Utilizando redes neurais informadas pela física para encontrar hamiltonianos efetivos em sistemas quânticos

Abstract

Effective Hamiltonians are a cornerstone of quantum optics and cavity quantum electro-dynamics (QED). By identifying and discarding terms that do not contribute significantly to the overall dynamics in a given parameter regime, they provide tractable models that still reproduce the experimentally relevant dynamics. In light-matter systems, a canonical example is the reduction of the quantum Rabi Hamiltonian (RH) to the effective Jaynes-Cummings Hamiltonian (JCH) under the rotating-wave approximation (RWA), where fast-oscillating terms are discarded from the system's Hamiltonian due to little contribution to dynamics. While effective-model selection is often justified analytically, as in the RWA, this approach requires a deep understanding of the physical system at hand, and is often heavily handmade. Hence, the central problem addressed in this dissertation is whether one can infer an effective Hamiltonian directly from limited dynamical data and physics a priori knowledge shared by many common physical systems. To this end, we leverage an inverse physics-informed neural network (PINN) framework to reproduce the passage of the RH to the JCH. In this setting, the complex-valued vector state of the composed atom-cavity system is represented by two real-valued neural networks (for the real and imaginary components) trained using a loss function with terms that (i) enforces Schrödinger dynamics through an ordinary differential equation (ODE) residual evaluated at fixed time instants, (ii) matches data observation from experimentally accessible expectation values (photon number and excited-state population), and (iii) imposes physical constraints through initial-condition and normalization penalties. Unknown atom-cavity coupling parameters present in the Hamiltonian are treated as trainable neural network parameters and learned jointly with the state trajectory. After validating the approach in a matched-model baseline (simulating train data with JCH and using the Jaynes-Cummings ODE residual) across distinct cavity initial states, we tackle effective-model identification by training with a split Rabi interaction containing separate rotating and counter-rotating couplings. Results show that in a parameter regime where RWA is feasible, training consistently recovers the Jaynes-Cummings coupling while suppressing the counter-rotating coupling toward zero, both when data are generated by full Rabi dynamics and when Jaynes-Cummings data are fit using the more expressive RH. We further show that the chosen non-negativity parameterization of the learned coupling parameter (absolute value versus softplus functions) influences how readily weakly identifiable terms collapse toward zero. These results support inverse PINNs as a practical workflow for testing whether candidate terms in full Hamiltonians are warranted by data and for extracting effective Hamiltonians from partial expectation-value measurements.