Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.04187

Distributed optimization of Lindblad equations for large-scale cavity QED systems

Hui-hui Miao

quant-phSubmitted Mar 4, 2026• Announced Mar 5, 2026• 0 citations
v1 · Mar 4, 2026v2 · Jun 7, 2026
arXivHTMLPDF

Abstract

This paper proposes a distributed computing framework for solving the Lindblad master equation in large-dimensional cavity QED systems. By leveraging the sparsity of the jump operator and combining this approach with the Cannon algorithm, the computational complexity of non-unitary terms is reduced from $O(MN^3)$ to $O(MN)$. For unitary terms, a combination of Taylor series approximation and the Cannon algorithm enables distributed matrix exponentiation, though scalability is limited by cross-processor communication. The proposed dynamic subspace construction method further reduces the Hamiltonian dimension: when $n_{\text{at}}=10$, the dimension is reduced to $5.63\%$ of the full Hamiltonian, with a memory footprint of only $0.32\%$. Results show that this framework significantly accelerates non-unitary evolution, providing a feasible solution for simulating large-scale open quantum systems where the number of dissipative channels $M$ is much larger than the Hamiltonian dimension $N$.

Comments
8 pages, 8 figures; Supplementary Information: 23 pages, 12 figures, 5 tables

Related papers

Sobre la solución del Hamiltoniano de Tavis-Cummings en Óptica Cuántica: dinámica y estructura basada en la base colectivas de spines

Mario José Félix Rojas

We present a comprehensive theoretical and computational study of the Tavis–Cummings model in the collective-spin basis, addressing both its static spectral properties and dynamic behavior under unitary and open-system evolution. By exploiting the commutation of the total-excitation operator with the Hamiltonian, we construct block-diagonal representations for systems of two-level molecules coupled to a single quantized cavity mode. We derive analytic expressions for the degeneracies, block dimensions, and secular polynomials, introducing a family of orthogonal “Félix–Sanz-Vicario” new class of orthogonal polynomials (here baptized Félix-Sanz-Vicario polynomials) whose roots furnish the energy eigenvalues and whose finite-degree structure allows precise control over photon-number truncation in small manifolds. Numerical implementations in Python (QuTiP) confirm exact agreement with the original Tavis–Cummings solution in each excitation manifold, while also highlighting the role of lower-spin subspaces—absent in the bosonized highest-spin description—in mediating configurational molecular interactions via a rank-1 spherical tensor operator. Time-dependent simulations compare dynamics under the rotating wave aproximation (Tavis-Cummings model) and without it (Dicke model), demonstrating population transfer, Rabi oscillations, and collapse–revival phenomena. The dynamics of open quantum system, via a Lindblad master equation, illustrate cavity-loss effects on photon-number decay, von Neumann entanglement entropy, and coherent exchange under dissipation. Our results establish that the collective-spin framework, using a set of solutions in terms of orthogonal polynomials, provides both analytical insight and computational efficiency for spectroscopic modeling, ultrastrong coupling studies, and the design of low-photon-number quantum simulations.

•Jan 1, 2025•0 citations

Emergence of a quasiergodic steady state in a dissipative Tavis-Cummings array

Debabrata Mondal, K. Sengupta, Subhasis Sinha

In an atom-photon interacting system described by the Tavis-Cummings-Hubbard (TCH) model, we demonstrate the emergence of a quasi-steady state in a dissipative environment that exhibits intriguing ergodic behavior. The TCH model undergoes a dissipative transition from normal to superradiant phase hosting a gapped Higgs and gapless Goldstone modes. However, in a large region of the phase diagram, the instability of the Goldstone mode leads to the disappearance of the stable superradiant phase. In this regime, the decorrelator dynamics reveals light cone spreading of the perturbations and positive Lyapunov exponent, indicating enhanced fluctuations. Remarkably, a quasi-steady state emerges under quench dynamics in this unstable regime; in this state, a class of collective quantities such as site-averaged photon number and atomic excitations approach a steady value, in spite of the large temporal fluctuations in corresponding microscopic variables. This quasi-steady state describes an incoherent fluid of photons with significant phase fluctuation. The phase space dynamics reveals a fascinating ergodic behavior in the presence of dissipation leading to the characterization of the dynamical variables into two distinct classes. The first class includes site-averaged photon numbers and atomic excitations; these exhibit a stationary distribution regardless of the initial condition indicating ergodic behavior. The second class of variables, particularly those related to photon or atom phases, in contrast, retain information about the initial conditions, resulting in a violation of ergodicity for finite size systems. Additionally, the dynamical variables of the ergodic class exhibit a fascinating collective scarring phenomenon as the peak of their distribution is attracted towards the unstable steady state, analogous to the single-particle quantum scar. We discuss the relevance of our findings in the current experiments.

arXiv:2310.12779•Physical Review A•Oct 7, 2024•6 citations

Dissipative phase transition in an open Tavis–Cummings model with a two-photon drive

Kingshuk Adhikary, Özgür Esat Müstecaplıoğlu, Bimalendu Deb

Abstract We investigate the steady-state quantum phases and associated collective phenomena of an open Tavis–Cummings (TC) model driven by a two-photon source. The standard (non-dissipative) TC model describes the quantum phases of an ensemble of N q two-level quantum emitters interacting with a single-mode electromagnetic field. The interplay among coherent driving, dissipation, and dipole interactions of the open TC model results in emergent collective phenomena, leading to a dissipative or non-equilibrium phase transition from the normal to the superradiant phase. We solve the Liouvillian equation analytically using the semi-classical mean-field approximation. We carry out stability analysis of the steady-state phases and determine the phase boundary. The use of Holstein–Primakoff transformation in the thermodynamic limit N q → ∞ reduces the system to an effective coupled-oscillator model, the solution of which yields collective modes.

Journal of Physics B Atomic Molecular and Optical Physics•Jun 7, 2023•6 citations

Non-equilibrium many-body effects in driven nonlinear resonator arrays

Thomas Grujic, Stephen R. L. Clark, Dieter Jaksch +1 more

We study the non-equilibrium behavior of optically driven dissipative coupled resonator arrays. Assuming each resonator is coupled with a two-level system via a Jaynes–Cummings interaction, we calculate the many-body steady state behavior of the system under coherent pumping and dissipation. We propose and analyze the many-body phases using experimentally accessible quantities such as the total excitation number, the emitted photon spectra and photon coherence functions for different parameter regimes. In parallel, we also compare and contrast the expected behavior of this system assuming the local nonlinearity in the cavities is generated by a generic Kerr effect as described by the Bose–Hubbard (BH) model rather than a Jaynes–Cummings interaction. We find that the behavior of the experimentally accessible observables produced by the two models differs for realistic regimes of interactions even when the corresponding nonlinearities are of similar strength. We analyze in detail the extra features available in the Jaynes–Cummings–Hubbard (JCH) model originating from the mixed nature of the excitations and investigate the regimes where the BH approximation would faithfully match the JCH physics. We find that the latter is true for values of the light–matter coupling and losses beyond the reach of current technology. Throughout the study we operate in the weak pumping, fully quantum mechanical regime where approaches such as mean field theory fail, and instead use a combination of quantum trajectories and the time evolving block decimation algorithm to compute the relevant steady state observables. In our study we have assumed small to medium size arrays (from 3 up to 16 sites) and values of the ratio of coupling to dissipation rate g / γ ∼ 20, which makes our results implementable with current designs in circuit QED and with near future photonic crystal set ups.

arXiv:1205.0994•New Journal of Physics•Oct 16, 2012•56 citations

The role of dephasing for dark state coupling in a molecular Tavis–Cummings model

Eric Davidsson, Markus Kowalewski

The collective coupling of an ensemble of molecules to a light field is commonly described by the Tavis-Cummings model. This model includes numerous eigenstates that are optically decoupled from the optically bright polariton states. Accessing these dark states requires breaking the symmetry in the corresponding Hamiltonian. In this paper, we investigate the influence of non-unitary processes on the dark state dynamics in the molecular Tavis-Cummings model. The system is modeled with a Lindblad equation that includes pure dephasing, as it would be caused by weak interactions with an environment, and photon decay. Our simulations show that the rate of pure dephasing, as well as the number of two-level systems, has a significant influence on the dark state population.

arXiv:2304.09583•The Journal of Chemical Physics•Jul 26, 2023•22 citations

On the Dynamics of the Tavis–Cummings Model

Zhiyuan Dong, Guofeng Zhang, Ai‐Guo Wu +1 more

The purpose of this article is to present a comprehensive study of the Tavis–Cummings model from a system-theoretic perspective. A typical form of the Tavis–Cummings model is composed of an ensemble of noninteracting two-level systems (TLSs) that are collectively coupled to a common cavity resonator. The associated quantum linear passive system is proposed, whose canonical form reveals typical features of the Tavis–Cummings model, including$\sqrt{N}$- scaling, dark states, bright states, single-excitation superradiant, and subradiant states. The passivity of this linear system is related to the vacuum Rabi mode splitting phenomenon in Tavis–Cummings systems. On the basis of the linear model, an analytic form is presented for the steady-state output state of the Tavis–Cummings model driven by a single-photon state. Master equations are used to study the excitation properties of the Tavis–Cummings model in the multiexcitation scenario. Finally, in terms of the transition matrix for a linear time-varying system, a computational framework is proposed for calculating the state of the Tavis–Cummings model, which is applicable to the multiexcitation case.

arXiv:2110.14174•IEEE Transactions on Automatic Control•Apr 26, 2022•17 citations
This paper proposes a distributed computing framework for solving the Lindblad master equation in large-dimensional cavity QED systems. By leveraging the sparsity of the jump operator and combining this approach with the Cannon algorithm, the computational complexity of non-unitary terms is reduced from O(MN3)O(MN^3)O(MN3) to O(MN)O(MN)O(MN). For unitary terms, a combination of Taylor series approximation and the Cannon algorithm enables distributed matrix exponentiation, though scalability is limited by cross-processor communication. The proposed dynamic subspace construction method further reduces the Hamiltonian dimension: when nat=10n_{\text{at}}=10nat​=10, the dimension is reduced to 5.63%5.63\%5.63% of the full Hamiltonian, with a memory footprint of only 0.32%0.32\%0.32%. Results show that this framework significantly accelerates non-unitary evolution, providing a feasible solution for simulating large-scale open quantum systems where the number of dissipative channels MMM is much larger than the Hamiltonian dimension NNN.
We present a comprehensive theoretical and computational study of the Tavis–Cummings model in the collective-spin basis, addressing both its static spectral properties and dynamic behavior under unitary and open-system evolution. By exploiting the commutation of the total-excitation operator with the Hamiltonian, we construct block-diagonal representations for systems of two-level molecules coupled to a single quantized cavity mode. We derive analytic expressions for the degeneracies, block dimensions, and secular polynomials, introducing a family of orthogonal “Félix–Sanz-Vicario” new class of orthogonal polynomials (here baptized Félix-Sanz-Vicario polynomials) whose roots furnish the energy eigenvalues and whose finite-degree structure allows precise control over photon-number truncation in small manifolds. Numerical implementations in Python (QuTiP) confirm exact agreement with the original Tavis–Cummings solution in each excitation manifold, while also highlighting the role of lower-spin subspaces—absent in the bosonized highest-spin description—in mediating configurational molecular interactions via a rank-1 spherical tensor operator. Time-dependent simulations compare dynamics under the rotating wave aproximation (Tavis-Cummings model) and without it (Dicke model), demonstrating population transfer, Rabi oscillations, and collapse–revival phenomena. The dynamics of open quantum system, via a Lindblad master equation, illustrate cavity-loss effects on photon-number decay, von Neumann entanglement entropy, and coherent exchange under dissipation. Our results establish that the collective-spin framework, using a set of solutions in terms of orthogonal polynomials, provides both analytical insight and computational efficiency for spectroscopic modeling, ultrastrong coupling studies, and the design of low-photon-number quantum simulations.
In an atom-photon interacting system described by the Tavis-Cummings-Hubbard (TCH) model, we demonstrate the emergence of a quasi-steady state in a dissipative environment that exhibits intriguing ergodic behavior. The TCH model undergoes a dissipative transition from normal to superradiant phase hosting a gapped Higgs and gapless Goldstone modes. However, in a large region of the phase diagram, the instability of the Goldstone mode leads to the disappearance of the stable superradiant phase. In this regime, the decorrelator dynamics reveals light cone spreading of the perturbations and positive Lyapunov exponent, indicating enhanced fluctuations. Remarkably, a quasi-steady state emerges under quench dynamics in this unstable regime; in this state, a class of collective quantities such as site-averaged photon number and atomic excitations approach a steady value, in spite of the large temporal fluctuations in corresponding microscopic variables. This quasi-steady state describes an incoherent fluid of photons with significant phase fluctuation. The phase space dynamics reveals a fascinating ergodic behavior in the presence of dissipation leading to the characterization of the dynamical variables into two distinct classes. The first class includes site-averaged photon numbers and atomic excitations; these exhibit a stationary distribution regardless of the initial condition indicating ergodic behavior. The second class of variables, particularly those related to photon or atom phases, in contrast, retain information about the initial conditions, resulting in a violation of ergodicity for finite size systems. Additionally, the dynamical variables of the ergodic class exhibit a fascinating collective scarring phenomenon as the peak of their distribution is attracted towards the unstable steady state, analogous to the single-particle quantum scar. We discuss the relevance of our findings in the current experiments.
Abstract We investigate the steady-state quantum phases and associated collective phenomena of an open Tavis–Cummings (TC) model driven by a two-photon source. The standard (non-dissipative) TC model describes the quantum phases of an ensemble of N q two-level quantum emitters interacting with a single-mode electromagnetic field. The interplay among coherent driving, dissipation, and dipole interactions of the open TC model results in emergent collective phenomena, leading to a dissipative or non-equilibrium phase transition from the normal to the superradiant phase. We solve the Liouvillian equation analytically using the semi-classical mean-field approximation. We carry out stability analysis of the steady-state phases and determine the phase boundary. The use of Holstein–Primakoff transformation in the thermodynamic limit N q → ∞ reduces the system to an effective coupled-oscillator model, the solution of which yields collective modes.
We study the non-equilibrium behavior of optically driven dissipative coupled resonator arrays. Assuming each resonator is coupled with a two-level system via a Jaynes–Cummings interaction, we calculate the many-body steady state behavior of the system under coherent pumping and dissipation. We propose and analyze the many-body phases using experimentally accessible quantities such as the total excitation number, the emitted photon spectra and photon coherence functions for different parameter regimes. In parallel, we also compare and contrast the expected behavior of this system assuming the local nonlinearity in the cavities is generated by a generic Kerr effect as described by the Bose–Hubbard (BH) model rather than a Jaynes–Cummings interaction. We find that the behavior of the experimentally accessible observables produced by the two models differs for realistic regimes of interactions even when the corresponding nonlinearities are of similar strength. We analyze in detail the extra features available in the Jaynes–Cummings–Hubbard (JCH) model originating from the mixed nature of the excitations and investigate the regimes where the BH approximation would faithfully match the JCH physics. We find that the latter is true for values of the light–matter coupling and losses beyond the reach of current technology. Throughout the study we operate in the weak pumping, fully quantum mechanical regime where approaches such as mean field theory fail, and instead use a combination of quantum trajectories and the time evolving block decimation algorithm to compute the relevant steady state observables. In our study we have assumed small to medium size arrays (from 3 up to 16 sites) and values of the ratio of coupling to dissipation rate g / γ ∼ 20, which makes our results implementable with current designs in circuit QED and with near future photonic crystal set ups.
The collective coupling of an ensemble of molecules to a light field is commonly described by the Tavis-Cummings model. This model includes numerous eigenstates that are optically decoupled from the optically bright polariton states. Accessing these dark states requires breaking the symmetry in the corresponding Hamiltonian. In this paper, we investigate the influence of non-unitary processes on the dark state dynamics in the molecular Tavis-Cummings model. The system is modeled with a Lindblad equation that includes pure dephasing, as it would be caused by weak interactions with an environment, and photon decay. Our simulations show that the rate of pure dephasing, as well as the number of two-level systems, has a significant influence on the dark state population.
The purpose of this article is to present a comprehensive study of the Tavis–Cummings model from a system-theoretic perspective. A typical form of the Tavis–Cummings model is composed of an ensemble of noninteracting two-level systems (TLSs) that are collectively coupled to a common cavity resonator. The associated quantum linear passive system is proposed, whose canonical form reveals typical features of the Tavis–Cummings model, includingN\sqrt{N}N​- scaling, dark states, bright states, single-excitation superradiant, and subradiant states. The passivity of this linear system is related to the vacuum Rabi mode splitting phenomenon in Tavis–Cummings systems. On the basis of the linear model, an analytic form is presented for the steady-state output state of the Tavis–Cummings model driven by a single-photon state. Master equations are used to study the excitation properties of the Tavis–Cummings model in the multiexcitation scenario. Finally, in terms of the transition matrix for a linear time-varying system, a computational framework is proposed for calculating the state of the Tavis–Cummings model, which is applicable to the multiexcitation case.