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arXiv:0904.0060 · Physical Review A

Modeling two-spin dynamics in a noisy environment

Matthew J. Testolin, Jared H. Cole, Lloyd C. L. Hollenberg

Published Oct 23, 2009• 6 citations
arXivHTMLPDF

Abstract

We describe how the effect of charge noise on a pair of spins coupled via the exchange interaction can be calculated by modeling charge fluctuations as a random telegraph noise process using probability density functions. We develop analytic expressions for the time-dependent superoperator of a pair of spins as a function of fluctuation amplitude and rate. We show that the theory can be extended to include multiple fluctuators, in particular, spectral distributions of fluctuators. These superoperators can be included in time-dependent analyses of the state of spin systems designed for spintronics or quantum information processing to determine the decohering effects of exchange fluctuations.

DOI
10.1103/physreva.80.042326

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We describe how the effect of charge noise on a pair of spins coupled via the exchange interaction can be calculated by modeling charge fluctuations as a random telegraph noise process using probability density functions. We develop analytic expressions for the time-dependent superoperator of a pair of spins as a function of fluctuation amplitude and rate. We show that the theory can be extended to include multiple fluctuators, in particular, spectral distributions of fluctuators. These superoperators can be included in time-dependent analyses of the state of spin systems designed for spintronics or quantum information processing to determine the decohering effects of exchange fluctuations.
Charge noise remains the primary obstacle to the development of quantum information technologies with semiconductor spin qubits. We use an exact analytical calculation to determine the effects of quasistatic charge noise on a ring of three equally spaced exchange-coupled quantum dots. We calculate the disorder-averaged return probability from a specific initial state and use it to determine the coherence time T2∗{T}_{2}^{*}T2∗​ and show that it depends on only the disorder strength and not the mean interaction strength. We also use a perturbative approach to investigate other arrangements of three or four qubits, finding that the return probability contains multiple oscillation frequencies. These oscillations decay in a Gaussian manner, determined by differences in energy levels of the Hamiltonian. We give quantitative values for gate times resulting in several target fidelities. We find that the decoherence time decreases with an increasing number of qubits. Our work provides useful analytical insight into the charge noise dynamics of coupled spin qubits.
We study the dynamics of dephasing in a quantum two-level system by modeling both 1/f1/f1/f and high-frequency noise by random telegraph processes. Our approach is based on a so-called spin-fluctuator model in which a noisy environment is modeled by a large number of fluctuators. In the continuous limit we obtain an effective random process (ERP) that is described by a distribution function of the fluctuators. In a simplified model, we reduce the ERP to the two (slow and fast) ensembles of fluctuators. Using this model, we study decoherence in a superconducting flux qubit and we compare our theoretical results with the available experimental data. We demonstrate good agreement of our theoretical predictions with the experiments. Our approach can be applied to many quantum systems, such as biological complexes, semiconductors, superconducting, and spin qubits, where the effects of interaction with the environment are essential.
The effects of a collection of classical two-level charge fluctuators on the coherence of a dynamically decoupled qubit are studied. Distinct dynamics is found at different qubit working positions. Exact analytical formulas are derived at pure dephasing and approximate solutions are found at the general working position, for weakly and strongly coupled fluctuators. Analysis of these solutions, combined with numerical simulations of the multiple random telegraph processes, reveal the scaling of the noise with the number of fluctuators and the number of control pulses, as well as dependence on other parameters of the qubit-fluctuators system. These results can be used to determine potential microscopic models for the charge environment by performing noise spectroscopy.
We study the domain of applicability of the classical telegraph noise model in the study of decoherence in qubits. We investigate the decoherence of a qubit coupled to either a quantum fluctuator, a quantum two-level system (TLS) again coupled to an environment, or to a classical fluctuator modeled by random telegraph noise. In order to do this, we construct a model for the quantum fluctuator where we can adjust the temperature of its environment, and the decoherence rate independently. The model has a well-defined classical limit at any temperature and this corresponds to the appropriate random telegraph process, which is symmetric at high temperatures and becomes asymmetric at low temperatures. We find that the difference in the qubit decoherence rates predicted by the two models depends on the ratio between the qubit-fluctuator coupling and the decoherence rate in the pointer basis of the fluctuator. This is then the relevant parameter, which determines whether the fluctuator, has to be treated quantum mechanically or can be replaced by a classical telegraph process. We also compare the mutual information between the qubit and the fluctuator in the classical and the quantum model.
Abstract The electron exchange interaction is a promising medium for the entanglement of single-spin qubits in semiconductors as it results in high-speed two-qubit gates. The quality of such entangling gates is reduced by the presence of noise caused by nearby defects acting as two-level fluctuators. To date, the effect of charge noise has been calculated assuming a Gaussian distribution of exchange interaction frequencies between the qubits equivalent to a linear coupling of charge noise with the exchange interaction. In reality the coupling can differ significantly from this linear-coupling approximation depending on the inter-qubit tunnel coupling, detuning of the qubit system, and the magnitude of charge noise. We derive analytical expressions for the frequency spectra of exchange oscillations that encompasses both linear and non-linear coupling to charge-noise. The resulting decoherence times and decay profiles of the exchange oscillations vary considerably. When compared with recent experiments our model shows that non-linear charge-noise coupling is significant and requires consideration to characterise and optimise exchange-based entangling gates.
We investigate the effect of magnetic field and charge noise on the generation of entanglement between two Heisenberg exchange-coupled electron spins in a double quantum dot. We focus on exchange-driven evolution that would ideally take an initial unentangled tensor product state to a maximally entangled state in the absence of noise. The presence of noise obviously adversely affects the attainment of maximal entanglement, which we study quantitatively and exactly. To quantify the effects of noise, we calculate two-qubit coherence times and entanglement fidelity, both of which can be extracted from simulations or measurements of the return probability as a function of interaction time, i.e., the time period during which the exchange coupling remains effective between the two spins. We perform these calculations for a broad range of noise strengths that includes the regime of recent experiments. We find that the two types of noise reduce the amount of entanglement in qualitatively distinct ways and that, although charge noise generally leads to faster decoherence, the relative importance of the two types of noise in entanglement creation depends sensitively on the strength of the exchange coupling. Our results can be used to determine the level of noise suppression needed to reach quantum error correction thresholds. We provide quantitative guidance for the requisite noise constraints necessary to eventually reach the >99>99%>99 fidelity consistent with the quantum error correction threshold.