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arXiv:2009.13932 · Physical Review Letters

Exactness of Mean-Field Equations for Open Dicke Models with an Application to Pattern Retrieval Dynamics

Federico Carollo, Igor Lesanovsky

Published Jun 8, 2021• 74 citations
arXivHTMLPDF

Abstract

Open quantum Dicke models are paradigmatic systems for the investigation of light-matter interaction in out-of-equilibrium quantum settings. Albeit being structurally simple, these models can show intriguing physics. However, obtaining exact results on their dynamical behavior is challenging, since it requires the solution of a many-body quantum system with several interacting continuous and discrete degrees of freedom. Here, we make a step forward in this direction by proving the validity of the mean-field semiclassical equations for open multimode Dicke models, which, to the best of our knowledge, so far has not been rigorously established. We exploit this result to show that open quantum multimode Dicke models can behave as associative memories, displaying a nonequilibrium phase transition toward a pattern-recognition phase.

DOI
10.1103/physrevlett.126.230601

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Open quantum Dicke models are paradigmatic systems for the investigation of light-matter interaction in out-of-equilibrium quantum settings. Albeit being structurally simple, these models can show intriguing physics. However, obtaining exact results on their dynamical behavior is challenging, since it requires the solution of a many-body quantum system with several interacting continuous and discrete degrees of freedom. Here, we make a step forward in this direction by proving the validity of the mean-field semiclassical equations for open multimode Dicke models, which, to the best of our knowledge, so far has not been rigorously established. We exploit this result to show that open quantum multimode Dicke models can behave as associative memories, displaying a nonequilibrium phase transition toward a pattern-recognition phase.
Dicke-like models can describe a variety of physical systems, such as atoms in a cavity or vibrating ion chains. In equilibrium these systems often feature a radical change in their behavior when switching from weak to strong spin-boson interaction. This usually manifests in a transition from a "dark" to a "superradiant" phase. However, understanding the out-of-equilibrium physics of these models is extremely challenging, and even more so for strong spin-boson coupling. Here we show that the nonequilibrium strongly interacting multimode Dicke model can mimic some fundamental properties of an associative memory-a system which permits the recognition of patterns, such as letters of an alphabet. Patterns are encoded in the couplings between spins and bosons, and we discuss the dynamics of the spins from the perspective of pattern retrieval in associative memory models. We identify two phases, a "paramagnetic" and a "ferromagnetic" one, and a crossover behavior between these regimes. The "ferromagnetic" phase is reminiscent of pattern retrieval. We highlight similarities and differences with the thermal dynamics of a Hopfield associative memory and show that indeed elements of "machine learning behavior" emerge in the strongly coupled multimode Dicke model.
Nowadays, the research field of Machine Learning (ML) is undergoing a rapid expansion. In addition, in the last decades, within the research area of quantum computation, several quantum protocols and algorithms have shown the ability to outperform their classical counterparts. In the light of these twofold advances, a frontier field investigating the potentiality of quantum implementations of ML architectures arose. Among the latter, many efforts are concentrated on Artificial Neural Networks (ANNs), which represent a highly successful ML architecture. This thesis focuses on characterizing and understanding the role of quantum effects on the behavior of an instance of ANN, the so-called Hopfield Neural Network (HNN). The HNN is an example of an associative memory model and it permits the recognition, or retrieval, of patterns, such as letters of an alphabet. Formally, it can be described in terms of a classical spin network with an all-to-all interaction, ruled by a stochastic dynamics. Its relatively intuitive behavior, and its physical description make the HNN being a good candidate for a theoretical investigation of its possible quantum generalizations. In the first part of our work we deal with the implementation of fully connected Ising models, such as the HNN one, in terms of quantum systems. To this end, we refer to the cases where the theory of Open Quantum System (OQS) is exploited for engineering steady states of out-of-equilibrium dynamics. This paradigm allows us to build a dissipative - yet quantum - dynamics which encodes in its stationary state the thermal state of the original (classical) problem. Such a construction gives us the chance to address the question as to whether the quantum description could lead to a speed-up in reaching the shared stationary state with respect to the classical counterpart. We show that a suitable choice of the parameters allows the system to switch between a classical regime and a quantum one, the latter being characterized by an accelerated approach towards the long-time limit state. In the second part of our work we focus on practically relevant implementations of quantum systems that can show analogies with the HNN model. To this end, we consider spin-boson models as described by the Dicke Hamiltonian. Indeed, at strong spin-boson interaction couplings, and when the latter are additionally characterized by disorder, the equilibrium Dicke model is able to reproduce the retrieval mechanism typical of HNNs. When dissipation is included, the theoretical description becomes more challenging. We analyze the open, disordered Dicke model by employing some perturbative techniques in order to deal with the out-of-equilibrium dynamics at strong coupling. As a result, we are able to highlight similarities and differences between the quantum model and HNNs.
We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ``state-dependent" rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-)local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.
Abstract We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ‘state-dependent’ rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-) local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.
The Dicke model with a weak dissipation channel is realized by coupling a Bose-Einstein condensate to an optical cavity with ultranarrow bandwidth. We explore the dynamical critical properties of the Hepp-Lieb-Dicke phase transition by performing quenches across the phase boundary. We observe hysteresis in the transition between a homogeneous phase and a self-organized collective phase with an enclosed loop area showing power-law scaling with respect to the quench time, which suggests an interpretation within a general framework introduced by Kibble and Zurek. The observed hysteretic dynamics is well reproduced by numerically solving the mean-field equation derived from a generalized Dicke Hamiltonian. Our work promotes the understanding of nonequilibrium physics in open many-body systems with infinite range interactions.
Ultracold atoms in multimode optical cavities provide a rich testbed for many-body phenomena enabled by light-mediated interactions. Recent experiments include realizations of spin glasses and associative memories, as described by multimode Dicke models with disordered couplings. However, the properties of multimode Dicke models with ordered coupling geometries remain largely unexplored. In this work, we investigate the stable steady-states of the multimode Dicke model with an ordered nearest-neighbor coupling geometry, where ncn_cnc​ atomic clusters are coupled via nc−1n_c-1nc​−1 cavity modes. We show that the number of mean-field stable steady-states in the superradiant phase exhibits Fibonacci scaling with the number of atomic clusters, and that a subset of these steady-states exhibit persistent oscillations. Using both the truncated Wigner approximation and the numerically-exact hierarchy of pure states, we further demonstrate that these features of the stable steady-state solutions persist for finite cluster sizes. Ordered multimode Dicke models, such as the nearest-neighbor coupling geometry considered here, are accessible with current experimental technologies and point toward a broader class of strongly interacting dissipative systems with similarly rich behavior.