Optical Tautochrone and Squeezing Dynamics in Non-uniform Lattices

Abstract

We present exact analogies between the tautochrone problem of mechanics and the squeezed states of quantum optics, to optical lattices. Both phenomena emerge in the same physical system, that of waveguide arrays with non-uniform couplings. Extension to two dimensions yields Lissajous-type trajectories and multidirectional tautochrone focusing. Furthermore, we investigate the impact of Kerr nonlinearity and show that it determines the diffraction behavior, namely coherent-state-like or squeezed propagation. These quantum inspired classical lattices highlight the role of the coupling coefficients to beam engineering and light control in complex media.

Figures (7)

• Figure 1: Optical tautochrone and squeezing dynamics in non-uniform photonic lattices. (a) Schematic illustration of the tautochrone effect: two Gaussian inputs are launched at different positions in the lattice and focus at the same site after a fixed propagation distance $z$. For clarity, each input is drawn as exciting a single waveguide, though in practice the beams are extended. Inset: classical tautochrone, where particles sliding along a cycloid under gravity reach the bottom simultaneously, regardless of their starting points. (b) Schematic illustration of squeezing dynamics: a broad Gaussian input undergoes periodic squeezing during propagation. Inset: the wavepacket is viewed as a collection of particles that all arrive together at the bottom of the curve, resulting in compression.
• Figure 2: Tautochrone effect and oscillatory dynamics in optical lattices. (a1) Intensity evolution of three non-interfering Gaussian wavepackets with zero initial momentum ($p_0=0$) in a one-dimensional lattice with $N=299$ sites. We chose the following parameters in Eq. \ref{['couplings']}: $C = \tfrac{4N}{7}$ and $\omega = \tfrac{2\pi}{100}$. The first wavepacket is centered at site $n_0 = 40$ with width $w_0 \approx 11.4$, the second at $n_0 = 130$ with $w_0 \approx 13$, and the third at $n_0 = 225$ with $w_0 \approx 12.4$. (a2) Same 1D setup as (a1) but with the three beams interfering; we display the site-resolved intensity $I_n(z) = |\psi_n(z)|^2$. (b1)–(b3) Extension to a two-dimensional lattice ($35 \times 35$ sites). Shown is the intensity evolution of three interfering wavepackets. The first wavepacket is centered at $(n_0 = 10, m_0 = 10)$ with equal widths $w_0 \approx 4.3$ in both directions. The second wavepacket is centered at $(n = 26, m = 12)$ with widths $w_{0,x} \approx 4.3$ along the $x$-direction and $w_{0,y} \approx 4.4$ along the $y$-direction. The third wavepacket is centered at $(n_0 = 12, m_0 = 27)$, with widths $w_{0,x} \approx 4.4$ and $w_{0,y} \approx 4.2$. (d1)–(d3) Evolution of a single wavepacket in a 2D lattice, illustrating the effect of momentum and frequency variations: (d1) zero initial momentum, equal frequencies ($\omega_x = \omega_y$); (d2) nonzero momentum in $y$-direction ($p_y = 0.4$), equal frequencies; (d3) zero momentum, unequal frequencies ($\omega_x \neq \omega_y$).
• Figure 3: Coherent-state-like and squeezing evolution. (a1) Evolution of two Gaussian wavepackets symmetrically placed around the lattice center, forming a quantum cat state. The first (second) wavepacket is centered at $n_0 = 100$ ($n_0 = 200$) with equal widths $w \approx 12.8$. Lattice parameters: $C = \tfrac{4N}{7}$ and $\omega = \tfrac{2\pi}{100}$. (a2)–(a4) Discrete Wigner distribution of the state in (a1) at propagation distances $z = 0$, $z = 8$, and $z = 12.5$. (b1)–(b3) Evolution of the wavepacket magnitude $|\psi_n(z)|$ for a Gaussian input centered at $n_0 = 100$, with $p = 0$ and initial widths $w_0 \approx 4.8$, $12.8$, and $20.8$, respectively. (c1)–(c3) Evolution of a Wigner-function contour for the same parameters as in (b1)–(b3). (d) Amplitude of width oscillations $A$ as a function of $w_0$ and $C$ [see Eq. \ref{['couplings']}].
• Figure 4: Impact of Kerr nonlinearity on beam squeezing. (a) Evolution of the wavepacket width from Fig. \ref{['fig3']}(b3) for four different values of the nonlinearity parameter $\gamma$. (b) Wavepacket evolution for $\gamma = 0.4$, illustrating near coherent-state-like propagation. (c) Variance of the wavepacket width as a function of the initial width $w_0$ and the nonlinearity strength $\gamma$. Regions of minimal variance indicate conditions for approximately coherent-state-like evolution.
• Figure 5: (a) Eigenvalues of the Hamiltonian $\mathbf{H}$ for a one-dimensional non-uniform lattice consisting of $N=23$ sites and with couplings $J_n$ defined by Eq. \ref{['couplings']}. Two values of the parameter $C$ are considered (blue and yellow dots). $\omega$ is set at $\frac{2\pi}{100}$. Inset: Schematic showing a longitudinal cross section of the waveguide array, with black dots indicating the positions of individual waveguides. (b) Same as in (a), but for the corresponding two-dimensional non-uniform lattice consisting of $N=11\times11$ sites.