Unveiling coherent dynamics in non-Markovian open quantum systems: exact expression and recursive perturbation expansion

Technical Summary: Unveiling Coherent Dynamics in Non-Markovian Open Quantum Systems

Problem Statement
The dynamics of open quantum systems are governed by the interplay between coherent evolution and environmental interactions. While the Lindblad framework successfully describes energy renormalization (such as Lamb and AC Stark shifts) under weak-coupling and Markovian assumptions, extending this understanding to strong-coupling and non-Markovian regimes remains a significant challenge. In the time-convolutionless (TCL) master equation formalism, the generator of reduced dynamics, $L_t$, can be decomposed into a coherent part (effective Hamiltonian $K(t)$) and a dissipative part ($D_t$). However, this decomposition is not unique due to the intrinsic symmetry of the generator. Resolving this ambiguity is essential for consistently quantifying coherent energy shifts and understanding energy renormalization beyond the Markovian limit. While exact expressions for $K(t)$ exist for specific integrable models, a general, systematic framework for deriving $K(t)$ in arbitrary non-Markovian regimes is lacking.

Methodology
The authors introduce a systematic framework to derive the effective Hamiltonian $K(t)$ by applying the minimal dissipation principle, a foundational approach recently proposed by Hayden and Sorce. This principle uniquely isolates the coherent component by projecting the generator $L_t$ onto the subspace of superoperators that act as commutators with Hermitian operators.

The methodology proceeds in three main stages:

1. Operational Definition: The authors define a scalar product between superoperators based on averaging over Haar-random pure states. This allows for the unique identification of the effective Hamiltonian $K$ that minimizes the dissipative contribution. They derive a general, operational expression for $K$ in terms of an orthonormal basis of the Hilbert space, valid for any linear, Hermiticity-preserving, trace-annihilating generator.
2. Recursive Perturbative Expansion: Recognizing that explicit forms of $L_t$ are often inaccessible in closed form, the authors develop a recursive perturbative expansion for $K(t)$ in powers of the system-bath coupling strength $\lambda$. This expansion expresses $K(t)$ directly in terms of system interaction operators and bath correlation functions (specifically, ordered cumulants).
3. Symmetry Analysis: The framework analyzes the structure of the perturbative series by decomposing bath correlation functions and system operators into real and imaginary parts, which exhibit distinct symmetries under time-ordering reversal.

Key Contributions and Results

• General Expression for $K(t)$: The paper provides a closed-form expression (Eq. 6) to extract the effective Hamiltonian from any known time-local generator $L_t$. This expression, $K = \frac{1}{2id} \sum [F^\dagger_\alpha, L[F_\alpha]]$, offers a practical tool for computing coherent shifts once the generator is known.
• Recursive Perturbative Series: The authors derive a systematic expansion (Eq. 16) for the $n$-th order contribution to the effective Hamiltonian, $K_n$. This formula explicitly links energy renormalization to $n$-point bath correlation functions and system operators in the interaction picture.
• Distinction Between Even and Odd Orders: A central finding is the fundamental symmetry difference between even and odd orders in the perturbative expansion:
  • Odd-order terms ($K_{2m+1}$): These terms are overall symmetric under time-ordering reversal. They couple system and bath contributions of the same symmetry class. In spin systems, odd-order terms do not commute with the free Hamiltonian (unless the interaction commutes with it), leading to rotations of the Hamiltonian eigenbasis.
  • Even-order terms ($K_{2m}$): These terms are overall antisymmetric under time-ordering reversal. They couple contributions of opposing symmetry classes. In spin systems, even-order terms commute with the free Hamiltonian, resulting in energy level shifts without rotating the eigenbasis.
• Application to Spin Systems: The framework is applied to paradigmatic spin models:
  • For pure dephasing (interaction $A=\sigma_z$), odd terms vanish, and the effective Hamiltonian results only in a time-dependent energy shift (renormalization of frequency).
  • For orthogonal coupling (interaction $A=\sigma_x$), odd terms do not vanish and induce eigenbasis rotations, while even terms contribute to energy shifts.
  • The analysis explains why standard Gaussian bath models (characterized by two-point correlations) exhibit simpler renormalization (no eigenbasis rotation for pure dephasing), whereas baths with time-asymmetry (non-zero odd-order cumulants) lead to more complex behaviors.

Significance
The paper claims that this framework provides a systematic tool for analyzing energy renormalization effects across different coupling regimes, bridging the gap between foundational theory and practical calculation in non-Markovian quantum thermodynamics.

• Conceptual Clarity: It resolves the ambiguity in defining the effective Hamiltonian in strong-coupling regimes, offering a consistent quantification of coherent shifts.
• Thermodynamic Relevance: The effective Hamiltonian $K(t)$ is identified as a key link to quantum thermodynamics in the strong-coupling domain, capturing energy renormalization that influences efficiency and work extraction in quantum thermal machines.
• Analytical Guidance: The perturbative series serves as a complementary tool to numerical methods like Hierarchical Equations of Motion (HEOM). It provides analytical insight into the structure of numerical results, helping to interpret phenomena such as eigenbasis rotations and time-dependent shifts observed in simulations.
• Physical Interpretation: By distinguishing the roles of even and odd orders, the work clarifies how specific environmental features (e.g., time-asymmetry in bath distributions) drive distinct dynamical behaviors, such as the rotation of the system's eigenbasis.