An integral-free representation of the Dyson series using divided differences

Abstract

The Dyson series is an infinite sum of multi-dimensional time-ordered integrals, which serves as a formal representation of the quantum time-evolution operator in the interaction-picture. Using the mathematical tool of divided differences, we introduce an alternative representation for the series that is entirely free from both time ordering and integrals. In this new formalism, the Dyson expansion is given as a sum of efficiently-computable divided differences of the exponential function, considerably simplifying the calculation of the Dyson expansion terms, while also allowing for time-dependent perturbation calculations to be performed directly in the Schr{ö}dinger-picture. We showcase the utility of this novel representation by studying a number of use cases. We also discuss several immediate applications.

Figures (2)

• Figure 1: The probability to remain in the initial state $|0\rangle$ (eigenstate of $\sigma_z$ with eigenvalue $+1$) for a highly oscillating (here, ${\omega_0=2 \omega=200}$) two-level system for different values of time $t$. The figure shows numerically-exact calculations using the proposed method for a wide range of coupling strengths. The black lines passing through the data points correspond to numerically-exact solutions obtained using 4th-order Runge-Kutta simulations carried out independently to verify the correctness of the our results.
• Figure 2: (a) Probabilities of the various modes at $t=0.04$ for the anharmonic Hamiltonian, Eq. (\ref{['eq:HanharApp']}). Here, $\omega=1, \Omega=2, \gamma=0.02$ and the initial state is $|n=4\rangle$. The blue circles are obtained from numerical integration of the Schrödinger equation while the orange triangles correspond to the divided-differences expansion up to order $Q=5$. (b) Infidelity $1-|\langle \psi(t)|\psi_Q(t)\rangle|^2$ as a function of time The various curves correspond to different truncation orders $Q=0,\ldots,3$. As we expect, the higher the expansion order is, the better the approximation becomes.