# Exact dynamics of quantum systems driven by time-varying Hamiltonians: solution for the Bloch-Siegert Hamiltonian and applications to NMR

## TL;DR

This paper introduces a novel non-perturbative path-sum method for exactly solving the dynamics of quantum systems with time-varying Hamiltonians, demonstrated on two-level and many-body NMR systems.

## Contribution

The paper presents the path-sum approach, a new mathematical technique that guarantees convergence and provides exact solutions for finite quantum systems with time-dependent Hamiltonians.

## Key findings

- Exact solutions for two-level quantum systems.
- Analytical results for NMR-related many-body Hamiltonians.
- Demonstrated applicability to Bloch-Siegert effect and spin diffusion.

## Abstract

Comprehending the dynamical behaviour of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood as the usual methods including diagonalisation of the Hamiltonian do not work in this setting. In fact, since the inception of Magnus' expansion in 1954, no fundamentally novel mathematical approach capable of solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here of an entirely different non-perturbative approach, termed path-sum, which is always guaranteed to converge, yields the exact analytical solution in a finite number of steps for finite systems and is invariant under scale transformations of the quantum state space. Path-sum can be combined with any state-space reduction technique and can exactly reconstruct the dynamics of a many-body quantum system from the separate, isolated, evolutions of any chosen collection of its sub-systems. As examples of application, we solve analytically for the dynamics of all two-level systems as well as of a many-body Hamiltonian with a particular emphasis on NMR (Nuclear Magnetic Resonance) applications: Bloch-Siegert effect, coherent destruction of tunneling and $N$-spin systems involving the dipolar Hamiltonian and spin diffusion.