Novel frame changes for quantum physics

TL;DR

The paper develops a $igstar$-algebraic framework for frame changes of linear non-autonomous ODEs, including quantum evolution, by using a two-time Green's function and resolvent. It introduces the biframe change, a symmetric two-part frame transformation, and proves that it accelerates perturbative Dyson expansions, achieving an error of order $ ext{$oldsymbol{ extepsilon}$}^{2n+1}$ for a cost comparable to standard frames. Extending to tri-frames, the authors show cubic acceleration, and demonstrate the concepts with a time-dependent Hamiltonian where numerical Dyson-order estimates converge faster in the biframe than in laboratory or standard frames. The results cast frame changes as purely linear-$igstar$ algebraic operations, suggesting infinitely many such frames with potential applications beyond QM, including Heun-function perturbations.

Abstract

We present novel, exotic types of frame changes for the calculation of quantum evolution operators. We detail in particular the biframe, in which a physical system's evolution is seen in an equal mixture of two different standard frames at once. We prove that, in the biframe, convergence of all series expansions of the solution is quadratically faster than in `conventional' frames. That is, if in laboratory frame or after a standard frame change the error at order $n$ of some perturbative series expansion of the evolution operator is on the order of $ε^n$, $0<ε<1$, for a computational cost $C(n)$ then it is on the order of $ε^{2n+1}$ in the biframe for the same computational cost. We demonstrate that biframe is one of an infinite family of novel frames, some of which lead to higher accelerations but require more computations to set up initially, leading to a trade-off between acceleration and computational burden.

Figures (1)

• Figure 1: Logarithm of the relative error $\epsilon$ of Eq. (\ref{['eq:froerr']}) for the first 12 orders of the Dyson series expansion in the laboratory frame (red circles and line), after a standard frame change (black squares and line) and in the biframe (blue stars and line). Scaled parameters: $\omega_0/\omega\simeq 0.67$, $\beta/\omega\simeq 0.53$, total scaled simulation time $\omega T=6$. The dashed straight line at the bottom represents machine precision. The poor performance of the laboratory frame is due to the inability of its low orders to correctly fit the behavior at long times $t\sim T$, which heavily degrades the overall error measure $\epsilon$.