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arXiv:1304.6711 · Physical Review B

Hyperfine interaction induced dephasing of coupled spin qubits in semiconductor double quantum dots

Jo-Tzu Hung, Łukasz Cywiński, Xuedong Hu, S. Das Sarma

Published Aug 19, 2013• 26 citations
arXivHTMLPDF

Abstract

We investigate theoretically the hyperfine-induced dephasing of two-electron-spin states in a double quantum dot with a finite singlet-triplet splitting $J$. In particular, we derive an effective pure dephasing Hamiltonian, which is valid when the hyperfine-induced mixing is suppressed due to the relatively large $J$ and the external magnetic field. Using both a quantum theory based on resummation of ring diagrams and semiclassical methods, we identify the dominant dephasing processes in regimes defined by values of the external magnetic field, the singlet-triplet splitting, and inhomogeneity in the total effective magnetic field. We address both free induction and Hahn echo decay of superposition of singlet and unpolarized triplet states (both cases are relevant for singlet-triplet qubits realized in double quantum dots). We also study hyperfine-induced exchange gate errors for two single-spin qubits. Results for III-V semiconductors as well as silicon-based quantum dots are presented.

DOI
10.1103/physrevb.88.085314

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We investigate theoretically the hyperfine-induced dephasing of two-electron-spin states in a double quantum dot with a finite singlet-triplet splitting JJJ. In particular, we derive an effective pure dephasing Hamiltonian, which is valid when the hyperfine-induced mixing is suppressed due to the relatively large JJJ and the external magnetic field. Using both a quantum theory based on resummation of ring diagrams and semiclassical methods, we identify the dominant dephasing processes in regimes defined by values of the external magnetic field, the singlet-triplet splitting, and inhomogeneity in the total effective magnetic field. We address both free induction and Hahn echo decay of superposition of singlet and unpolarized triplet states (both cases are relevant for singlet-triplet qubits realized in double quantum dots). We also study hyperfine-induced exchange gate errors for two single-spin qubits. Results for III-V semiconductors as well as silicon-based quantum dots are presented.
We have evaluated hyperfine-induced electron spin dynamics for two electrons confined to a double quantum dot. Our quantum solution accounts for decay of a singlet-triplet correlator even in the presence of a fully static nuclear spin system, with no ensemble averaging over initial conditions. In contrast to an earlier semiclassical calculation, which neglects the exchange interaction, we find that the singlet-triplet correlator shows a long-time saturation value that differs from 12\frac{1}{2}21​, even in the presence of a strong magnetic field. Furthermore, we find that the form of the long-time decay undergoes a transition from a rapid Gaussian to a slow power law $(\ensuremath{\sim}1∕{t}^{3∕2})$ when the exchange interaction becomes nonzero and the singlet-triplet correlator acquires a phase shift given by a universal (parameter independent) value of $3\ensuremath{\pi}∕4$ at long times. The oscillation frequency and time-dependent phase shift of the singlet-triplet correlator can be used to perform a precision measurement of the exchange interaction and Overhauser field fluctuations in an experimentally accessible system. We also address the effect of orbital dephasing on singlet-triplet decoherence and find that there is an optimal operating point where orbital dephasing becomes negligible.
We show that singlet-triplet superpositions of two-electron spin states in a double quantum dot undergo a phonon-induced pure dephasing which relies only on the tunnel coupling between the dots and on the Pauli-exclusion principle. As such, this dephasing process is independent of spin-orbit coupling or hyperfine interactions. The physical mechanism behind the dephasing is elastic phonon scattering, which persists to much lower temperatures than real phonon-induced transitions. Quantitative calculations performed for a lateral GaAs/AlGaAs gate-defined double quantum dot yield microsecond dephasing times at sub-Kelvin temperatures, which is consistent with experimental observations.
We study theoretically the phonon-induced relaxation (T1{T}_{1}T1​) and decoherence times (T2{T}_{2}T2​) of singlet-triplet qubits in lateral GaAs double quantum dots (DQDs). When the DQD is biased, Pauli exclusion enables strong dephasing via two-phonon processes. This mechanism requires neither hyperfine nor spin-orbit interaction and yields ${T}_{2}\ensuremath{\ll}{T}_{1}$, in contrast to previous calculations of phonon-limited lifetimes. When the DQD is unbiased, we find ${T}_{2}\ensuremath{\simeq}2{T}_{1}$ and much longer lifetimes than in the biased DQD. For typical setups, the decoherence and relaxation rates due to one-phonon processes are proportional to the temperature TTT, whereas the rates due to two-phonon processes reveal a transition from T2{T}^{2}T2 to higher powers as TTT is decreased. Remarkably, both T1{T}_{1}T1​ and T2{T}_{2}T2​ exhibit a maximum when the external magnetic field is applied along a certain axis within the plane of the two-dimensional electron gas. We compare our results with recent experiments and analyze the dependence of T1{T}_{1}T1​ and T2{T}_{2}T2​ on system properties such as the detuning, the spin-orbit parameters, the hyperfine coupling, and the orientation of the DQD and the applied magnetic field with respect to the main crystallographic axes.
We study electron-phonon interaction induced decoherence between two-electron singlet and triplet states in a semiconductor double quantum dot using a spin-boson model. We investigate the onset and time evolution of this dephasing, and study its dependence on quantum dot parameters such as dot size and double dot separations, as well as the host materials (GaAs and Si). At the short time limit, electron-phonon interaction only causes an incomplete initial Gaussian decay of the off-diagonal density matrix element in the singlet-triplet Hilbert space; a complete long-time exponential decay due to phonon relaxation would eventually dominate over two-spin decoherence. We analyze two-spin decoherence in both symmetric and biased double quantum dots, identifying their difference in electron-phonon coupling and the relevant consequences.
In a coupled double-quantum-dot system, we present a theory for the interplay between electron and nuclear spins when the two-electron singlet state is brought into resonance with one triplet state in moderate external magnetic field. We show that the quantum interference between first-order and second-order hyperfine processes can lead to a feedback mechanism for manipulating the nuclear hyperfine fields. In a uniform external field, positive- and negative-feedback controls can be realized for the gradient of the longitudinal hyperfine field as well as the average transverse hyperfine field in the double dot. The negative feedback which suppresses fluctuations in the longitudinal nuclear field gradient can enhance the decoherence time of a singlet-triplet qubit to microsecond regime. We discuss the possibility of enhancing the decoherence time of each individual spin in a cluster of dots using the negative-feedback control on the transverse nuclear field.