We introduce Liouville-Fock-state lattices (LFSLs) as a framework for visualizing open quantum systems through matrix representations of the Lindblad master equation. These generalize Hamiltonian Fock-state lattices, which have been considered for realizing synthetic dimensions in quantum simulators. By vectorizing the Lindblad master equation, the dynamics map to a doubled Hilbert space, naturally forming a synthetic lattice. Unlike the unitary evolution of pure states, LFSLs feature non-Hermitian dynamics with drifts, sources, and sinks—paralleling stochastic classical lattices. As such, we suggest that these systems can serve as potential quantum simulators of classical lattice models, with a computational advantage arising from large state spaces encoded in quantum models. For this purpose, we analyze both the Fock representation and alternative positive-semidefinite representations that yield real, non-negative populations, akin to classical probability distributions. Within this formalism, we also demonstrate a phenomenon, which we term environment-induced frustration, in which competing terms in the LFSL give rise to geometric frustration and a massively degenerate steady-state manifold.

With the objective of simulating the physical behavior of electrons in a dynamic background, we investigate a cold atomic Bose-Fermi mixture confined in an optical lattice potential solely affecting the bosons. The bosons, residing in the deep superfluid regime, inherit the periodicity of the optical lattice, subsequently serving as a dynamic potential for the polarized fermions. Owing to the atom-phonon interaction between the fermions and the condensate, the coupled system exhibits a Berezinskii-Kosterlitz-Thouless transition from a Luttinger liquid to a Peierls phase. However, under sufficiently strong Bose-Fermi interaction, the Peierls phase loses stability, leading to either a collapsed or a separated phase. We find that the primary function of the optical lattice is to stabilize the Peierls phase. Furthermore, the presence of a confining harmonic trap induces a diverse physical behavior, surpassing what is observed for either bosons or fermions individually trapped. Notably, under attractive Bose-Fermi interaction, the insulating phase may adopt a fermionic wedding-cake-like configuration, reflecting the dynamic nature of the underlying lattice potential. Conversely, for repulsive interaction, the trap destabilizes the Peierls phase, causing the two species to separate.

In this paper we present a concrete example comparing the results predicted by non -Hermitian quantum mechanics with those of a more comprehensive description that considers environment -induced fluctuations. Our results highlight inaccuracies in the non -Hermitian model. Specifically, we investigate the non -Hermitian skin effect and sensor in the Hatano-Nelson model, contrasting it with a more precise Lindblad description. Our analysis reveals that these phenomena can undergo breakdown when environmental fluctuations come to the forefront, resulting in a nonequilibrium phase transition from a localized skin phase to a delocalized phase. Beyond this specific case study, we engage in a broader discussion regarding the interpretations and implications of non -Hermitian quantum mechanics. This examination serves to broaden our understanding of these phenomena and their potential consequences.

In this paper, we analyze the harmonically driven Jaynes–Cummings and Lipkin–Meshkov–Glick models using both numerical integration of time-dependent Hamiltonians and Floquet theory. For a separation of time scales between the drive and intrinsic Rabi oscillations in the former model, the driving results in an effective periodic reversal of time. The corresponding Floquet Hamiltonian is a Wannier–Stark model, which can be analytically solved. Despite the chaotic nature of the driven Lipkin–Meshkov–Glick model, moderate system sizes can display qualitatively different behaviors under varying system parameters. Ergodicity arises in systems that are neither adiabatic nor diabatic, owing to repeated multi-level Landau–Zener transitions. Chaotic behavior, observed in slow driving, manifests as random jumps in the magnetization, suggesting potential utility as a random number generator. Furthermore, we discuss both models in terms of a Floquet Fock state lattice.

2023 marked the 60th anniversary of the Jaynes–Cummings model, a foundational model in quantum optics. Over the years, its importance has expanded beyond traditional light–matter interaction systems, such as cavity QED. This special issue presents a collection of articles that showcase the evolution of the model’s applications, blending traditional topics with contemporary developments.

The Jaynes-Cummings Model (JCM) has been at the forefront of modern physics as one of the simplest, yet intricately nonlinear, models of light-matter interaction. Focusing on the omnipresence of the JCM across a range of disciplines, this significantly updated and comprehensive review conveys to the reader the fundamental generality of its formalism, looking at a wide range of applications in specific physical systems and across disciplines including atomic physics, quantum optics, solid-state physics and quantum information sciences. An ideal reference for researchers in quantum physics and quantum optics, the book also comprises an accessible introduction for students engaged with non-equilibrium quantum phase transitions, quantum computing and simulation, quantum many-body physics, cavity, circuit and waveguide quantum electrodynamics. Part of IOP Series in Quantum Technology. Full abstract Key features • Collects JCM physics and applications scattered across literature and different applications. • The exposition guides the reader through a rich and appealing landscape interlacing quantum optics and condensed-matter physics. • All chapters discuss theory and experiment, linked historically to the development of the various directions stemming from JC physics. This is accompanied by a thorough list of references to the key publications. • The presentation is kept concise, while continuous text is interspersed with various illustrations and an economical use of mathematical expressions.

In this contribution to the memorial issue of Göran Lindblad, we investigate the periodically driven Lindblad equation for a two-level system. We analyze the system using both adiabatic diagonalization and numerical simulations of the time-evolution, as well as Floquet theory. Adiabatic diagonalization reveals the presence of exceptional points in the system, which depend on the system parameters. We show how the presence of these exceptional points affects the system evolution, leading to a rapid dephasing at these points and a staircase-like loss of coherence. This phenomenon can be experimentally observed by measuring, for example, the population inversion. We also observe that the presence of exceptional points seems to be related to which underlying Lie algebra the system supports. In the Floquet analysis, we map the time-dependent Liouvillian to a non-Hermitian Floquet Hamiltonian and analyze its spectrum. For weak decay rates, we find a Wannier-Stark ladder spectrum accompanied by corresponding Stark-localized eigenstates. For larger decay rates, the ladders begin to dissolve, and new, less localized states emerge. Additionally, their eigenvalues are exponentially sensitive to perturbations, similar to the skin effect found in certain non-Hermitian Hamiltonians.

We analyze a set of models frequently appearing in quantum optical settings by expressing their Hamiltonians in terms of Fock-state lattices (FSLs). The few degrees-of-freedom of such models, together with the system symmetries, make the emerging FSLs relatively simple such that they can be linked to known lattice models from the condensed-matter community. Thus, the FSLs may shed new light on known quantum optical systems. While we provide a rather long list of models and their corresponding FSLs, we pick a few to demonstrate the method's strength. The three-mode boson model, for example, is shown to display a fractal spectrum and chiral evolution in the FSL characterized by localized distributions traversing along symmetric trajectories. In a second example, we consider the central spin model, which generates an FSL reminiscent of the Su-Schrieffer-Heeger model hosting topological edge states. We further demonstrate how the phenomenon of flat bands in lattice models can manifest in related FSLs, which can be linked to so-called dark states.

We propose and experimentally explore a method for realizing frustrated lattice models using a Bose-Einstein condensate held in an optical square lattice. A small lattice distortion opens up an energy gap such that the lowest band splits into two. Along the edge of the first Brillouin zone for both bands, a nearly flat energy-momentum dispersion is realized. For the excited band, a highly degenerate energy minimum arises. By loading ultracold atoms into the excited band, a classically frustrated XY model is formed, describing rotors on a square lattice with competing nearest and next-nearest tunneling couplings. Our experimental optical lattice provides a regime where a fully coherent Bose-Einstein condensate is observed and a regime where frustration is expected. If we adiabatically tune from the condensate regime to the regime of frustration, the momentum spectra show a complete loss of coherence. Upon slowly tuning back to the condensate regime, coherence is largely restored. Good agreement with model calculations is obtained.

The Jaynes–Cummings (JC) model has been at the forefront of quantum optics for almost six decades to date, providing one of the simplest yet intricately nonlinear formulations of light-matter interaction in modern physics. Laying most of the emphasis to the omnipresence of the model a crossa range of disciplines, this monograph brings up the fundamental generality of its formalism, looking at a wide gamut of applications in specific physical systems among several realms, including atomic physics, quantum optics, solid-state physics and quantum information science. When bringing the various pieces together to assemble our narrative, we have primarily targeted researchers in quantum physics and quantum optics. The monograph also comprises an accessible introduction for graduate students engaged with non-equilibrium quantum phase transitions, quantum computing and simulation, and quantum many-body physics. In that framework, we aim to reveal the common ground between physics and applications scattered across literature and different technological advancements. The exposition guides the reader through a vibrant field interlacing quantum optics and condensed-matter physics. All sections are devoted to the strong interconnection between theory and experiment, historically linked to the development of the various modern research directions stemming from JC physics. This is accompanied by a comprehensive list of references to the key publications that have shaped its evolution since the early 1960s. Finally, we have endeavoured to keep the presentation of such a multi-sided material as concise as possible, interspersing continuous text with various illustrations alongside an economical use of mathematical expressions.