Quantum theory of spectral-diffusion-induced electron spin decoherence

Abstract

A quantum cluster expansion method is developed for the problem of localized electron spin decoherence due to dipolar fluctuations of lattice nuclear spins. At the lowest order it provides a microscopic explanation for the Lorentzian diffusion of Hahn echoes without resorting to any phenomenological Markovian assumption. Our numerical results show remarkable agreement with recent electron spin echo experiments in phosphorus doped silicon.PACS numbers: 76.60.Lz; 03.65.Yz; 03.67.Lx It was realized a long time ago that spectral diffusion due to the dipolar fluctuations of nuclear spins often dominates the coherence decay in electron spin echo experiments [1,2]. The recent advent of spin-based quantum computation in semiconductor nanostructures revived the interest in spectral diffusion, which is expected to be the dominant channel for low-temperature spin decoherence in several spin-based quantum computer architectures [3]. In spectral diffusion, the electron spin Zeeman frequency diffuses in time due to the noise produced by the nuclear spin bath. Dipolar fluctuations in the nuclear spins give rise to a temporally random effective magnetic field at the localized electron spin, leading to irreversible decoherence (i.e. a T 2 -process). All available theories to date are based on classical stochastic modeling of the nuclear field, a Markovian theoretical framework which is inevitably phenomenological since it requires an arbitrary choice for the spectrum of nuclear fluctuations. Such a classical Markovian modeling is arguably incompatible with the strict requirements of spin coherence and control in a quantum information device. In addition, recent rapid experimental progress in single spin measurements [4], which in the near future promise sensitive measurements of quantum effects in spin resonance, also warrant a quantum theory of spectral diffusion. Here we propose a quantum theoretical framework for spectral diffusion which is non-stochastic and fully microscopic. In addition, our theory produces an accurate quantitative prediction for the initial decoherence, which is the most important regime for quantum computation. To the best of our knowledge, ours is the first quantum theory for electron spin spectral diffusion.Spectral diffusion is not a limiting decoherence process for silicon or germanium based quantum computer proposals because these can, in principle, be fabricated free of nuclear spins using isotopic purification. Unfortunately this is not true for the important class of materials based on III-V compounds, where spectral diffusion has been shown to play a major role [3,5]. There is as yet no experimental measurement of localized spin decoherence (echo decay) in III-V materials, but such experimental results are anticipated in the near future.Our theory reveals that the inclusion of quantum corrections to nuclear spin fluctuation increases the degree of decoherence, as is best evidenced from our explanation of the existing factor of three discrepancy between the Markovian stochastic theory [5] and...

Electron Spin as a Spectrometer of Nuclear-Spin Noise and Other Fluctuations

“…Our explicit numerical calculations for the echo decay without broadening [Eq. (96)] reproduced the equivalent calculation of Witzel et al [21] with no visible deviation. For k B T ≫ γ n B we may assume that the nuclear spins are completely unpolarized (the experimental data was taken at T = 4 K and B = 0.3 T [18]).…”

“…The nuclear spin noise spectrum was calculated from a pair-flip-flop model, resulting in a linear combination of sharp transitions (delta functions). The echo decay due to these sharp transitions is identical to the one derived by the lowest order cluster expansion [21]. Next, we showed how to obtain a smooth noise spectrum by adding broadening to these transitions using a mean-field approach.…”

“…First, what is the contribution of higher order nuclear spin transitions to the noise spectrum? This question may be answered by going beyond the simple pair flip-flop model assumed here, in a similar fashion as the cluster expansion developed in [21], or using an alternative linked cluster expansion for the spin Green's function [39]. Another interesting open question is the design of optimal sequences for suppressing the effects of nuclear spin noise in electron spin evolution, as was done for the random telegraph noise model in Ref.…”

“…This allows us to give an elegant and simple derivation of the lowest order cluster expansion results of Ref. [21] and to interpret these results from the point of view of non-equilibrium statistical mechanics. The full noise spectrum is expressed as a sum of delta-function contributions corresponding to isolated pair flip-flop transitions.…”

“…The next step was to develop a full quantum theory for the nuclear spin dynamics affecting the electron spin. In reference [21] a cluster expansion method was developed to calculate echo decay due to the closed-system dynamics of a group of dipolar coupled nuclear spins, without any stochastic assumption about the nuclear spin dynamics. At lowest order in this cluster expansion the qualitative and quantitative agreement with experimental data was quite good.…”