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.
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.
This work presents a systematic numerical study of the dissipative Jaynes-Cummings model, consisting of a two-level system (qubit) coupled to a truncated harmonic oscillator, both subject to Markovian dissipation. The model is standard in quantum optics and open quantum systems. The master equation is solved using the Lindblad formalism with BDF integration. The study focuses on two standard observables: the qubit excited-state population P_exc(t) and the qubit Bloch vector length |r(t)|. A three-dimensional parameter scan is performed over the coupling g, the qubit dephasing rate gamma_phi, and the oscillator relaxation rate gamma_a, giving 216 cells in total. The parameters used are omega_0 = 6 Hz, omega_1 = 40 Hz, giving a detuning delta = 34 Hz, and thermal occupation n_th = 0.05. MAIN FINDINGS: The single-exponential description of the excited-state population is valid only at intermediate coupling (g approximately 4 to 8 Hz). Outside this region, the dynamics exhibits a multi-timescale structure. The decay time T is non-monotonic in g, with a minimum at g approximately 7 Hz. The location of the minimum depends only weakly on gamma_a, following a power law with exponent 0.127. At strong coupling (g greater than 15 Hz), a two-exponential model is strongly preferred by the Akaike Information Criterion in 43 out of 45 cells. The two timescales are separated by a factor of approximately 150. The fast timescale T_1 is best approximated by the Purcell formula (median ratio of numerical to theoretical value 0.68). The Fermi golden rule overestimates the decay rate by a factor of approximately 4.5. Neither the dressed-state correction nor the Bloch-Redfield calculation improves on the Purcell formula. The Bloch vector length does not follow any simple functional form (single exponential, stretched exponential, power law, or double exponential). All models have mean R-squared in the range 0.31 to 0.36. Its time derivative is, however, almost perfectly correlated with that of the population (mean correlation 0.91). The results are converged with respect to the Fock-space truncation at N_0 = 12, with identical observables for N_0 between 12 and 20. SCOPE: This is a purely numerical study of a well-defined quantum-optical model. It contains no empirical data, no claims about biological or cognitive systems, and no speculative ontology. The model, the numerical method, and the observables are standard. The results include several negative findings (the failure of perturbative formulas, the lack of a simple functional form for the Bloch vector) which are reported honestly and which identify the regime where non-perturbative methods are required. The purpose of this release is to provide reproducible numerical data and a benchmark for future analytical work on the dissipative Jaynes-Cummings model in the strong-coupling regime. CONTENTS: The LaTeX technical note (approximately 20 pages), the Python scripts for the numerical simulations, and the CSV files with the fit results, regime classification, and theoretical comparison. REFERENCES: E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).G. Lindblad, Commun. Math. Phys. 48, 119 (1976).H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).E. M. Purcell, Phys. Rev. 69, 681 (1946).C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley (1992).W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH (2001).
The discretization approximation method commonly used to simulate the dynamics of quantum systems coupled to the environment in continuum often suffers from the periodically partial recovery of the initial state because of the effect of finite dimension, dubbed the recurrence. To address this issue, we propose a generalization of the discretization approximation method into the complex frequency space based on complex Gauss quadratures. An effective Hamiltonian can be established by this way, which is non-Hermitian and demonstrates the complex energy modes with a negative imaginary part, describing the dissipation of the system. This method is applied to examine the dynamics in two exactly solvable models, the dephasing model and the single-excitation dissipative dynamics in the Aubry–André–Harper model. By comparison with the exact numerics and analytical results, it is found that our approach not only significantly reduces the effect of recurrence and improves the effectiveness of calculation but also provides a unique perspective into the dynamics of open systems from the point of complex energy levels. Furthermore, we establish a simple relationship between the parameters in computation and the effectiveness of simulation by analyzing the computational error.
Dissipative properties of a composite system interacting with an environment are studied in order to obtain insight into relaxation and decoherence phenomena. As a typical system with internal degree of freedom, a Jaynes–Cummings (JC) model is considered where a spin (qubit) interacts with a single bosonic mode. When the whole system is initially in a state of at most single excitation, the model becomes solvable for excitation number conserving Hamiltonians, even if the JC system interacts with its environment composed of harmonic oscillators. Effects of initial correlations between the system and the environment are also taken into account. Basic equations for the density matrix are solved exactly to determine dynamics of the reduced systems. Mutually related problems of initial correlations, entanglement and non-Markovian property are studied explicitly by examining the time evolution of the reduced density matrices, the concurrence and the trace distance.
Phase transitions are commonly held to occur only in the thermodynamical limit of a large number of system components. Here, we exemplify at the hand of the exactly solvable Jaynes-Cummings (JC) model and its generalization to finite JC lattices that finite component systems of coupled spins and bosons may exhibit quantum phase transitions (QPTs). For the JC model we find a continuous symmetry-breaking QPT, a photonic condensate with a macroscopic occupation as the ground state, and a Goldstone mode as a low-energy excitation. For the two site JC lattice we show analytically that it undergoes a Mott-insulator to superfluid QPT. We identify as the underlying principle of the emergence of finite system QPTs the combination of increasing atomic energy and increasing interaction strength between the atom and the bosonic mode, which allows for the exploration of an increasingly large portion of the infinite dimensional Hilbert space of the bosonic mode. This suggests that finite system phase transitions will be present in a broad range of physical systems.
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.
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.