Quantum theory for electron spin decoherence induced by nuclear spin dynamics in semiconductor quantum computer architectures: Spectral diffusion of localized electron spins in the nuclear solid-state environment

Abstract

We consider the decoherence of a single localized electron spin due to its coupling to the lattice nuclear spin bath in a semiconductor quantum computer architecture. In the presence of an external magnetic field and at low temperatures, the dominant decoherence mechanism is the spectral diffusion of the electron spin resonance frequency due to the temporally fluctuating random magnetic field associated with the dipolar interaction induced flip-flops of nuclear spin pairs. The electron spin dephasing due to this random magnetic field depends intricately on the quantum dynamics of the nuclear spin bath, making the coupled decoherence problem difficult to solve. We provide a formally exact solution of this non-Markovian quantum decoherence problem which numerically calculates accurate spin decoherence at short times, which is of particular relevance in solid-state spin quantum computer architectures. A quantum cluster expansion method is developed, motivated, and tested for the problem of localized electron spin decoherence due to dipolar fluctuations of lattice nuclear spins. The method is presented with enough generality for possible application to other types of spin decoherence problems. We present numerical results which are in quantitative agreement with electron spin echo measurements in phosphorus doped silicon. We also present spin echo decay results for quantum dots in GaAs which differ qualitatively from that of the phosphorus doped silicon system. Our theoretical results provide the ultimate limit on the spin coherence (at least, as characterized by Hahn spin echo measurements) of localized electrons in semiconductors in the low temperature and the moderate to high magnetic field regime of interest in scalable semiconductor quantum computer architectures.

Universal quantum control of two-electron spin quantum bits using dynamic nuclear polarization

“…Here we take advantage of the interaction of 1 the electrons with the nuclear magnetic field of the Ga and As sublattices of the host material in order to generate the required magnetic field gradient. While fluctuations of this hyperfine field are known to be a major source of decoherence [8][9][10][11][12] , in this letter we demonstrate the possibility of building up a gradient in the hyperfine field that significantly exceeds the fluctuations and can be sustained for times longer than 30 min. This is done by employing pumping schemes that transfer spin and thus magnetic moment from the electronic system to the nuclei.…”

Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot

“…Most progress was made in well-controlled III-V quantum dots, where spin manipulation with two 6,11 , three 12 and four 13 dots has been realized, but gate fidelities and spin coherence times are limited by the unavoidable interaction with the fluctuating nuclear spins in the host substrate 5,6 . While the randomness of the nuclear spin bath could be mitigated to some extent by feedback techniques 14 , eliminating the nuclear spins by using group IV host materials offers the potential for extremely long electron spin coherence times that exceed one second in P impurities in bulk 28 Si 15,16 .Much effort has been made to develop stable spin qubits in quantum dots defined in carbon nanotubes 17,18 , Ge/Si core/shell nanowires 19 , Si MOSFETs 20,21 and Si/SiGe 2D electron gases 16,22,23 . However, coherent control in these group IV quantum dots is so far limited to a Si/SiGe singlet-triplet qubit with only single-axis control 23 and a carbon nanotube single-electron spin qubit, with a Hahn echo decay time of 65 ns 17 .…”

“…Most progress was made in well-controlled III-V quantum dots, where spin manipulation with two 6,11 , three 12 and four 13 dots has been realized, but gate fidelities and spin coherence times are limited by the unavoidable interaction with the fluctuating nuclear spins in the host substrate 5,6 . While the randomness of the nuclear spin bath could be mitigated to some extent by feedback techniques 14 , eliminating the nuclear spins by using group IV host materials offers the potential for extremely long electron spin coherence times that exceed one second in P impurities in bulk 28 Si 15,16 .…”

The Decoherence of the Electron Spin and Meta-Stability of 13C Nuclear Spins in Diamond

“…Consequently, it has been reported that (inhomogeneous) electron spin-dephasing times of T * 2 = 0.5 − 6 μs can be achieved in Diamond, at 300 K-without the need for cryogenic cooling [1,4,18], which makes Diamond an attractive candidate for Quantum Information Processing. Following [12], however, we argue that whilst it is clear that a successful qubit can be constructed from the spin polarization the 3 A 2 state (as has been successfully demonstrated in [1,4,9,16,17]) it is not at all clear why this state should be so highly stable from a calculational standpoint, with such a relatively long decoherence time, given the current understanding of the mechanism of spectral diffusion [19,20].…”

“…However, it is our new argument that this qubit does not simply decohere to become entangled with the bath, but with a bath which is also decohering to become entangled with itself and, therefore, that this slowly changes the definition of the SO(3) rotational symmetry of the electron spin to make it evolve into an oblate spheroid (relative to its initial spin projection). Whilst previous calculational schemes have treated the nuclear spin bath dynamics [19,20], these schemes have been restricted to short-time regimes and localised electron states, which we are now able to go beyond in our new approach. The limitation we have resolved is that there is generally a two-scale process involved in bath dynamics: The range of the nuclear dipole interaction strength and the cluster size of the frozen cores of nuclear spins that form through the flip-flop processes [19], and it was not previously understood to be possible to define a expansion program that is simultaneously valid at two very different scales.…”