Abstract

We revisit electromagnetic field propagation through tight-binding arrays of coupled photonic waveguides, with properties independent of the propagation distance, and recast it as a symmetry problem. We focus our analysis on photonic lattices with underlying symmetries given by three well-known groups, SU(2), SU(1, 1) and Heisenberg-Weyl, to show that disperssion relations, normal states and impulse functions can be constructed following a Gilmore-Perelomov coherent state approach. Furthermore, this symmetry based approach can be followed for each an every lattice with an underlying symmetry given by a dynamical group.

© 2015 Optical Society of America

1. Introduction

Photonic lattices, that is, tight-binding arrays of photonic waveguides, have proved a valuable tool in the classical simulation of quantum physics [1, 2]. Vice versa, the insight gained from these quantum-classical analogies has been useful as a tool for photonic lattice design [3]. This comes from the fact that the differential equation set describing the propagation of field amplitudes through an array of nearest neighbor coupled waveguides [4–10 ],

iddzl(z)=ωf(l,z)l(z)+λ[g(l,z)l1(z)+g(l+1,z)l+1(z)],
with l = 0, 1, 2,... and the restriction g(0, z) = 0, can be cast in a vector differential equation form similar to Schrödinger equation,
iddz|(z)=H^(z)|(z).
Here, the field at the l-th waveguide is l(z), the effective waveguide refractive index is ωf(l, z), and the effective nearest neighbor coupling λg(l, z) [3]. We have borrowed Dirac notation from quantum mechanics, such that vectors are written as kets,
|(z)=(0(z)1(z)2(z)),
l=0l(z)|j,
that collect all the information from the field amplitudes, and the real-symmetric tridiagonal matrices that collect all the properties of the tight binding array are written as operators,
H^(z)=(ωf(0,z)λg(0,z)0λg(0,z)ωf(1,z)λg(1,z)0λg(1,z)ωf(2,z)λg(2,z)),
ωA^0(z)+λ[A^+(z)+A^(z)].
The operator  0(z) = f(, z) is a diagonal matrix, the operators  +(z) = g(, z) and  (z) = V̂g(, z) are lower- and upper-diagonal matrices in terms of the step, |j〉 = j|j〉, up-step, |j〉 = |j + 1〉, and down-step, |j〉 = |j − 1〉 operators. Note that the functions related to the effective index and coupling parameters, f(l, z) and g(l, z), have been rewritten in terms of the step operator, with action f(, z)|l〉 = f(l, z)|l〉 and g(, z)|l〉 = g(l, z)|l〉, such that it is straightforward to recover the original differential set. At this point, it is clear that the field amplitude distribution at a point z in the photonic lattices is given by the solution to the Schrödinger-like equation in (2),
|(z)=ei0zH^(x)dx|(0),
where the information from the initial field amplitudes distribution impinging the lattice is collected in the vector |(0)〉. It is straightforward to connect this result with the impulse function, m,n(ω, λ, z), that provides the field amplitude at the n-th waveguide given that the initial field impinged just the m-th waveguide,
m,n(ω,λ,z)=n|ei0zH^(x)dx|m.
Note that it is typically hard to calculate the propagator ei0zH^(x)dx but underlying symmetries may provide help [11, 12].

Our goal here is to bring forward the fact that underlying symmetries may simplify greatly the calculations of dispersion relations, normal modes and impulse functions for photonic lattices. We will focus in the simplest case of lattices which parameters do not depend on the propagation distance. In this case, if the operators  0 and  ± form a closed group under commutation [Â, ] = ÂB̂B̂Â, it is possible to use so-called disentangling formulas [13–15 ] to calculate the corresponding impulse function,

m,n(ω,λ,z)=n|eizH^|m,
=n|eiz[ωA^0+λ(A^++A^)]|m,
=n|ea+(z)A^+ea0(z)A^0ea(z)A^|m,
in terms of a particular class of generalized Gilmore-Perelomov coherent states [16–18 ],
D^[a0(z),a±(z)]|mea+(z)A^+ea0(z)A^0ea(z)A^|m.
Thus, these photonic lattices can be seen as optical simulators of generalized Gilmore-Perelomov coherent states. In the following sections, we will provide three working examples of lattices with an underlying symmetry and show that a symmetry based, Gilmore-Perelomov generalized coherent state approach can help us providing dispersion relations, normal modes and impulse functions. First we will work with finite lattices with underlying SU(2) symmetry; the so-called Jx photonic lattices used to produce coherent transfer between input and output waveguides [19, 20] are a particular example of this underlying symmetry. Then we will move to semi-infinite lattices described by the SU(1, 1) group which have as particular realization in the literature the so-called Jacobi lattices [21]. Finally, we will explore lattices with underlying Heisenberg-Weyl symmetry that includes as a specific example the so-called Glauber-Fock lattices [22–26 ] and close with a summary. In all the following cases, the closed forms for the respective impulse functions, in terms of the corresponding generalized Gilmore-Perelomov coherent states, are new in both the optics and quantum mechanics literature, to the best of our knowledge. In order to corroborate them, all the analytic results were compared with numeric diagonalization, for normal modes, and with results from a normal-modes-decomposition and small-step propagation, for the impulse functions. In all cases analytic and numeric results are in good agreement.

2. Lattices with underlying SU(2) symmetry

Let us start with a finite array of coupled waveguides described by the matrix,

H^SU(2)(ω,λ,j)=ω(n^j)+λ[V^n^(2j+1n^)+n^(2j+1n^)V^],
ωJ^0+λ(J^+J^+),
in terms of the SU(2) group [Ĵ +, Ĵ ] = 2Ĵ 0 and [Ĵ 0, Ĵ ±] = ±Ĵ ± [27] and j a positive integer such that 2j + 1 is the size of the lattice. It is straightforward to construct a Gilmore-Perelomov displacement operator [16–18 ],
D^SU(2)(θ)=eθJ^+θ*J^,
=eθ|θ|tan|θ|J^+elnsec2|θ|J^0eθ*|θ|tan|θ|J^,
that diagonalizes this matrix,
D^SU(2)(θ)H^SU(2)(ω,λ)D^SU(2)(θ)=ω2+4λ2(n^j),
with
tan2θ=2λω.
Thus, we can immediately construct the dispersion relation for this photonic lattice,
ΩSU(2)(m,ω,λ,j)=ω2+4λ2(mj),
and its normal modes, given by what we are going to call SU(2) displaced number states, such that the n-th component of the m-th normal mode is given by the expression,
n|ΩSU(2)(m,ω,λ,j)n|D^SU(2)(θ)|m,
=(1)n(2jm)(2jn)(cos|θ|)2jmn(sin|θ|)m+n××Km(n,sin2|θ|,2j),
where we have used the fact that the lattice parameters are real and positive, λ > 0 and ω ≥ 0. The notation (ab) and Kn(x, p, N) stand for the binomial coefficient and Krawtchouk polynomials [28], which are well known in the discrete optics community [29–32 ], in that order. Now, it is also possible to compute the impulse function of this array by using standard disentangling formulas for SU(2) [13–15 ], and some algebra, in terms of generalized Gilmore-Perelomov coherent states,
m,n(SU(2))(ω,λ,j,z)=n|eizH^SU(2)|m,
=(2jm)(2jn)(2iλsinz2ω2+4λ2)m+n(ω2+4λ2)j××(ω2+4λ2cosz2ω2+4λ2iωsinz2ω2+4λ2)2jmn××Km(n,4λ2ω2+4λ2sin2z2ω2+4λ2,2j).
Note that the impulse function will show a 2π/ω2+4λ2 periodicity due to the squared sine argument in the Krawtchouk polynomials. Figure 1 shows the periodicity discussed above in the intensity, that is, the squared impulse function, |m,n(SU(2))(ω,λ,j,z)|2, of an initial field amplitude impinging just the central waveguide of an SU(2) lattice with parameters λ = 0.4ω and j = 5.

 figure: Fig. 1

Fig. 1 Intensity at the n-th waveguide, |m,n(SU(2))(ω,λ,j,z)|2, for light initially impinging the middle waveguide, m = 5, of an SU(2) array with parameters λ = 0.4ω and j = 5.

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In the specific case provided by waveguides with identical effective refractive index, such that the inclusion of an overall constant phase leads to ω = 0, the Hamiltonian-like matrix becomes,

H^SU(2)(0,λ,j)=2λJ^x,J^x=12(J^++J^).
This may be the origin of the Jx lattice moniker for such restriction [19, 20]. In this case it is straightforward to take the limit,
limω012arctan2λω=π4
and use this value to find the normal modes,
n|Ω(m,0,λ,j)SU(2)=12jKm(n,12,2j),
and the impulse function,
m,n(SU(2))(0,λ,j,z)=(2jm)(2jn)(isinλz)m+n(cosλz)2jmn××Km(n,sin2λz,2j).
It is straightforward to notice that the lattice shows a π/λ periodicity. It is also simple to check the properties of Krawtchouk polynomials and realize that the initial field amplitude at the m-th waveguide will be transferred to the (2jm)-th waveguide at half the periodicity, λz = π/2,
m,n(SU(2))(0,λ,jπ2λ)=δn,2jm,
where the notation δa,b stands for Kronecker delta that is equal to one when a = b and zero for all other cases. This has been used for coherent transfer in the so-called Jx lattices [19, 20] and is an immediate consequence of the impulse function given in terms of trigonometric functions and Krawtchouk polynomials. Figure 2 shows this coherent transfer and the periodicity discussed above in the intensity, |m,n(SU(2))(ω,λ,j,z)|2, of an initial field amplitude impinging just the m = 2 waveguide of an SU(2) lattice with parameters ω = 0 and j = 5.

 figure: Fig. 2

Fig. 2 Intensity at the n-th waveguide, |m,n(SU(2))(ω,λ,j,z)|2, for light initially impinging the m = 2 waveguide of an SU(2) array with parameters ω = 0 and j = 5.

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3. Lattices with underlying SU(1, 1) symmetry

There is another three-element closed group that can be used to describe photonic lattices,

H^SU(1,1)(ω,λ,k)=ω(n^+k)+λ[V^n^(2k1+n^)+n^(2k1+n^)V^],
ωK^0+λ(K^+K^+).
Here the operators form the SU(1, 1) group that satisfy the commutation relations, [ +, ] = −2 0 and [ 0, ±] = ± ±. Here, the ideal lattice has infinite size but these results will describe a real-world finite lattice as long as the propagated electromagnetic field amplitudes stay far from the edge of the array. The real positive parameter k > 0 is the Bargmann parameter of the SU(1,1) group [27]. Again, this effective Hamiltonian-like matrix can be diagonalized,
D^SU(1,1)(ξ)H^SU(1,1)(ω,λ,k)D^SU(1,1)(ξ)=ω24λ2(n^+k),
with the difference that a strong restriction appears,
tanh2ξ=2λω,ω>2λ.
Here the displacement operator has the form,
D^SU(1,1)(ξ)=eξK^+ξ*K^,
=eξ|ξ|tanh|ξ|K^+elnsech2|ξ|K^0eξ*|ξ|tanh|ξ|K^,
where we have used the standard disentangling formulas for SU(1, 1) [13–15 ]. Under the restriction, ω > 2λ, the dispersion relation is:
ΩSU(1,1)(m,ω,λ,k)=ω24λ2(m+k),
with the n-th component of the m-th normal states given by the SU(1, 1) displaced number states,
n|ΩSU(1,1)(m,ω,λ,k)=n|D^SU(1,1)(ξ)|m,
=(1)nΓ(2k)Γ(m+2k)Γ(n+2k)m!n!(sinh|ξ|)m+n(cosh|ξ|)2kmn××F21(m,n,2k,csch2|ξ|),
where the notation Γ(x) and 2 F 1(a, b; c; x) stand for the Gamma and Gauss hypergeometric functions, in that order, and we have accounted for the fact that the paremeters ω and λ are real and ω > 2λ. It is quite important to note that, here, it is only possible to calculate the normal states for the restriction ω > 2λ with the Gilmore-Perelomov coherent approach due to the characteristics of the SU(1, 1) group [33].

Again, it is possible to work out an impulse function in terms of what we will call SU(1, 1) generalized Gilmore-Perelomov coherent states,

m,n(SU(1,1))(ω,λ,k,z)=n|eizH^SU(2)|m,
=(4λ2ω2)Γ(2k)Γ(m+2k)Γ(n+2k)m!n!(2iλsinhz24λ2w2)m+n××(4λ2ω2coshz24λ2w2iωsinhz24λ2w2)2kmn××F21(n,m;2k;(ω24λ24λ2)csch2z24λ2w2).
Here it is of great importance to note that this impulse function works for any given case of ω and λ as the restriction ω > 2λ appears only for the dispersion relation and normal states. Thus, we have two types of propagation regimes. The first one is given by the relation ω > 2λ that transforms the hyperbolic functions into trigonometric functions. Thus, the lattice will be 2π/ω22λ2 periodical. Figure 3 shows this periodicity in the squared impulse function, |m,n(SU(2))(ω,λ,j,z)|2, of an initial field amplitude impinging just the m = 15 waveguide of an SU(1, 1) lattice with parameters λ = 0.4ω and k = 1/4.

 figure: Fig. 3

Fig. 3 Intensity at the n-th waveguide, |m,n(SU(1,1))(ω,λ,k,z)|2, for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters λ = 0.4ω and k = 1/4.

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The other will be defined by ω ≤ 2λ, where the lattice is not periodical and light impinging the m-th lattice will disperse with propagation due to the hyperbolic functions. In the special cases, ω = 2λ and ω = 0, the impulse function can be calculated via the limits of the hyperbolic functions. The impulse function for ω = 2λ, shown in Fig. 4 for Bargmann parameter k = 1/4, is given by the expression,

m,n(SU(1,1))(2λ,λ,k,z)=Γ(m+2k)Γ(n+2k)m!n!(1)k+m+n(λz)m+n(λz+i)2kmnΓ(2k)××F21(n,m;2k;1λ2z2).
and for ω = 0, shown in Fig. 5 for k = 1/4, by the expression,
m,n(SU(1,1))(0,λ,k,z)=Γ(m+2k)Γ(n+2k)m!n!(isinhλz)m+n(coshλz)2kmnΓ(2k)××F21(n,m;2k;csch2λz).
Note that a particular realization of lattices with this underlying symmetry described by the parameters ω = 1 + β 2, λ = β with the parameter β such that ω > 2λ for the Bargmann parameters k = 1/2 and k = 1, have been used to report pairs of isospectral arrays of photonic lattices as an analogy to supersymmetric quantum mechanics. In these cases the impulse function reduces to Jacobi polynomials and gave rise to the moniker of Jacobi photonic lattices [21].

 figure: Fig. 4

Fig. 4 Intensity at the n-th waveguide, |m,n(SU(1,1))(ω,λ,k,z)|2, for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters ω = 2λ and k = 1/4.

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 figure: Fig. 5

Fig. 5 Intensity at the n-th waveguide, |m,n(SU(1,1))(ω,λ,k,z)|2, for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters ω = 0 and k = 1/4.

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4. Lattices with underlying Heisenberg-Weyl symmetry

The Heisenber-Weyl lattice has an effective coupling that increases as the square root of the position of the waveguide,

H^HW(ω,λ,z)=ωn^+λ(V^n^+n^V^),
=ωn^+λ(a^+a^),
where we have used the creation (annihilation) operators â (â) that satisfy the commutation relations [â , â] = −1, [, â ] = â , and [, â] = −â. This matrix can be diagonalized,
D^HW(α)H^HWD^HW(α)=ωn^λ2ω,α=λω.
by the Glauber displacement operator [34, 35],
D^HW(α)=eαa^α*a^.
Thus, the dispersion relation,
ΩHW(m,ω,λ)=ωmλ2ω,
and the normal modes are given by the displaced Fock states [36, 37],
n|ΩHW(m,ω,λ)=n|D^HW(α)|m,
=(1)mm!n!e12|λω|2(λω)nmU(m,nm+1,|λω|2),
where the U(a, b, x) stands for Tricomi confluent hypergeometric function [38]. Typically, displaced number states are given in terms of generalized Laguerre polynomilas, but this requires to split the result into two cases, one for nm and another for n < m. We favor Tricomi confluent hypergemetric functions because it is straightforward to use them in numeric simulations, rather than defining the cases required with generalized Laguerre polynomials.

The impulse function of this array is given by the following expression,

m,n(HW)(ω,λ,z)=n|eizH^HW|m,
=(1)mm!n!eiz(mωλ2ω)e|λω|2(1eiwz)[λω(eiwz1)]nm××U(m,nm+1,4|λω|2sin2ωz2),
where we have used the properties of Glauber displacement operator [39]. Thus, light impinging just at just the m-th waveguide will produce a field amplitude distribution at the waveguides that is equivalent to the amplitude distribution of the quantum optics displaced number state. Note that this impulse function has a periodicity of 2π/ω due to the argument in the exponential functions and the π/ω periodicity of the squared sine argument in Tricomi confluent hypergeometric function. This is shown in Fig. 6 for an initial field impinging the m = 24 waveguide in a Heisenberg-Weyl lattice with parameter λ = 0.4ω. A displaced number state is nothing else than the state obtained by the action of Glauber displacement operator over a Fock state, which may be the origin of the Glauber-Fock lattice moniker given to specific realizations of lattices with underlying Heisenberg-Weyl symmetry [23–26 ]. Note that these well known results for ω = 0 are recovered by taking the limit of the impulse function,
m,n(HW)(0,λ,z)=limω0m,n(HW)(ω,λ,z),
=(1)mm!n!e12|λz|2(iλz)nmU(m,nm+1,|λz|2),
and in this particular realization the periodicity is lost as shown in Fig. 7. A more complicated realization belonging to this symmetry class is provided by what we will call a Lewis-Ermakov lattice, where first and second neighbors couple and effective refractive indices and couplings may include propagation distance dependence [11].

 figure: Fig. 6

Fig. 6 Intensity at the n-th waveguide, |m,n(HW)(ω,λ,z)|2, for light initially impinging the m = 24 waveguide of a Heisenberg-Weyl array with parameter λ = 0.4 ω.

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 figure: Fig. 7

Fig. 7 Intensity at the n-th waveguide, |m,n(HW)(ω,λ,z)|2, for light initially impinging the m = 24 waveguide of a Heisenberg-Weyl array with parameter ω = 0.

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5. Conclusion

We have used the well known Gilmore-Perelomov coherent states from quantum mechanics [16–18 ] to provide closed form analytic dispersion relations, normal modes and impulse functions for three types of photonic lattices with underlying SU(2), SU(1, 1) and Heisenberg-Weyl symmetries. This symmetry based approach allowed us to analyzed the characteristics of these photonic lattices straight from the impulse functions; in particular, their periodicity or lack of it. We want to emphasize the fact that the propagator for any given tight-binding array of photonic waveguides may be calculated in this way as long as the dynamical group is found, even in the case of arrays were the parameters depend on the propagation distance [11, 12]. Furthermore, these photonic lattices can be seen as optical simulators of a class of Gilmore-Perelomov generalized coherent states.

Acknowledgments

BMRL acknowledges fruitful discussion with Miguel A. Bandres and LVV support from CONACYT master studies grant #294880.

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References

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  1. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
    [Crossref]
  2. S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
    [Crossref]
  3. B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
    [Crossref]
  4. A. L. Jones, “Coupling of optical fibers and scattering in fibers,” J. Opt. Soc. Am. 55, 261–271 (1965).
    [Crossref]
  5. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
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  6. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [Crossref]
  7. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
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  8. W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
    [Crossref]
  9. H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
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  10. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [Crossref]
  11. B. M. Rodríguez-Lara, P. Aleahmad, H. M. Moya-Cessa, and D. N. Christodoulides, “Ermakov-lewis symmetry in photonic lattices,” Opt. Lett. 39, 2083–2085 (2014).
    [Crossref] [PubMed]
  12. B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
    [Crossref]
  13. R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
    [Crossref]
  14. K. Wódkiewicz and J. H. Eberly, “Coherent states, squeezed fluctuations, and the SU(2) and SU(1, 1) groups in quantum-optics applications,” J. Opt. Soc. Am. B 2, 458–466 (1985).
    [Crossref]
  15. M. Ban, “Decomposition formulas for su(1, 1) and su(2, 2) Lie algebras and their applications in quantum optics,” J. Opt. Soc. Am. B 10, 1347–1359 (1993).
    [Crossref]
  16. R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. 74, 391–463 (1972).
    [Crossref]
  17. A. M. Perelomov, “Coherent states for arbitrary Lie group,” Comm. Math. Phys. 26, 222–236 (1972).
    [Crossref]
  18. W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
    [Crossref]
  19. A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
    [Crossref]
  20. A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
    [Crossref]
  21. A. Zuñiga-Segundo, B. M. Rodríguez-Lara, D. J. Fernández C., and H. M. Moya-Cessa, “Jacobi photonic lattices and their SUSY partners,” Opt. Express 22, 987–994 (2014).
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  22. A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
    [Crossref] [PubMed]
  23. R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
    [Crossref] [PubMed]
  24. B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
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    [Crossref] [PubMed]
  26. A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
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  32. L. E. Vicent and K. B. Wolf, “Analysis of digital images into energy-angular momentum modes,” J. Opt. Soc. Am. A 28, 808–814 (2011).
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  33. R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).
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  34. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
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  35. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
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2015 (1)

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

2014 (3)

2013 (2)

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

2012 (2)

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

2011 (4)

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
[Crossref]

L. E. Vicent and K. B. Wolf, “Analysis of digital images into energy-angular momentum modes,” J. Opt. Soc. Am. A 28, 808–814 (2011).
[Crossref]

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[Crossref]

2010 (1)

2009 (2)

K. B. Wolf, “Mode analysis and signal restoration with Kravchuk functions,” J. Opt. Soc. Am. A 26, 509–516 (2009).
[Crossref]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

2008 (1)

2007 (1)

1994 (1)

1993 (1)

1991 (1)

A. Wünsche, “Displaced Fock states and their connection to quasiprobabilities,” Quantum Opt. 3, 359–383 (1991).
[Crossref]

1990 (1)

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

1987 (2)

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

1985 (2)

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

1972 (4)

R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. 74, 391–463 (1972).
[Crossref]

A. M. Perelomov, “Coherent states for arbitrary Lie group,” Comm. Math. Phys. 26, 222–236 (1972).
[Crossref]

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[Crossref]

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

1967 (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
[Crossref]

1965 (1)

1963 (2)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

Abouraddy, A. F.

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

Aleahmad, P.

Askey, R.

R. Askey, Orthogonal Polynomials and Special Functions, in CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1975).
[Crossref]

Ban, M.

Barut, A. O.

A. O. Barut and R. Raczka, Theory of group representations and applications (PWN Polish Scientific Publishers, 1980).

Christodoulides, D.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Christodoulides, D. N.

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

B. M. Rodríguez-Lara, P. Aleahmad, H. M. Moya-Cessa, and D. N. Christodoulides, “Ermakov-lewis symmetry in photonic lattices,” Opt. Lett. 39, 2083–2085 (2014).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

Dreisow, F.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Eberly, J. H.

Feng, D. H.

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

Fernández C., D. J.

Gilmore, R.

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. 74, 391–463 (1972).
[Crossref]

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Hardy, A.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Haus, H. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Heinrich, M.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Huang, W. P.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Huang, W.-P.

Jones, A. L.

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Kay, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Keil, R.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Klauder, J. R.

J. R. Klauder and B.-S. Skagerstam, Coherent States. Applications in Physics and Mathematical Physics (World Scientific, 1985).
[Crossref]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

Krötzsch, G.

Kwek, L.-C.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Lebedev, N. N.

N. N. Lebedev, Special Functions and their Applications (Prentice-Hall, 1965).

Longhi, S.

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[Crossref]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

Moya-Cessa, H.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

Moya-Cessa, H. M.

Nolte, S.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Osinski, M.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Perelomov, A. M.

A. M. Perelomov, “Coherent states for arbitrary Lie group,” Comm. Math. Phys. 26, 222–236 (1972).
[Crossref]

Perez-Leija, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

Puri, R. R.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).
[Crossref]

Raczka, R.

A. O. Barut and R. Raczka, Theory of group representations and applications (PWN Polish Scientific Publishers, 1980).

Rodríguez-Lara, B.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Rodríguez-Lara, B. M.

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

B. M. Rodríguez-Lara, P. Aleahmad, H. M. Moya-Cessa, and D. N. Christodoulides, “Ermakov-lewis symmetry in photonic lattices,” Opt. Lett. 39, 2083–2085 (2014).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

A. Zuñiga-Segundo, B. M. Rodríguez-Lara, D. J. Fernández C., and H. M. Moya-Cessa, “Jacobi photonic lattices and their SUSY partners,” Opt. Express 22, 987–994 (2014).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
[Crossref]

Satyanarayana, M. V.

M. V. Satyanarayana, “Generalized coherent states and generalized squeezed coherent states,” Phys. Rev. D 32, 400–404 (1985).
[Crossref]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

Skagerstam, B.-S.

J. R. Klauder and B.-S. Skagerstam, Coherent States. Applications in Physics and Mathematical Physics (World Scientific, 1985).
[Crossref]

Snyder, A. W.

Soto-Eguibar, F.

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

Streifer, W.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

Szameit, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

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H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

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[Crossref]

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A. Wünsche, “Displaced Fock states and their connection to quasiprobabilities,” Quantum Opt. 3, 359–383 (1991).
[Crossref]

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A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

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W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

Zuñiga-Segundo, A.

Ann. Phys. (1)

R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. 74, 391–463 (1972).
[Crossref]

Appl. Phys. B (1)

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[Crossref]

Comm. Math. Phys. (1)

A. M. Perelomov, “Coherent states for arbitrary Lie group,” Comm. Math. Phys. 26, 222–236 (1972).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

J. Lightwave Technol. (2)

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

J. Math. Phys. (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (2)

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S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
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Phys. Rev. (1)

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[Crossref]

Phys. Rev. A (5)

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
[Crossref]

Phys. Rev. D (1)

M. V. Satyanarayana, “Generalized coherent states and generalized squeezed coherent states,” Phys. Rev. D 32, 400–404 (1985).
[Crossref]

Phys. Rev. Lett. (2)

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Phys. Scr. (1)

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

Quantum Opt. (1)

A. Wünsche, “Displaced Fock states and their connection to quasiprobabilities,” Quantum Opt. 3, 359–383 (1991).
[Crossref]

Rev. Mod. Phys. (1)

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

Other (5)

N. N. Lebedev, Special Functions and their Applications (Prentice-Hall, 1965).

J. R. Klauder and B.-S. Skagerstam, Coherent States. Applications in Physics and Mathematical Physics (World Scientific, 1985).
[Crossref]

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R. Askey, Orthogonal Polynomials and Special Functions, in CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1975).
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R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Intensity at the n-th waveguide, | m , n ( SU ( 2 ) ) ( ω , λ , j , z ) | 2 , for light initially impinging the middle waveguide, m = 5, of an SU(2) array with parameters λ = 0.4ω and j = 5.
Fig. 2
Fig. 2 Intensity at the n-th waveguide, | m , n ( SU ( 2 ) ) ( ω , λ , j , z ) | 2 , for light initially impinging the m = 2 waveguide of an SU(2) array with parameters ω = 0 and j = 5.
Fig. 3
Fig. 3 Intensity at the n-th waveguide, | m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) | 2 , for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters λ = 0.4ω and k = 1/4.
Fig. 4
Fig. 4 Intensity at the n-th waveguide, | m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) | 2 , for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters ω = 2λ and k = 1/4.
Fig. 5
Fig. 5 Intensity at the n-th waveguide, | m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) | 2 , for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters ω = 0 and k = 1/4.
Fig. 6
Fig. 6 Intensity at the n-th waveguide, | m , n ( HW ) ( ω , λ , z ) | 2 , for light initially impinging the m = 24 waveguide of a Heisenberg-Weyl array with parameter λ = 0.4 ω.
Fig. 7
Fig. 7 Intensity at the n-th waveguide, | m , n ( HW ) ( ω , λ , z ) | 2 , for light initially impinging the m = 24 waveguide of a Heisenberg-Weyl array with parameter ω = 0.

Equations (52)

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i d d z l ( z ) = ω f ( l , z ) l ( z ) + λ [ g ( l , z ) l 1 ( z ) + g ( l + 1 , z ) l + 1 ( z ) ] ,
i d d z | ( z ) = H ^ ( z ) | ( z ) .
| ( z ) = ( 0 ( z ) 1 ( z ) 2 ( z ) ) ,
l = 0 l ( z ) | j ,
H ^ ( z ) = ( ω f ( 0 , z ) λ g ( 0 , z ) 0 λ g ( 0 , z ) ω f ( 1 , z ) λ g ( 1 , z ) 0 λ g ( 1 , z ) ω f ( 2 , z ) λ g ( 2 , z ) ) ,
ω A ^ 0 ( z ) + λ [ A ^ + ( z ) + A ^ ( z ) ] .
| ( z ) = e i 0 z H ^ ( x ) d x | ( 0 ) ,
m , n ( ω , λ , z ) = n | e i 0 z H ^ ( x ) d x | m .
m , n ( ω , λ , z ) = n | e i z H ^ | m ,
= n | e i z [ ω A ^ 0 + λ ( A ^ + + A ^ ) ] | m ,
= n | e a + ( z ) A ^ + e a 0 ( z ) A ^ 0 e a ( z ) A ^ | m ,
D ^ [ a 0 ( z ) , a ± ( z ) ] | m e a + ( z ) A ^ + e a 0 ( z ) A ^ 0 e a ( z ) A ^ | m .
H ^ SU ( 2 ) ( ω , λ , j ) = ω ( n ^ j ) + λ [ V ^ n ^ ( 2 j + 1 n ^ ) + n ^ ( 2 j + 1 n ^ ) V ^ ] ,
ω J ^ 0 + λ ( J ^ + J ^ + ) ,
D ^ SU ( 2 ) ( θ ) = e θ J ^ + θ * J ^ ,
= e θ | θ | tan | θ | J ^ + e ln sec 2 | θ | J ^ 0 e θ * | θ | tan | θ | J ^ ,
D ^ SU ( 2 ) ( θ ) H ^ SU ( 2 ) ( ω , λ ) D ^ SU ( 2 ) ( θ ) = ω 2 + 4 λ 2 ( n ^ j ) ,
tan 2 θ = 2 λ ω .
Ω SU ( 2 ) ( m , ω , λ , j ) = ω 2 + 4 λ 2 ( m j ) ,
n | Ω SU ( 2 ) ( m , ω , λ , j ) n | D ^ SU ( 2 ) ( θ ) | m ,
= ( 1 ) n ( 2 j m ) ( 2 j n ) ( cos | θ | ) 2 j m n ( sin | θ | ) m + n × × K m ( n , sin 2 | θ | , 2 j ) ,
m , n ( SU ( 2 ) ) ( ω , λ , j , z ) = n | e i z H ^ SU ( 2 ) | m ,
= ( 2 j m ) ( 2 j n ) ( 2 i λ sin z 2 ω 2 + 4 λ 2 ) m + n ( ω 2 + 4 λ 2 ) j × × ( ω 2 + 4 λ 2 cos z 2 ω 2 + 4 λ 2 i ω sin z 2 ω 2 + 4 λ 2 ) 2 j m n × × K m ( n , 4 λ 2 ω 2 + 4 λ 2 sin 2 z 2 ω 2 + 4 λ 2 , 2 j ) .
H ^ SU ( 2 ) ( 0 , λ , j ) = 2 λ J ^ x , J ^ x = 1 2 ( J ^ + + J ^ ) .
lim ω 0 1 2 arctan 2 λ ω = π 4
n | Ω ( m , 0 , λ , j ) SU ( 2 ) = 1 2 j K m ( n , 1 2 , 2 j ) ,
m , n ( SU ( 2 ) ) ( 0 , λ , j , z ) = ( 2 j m ) ( 2 j n ) ( i sin λ z ) m + n ( cos λ z ) 2 j m n × × K m ( n , sin 2 λ z , 2 j ) .
m , n ( SU ( 2 ) ) ( 0 , λ , j π 2 λ ) = δ n , 2 j m ,
H ^ SU ( 1 , 1 ) ( ω , λ , k ) = ω ( n ^ + k ) + λ [ V ^ n ^ ( 2 k 1 + n ^ ) + n ^ ( 2 k 1 + n ^ ) V ^ ] ,
ω K ^ 0 + λ ( K ^ + K ^ + ) .
D ^ SU ( 1 , 1 ) ( ξ ) H ^ SU ( 1 , 1 ) ( ω , λ , k ) D ^ SU ( 1 , 1 ) ( ξ ) = ω 2 4 λ 2 ( n ^ + k ) ,
tanh 2 ξ = 2 λ ω , ω > 2 λ .
D ^ SU ( 1 , 1 ) ( ξ ) = e ξ K ^ + ξ * K ^ ,
= e ξ | ξ | tanh | ξ | K ^ + e ln sech 2 | ξ | K ^ 0 e ξ * | ξ | tanh | ξ | K ^ ,
Ω SU ( 1 , 1 ) ( m , ω , λ , k ) = ω 2 4 λ 2 ( m + k ) ,
n | Ω SU ( 1 , 1 ) ( m , ω , λ , k ) = n | D ^ SU ( 1 , 1 ) ( ξ ) | m ,
= ( 1 ) n Γ ( 2 k ) Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( sinh | ξ | ) m + n ( cosh | ξ | ) 2 k m n × × F 2 1 ( m , n , 2 k , csch 2 | ξ | ) ,
m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) = n | e i z H ^ SU ( 2 ) | m ,
= ( 4 λ 2 ω 2 ) Γ ( 2 k ) Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( 2 i λ sinh z 2 4 λ 2 w 2 ) m + n × × ( 4 λ 2 ω 2 cosh z 2 4 λ 2 w 2 i ω sinh z 2 4 λ 2 w 2 ) 2 k m n × × F 2 1 ( n , m ; 2 k ; ( ω 2 4 λ 2 4 λ 2 ) csch 2 z 2 4 λ 2 w 2 ) .
m , n ( SU ( 1 , 1 ) ) ( 2 λ , λ , k , z ) = Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( 1 ) k + m + n ( λ z ) m + n ( λ z + i ) 2 k m n Γ ( 2 k ) × × F 2 1 ( n , m ; 2 k ; 1 λ 2 z 2 ) .
m , n ( SU ( 1 , 1 ) ) ( 0 , λ , k , z ) = Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( i sinh λ z ) m + n ( cosh λ z ) 2 k m n Γ ( 2 k ) × × F 2 1 ( n , m ; 2 k ; csch 2 λ z ) .
H ^ HW ( ω , λ , z ) = ω n ^ + λ ( V ^ n ^ + n ^ V ^ ) ,
= ω n ^ + λ ( a ^ + a ^ ) ,
D ^ HW ( α ) H ^ HW D ^ HW ( α ) = ω n ^ λ 2 ω , α = λ ω .
D ^ HW ( α ) = e α a ^ α * a ^ .
Ω HW ( m , ω , λ ) = ω m λ 2 ω ,
n | Ω HW ( m , ω , λ ) = n | D ^ HW ( α ) | m ,
= ( 1 ) m m ! n ! e 1 2 | λ ω | 2 ( λ ω ) n m U ( m , n m + 1 , | λ ω | 2 ) ,
m , n ( HW ) ( ω , λ , z ) = n | e i z H ^ HW | m ,
= ( 1 ) m m ! n ! e i z ( m ω λ 2 ω ) e | λ ω | 2 ( 1 e i w z ) [ λ ω ( e i w z 1 ) ] n m × × U ( m , n m + 1 , 4 | λ ω | 2 sin 2 ω z 2 ) ,
m , n ( HW ) ( 0 , λ , z ) = lim ω 0 m , n ( HW ) ( ω , λ , z ) ,
= ( 1 ) m m ! n ! e 1 2 | λ z | 2 ( i λ z ) n m U ( m , n m + 1 , | λ z | 2 ) ,

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