Pure quantum dephasing of a solid-state electron spin qubit in a large nuclear spin bath coupled by long-range hyperfine-mediated interactions

Abstract

We investigate decoherence due to pure dephasing of a localized spin qubit interacting with a nuclear spin bath. Although in the limit of a very large magnetic field the only decoherence mechanism is spectral diffusion due to dipolar flip-flops of nuclear spins, with decreasing field the hyperfine-mediated interactions between the nuclear spins become important. We take advantage of their long-range nature, and resum the leading terms in an 1/N expansion of the decoherence time-evolution function (N , being the number of nuclear spins interacting appreciably with the electron spin, is large). For the case of the thermal uncorrelated bath we show that our theory is applicable down to low magnetic fields (∼ 10 mT for a large dot with N = 10 6 ) allowing for comparison with recent experiments in GaAs quantum dot spin qubits. Within this approach we calculate the free induction decay and spin echo decoherence in GaAs and InGaAs as a function of the number of the nuclei in the bath (i.e. the quantum dot size) and the magnetic field. Our theory for free induction decay in a narrowed nuclear bath is shown to agree with the exact solution for decoherence due to hyperfine-mediated interaction which can be obtained when all the nucleielectron coupling constants are identical. For the spin echo evolution we show that the dominant decoherence process at low fields is due to interactions between nuclei having significantly different Zeeman energies (i.e. nuclei of As and two isotopes of Ga in GaAs), and we compare our results with recent measurements of spin echo signal of a single spin confined in a GaAs quantum dot. For the same set of parameters we perform calculations of decoherence under various dynamical decoupling pulse sequences, and predict the effect of these sequences in low B regime in GaAs. arXiv:0903.2256v3 [cond-mat.mes-hall]

Semiclassical model for the dephasing of a two-electron spin qubit coupled to a coherently evolving nuclear spin bath

“…For the simple model in section V, in which we ignored the nuclear dipole-dipole interaction and the Knight shift of the nuclear Larmor frequencies, we demonstrated the equivalence of the semi-classical treatment and the quantum diagrammatic summation of Ref. 32. The semi-classical treatment quantitatively captures the observed phenomena of monotonic decay of the echo signal in strong magnetic fields, and the collapses and revivals of the echo signal in weaker magnetic fields.…”

“…(8), an identical expression for H el,z was formally derived in Ref. 32 from the full quantum Hamiltonian, Eq. (1), using a Schrieffer-Wolff transformation.…”

“…(22) and (23), which describe the dephasing of a single electron spin in an isolated quantum dot, were derived previously from a fully quantum mechanical treatment in Ref. 32. A key element of the derivation in that work was the vanishing of the contribution of the commutator [I + k , I − l ] between nuclear spin operators in the low order perturbation expansion of the evolution operator.…”

“…We first show that, within a simple model which neglects the effects of the Knight shift and the dipolar interaction between nuclear spins, the semi-classical treatment reproduces the expressions for the spin echo signal obtained in Refs. 32,33, where a summation of diagrams in a perturbative quantum mechanical treatment was used. We then present a more detailed microscopic model, which incorporates the Knight shift and the nuclear dipole-dipole interactions, as well as inhomogeneous hyperfine and Zeeman couplings.…”

Decay of the rotary echoes for the spin of a nitrogen-vacancy center in diamond

“…When the electronic spin is decohered by the bath of nuclear spins, the back-action from the central spin to many bath spins may become strong compared to the coupling within the bath. In this case decoherence of the central spin may be governed by a mechanism similar to electronic spin echo envelope modulation (ESEEM) 60 , modified by the many-spin nature of the bath, and other methods must be used to analyze the decoherence dynamics [75][76][77][78][79][80] As a result, the influence of the N atom bath on the NV spin is described with the simplified dephasing Hamiltonian B(t)S z 0 , where the random field B(t) is assumed to be Gaussian (because many nitrogen atoms contribute to the field B(t)), stationary (since the back-action on the bath is negligible), and Markovian (due to the large size of the bath and negligible back-action) stochastic process. Such a noise field is represented by an OrnsteinUhlenbeck (O-U) random process 81 , with the correlation function…”