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arXiv:1206.0075 · Physical Review B

Dynamical decoupling of a singlet-triplet qubit afflicted by a charge fluctuator

Guy Ramon

Published Sep 21, 2012• 12 citations
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Abstract

The efficiency of dynamical decoupling pulse sequences in removing noise due to a nearby charge fluctuator is studied for a singlet-triplet spin qubit. We develop a numerical method to solve the dynamical equations for all three components of the Bloch vector under a general pulse protocol, where pulses are applied along an arbitrary rotation axis. The qubit is shown to undergo both dephasing and dissipative dynamics, pending on its working position. Analytical solutions are found for the limits of weakly and strongly coupled fluctuators, shedding light on the distinct dynamics in the different parameter regimes. Scaling of the qubit decay time with the number of control pulses is found to follow a power law over a wide range of parameters and qubit bias points.

DOI
10.1103/physrevb.86.125317

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The efficiency of dynamical decoupling pulse sequences in removing noise due to a nearby charge fluctuator is studied for a singlet-triplet spin qubit. We develop a numerical method to solve the dynamical equations for all three components of the Bloch vector under a general pulse protocol, where pulses are applied along an arbitrary rotation axis. The qubit is shown to undergo both dephasing and dissipative dynamics, pending on its working position. Analytical solutions are found for the limits of weakly and strongly coupled fluctuators, shedding light on the distinct dynamics in the different parameter regimes. Scaling of the qubit decay time with the number of control pulses is found to follow a power law over a wide range of parameters and qubit bias points.
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 dissipative dynamics of a qubit that is afflicted by classical random telegraph noise and is subject to dynamical decoupling. We derive exact formulas for the qubit dynamics at arbitrary working points in the limit of infinitely strong control pulses (bang-bang control), and we investigate in great detail the efficiency of the dynamical decoupling techniques for both Gaussian and non-Gaussian (slow) noise at qubit pure dephasing and at optimal point. We demonstrate that control sequences can be successfully implemented as diagnostic tools to infer spectral properties of a few fluctuators interacting with the qubit. The analysis is extended in order to include the effect of noise in the pulses, and we give upper bounds on the noise levels that can be tolerated in the pulses while still achieving an efficient dynamical decoupling performance.
Many solid-state qubit systems are afflicted by low-frequency noise mechanisms that operate along two perpendicular axes of the Bloch sphere. Depending on the qubit control fields, either noise can be longitudinal or transverse to the quantization axis of the qubit, thus affecting its dynamics in distinct ways, generally contributing to decoherence that goes beyond pure dephasing. Here, we present a theory that provides a unified platform to study dynamics of a qubit subjected to two perpendicular low-frequency noises (assumed to be Gaussian and uncorrelated) under dynamical decoupling pulse sequences. The theory is demonstrated by the commonly encountered case of power law noise spectra, where approximate analytical results can be obtained.
In addition to magnetic field and electric charge noise adversely affecting spin-qubit operations, performing single-qubit gates on one of multiple coupled singlet-triplet qubits presents a new challenge: crosstalk, which is inevitable (and must be minimized) in any multiqubit quantum computing architecture. We develop a set of dynamically corrected pulse sequences that are designed to cancel the effects of both types of noise (i.e., field and charge) as well as crosstalk to leading order, and provide parameters for these corrected sequences for all 24 of the single-qubit Clifford gates. We then provide an estimate of the error as a function of the noise and capacitive coupling to compare the fidelity of our corrected gates to their uncorrected versions. Dynamical error correction protocols presented in this work are important for the next generation of singlet-triplet qubit devices where coupling among many qubits will become relevant.
We investigate the scaling of coherence time T2{T}_{2}T2​ with the number of $\ensuremath{\pi}$ pulses ${n}_{\ensuremath{\pi}}$ in a singlet-triplet spin qubit using Carr-Purcell-Meiboom-Gill (CPMG) and concatenated dynamical decoupling (CDD) pulse sequences. For an even numbers of CPMG pulses, we find a power law ${T}_{2}\ensuremath{\propto}({n}_{\ensuremath{\pi}}{)}^{{\ensuremath{\gamma}}_{e}}$, with ${\ensuremath{\gamma}}_{e}=0.72\ifmmode\pm\else\textpm\fi{}0.01$, essentially independent of the envelope function used to extract T2{T}_{2}T2​. From this surprisingly robust value, a power-law model of the noise spectrum of the environment, $S(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}\ensuremath{\beta}}$, yields $\ensuremath{\beta}={\ensuremath{\gamma}}_{e}/(1\ensuremath{-}{\ensuremath{\gamma}}_{e})=2.6\ifmmode\pm\else\textpm\fi{}0.1$. Model values for ${T}_{2}\left({n}_{\ensuremath{\pi}}\right)$ using $\ensuremath{\beta}=2.6$ for CPMG with both even and odd ${n}_{\ensuremath{\pi}}$ up to 32 and CDD orders 3 through 6 compare very well with the experiment.
We experimentally demonstrate coherence recovery of singlet-triplet superpositions by interlacing qubit rotations between Carr-Purcell (CP) echo sequences. We then compare the performance of Hahn, CP, concatenated dynamical decoupling (CDD), and Uhrig dynamical decoupling for singlet recovery. In the present case, where gate noise and drift combined with spatially varying hyperfine coupling contribute significantly to dephasing, and pulses have limited bandwidth, CP and CDD yield comparable results, with T(2)∼80 μs.