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arXiv:0909.3439 · Physical Review A

Optimized dynamical decoupling for power-law noise spectra

Stefano Pasini, Götz Silvester Uhrig

Published Jan 14, 2010• 84 citations
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Abstract

We analyze the suppression of decoherence by means of dynamical decoupling in the pure-dephasing spin-boson model for baths with power law spectra. The sequence of ideal $\ensuremath{\pi}$ pulses is optimized according to the power of the bath. We expand the decoherence function and separate the canceling divergences from the relevant terms. The proposed sequence is chosen to be the one minimizing the decoherence function. By construction, it provides the best performance. We analytically derive the conditions that must be satisfied. The resulting equations are solved numerically. The solutions are very close to the Carr-Purcell-Meiboom-Gill sequence for a soft cutoff of the bath while they approach the Uhrig dynamical-decoupling sequence as the cutoff becomes harder.

DOI
10.1103/physreva.81.012309

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We analyze the suppression of decoherence by means of dynamical decoupling in the pure-dephasing spin-boson model for baths with power law spectra. The sequence of ideal \ensuremathπ\ensuremath{\pi}\ensuremathπ pulses is optimized according to the power of the bath. We expand the decoherence function and separate the canceling divergences from the relevant terms. The proposed sequence is chosen to be the one minimizing the decoherence function. By construction, it provides the best performance. We analytically derive the conditions that must be satisfied. The resulting equations are solved numerically. The solutions are very close to the Carr-Purcell-Meiboom-Gill sequence for a soft cutoff of the bath while they approach the Uhrig dynamical-decoupling sequence as the cutoff becomes harder.
We study the performance of dynamical decoupling in suppressing decoherence caused by soft-cutoff Gaussian noise, using short-time expansion of the noise correlations and numerical optimization. For the noise with soft cutoff at high frequencies, there exists no dynamical decoupling scheme to eliminate the decoherence to arbitrary orders of the short time, regardless of the timing or pulse shaping of the control under the population conserving condition. We formulate the equations for optimizing pulse sequences that minimize decoherence up to the highest possible order of the short time for the noise correlations with odd power terms in the short-time expansion. In particular, we show that the Carr-Purcell-Meiboom-Gill sequence is optimal in the short-time limit for the noise correlations with a linear order term in the time expansion.
Maintaining quantum coherence is a crucial requirement for quantum computation; hence protecting quantum systems against their irreversible corruption due to environmental noise is an important open problem. Dynamical decoupling (DD) is an effective method for reducing decoherence with a low control overhead. It also plays an important role in quantum metrology, where, for instance, it is employed in multiparameter estimation. While a sequence of equidistant control pulses [the Carr-Purcell-Meiboom-Gill (CPMG) sequence] has been ubiquitously used for decoupling, Uhrig recently proposed that a nonequidistant pulse sequence [the Uhrig dynamic decoupling (UDD) sequence] may enhance DD performance, especially for systems where the spectral density of the environment has a sharp frequency cutoff. On the other hand, equidistant sequences outperform UDD for soft cutoffs. The relative advantage provided by UDD for intermediate regimes is not clear. In this paper, we analyze the relative DD performance in this regime experimentally, using solid-state nuclear magnetic resonance. Our system qubits are 13C^{13}\mathrm{C}13C nuclear spins and the environment consists of a 1H^{1}\mathrm{H}1H nuclear spin bath whose spectral density is close to a normal (Gaussian) distribution. We find that in the presence of such a bath, the CPMG sequence outperforms the UDD sequence. An analogy between dynamical decoupling and interference effects in optics provides an intuitive explanation as to why the CPMG sequence performs better than any nonequidistant DD sequence in the presence of this kind of environmental noise.
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.
Dynamical decoupling (DD) emerges as a promising tool to suppress decoherence of quantum systems by eliminating the system-environment coupling. We investigate its power in combating pure dephasing of a qubit under random classical noise. Interestingly, the decoherence rate can be viewed as the interplay between noise spectrum and a controllable filter function. More specifically, the filter function is determined by the DD sequence we employed. Here we take Uhrig DD (UDD) as a benchmark sequence to illustrate the filtering process. We find that UDD sequence works as a perfect dynamical filter for low-frequency noise. For spectrum with hard cutoff, UDD increases the performance order by order with linearly increased number of pulses. However, for noise spectrum with soft cutoff, the required pulse number grows exponentially.
The optimal N-pulse dynamical decoupling discovered by Uhrig for a spin-boson model [Phys. Rev. Lett. 98, 100504 (2007)10.1103/PhysRevLett.98.100504] is proved to be universal in suppressing to O(T;{N+1}) the pure dephasing or the longitudinal relaxation of a qubit (or spin 1/2) coupled to a generic bath in a short-time evolution of duration T. For suppressing the longitudinal relaxation, a Uhrig pi-pulse sequence can be generalized to be a superposition of the ideal Uhrig pi-pulse sequence as the core and an arbitrarily shaped pulse sequence satisfying certain symmetry requirements. The generalized Uhrig dynamical decoupling offers the possibility of manipulating the qubit while simultaneously combating the longitudinal relaxation.
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.