Universal Lindblad equation for open quantum systems (2004.01469v2)

Abstract: We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.

• The paper introduces a universal Lindblad equation that overcomes limitations of prior methods, including the Bloch-Redfield formalism.
• It employs bath correlation decomposition and the Markovian parameter Γτ to derive an error bound of order Γ²τ while maintaining state positivity.
• This approach enables efficient numerical simulations of complex systems such as many-body spin chains and Floquet systems in quantum computation.

Universal Lindblad Equation for Open Quantum Systems

The paper of open quantum systems, where a quantum system interacts with its surrounding environment or bath, is pivotal in areas ranging from atomic physics to quantum information processing. Various theoretical frameworks are available for modeling open quantum systems; however, each method has its limitations, particularly the challenge of preserving the complete physicality of a quantum state during its evolution. The paper by Frederik Nathan and Mark S. Rudner introduces a Universal Lindblad Equation (ULE) designed to efficiently describe a wide variety of open quantum systems, overcoming many existing constraints found in prior methodologies like the Bloch-Redfield (BR) formalism.

Key Contributions

Nathan and Rudner present a derivation of a Markovian master equation of Lindblad form, which is significant because it accommodates the dynamics of open quantum many-body systems without stringent conditions on system level spacings. This universality, as provided by the ULE, is not commonly achievable by earlier approaches such as the quantum optical master equation, which relies on the rotating-wave approximation (RWA) valid only under specific conditions like sparse energy level distributions.

The Lindblad form guarantees the preservation of quantum state's positivity and trace during evolution, sidestepping the instabilities sometimes encountered with BR equations over long timescales. The ULE, as derived in the paper, maintains accuracy at a level comparable to the BR approach but with enhanced robustness, making it amenable to numerical analysis involving stochastic evolution of pure states rather than density matrices, thereby optimizing computational resources.

Theoretical Foundation and Methodology

The approach begins with the decomposition of the bath correlation function, encapsulated by a "jump correlator," which leads to a definition of bath characteristics in terms of timescales, namely $\Gamma^{-1}$ (interaction times) and $\tau$ (correlation times). The interplay between these timescales determines the Markovianity parameter $\Gamma\tau$ whose smallness ensures the validity of the ULE, which implies the evolution of the system is independent of specific system detail aside from system-bath interactions.

Notably, the authors postulate that one can construct a continuous family of Markovian approximations beyond what is typically applied in deriving BR equations. By choosing an appropriate member from this family, the authors derive the ULE demonstrating that it has an error bound correlated to $\Gamma^2\tau$. Additionally, the error is smaller than the order of bath-induced system evolution characterized by $\Gamma$, emphasizing the relevance and precision of the ULE in describing quantum dynamics.

Implications and Applications

The ULE is versatile, applicable to static and dynamic systems including Floquet systems, allowing simulations that reveal the behavior of complex systems such as many-body spin chains coupled to baths. The implications are broad, ranging from improved understanding of quantum thermodynamics to laying groundwork for future advancements in quantum computation and noise-resilient quantum technologies.

Future Directions

While the current work provides substantial improvements over existing approaches, future research could explore higher-order corrections beyond the existing perturbative framework, optimizing the characterization of non-Markovian effects which could be significant in some quantum many-body systems. Another avenue includes leveraging this formalism to explore system-bath interactions beyond Gaussian statistics, widening the applicability of the ULE.

Overall, Nathan and Rudner's work provides a robust framework for simulating open quantum systems, balancing theoretical elegance with practical utility, and promises to be a critical tool for researchers exploring the complexities of quantum systems interacting with their environments.