Magnus expansion is used to identify effective Hamiltonians describing the coarse-grained dynamics of more complex problems. Here, we apply this method to a two-level system driven by an and AC field. We derive Stark and Bloch-Siegert shifs of both diagonal and off-diagonal entries of the Hamiltonian as a result of coarse-graining only.
We derive a Magnus expansion for a frequency chirped quantum two-level system. We obtain a time-independent effective Hamiltonian which generates a stroboscopic time evolution. At lowest order the according dynamics is identical to results from using a rotating wave approximation. We determine, furthermore, also the next higher-order corrections within our expansion scheme in correspondence to the Bloch-Siegert shifts for harmonically driven systems. Importantly, our scheme can be extended to more complicated systems, i.e., even many-body systems.
We formulate an effective-description framework for the dynamics of open quantum systems by extending the time-coarse-graining formalism to open systems. Our coarse-graining procedure efficiently removes high-frequency processes which are responsible for coherences between lower- and upper-manifold states and are irrelevant when considering only low-energy dynamics. We investigate the regime of validity of the resulting coarse-grained master equation by applying it to multilevel atoms driven by far-detuned lasers. Except for such high-frequency coherences, we find good agreement between the exact and coarse-grained dynamics unless the driving lasers are too strong or the initial high-frequency coherences are sizable.
One of the most powerful and potentially useful methods of time-dependent matrix differential equations is the Magnus expansion. In this work, we applied the Magnus expansion for Schrödinger and master equations of a dissipative two-level system interacting with a time-dependent π pulse. Two different models of dissipative two-level systems are considered, namely (i) the system decays from two considered levels into some other levels with given decay rates; (ii) the system decays from its excited level to its ground level. The obtained approximate solutions of a time dynamics by the first-order Magnus expansion are well consistent with the results of numerical calculations obtained by the fourth-order Runge-Kutta method.
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. Since the inception of Magnus’ expansion in 1954, no fundamentally novel mathematical method for 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. We solve for the dynamics of all two-level systems as well as of many-body Hamiltonians with a particular emphasis on NMR applications (Bloch-Siegert effect and N-spin systems involving the dipolar Hamiltonian and spin diffusion).
Inspired by holographic Wilsonian renormalization, we consider coarse graining a quantum system divided between short-distance and long-distance degrees of freedom (d.o.f.), coupled via the Hamiltonian. Observations using purely long-distance observables are described by the reduced density matrix that arises from tracing out the short-distance d.o.f. The dynamics of this density matrix is non-Hamiltonian and nonlocal in time, on the order of some short time scale. We describe this dynamics in a model system with a simple hierarchy of energy gaps $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}>\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}$, in which the coupling between high- and low-energy d.o.f. is treated to second order in perturbation theory. We then describe the equations of motion under suitable time averaging, reflecting the limited time resolution of actual experiments, and find an expansion of the master equation in powers of $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}/\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}$, after the fashion of effective field theory. The failure of the system to be Hamiltonian or even Markovian appears at higher orders in this ratio. We compute the evolution of the density matrix in three specific examples: coupled spins, linearly coupled simple harmonic oscillators, and an interacting scalar quantum field theory. Finally, we argue that the logarithm of the Feynman-Vernon influence functional is the correct analog of the Wilsonian effective action for this problem.