d. (7 The integral (4 can be expressed through the Hermite polynomial, depending on two variables: n n = ( n+n n!n! ( 1 l(ωl(ω l(ω + l(ω exp d H {R} (l(ω + l(ω nn (y 1, y. (8 The arguments of the Hermite polynomial are ( l(ω R = l(ω l(ωl(ω l(ω +l(ω l(ωl(ω l(ω + l(ω ( ( y1 = dl(ω 1 (9 y l(ω +l(ω l(ω l(ω The probability of transition from the state n to n is equal to P n n = ( n n. (10 To illustrate the distribution of probability we will construct a graph of the function Pn n. On the Fig. 1 the plot for the initial level n = 0, potential stretching ω = 3 and shift ω ωd = 9 is shown. 4

5 Fig. 3. The plot of P n x,n y 0,0 for ω x ω x =, ω y ω y = 3, ω x d x = 9, ω y d y = 16. The maximum probability is observed for the 13th level. This result is in chime with the number, calculated directly from the Frank-Condon principle. The graph for transition from n = 3 with ω ω = 3 and ωd = 16 is shown. In this case the most probable final state is Elliptic Waveguides Now let us extend our considerations to the case of elliptical waveguides with quadratic refractive index, which corresponds to one-electron atom with a 3D parabolic potential: n(x, y = k x x + k y y U(x, y = ω xx + ω y y. (11 5

6 Fig. 4. The plot of P n x,n y,1 for ω x ω x Analogously, the shifted and deformed potential is =, ω y ω y = 3, ω x d x = 16, ω y d y = 16. U (x, y = ω x(x d x + ω y(y d y. (1 The unshifted wave function of electron in potential (11 is ψ (x, y, n x, n y, ω x, ω y, 0, 0 = exp ( ( 1 x + y ( ( l (ω x l (ω y x y H (π nx+ny n x!n y!l(ω x l(ω y 1 nx H ny. (13 l(ω x l(ω y In the same way, we can find the probability distribution in D case for the transition ψ (x, y, n x, n y, ω x, ω y, 0, 0 ψ ( x, y, n x, n y, ω x, ω y, d x, d y by calculating the overlap integral n n = ψ (x, y, n x, n y, ω x, ω y, 0, 0 ψ ( x, y, n x, n y, ω x, ω y, d x, d y dxdy. (15 6 (14

7 The Fig. 3 presents the distribution for the transition from the basic level for ω x ω x =, ω y ω y = 3, ω x d x = 9, ω y d y = 16. The maximum of this function is observed for the final state 3, 8. In the Fig. 4 the 3D plot of transition probability for the initial state, 1 and ω x ω y ω y ω x =, = 3, and ω x d x = ω y d y = 16 is depicted. The state 8, 4 is the most probable final state. The pictures are obviously very similar to the planar case. The transition probability distribution for the initial state of 0, 0 has the form reminding Gaussian. For the transitions from not the base state, we also can observe a multi-maximum surface. Since the potential (1 has no the xy component, the wave created as a result of transition doesn t contain the entanglement term. To get the state analogous to entangled, we should actually consider the potential with this term. U(x, y = ω xx + ω y y + γxy The wave function after the transition will have the following form. ψ (x, y = exp ( ( 1 x + y l (ω x l (ω y+γxy H (π nx+ny n x!n y!l(ω x l(ω y 1 nx (ax + by H ny (cx + dy. (17 But the resulting picture for probability distribution will change slightly in this case, so we are presenting just plots for potential (1 with γ = 0, keeping in mind that the actual graphs are very close to these. 4. Acknowledgments V.I.M. acknowledges the support of the Russian Foundation for Basic Research under Project No (16 References 1. M. A. Man ko and G. T. Mikaelyan, Sov. J. Quantum Electron., 13, 1506 ( M. A. Man ko, V. I. Man ko and R. V. Mendes, Phys. Lett. A, 88, 13 ( M. A. Leontovich, Izv. Akad. Nauk SSSR, Ser. Fiz., 8, 16 ( V. A. Fock and M. A. Leontovich, Zh. Eksp. Teor. Fiz., 16, 557 ( E. V. Doktorov, I. A. Malkin and V. I. Man ko, J. Mol. Spectr., 56, 1 ( E. V. Doktorov, I. A. Malkin and V. I. Man ko, J. Mol. Spectr., 64, 30 (

8 7. I. A. Malkin and V. I. Man ko, Dynamical Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow ( L. D. Mikheev, Laser Part. Beams, 10, 473 ( V. I. Tcheremiskine, M. L. Sentis, L. D. Mikheev, Appl. phys. lett., 81, 403 ( S. Jackel, R. Burris, J. Grun, A. Ting, C. Manka, K. Evans and J. Kosakowskii, Opt. Lett., 10, 1086 ( F. Gerome, K. Cook, A. K. George, W. J. Wadsworth and J. C. Knight, Optics Express, 1, 835 ( A. Toniolo, M. Persico, J. Comput. Chem.,, 968 ( H. Kikuchi, M. Kubo, N. Watanabe, and H. Suzuki, J. Chem. Phys., 119, 79 ( P. T. Ruhoff, Chem. Phys., 186, 355 ( P. T. Ruhoff, M. A. Ratner, Int. J. Quant. Chem., 77, 383 ( E. Schrödinger, Proc. Camb. Phil. Soc., 31, 555 ( A. Einstein, B. Podolsky, N. Rosen, Phys. Rev., 47, 777 ( V. I. Manko, A. A. Sergeevich, J. Russ. Laser Res., 8, 516 ( R. Simon, Phys. Rev. Lett., 84, 76 ( A. S. Chirkin and M. Yu. Saigin, Acta Phys. Hung. B, 6/1-, 63 ( M. Yu. Saigin and A. S. Chirkin, Mod. Probl. Stat. Phys., 5, 169 (006.. A. S. Chirkin and M. Yu. Saigin, J. Russ. Laser Res., 8, 505 (007. 8