We formulate a set of conditions under which the nonstationary Schrödinger equation with a time-dependent Hamiltonian is exactly solvable analytically. The main requirement is the existence of a non-Abelian gauge field with zero curvature in the space of system parameters. Known solvable multistate Landau-Zener models satisfy these conditions. Our method provides a strategy to incorporate time dependence into various quantum integrable models while maintaining their integrability. We also validate some prior conjectures, including the solution of the driven generalized Tavis-Cummings model.
We analyze Hamiltonians linear in the time variable for which the multistate Landau-Zener problem is known to have an exact solution. We show that they either belong to families of mutually commuting Hamiltonians polynomial in time or reduce to the 2 × 2 Landau-Zener problem, which is considered trivially integrable. The former category includes the equal slope, bow-tie, and generalized bow-tie models. For each of these models we explicitly construct the corresponding families of commuting matrices. The equal slope model is a member of an integrable family that consists of the maximum possible number (for a given matrix size) of commuting matrices linear in time. The bow-tie model belongs to a previously unknown, similarly maximal family of quadratic commuting matrices. We thus conjecture that quantum integrability understood as the existence of nontrivial parameter-dependent commuting partners is a necessary condition for the Landau-Zener solvability. Descendants of the 2 × 2 Landau-Zener Hamiltonian are e.g. general SU (2) and SU (1, 1) Hamiltonians, time-dependent linear chain, linear, nonlinear, and double oscillators. We explicitly obtain solutions to all these Landau-Zener problems from the 2 × 2 case.
Innovative goods authentication strategies are of fundamental importance considering the increasing counterfeiting levels. Such a task has been effectively addressed with the so-called physical unclonable functions (PUFs), being physical properties of a system that characterize it univocally. PUFs are commonly implemented by exploiting naturally occurring nonidealities in clean-room fabrication processes. The broad availability of classic paradigm PUFs, however, makes them vulnerable. Here, we propose a hybrid plasmonic/photonic multilayered structure working as a three-level strong PUF. Our approach leverages on the combination of a functional nanostructured surface, a resonant response, and a unique chromatic signature all together in one single device. The structure consists of a resonant cavity, where the top mirror is replaced with a layer of plasmonic Ag nanoislands. The naturally random spatial distribution of clusters and nanoparticles formed by this deposition technique constitutes the manufacturer-resistant nanoscale morphological fingerprint of the proposed PUF. The presence of Ag nanoislands allows us to tailor the interplay between the photonic and plasmonic modes to achieve two additional security levels. The first one is constituted by the chromatic response and broad iridescence of our structures, while the second by their rich spectral response, accessible even through a common smartphone lightemitting diode. We demonstrate that the proposed architectures could also be used as an irreversible and quantitative temperature exposure label. The proposed PUFs are inexpensive, chip-to-wafer-size scalable, and can be deposited over a variety of substrates. They also hold a great promise as an encryption framework envisioning morpho-cryptography applications.
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