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

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.

arXiv commentAccepted for publication in Physical Review Research

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• Floquet Analysis of Frequency Collisions
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• A Lanczos-like method for non-autonomous linear ordinary differential equations
• Spin dynamics under the influence of elliptically rotating fields: Extracting the field topology from time-averaged quantities
• An integral-free representation of the Dyson series using divided differences
• A continuous approach to Floquet theory for pulse-sequence optimization in solid-state NMR
• Error bounds for the Floquet-Magnus expansion and their application to the semiclassical quantum Rabi model
• Density operator approach to turbulent flows in plasma and atmospheric fluids
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• Refuting a Proposed Axiom for Defining the Exact Rotating Wave Approximation
• Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory