Computing time-dependent reduced models for classical and quantum dynamics

TL;DR

This work develops a time-dependent reduced-model framework for large autonomous dynamics projected onto a fixed subspace, motivated by time-convolutionless master equations in quantum physics. It introduces a recursive, polynomial-in-time expansion for the reduced generator $F_t$, with $F_{t,N} = \sum_{k=0}^N t^k F_{(k+1)}$, and provides a closed recursion $F_{(k)} = k \frac{R L^k J}{k!} - \sum_{h=1}^{k-1} F_{(k-h)} \frac{R L^h J}{h!}$ that yields $R e^{L t} J - \mathbb{T} e^{\int_0^t F_{s,N} ds} = O(t^{N+1})$. The approach avoids weak-coupling assumptions and, for quantum dynamics, ensures complete positivity and trace preservation at low orders under mild conditions; it is validated on dephasing spin-boson, dissipative central spin, and Ising spin-chain models. The results offer a tractable path to accurate short-time reduced dynamics in both classical and quantum settings, with potential impact on simulating very large systems and understanding environment-induced effects. Limitations include guaranteed accuracy only for small times, and CP properties may degrade at higher orders in some non-bipartite cases.

Abstract

This paper introduces a novel method for approximating the dynamics of a large autonomous system projected onto a fixed subspace. The core contribution is a novel recursive algorithm to construct an effective time-dependent generator that is polynomial in the time variable, ensuring accuracy for short time scales. The derivation is based on the Taylor expansion of the exponential map and a new result for computing the time-ordered exponential of polynomial generators. This work is motivated by the challenge of deriving time-convolutionless master equations in quantum physics and the proposed method offers an alternative to typical derivations based on expansions in the coupling strength. The resulting approximation is accurate for small times, does not require a weak-coupling assumption, performs better than a truncation of the exponential map at low orders, and crucially, guarantees a completely positive and trace-preserving map at the lowest orders. The proposed method is validated against several prototypical models: a dephasing spin-boson model, a central spin model, and an Ising spin chain.

Figures (7)

• Figure 1: Graphical representation of the approximation problem. The pale blue plane represents the subspace $\mathscr{V}$ onto which $P$ projects. The red curve represents the trajectory of the original model $x_t$, which starts in $\mathscr{V}$ at $t=0$. The blue curve represents the projection of the true trajectory $Px_t$, i.e. the one we want to approximate. Finally, the orange curve represents the approximate evolution we aim to find, i.e. $Jz_t$.
• Figure 2: Operator and Hilbert-Schmidt norm of the first 20 $F_{(k)}$ terms (left) and of the first 50 $E_{(k)}$ terms when $N=20$ (right).
• Figure 3: The continuous lines represent the error norm between the exact reduced evolution $R x_t$ and the approximated reduced evolution $z_t$ for different values of the approximation order $N$. The dashed curves represent an approximation obtained by truncating the Taylor expansion of the exponential for different values of the truncation order $N$.
• Figure 4: Simulation of the dephasing spin-boson model for $s=0.5,1,1.5$ where we compare the exact solution $\rho_{S,t}$ given in Eq.\ref{['eq:exact_boson']}, the second order approximation $\mu_t$ derived in this work, given in Eq. \ref{['eq:time_ordered_boson']} and the coarse-grained dynamics $\tilde{\rho}_{S,t}$ given in Eq. \ref{['eq:coarse_grained_boson']}. Left: Evolution of the expectation value $\left< \sigma_x \right>$ for the exact solution (continuous curves), second order approximation (dashed curve) and coarse-grained dynamics (dotted curve). The expectations values of the cases $s=0.5,1.5$ have been shifted by $\pm0.5$ respectively for graphical purposes. Right: evolution of the trace-norm error with respect to the exact solution committed by the second order approximation (continuous curve) and coarse-grained dynamics (dashed curve).
• Figure 5: Simulations of the dissipative central spin model in the purely Hamiltonian setting ($\Lambda=0$) on the left and in the dissipative setting ($\Lambda=0.8$) on the right. The exact evolution and various degrees of approximations ($N=1,2,3,4,10,20$) are shown in each figure. The first row shows the trajectory of the state of the central state $\rho_S$ in the Bloch sphere; the second row shows the magnetization of the central spin in the $\sigma_z$ direction versus time; the third row shows the trace norm error between the exact trajectory $\eta_t$ and the reduced one $\mu_t$, i.e. $\left|\left| \eta_t-\mu_t \right|\right|_{{\rm tr}}$ versus time for different values of the approximation $N$ and with dotted curve representing a fitted $\alpha t^{N+1}$ curve. The black crosses denote points where the evolved state $\mu_t$ exits the set of density operators.

Theorems & Definitions (11)

• Proposition 1
• proof
• Theorem 1
• Theorem 2
• proof
• Proposition 2
• Theorem 3
• Theorem 4
• proof
• Example 1