Simple and general unitarity conserving numerical real time propagators of time dependent Schrödinger equation based on Magnus expansion

TL;DR

This work develops unitary real-time propagators for the time-dependent Schrödinger equation using Magnus expansion (ME) combined with explicit time interpolation. By deriving explicit ME terms $M_n$ up to $n=4$ and pairing them with polynomial approximations of $H(t)$ within each interval, the authors produce finite-time propagators of second to fifth order (and discuss sixth-order variants) that preserve unitarity at each truncation. Numerical tests on a two-state sinusoidally driven model show that simple fourth-order ME schemes (notably those leveraging a mid-interval evaluation) can achieve accuracy comparable to more complex Gauss-based schemes, while higher-order ME terms provide improved fidelity at increased computational cost. The results indicate that ME-based propagators are practical, general tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, and open-system dynamics, with clear guidance on method choice for efficiency vs. accuracy.

Abstract

Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.

Figures (4)

• Figure 1: Time-dependent eigenvalues of $H(t)$ for the four cases of parameters listed in Table \ref{['table-parameter']}. The time range $(0,2\pi)$ shown covers a period of the Hamiltonian in all four cases.
• Figure 2: Populations of the excited state $|2\rangle$ for the four cases of parameters listed in Table \ref{['table-parameter']}. At time zero, all the populations are at the state $|1\rangle$. Markers for case II indicate times where level crossings occur, initially at $t \approx 4.712$ and repeating every $2\pi$.
• Figure 3: Plots of $\log_{10} ({\rm error})$, where ${\rm error}$ is defined according to eq \ref{['eq:error']}, vs. $\log_{10} (\delta t/t_c)$, where $t_c$ is the unit time and is equal to one in the present work. Different panels (I-IV) represent different cases of the Hamiltonian eq \ref{['eq:Hamiltonian']} as listed in Table \ref{['table-parameter']}. Eight different approximations are compared as listed in the legends. Markers represent actual data points while the lines are best linear lines fitting these points. Filled markers represent methods based on the Gaussian quadrature. Insets show close-ups of fourth-order methods. We have also used alternating markers for some overlapping lines.
• Figure S1: Populations of the excited state $|1\rangle$ for high frequency cases III and IV with parameters listed in Table 1 of the main text. At time zero, all the populations are at the state $|0\rangle$. Black dashed lines represent populations with the high frequency term removed.