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arXiv:2412.20648

Dynamic transition of the generalized Jaynes-Cummings model: multi-particles and inter-particle interaction effects

Liang Wen, Zhenhua Yu

Published Dec 30, 2024• 0 citations
arXivHTMLPDF

Abstract

How environments affect dynamics of quantum systems remains a central question in understanding transitions between quantum and classical phenomena and optimizing quantum technologies. A paradigm model to address the above question is the generalized Jaynes-Cummings model, in which a two-level particle is coupled to its environment modeled by a continuum boson modes. Previous analytic solution shows that, starting from the initial state that the particle is in its excited state and the boson modes in their vacuum state, the time evolution of the probability that the particle occupies the excited state exhibits a dynamic transition as the system-environment coupling varies; when the coupling is weak, the probability decays to zero monotonically, while a finite weight of the particle is localized in the excited state when the coupling is sufficiently strong. Here, we study the dynamic transition for the case that $N$ particles are initially excited with the boson modes in their vacuum state. In particular, we access the effects of an all to all Ising type interaction we introduce between the particles. Our calculation is carried out by the non-perturbative time-dependent numerical renormalization group method. We find that the critical coupling for the transition decreases with $N$, and is suppressed (enlarged) by the anti-ferromagnetic (ferromagnetic) Ising interaction. Our results enrich understanding on environmental effects on interacting quantum systems.

DOI
10.48550/arxiv.2412.20648

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The study of phase transitions in dissipative quantum systems based on the Liouvillian is often hindered by the difficulty of constructing a time-local master equation when the system-environment coupling is strong. To address this issue, the complex discretization approximation for the environment is proposed to study the quantum phase transition in the extended Jaynes-Cumming model with an infinite number of boson modes. This approach yields a non-Hermitian effective Hamiltonian that can be used to simulate the dynamics of the spin. It is found that the ground state of this effective Hamiltonian determines the spin dynamics in the single-excitation subspace. Depending on the opening of the energy gap and the maximum population of excitations on the spin degree of freedom, three distinct phases can be identified: fast decaying, localized, and stretched dynamics of the spin. This approach can be extended to multiple excitations, and similar dynamics were found in the double-excitation subspace, indicating the robustness of the single-excitation phase.
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The interaction of a two-level XYn-spin system with a two-mode cavity field is investigated through a generalized Jaynes-Cummings model in the rotating wave approximation. The spontaneous decay of a spin level was treated by considering the interaction of the two-level spin system with the modes of the universe in the vacuum state. The different cases of interest, characterized in terms of a detuning parameter for each mode, which emerge from the nonvanishing of certain commutation relations between interaction picture Hamiltonians associated with each mode, were analytically implemented and numerically discussed for various values of the initial mean photon number and spin-photon coupling constants. Photon distribution, time evolution of the spin population inversion, as well as the statistical properties of the field leading to the possible production of nonclassical states, such as antibunched light and violations of the Cauchy-Schwartz inequality are examined for an excited initial state. It was assumed that the two modes are initially in coherent states and have the same photon distribution. The case of zero detuning of both modes was treated in terms of a linearization of the expansion of the time evolution operator, while in other three cases, the computations were conducted via second- and third-order Dyson perturbation expansion of the time evolution operator matrix elements for the excited and ground states respectively.
How to analytically deal with the general entanglement dynamics of separate Jaynes–Cummings nodes with continuous-variable fields is still an open question, and few analytical approaches can be used to solve their general entanglement dynamics. Entanglement dynamics between two separate Jaynes–Cummings nodes are examined in this article. Both vacuum state and coherent state in the initial fields are considered through the numerical and analytical methods. The gap between two nonidentical qubit-field coupling strengths shifts the revival period and changes the revival amplitude of two-qubit entanglement. For vacuum-state fields, the maximal entanglement is fully revived after a gap-dependence period, within which the entanglement nonsmoothly decreases to zero and partly recovers without exhibiting sudden death phenomenon. For strong coherent-state fields, the two-qubit entanglement decays exponentially as the evolution time increases, exhibiting sudden death phenomenon, and the increasing gap accelerates the revival period and amplitude decay of the entanglement, where the numerical and analytical results have an excellent coincidence.
The paradigmatic spin-boson model considers a spin degree of freedom interacting with an environment typically constituted by a continuum of bosonic modes. This ubiquitous model is of relevance in a number of physical systems where, in general, one has neither control over the bosonic modes, nor the ability to tune distinct interaction mechanisms. Despite this apparent lack of control, we present a suitable transformation that approximately maps the spin-boson dynamics into that of a tunable multiphoton Jaynes-Cummings model undergoing dissipation. Interestingly, the latter model describes the coherent interaction between a spin and a single bosonic mode via the simultaneous exchange of n bosons per spin excitation. Resorting to the so-called reaction coordinate method, we identify a relevant collective bosonic mode in the environment, which is then used to generate multiphoton interactions following the proposed theoretical framework. Moreover, we show that spin-boson models featuring structured environments can lead to non-Markovian multiphoton Jaynes-Cummings dynamics. We discuss the validity of the proposed method depending on the parameters and analyse its performance, which is supported by numerical simulations. In this manner, the spin-boson model serves as a good analogue quantum simulator for the inspection and realization of multiphoton Jaynes-Cummings models, as well as the interplay of non-Markovian effects and, thus, as a simulator of light-matter systems with tunable interaction mechanisms.