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arXiv:quant-ph/0211081 · Physical Review A

Dynamical decoupling using slow pulses: Efficient suppression of 1/f noise

K. Shiokawa, Daniel A. Lidar

Published Mar 9, 2004• 83 citations
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Abstract

The application of dynamical decoupling pulses to a single qubit interacting with a linear harmonic oscillator bath with $1/f$ (more generally, strong low-frequency) spectral density is studied, and compared to the Ohmic case. Decoupling pulses that are slower than the fastest bath time scale are shown to drastically reduce the decoherence rate in the $1/f$ case. Contrary to conclusions drawn from previous studies, this shows that dynamical decoupling pulses do not always have to be ultrafast. Our results explain the recent experiment in which dephasing due to $1/f$ charge noise affecting a charge qubit in a small superconducting electrode was successfully suppressed using spin-echo-type gate-voltage pulses.

DOI
10.1103/physreva.69.030302

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The application of dynamical decoupling pulses to a single qubit interacting with a linear harmonic oscillator bath with 1/f1/f1/f (more generally, strong low-frequency) spectral density is studied, and compared to the Ohmic case. Decoupling pulses that are slower than the fastest bath time scale are shown to drastically reduce the decoherence rate in the 1/f1/f1/f case. Contrary to conclusions drawn from previous studies, this shows that dynamical decoupling pulses do not always have to be ultrafast. Our results explain the recent experiment in which dephasing due to 1/f1/f1/f charge noise affecting a charge qubit in a small superconducting electrode was successfully suppressed using spin-echo-type gate-voltage pulses.
To suppress decoherence of solid-state qubits induced by non-Markovian noises, we propose a strategy to couple the qubit with a chaotic device, whose broad power distribution in the high-frequency domain can be used to freeze the noises in the similar way of the dynamical decoupling control (DDC) method. Compared with the DDC, high-frequency components can be generated by the chaotic device even driven by a low-frequency field and thus the control fields do not need any optimization. Thus, our method is easier to be realized in experiments. As an application to superconducting circuits, we find that various noises in a broad band, including low-frequency 1/ƒ, high-frequency Ohmic, sub-Ohmic, and super-Ohmic noises, can be efficiently suppressed by coupling the qubit to a Duffing oscillator, and the decoherence rate of the qubit is efficiently decreased by about 100 times in magnitude. Using experimentally realizable parameters, the decoherence time of the superconducting charge qubit can reach 200 µs by our method, which is 10 ∼ 100 times longer in magnitude than that observed in experiments.
One of the simplest and least resource-intensive methods to suppress decoherence for qubit operations, namely, dynamical decoupling (DD), is investigated for a broad range of realistic noise sources with time-retarded feedback. By way of example, the Carr-Purcell-Meiboom-Gill sequence is analyzed in a numerically rigorous manner accounting also for correlations between qubits and environments. Since experimentally noise sources are characterized through spectral densities, we adopt the spin-boson model as a suitable framework to describe the qubit dynamics under DD for a given spectral density J ( ω ) ∝ ω s . Motivated by the situation for superconducting qubits, the spectral exponent s is varied from s = 1 (Ohmic bath) to a substantially small value 0 < s ≪ 1 (deep sub-Ohmic bath), in order to investigate the impact of time-nonlocal back action on DD performances for enhanced coherence times. As reference to the DD schemes, dynamics of a single qubit subject to Ramsey sequences without any pules and Hahn echo sequences are also investigated.
We study the suppression of qubit dephasing through Uhrig dynamical decoupling (UDD) in nontrivial environments modeled within the spin-boson formalism. In particular, we address the case of (i) a qubit coupled to a bosonic bath with power-law spectral density, and (ii) a qubit coupled to a single harmonic oscillator that dissipates energy into a bosonic bath, which embodies an example of a structured bath for the qubit. We then model the influence of random time jitter in the UDD protocol by sorting pulse-application times from Gaussian distributions centered at appropriate values dictated by the optimal protocol. In case (i) we find that, when few pulses are applied and a sharp cutoff is considered, longer coherence times and robust UDD performances (against random timing errors) are achieved for a super-Ohmic bath. On the other hand, when an exponential cutoff is considered, a super-Ohmic bath is undesirable. In case (ii) the best scenario is obtained for an overdamped harmonic motion. Our study provides relevant information for the implementation of optimized schemes for the protection of quantum states from decoherence.
To suppress decoherence of solid-state qubits which are coupled to the non-Markovian noises, we propose a strategy to couple the qubit with a chaotic device, of which the broad power distribution in the high-frequency domain can be used to freeze the noises just like the dynamical decoupling control (DDC) method. Compared with the DDC, high-frequency components can be generated by the chaotic device even driven by a low-frequency field and we do not need to optimize the control fields to generate complex control pulses. As an application to superconducting circuits, we find that various noises in a wide frequency domain, including low-frequency 1/f1/f1/f, high-frequency Ohmic, sub-Ohmic, and super-Ohmic noises, can be efficiently suppressed by coupling the qubit to a Duffing oscillator, and the decoherence rate of the qubit is efficiently decreased for about 100 times in magnitude.
Recent studies have shown that applying a sequence of Hahn spin-echo pulses to a qubit system at judiciously chosen intervals can, in certain noise environments, greatly improve the suppression of phase errors compared to traditional dynamical decoupling approaches. By enforcing a simple analytical condition, we obtain sets of dynamical decoupling sequences that are designed for optimized noise filtration, but are independent of the noise spectrum up to a single scaling factor set by the coherence time of the system. These sequences are tested in a model qubit system, ;{9}Be;{+} ions in a Penning trap. Our combined theoretical and experimental studies show that in high-frequency-dominated noise environments with sharp high-frequency cutoffs this approach may suppress phase errors orders of magnitude more efficiently than comparable techniques can.
We investigate the capability of dynamical decoupling techniques to reduce decoherence from a realistic environment generating 1/f noise. The predominance of low frequency modes in the noise profile allows for decoherence scenarios where relatively slow control rates suffice for a drastic improvement. However, the actual figure of merit is very sensitive to the details of the dynamics, with decoupling performance which may deteriorate for non-Gaussian noise and/or high frequency working points. Our results are promising for robust solid-state qubits and beyond.