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Optical Materials Express

Modelling non-Markovian dynamics in photonic crystals with recurrent neural networks

Adam Burgess, Marian Florescu A

Published May 30, 2021• 8 citations
Publisher

Abstract

We develop a recurrent neural network framework to model the non-Markovian dynamics exhibited by two-level atoms interacting with the radiation reservoir of a photonic crystal. Despite the strong non-Markovianity of the atomic dynamics induced by the rapid spectral variation in photonic density of states of the photonic reservoir, our recurrent neural network approach is able to capture precise details in the atomic evolution, including the fractional steady-state atomic population inversion and spectral splitting of the atomic transition. We demonstrate the robustness of the recurrent neural network setup against reduced data sets and its effectiveness to deal with systems of increased complexity.

DOI
10.1364/ome.425263

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We develop a recurrent neural network framework to model the non-Markovian dynamics exhibited by two-level atoms interacting with the radiation reservoir of a photonic crystal. Despite the strong non-Markovianity of the atomic dynamics induced by the rapid spectral variation in photonic density of states of the photonic reservoir, our recurrent neural network approach is able to capture precise details in the atomic evolution, including the fractional steady-state atomic population inversion and spectral splitting of the atomic transition. We demonstrate the robustness of the recurrent neural network setup against reduced data sets and its effectiveness to deal with systems of increased complexity.
Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini–Kossakowski–Sudarshan–Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks (RNNs) for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that RNNs are accurate over different times and regimes.
Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini–Kossakowski–Sudarshan–Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks (RNNs) for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that RNNs are accurate over different times and regimes.
We describe the role of first non-Markovian corrections to resonance fluorescence in photonic crystals, using a perturbative expansion of the Heisenberg equations of motion in powers of the atom-field reservoir coupling strength. Non-Markovian effects arise from the rapid variation of the photonic density of states with frequency. Our method recaptures the physics of the photon-atom bound state in the presence of a full photonic band gap. For the anisotropic three-dimensional photonic band gap, it predicts remarkable features in the resonance fluorescence, such as atomic population inversion and switching behavior in a two-level atom for moderate values of the applied laser field. The magnitude of the switching effect depends sensitively on the external laser intensity and its detuning frequency from the atomic transition. The robustness of this effect against nonradiative decay and dephasing mechanisms is also investigated.
We explore the dynamics of NNN coupled atomic two-level systems embedded within a generic bosonic reservoir under specific system symmetries. In the regime of many atoms identically coupled to a single reservoir, we identify remarkable effects, notably that the initial configuration of the atomic excited-state amplitudes strongly impacts the dynamics of the system and can even fully sever the system from its environment. Additionally, we find that steady-state amplitudes of the excited states become independent of the specific structure of the bosonic reservoirs considered. The framework introduced is deployed to model a structured photonic reservoir associated with a photonic crystal, where it recaptures previous theoretical and experimental results for the superradiance rates even within the single-excitation regime. For the photonic band-gap system, our formalism predicts the generation of pairwise entanglement between initially uncorrelated atomic systems. Furthermore, it suggests that---with respect to a non-Markovian metric---the non-Markovianity of the aggregated many-atom system is modulated by the total number of atoms. This is due to a stark interplay between the Lamb shifting of the atomic transition energies due to their varying number and the increased number of atomic systems with resonant transition energies.
Quantum reservoir computing has emerged as a promising paradigm within the field of quantum machine learning, harnessing the inherent properties of quantum systems to optimise and enhance information processing capabilities. Here, we explore the potential of quantum-inspired machine learning methodologies by leveraging the complex dynamics of quantum reservoirs to address computationally challenging tasks with enhanced efficiency and accuracy. To this end, we employ an open quantum system model comprising two-level atomic ensembles coupled to Lorentzian photonic cavities to construct a quantum physical reservoir computer layer for a recurrent neural network. We evaluate the effectiveness of this approach by applying it to a standard machine learning image-recognition problem and benchmarking its performance against a conventional neural network of similar architecture, but lacking the quantum physical reservoir computer layer. Remarkably, as the dataset size increases, the quantum physical reservoir computer outperforms the conventional neural network, requiring fewer training epochs and a smaller dataset to achieve comparable accuracy. Furthermore, we employ the quantum physical reservoir computing approach to model the dynamics of open quantum systems, focusing on atomic system ensembles interacting with a structured photonic reservoir associated with a photonic band-gap material. Our results reveal that the quantum reservoir computer provides equally powerful representations for quantum dynamical problems, maintaining effectiveness even under constraints of limited training data.
Abstract We present a novel methodology utilizing recurrent neural networks (RNNs) to classify Markovian and non-Markovian quantum processes, leveraging time series data derived from Choi states. The model exhibits exceptional accuracy, surpassing 95%, across diverse scenarios, including dephasing and Pauli channels in an arbitrary basis, generalized amplitude damping dynamics, and even in the presence of noise. Additionally, the developed model shows efficient forecasting capabilities for the analyzed time series data. These results suggest the potential of RNNs in discerning and predicting the Markovian nature of quantum processes.