L. Menon Jr. | Searcharxiv

arXiv2022

Unitary evolution for a two-level quantum system in fractional-time scenario

The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be non-unitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the non-unitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.

quant-ph↗

arXiv2022

Efficiency of random search with space-dependent diffusivity

We address the problem of random search for a target in an environment with space-dependent diffusion coefficient $D(x)$. From a general form of the diffusion differential operator that includes Itô, Stratonovich, and Hänggi-Klimontovich interpretations of the associated stochastic process, we obtain the first-passage time distribution and the search efficiency $\mathcal{E}=\langle 1/t \rangle$. For the paradigmatic power-law diffusion coefficient $D(x) = D_0|x|^α$, with $α<2$, which controls whether the mobility increases or decreases with the distance from a target at the origin, we show the impact of the different interpretations. For the Stratonovich framework, we obtain a closed expression of the search efficiency, valid for arbitrary diffusion coefficient $D(x)$. We show that a heterogeneous diffusivity profile leads to lower efficiency than the homogeneous average level, and the efficiency depends only on the distribution of diffusivity values and not on its spatial organization, features that breakdown under other interpretations.

cond-mat.stat-mech↗