Real-Time Dyson Expansion Achieves Accurate Correlation Effects in Driven Quantum Systems

Researchers are increasingly focused on understanding the behaviour of driven quantum systems, and a new study by Thomas Blommel, M. Rey Lambert, and Michael A. Kurniawan, all from the University of California, Santa Barbara, alongside Annabelle Canestraight and Vojtech Vlcek, details a systematic comparison of the real-time Dyson expansion with established methods for simulating these complex systems. This work is significant because it demonstrates how the real-time Dyson expansion accurately captures dynamical correlations, including subtle oscillations often lost in other calculations, and resolves rich non-equilibrium spectral structure in time-resolved photoemission. By bridging the gap between simpler mean-field approaches and full two-time simulations, the team’s findings offer a computationally efficient pathway to model ultrafast spectroscopic data and gain deeper insights into the behaviour of interacting quantum materials.

Focusing on density matrix dynamics, time-off-diagonal Green’s functions, and time-resolved photoemission spectra, the team benchmarked RTDE against fully self-consistent Kadanoff-Baym equation (KBE) calculations, the generalized Kadanoff-Baym ansatz (GKBA), and exact diagonalization for small systems using second order many-body perturbation theory. Employing a driven two-band Hubbard model, researchers show that mean-field single particle density matrix trajectories provide a reliable baseline for RTDE across a broad range of interaction strengths and excited-carrier populations. This establishes a robust foundation for more complex calculations by leveraging established mean-field techniques.

Further, the study reveals that RTDE accurately captures correlation effects in the Green’s functions, including long-lived oscillations and revivals that are strongly suppressed by the overdamping inherent to self-consistent KBE schemes. This is a significant advancement, as traditional KBE methods often struggle to represent these crucial dynamical features accurately. Consequently, RTDE resolves rich non-equilibrium spectral structure in time-resolved photoemission, such as interaction- and population-dependent quasiparticle splittings and bandgap renormalization, which are largely washed out in self-consistent approaches, yet are present in exact solutions. The ability to discern these subtle spectral features is critical for understanding the fundamental behaviour of materials under ultrafast excitation.
The research demonstrates that RTDE bridges the gap between mean-field propagation and full two-time KBE simulations, retaining favorable linear scaling while capturing essential dynamical correlations relevant for ultrafast spectroscopy. This linear scaling is a major computational advantage, enabling simulations of larger systems and longer timescales than previously possible with traditional KBE methods. Experiments show that this approach allows for a more efficient and accurate exploration of non-equilibrium many-body dynamics, offering a systematically improvable route to understanding complex quantum phenomena. The work opens new avenues for simulating the behaviour of materials under extreme conditions, potentially leading to the design of novel materials with tailored optical and electronic properties.

This breakthrough reveals that RTDE effectively reconstructs time-nonlocal information as a Markovian perturbation theory on top of a non-equilibrium mean-field trajectory for the density matrix. Practically, RTDE builds upon the ideas of the G1-G2 scheme by deriving linearly-scaling equations for integrating the Green’s function away from the t = t′ diagonal. By incorporating dynamical self-energy effects during a reconstruction step, the method avoids the full memory-dependent collision integral, significantly reducing computational cost. The study focused on density matrix dynamics, time-off-diagonal Green’s functions, and time-resolved photoemission spectra, benchmarking RTDE against fully self-consistent Kadanoff-Baym equation (KBE) calculations, the generalized Kadanoff-Baym ansatz (GKBA), and exact diagonalization for small systems using second order perturbation theory. Researchers employed a driven two-band Hubbard model to demonstrate that mean-field single particle density matrix trajectories provide a reliable baseline for RTDE across a broad range of interaction strengths and excited-carrier populations. The team engineered RTDE to accurately capture correlation effects in the Green’s functions, including long-lived oscillations and revivals that are strongly suppressed by the overdamping inherent to self-consistent KBE schemes.

This method achieves a resolution of rich non-equilibrium spectral structure in time-resolved photoemission, such as interaction- and population-dependent quasiparticle splittings and bandgap renormalization, which are largely washed out in self-consistent approaches but are present in exact solutions. The work details the calculation of the single-particle Green’s function, defined as a trace over time-ordered products of operators on the Keldysh contour, explicitly defining the lesser and greater Keldysh components based on contour argument placement. Scientists defined the retarded and advanced components as linear combinations of the lesser and greater components, utilising the Heaviside function to constrain calculations to the lower half of the time-diagonal. The single-particle reduced density matrix was then related to the lesser component of the Green’s function, establishing a crucial link between the Green’s function and observable density.

Researchers then implemented the KBE, acknowledging that closed-form solutions for interacting systems are generally unattainable and necessitating perturbative treatment of interactions via the self-energy, Σ(z, z′), which mirrors the Keldysh structure and symmetries of the Green’s function. The study pioneered RTDE as a reconstruction of time-nonlocal information as a Markovian perturbation theory on top of a non-equilibrium mean-field trajectory for the density matrix. Practically, RTDE builds upon the ideas of the G1-G2 scheme by deriving linearly-scaling equations for integrating the Green’s function away from the t = t′ diagonal, incorporating dynamical self-energy effects during a reconstruction step in the time-off-diagonal direction without evaluating the full memory-dependent collision integral. The research presents a systematic comparison of RTDE with established non-equilibrium Green’s function approaches, specifically focusing on density matrix dynamics, time-off-diagonal Green’s functions, and time-resolved photoemission spectra. Experiments revealed that RTDE accurately captures correlation effects in the Green’s functions, including long-lived oscillations and revivals, which are often suppressed by the overdamping inherent in self-consistent Kadanoff-Baym equation (KBE) schemes. Using a driven two-band Hubbard model, the team measured mean-field single particle density matrix trajectories, demonstrating their reliability as a baseline for RTDE across a broad range of interaction strengths and excited-carrier populations.

Results demonstrate that RTDE resolves rich non-equilibrium spectral structure in time-resolved photoemission, identifying interaction- and population-dependent quasiparticle splittings and bandgap renormalization. These spectral features are largely washed out in self-consistent approaches, yet are present in exact solutions obtained through second order perturbation theory. The study meticulously benchmarked RTDE against fully self-consistent KBE calculations and the generalized Kadanoff-Baym ansatz (GKBA) for small systems. Measurements confirm that RTDE bridges the gap between mean-field propagation and full two-time KBE simulations, retaining favorable linear scaling while capturing essential dynamical correlations relevant for ultrafast spectroscopy.

The breakthrough delivers a method for reconstructing time-nonlocal correlation functions as a Markovian perturbation theory, building upon non-equilibrium mean-field trajectories for the density matrix. Tests prove that this approach accurately integrates the Green’s function away from the t = t′ diagonal, incorporating dynamical self-energy effects during a reconstruction step. The work details the NEGF formalism and the approximations used, defining the single-particle Green’s function as capturing the evolution of a single quasiparticle probability amplitude for two times on the Keldysh contour. Scientists defined the lesser and greater Keldysh components, G (t, t′), and the retarded and advanced components, GR(t, t′) and GA(t, t′), establishing the foundational equations for the analysis. The research focused on density matrix dynamics, time-off-diagonal Green’s functions, and time-resolved photoemission spectra, benchmarking RTDE against Kadanoff-Baym equation (KBE) calculations, the generalized Kadanoff-Baym ansatz (GKBA), and exact diagonalization for small systems using second order perturbation theory. Using a driven two-band Hubbard model, the study demonstrated that mean-field single particle density matrix trajectories provide a reliable baseline for RTDE calculations across a broad range of interaction strengths and excited-carrier populations. Furthermore, RTDE accurately captures correlation effects in the Green’s functions, including long-lived oscillations and revivals that are suppressed by the overdamping inherent in self-consistent KBE schemes.

Consequently, RTDE resolves rich non-equilibrium spectral structure in time-resolved photoemission, such as interaction- and population-dependent quasiparticle splittings and bandgap renormalization, which are largely obscured in self-consistent approaches. The findings establish RTDE as a method bridging mean-field propagation and full two-time KBE simulations, offering favorable linear scaling while capturing essential dynamical correlations relevant for ultrafast spectroscopy. The authors acknowledge that the success of RTDE relies on building perturbatively on top of a mean-field reduced density matrix, representing a one-shot correction. Future research could explore the application of RTDE to more complex systems and investigate the limits of its validity as interaction strengths increase.