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arXiv:1401.7572 · Physical Review A

Arbitrary spin in a spin bath: Exact dynamics and approximation techniques

Vitalii Semin, Ilya Sinayskiy, Francesco Petruccione

Published Jan 13, 2014• 14 citations
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Abstract

A model of an arbitrary spin coupled to a bath of spins $\frac{1}{2}$ in a star configuration is considered. The exact reduced dynamics of the central spin is found for the case of noncorrelated initial conditions of the system and the bath. The exact solution is used to test two approximation techniques, namely, the Nakajima-Zwanzig projection operator technique and the time-convolutionless projection operator technique corresponding to the second order of the coupling constant. Two types of projection operators are used for deriving the master equations, and the results are compared with the exact solution for a central spin equal to one. It is shown that the approximate master equations reproduce the exact dynamics on time scales $1/(A\sqrt{N})$, where $A$ is the coupling constant and $N$ is the number of spins in the bath.

DOI
10.1103/physreva.89.012107

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The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath ϱB(t)\varrho_{B}(t)ϱB​(t) and the system ϱS(t)\varrho_{S}(t)ϱS​(t) are written down. A formal solution of the equation for ϱB(t)\varrho_{B}(t)ϱB​(t), having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in ϱS(t)\varrho_{S}(t)ϱS​(t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of ϱB(t)\varrho_{B}(t)ϱB​(t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.
The generalized master equation of Nakajima-Zwanzig (NZ) type has been used extensively to investigate the coherence dynamics of the central spin model with a nuclear bath in a narrowed state characterized by a well-defined value of the Overhauser field. We reconsider the perturbative NZ approach and apply it to the exactly solvable case of a system with uniform hyperfine couplings. This is motivated by the fact that the effective-Hamiltonian-based theory suggests that the dynamics of the realistic system at low magnetic fields and short times can be mapped onto the uniform coupling model. We show that the standard NZ approach fails to reproduce the exact solution of this model beyond very short times, while the effective-Hamiltonian calculation agrees very well with the exact result on time scales during which most of the coherence is lost. Our key finding is that, in order to extend the time scale of applicability of the NZ approach in this case, instead of using a single projection operator one has to use a set of correlated projection operators which properly reflect the symmetries of the problem and greatly improve the convergence of the theory. This suggests that the correlated projection operators are crucial for a proper description of narrowed-state free-induction decay at short times and low magnetic fields. Our results thus provide important insights toward the development of a more complete theory of central spin decoherence applicable in a broader regime of time scales and magnetic fields.