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arXiv:0903.5427 · The European Physical Journal B

Projection operator approach to transport in complex single-particle quantum systems

Robin Steinigeweg, Jochen Gemmer B, Heinz‐Peter Breuer, Heinz–Jürgen Schmidt

Published Apr 25, 2009• 6 citations
arXivHTMLPDF
DOI
10.1140/epjb/e2009-00143-2

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The work at hand presents a novel approach to transport in closed quantum systems. To this end a method is introduced which is essentially based on projection operator techniques, in particular on the time-convolutionless (TCL) technique. The projection onto local densities of quantities such as energy, magnetization, particles, etc. yields the reduced dynamics of the respective quantities in terms of a systematic perturbation expansion. Especially, the lowest order contribution of this expansion is used as a strategy for the analysis of transport in "modular" quantum systems. The term modular basically corresponds to (quasi-) one-dimensional structures consisting of identical or at least similar many-level subunits. Modular quantum systems are demonstrated to represent many physical situations and several examples are given. In the context of these quantum systems lowest order TCL is shown as an efficient tool which also allows to investigate the dependence of transport on the considered length scale. In addition an estimation for the validity range of lowest order TCL is derived. As a first application a "design" model is considered for which a complete characterization of all available transport types as well as the transitions to each other is possible. For this model the relationship to quantum chaos and the validity of the Kubo formula is further discussed. As an example for a "real" system the Anderson model is finally analyzed. The results are partially verified by the numerical solution of the full time-dependent Schroedinger equation which is obtained by exact diagonalization or approximative integrators.

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Czechoslovak Journal of Physics•Jun 1, 1998•0 citations
In this article, I provide a simple review on conventional methods and models on the transport phenomenon of nuclear many-body systems. By exploiting the basic idea of time-dependent projection operator, I recommend a novel method to derive the transport equation for collective motion which is embedded on the microscopic dynamics of timedependent single-particle motion. It is emphasized that the microscopic dynamics of single-particle motion should play an important role for understanding the dynamics of nuclear reaction and the synthesis mechanisms of new superheavy elements. The numerical results of energy dissipation and entropy production indicate that the collective motion passes through three stages, such as dephasing/decoherence, statistical relaxation and stationary state. The statistical relaxation is a complex anomalous diffusion process in general. With the aid of above analysis and results, a possible way to define the collective and single-particle variables for the realistic nuclear many-body systems is proposed.
It is remarkable that Heisenberg’s position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg’s uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time.
The work at hand presents a novel approach to transport in closed quantum systems. To this end a method is introduced which is essentially based on projection operator techniques, in particular on the time-convolutionless (TCL) technique. The projection onto local densities of quantities such as energy, magnetization, particles, etc. yields the reduced dynamics of the respective quantities in terms of a systematic perturbation expansion. Especially, the lowest order contribution of this expansion is used as a strategy for the analysis of transport in "modular" quantum systems. The term modular basically corresponds to (quasi-) one-dimensional structures consisting of identical or at least similar many-level subunits. Modular quantum systems are demonstrated to represent many physical situations and several examples are given. In the context of these quantum systems lowest order TCL is shown as an efficient tool which also allows to investigate the dependence of transport on the considered length scale. In addition an estimation for the validity range of lowest order TCL is derived. As a first application a "design" model is considered for which a complete characterization of all available transport types as well as the transitions to each other is possible. For this model the relationship to quantum chaos and the validity of the Kubo formula is further discussed. As an example for a "real" system the Anderson model is finally analyzed. The results are partially verified by the numerical solution of the full time-dependent Schroedinger equation which is obtained by exact diagonalization or approximative integrators.