Abstract
We derive general evolution equations describing the ensemble-average quantum dynamics generated by disordered Hamiltonians. The disorder average affects the coherence of the evolution and can be accounted for by suitably tailored effective coupling agents and associated rates which encode the specific statistical properties of the Hamiltonian's eigenvectors and eigenvalues, respectively. Spectral disorder and isotropically disordered eigenvector distributions are considered as paradigmatic test cases.
Highlights
Disorder is the expression of a lack of knowledge—e.g., on a physical system’s conformation or of a potential landscape, on scales that affect the system evolution
In the case of time-independent Lindblad operators and timeindependent positive rates, one usually speaks of a Markovian Lindblad master equation in the strict sense, describing a dynamical semigroup
In all situations considered in this article, the inverse map exhibits, at most, isolated divergences. We conjecture that this is a general property of dynamical ensemble averages on finite-dimensional Hilbert spaces as defined in Eq (3), due to the quasiperiodicity of the timeevolved state inherited from the discrete spectrum of Uλ
Summary
Disorder is the expression of a lack of knowledge—e.g., on a physical system’s conformation or of a potential landscape, on scales that affect the system evolution. No formalism is available that allows us to assess the effective dynamical quantum evolution of an arbitrary initial state in real time and to directly associate characteristic time scales and couplings thereof with the underlying disorder Note, that this represents a natural and—given the wealth of disordered quantum transport problems— substantial expansion of the generic playground of the theory of open quantum systems, by substituting an uncontrolled environment with a static, operator-valued random perturbation of the system Hamiltonian. We derive effective dynamical evolution equations for the ensemble-averaged state of disordered quantum systems, in the form of master equations, which, by their very structure, precisely meet the above purpose and show how the statistics of the disorder enters the unitary part, as well as the Lindblad operators and associated rates, as the equations’ specific ingredients We apply this theory to exemplary cases of quantum systems with random spectra and randomly distributed eigenvectors for which our method can be applied without any approximations.
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