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arXiv:1108.1199 · Physical Review B

Master equation approach to the central spin decoherence problem: Uniform coupling model and role of projection operators

Edwin Barnes, Łukasz Cywiński, S. Das Sarma

Published Oct 18, 2011• 37 citations

Abstract

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.

DOI: 10.1103/physrevb.84.155315

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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.

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.

The dynamics of an open quantum system is usually studied by performing a weak-coupling and weak-correlation expansion in the system-bath interaction. For systems exhibiting strong couplings and highly non-Markovian behavior this approach is not justified. We apply a recently proposed correlated projection superoperator technique to the model of a central spin coupled to a spin bath via full Heisenberg interaction. Analytical solutions to both the Nakajima-Zwanzig and the time-convolutionless master equation are determined and compared with the results of the exact solution. The correlated projection operator technique significantly improves the standard methods and can be applied to many physical problems such as the hyperfine interaction in a quantum dot.

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

\varrho_{B}(t)

and the system

\varrho_{S}(t)

are written down. A formal solution of the equation for

\varrho_{B}(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

\varrho_{S}(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

\varrho_{B}(t)

and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.

We present a modified approach for simulating electronically nonadiabatic dynamics based on the Nakajima-Zwanzig generalized quantum master equation (GQME). The modified approach utilizes the fact that the Nakajima-Zwanzig formalism does not require casting the overall Hamiltonian in system-bath form, which is arguably neither natural nor convenient in the case of the Hamiltonian that governs nonadiabatic dynamics. Within the modified approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density operator is fully captured by a memory kernel super-operator. A methodology for calculating the memory kernel from projection-free inputs is developed. Simulating the electronic dynamics via the modified approach, with a memory kernel obtained using exact or approximate methods, can be more cost effective and/or lead to more accurate results than direct application of those methods. The modified approach is compared to previously proposed GQME-based approaches, and its robustness and accuracy are demonstrated on a benchmark spin-boson model with a memory kernel which is calculated within the Ehrenfest method.

In spin-based nanosystems for quantum information processing, electron spin qubits are subject to decoherence due to their interactions with nuclear spin environments. In this paper, we present an exact master equation for a central spin-1/2 system in time-dependent external fields and coupled to a spin-half bath in terms of hyperfine interaction. The master equation provides a unified description for free and controlled dynamics of the central spin and is formally independent of the details and size of spin environments. Different from the previous approaches, the master equation remains exact even in the presence of external control fields. Using the parameters for realistic nanosystems with nonzero nuclear spins, such as GaAs, we investigate the Overhauser's effect on the decoherence dynamics of the central spin under different distributions of bath-spin frequencies and system-bath coupling strengths. Furthermore, we apply the leakage elimination operator, in a nonperturbative manner, to this system to suppress the decoherence induced by hyperfine interaction.