Perturbative regimes in central spin models

B. Erbe and J. Schliemann

Phys. Rev. B 85, 235423 – Published 11 June 2012

Abstract

Central spin models describe several types of solid state nanostructures which are presently considered as possible building blocks of future quantum information processing hardware. From a theoretical point of view, a key issue remains the treatment of the flip-flop terms in the Hamiltonian in the presence of a magnetic field. We consider homogeneous hyperfine and exchange coupling constants (which are different from each other) and systematically study the influence of these terms, both as a function of the field strength and the size of the spin baths. We find crucial differences between initial states with central spin configurations of high and low polarizations. This has strong implications with respect to the influence of a magnetic field on the flip-flop terms in central spin models of a single central spin and in those of more than one central spin. Furthermore, the dependencies on bath size and field differ from those anticipated so far. Our results might open the route for the systematic search for more efficient perturbative treatments of central spin problems.

Received 14 December 2011

DOI:https://doi.org/10.1103/PhysRevB.85.235423

Authors & Affiliations

B. Erbe and J. Schliemann

Institut für Theoretische Physik, Universität Regensburg, 93053 Regensburg, Germany

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Vol. 85, Iss. 23 — 15 June 2012

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Images

Figure 1
Spin dynamics for $N_{c} = 1, 2, 3$ and $N = 401$ . We choose an exchange coupling of $J_{e x} = (1 / 800) A$ . The magnetic field is fixed to $A / B = 4$ . We consider initial product states with $| α_{c} 〉 = | ⇓ 〉, | ⇓ ⇑ 〉, | ⇓ ⇓ ⇑ 〉$ and a very low bath polarization of $p_{b} = (1 / N)$ . In all cases, $N_{c} = 1, 2, 3$ , the amplitude of the oscillation is decaying to zero (followed by a series of revivals on longer time scales not shown). The influence of the magnetic field manifests itself in a decrease of the magnitude of the spin decay and an increase of the decoherence time as measured by the decay of the amplitude of the oscillation. The two quantities, denoted by $μ$ and $τ$ , are depicted in the first panel and the second panel, respectively. We investigate the scaling of the decoherence time by analyzing from which time on the amplitude of the oscillation falls under a threshold level. The concrete value is of no importance, as long as the amplitude is larger than the threshold level before the onset of the decay. For high values of $p_{c}$ the quantity $τ$ remains unaffected by the magnetic field and $μ$ is the relevant perturbative measure, whereas for small $p_{c}$ it is $τ$ which quantifies the influence of the flip-flop terms.Reuse & Permissions
Figure 2
$B$ field and $N$ scalings of $μ$ for $N_{c} = 1, 2$ and $N = 401, 801$ or $A / B = 8, 4$ , respectively. The exchange coupling in the case of $N_{c} = 2$ is fixed to $J_{e x} = (1 / 800) A$ . We consider initial product states with $| α_{c} 〉 = | ⇓ 〉, | ⇓ ⇓ 〉$ , corresponding to $p_{c} = 1$ , and a low bath polarization of $p_{b} = (1 / N)$ . The results are plotted on a double logarithmic scale. We find power laws $\sim$ $B^{- ν}$ with $ν \approx 2$ and $\sim$ $N^{- ν}$ with $ν \approx 1$ . In the first case, the exact values are given by $ν = 2.024 85, 2.097 05$ ( $N_{c} = 1, N = 401, 801$ ) and $ν = 1.955 65, 1.964 27$ ( $N_{c} = 2, N = 401, 801$ ). For the $N$ scaling we have $ν = 0.989 959, 1.017 76$ ( $N_{c} = 1, A / B = 8, 4$ ) and $ν = 0.925 328, 0.975 64$ ( $N_{c} = 2, A / B = 8, 4$ ). With respect to the magnetic field scaling, the fully analytical result, given in Eq. (24), is reproduced.Reuse & Permissions
Figure 3
$B$ field and $N$ scalings of $τ$ for $N_{c} = 2, 3$ and $N = 401, 801$ or $A / B = 8, 4$ , respectively. The exchange coupling is fixed to $J_{e x} = (1 / 800) A$ . We consider initial product states with $| α_{c} 〉 = | ⇓ ⇑ 〉, | ⇓ ⇓ ⇑ 〉$ , corresponding to $p_{c} = 0, \frac{1}{3}$ and a low bath polarization of $p_{b} = (1 / N)$ . The results are plotted on a double logarithmic scale. We find power laws $\sim$ $B^{ν}$ with $ν \approx 2$ and $N^{ν}$ with $ν \approx 2$ . In the first case, the exact values are given by $ν = 2.079 75, 1.995 43$ ( $N_{c} = 2, N = 401, 801$ ) and $ν = 1.977 42, 1.995 55$ ( $N_{c} = 3, N = 401, 801$ ). For the $N$ scaling we have $ν = 1.877 36, 1.977 22$ ( $N_{c} = 2, A / B = 8, 4$ ) and $ν = 1.770 72, 1.925 03$ ( $N_{c} = 3, A / B = 8, 4$ ). Note that an increase of $τ$ corresponds to a decrease of the flip-flop contributions to the dynamics.Reuse & Permissions

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