Quantum repeated interactions and the chaos game

Abstract

Inspired by the algorithm of Barnsley's chaos game, we construct an open quantum system model based on the repeated interaction process. We shown that the quantum dynamics of the appropriate fermionic/bosonic system (in interaction with an environment) provides a physical model of the chaos game. When considering fermionic operators, we follow the system's evolution by focusing on its reduced density matrix. The system is shown to be in a Gaussian state (at all time t) and the average number of particles is shown to obey the chaos game equation. Considering bosonic operators, with a system initially prepared in coherent states, the evolution of the system can be tracked by investigating the dynamics of the eigenvalues of the annihilation operator. This quantity is governed by a chaos game-like equation from which different scenarios emerge.

Mechanical harmonic chaos game

“…Inspired by the work of Barnsley, we proposed a mechanical harmonic version of the chaos game. As for the quantum chaos game presented in a previous study [2] the harmonic game provides a concrete physical system from which the chaos game equation emerges. While the quantum version requires an advanced understanding of quantum systems, building the harmonic game does not present any technical challenge.…”

“…This is confirmed by the scaling form of the distance to the origin (measured by E n ). In the presence of damping, the exact expression of second moments ⟨ ⟩ ⟨˙⟩ x x , 2 2 and ⟨ ˙⟩ xx are obtained and compared to numerical results. As in the original chaos game, those observables do not reveal the transition (from connected to disconnected) of the support of the distribution of points ȳn .…”

“…In previous work [2] the authors considered a quantum version of this game, in which a particle interacts stochastically with a reservoir. The quantum dynamics has led, for both fermionic and bosonic particles, to behaviour strongly similar to that of Barnsley's chaos game.…”

“…On one side (w<w c ), the set  (invariant under the iterated functions) is the interval [0, 1]. In this case, the distribution of points on  is in fact related to the Bernoulli convolution [2,8]. On the other hand, when (w>w c ) the invariant  is the Smith-Volterra-Cantor set: we say that the support of the distribution of points (x n ) is disconnected.…”

Quadratic Models for Engineered Control of Open Quantum Systems

“…Crucially, the other relevant limit of this technique allows for system-environment correlations to evolve freely. We show how exact results can be analytically obtained, for finite times between iterations and in the continuous limit, for the special case of quadratic Hamiltonian models (in which repeated interactions can be solved [22][23][24]).…”