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arXiv:0710.1416 · Physical Review Letters

Universal Pulse Sequence to Minimize Spin Dephasing in the Central Spin Decoherence Problem

B. Lee, Wayne M. Witzel, S. Das Sarma

Published Apr 23, 2008• 128 citations
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Abstract

We present a remarkable finding that a recently discovered [G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007)10.1103/PhysRevLett.98.100504] series of pulse sequences, designed to optimally restore coherence to a qubit in the spin-boson model of decoherence, is in fact completely model independent and generically valid for arbitrary dephasing Hamiltonians given sufficiently short delay times between pulses. The series maximizes qubit fidelity versus number of applied pulses for sufficiently short delay times because the series, with each additional pulse, cancels successive orders of a time expansion for the fidelity decay. The "magical" universality of this property, which was not appreciated earlier, requires that a linearly growing set of "unknowns" (the delay times) must simultaneously satisfy an exponentially growing set of nonlinear equations that involve arbitrary dephasing Hamiltonian operators.

DOI
10.1103/physrevlett.100.160505

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Publisher’s Note: Universal Pulse Sequence to Minimize Spin Dephasing in the Central Spin Decoherence Problem [Phys. Rev. Lett.100, 160505 (2008)]

B. Lee, Wayne M. Witzel, S. Das Sarma

Received 28 April 2008DOI:https://doi.org/10.1103/PhysRevLett.100.199901©2008 American Physical Society

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We present a remarkable finding that a recently discovered [G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007)10.1103/PhysRevLett.98.100504] series of pulse sequences, designed to optimally restore coherence to a qubit in the spin-boson model of decoherence, is in fact completely model independent and generically valid for arbitrary dephasing Hamiltonians given sufficiently short delay times between pulses. The series maximizes qubit fidelity versus number of applied pulses for sufficiently short delay times because the series, with each additional pulse, cancels successive orders of a time expansion for the fidelity decay. The "magical" universality of this property, which was not appreciated earlier, requires that a linearly growing set of "unknowns" (the delay times) must simultaneously satisfy an exponentially growing set of nonlinear equations that involve arbitrary dephasing Hamiltonian operators.
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 effectiveness of different dynamical decoupling protocols for storage of a single qubit in the presence of a purely dephasing bosonic bath, with emphasis on comparing quantum coherence preservation under uniform versus nonuniform delay times between pulses. In the limit of instantaneous bit-flip pulses, this is accomplished by establishing a different representation of the controlled qubit evolution, where the decoherence behavior after an arbitrary number of pulses is directly expressed in terms of the uncontrolled decoherence function. In particular, analytical expressions are obtained for approximation of the long- and short-term coherence behavior for both Ohmic and supra-Ohmic environments. By focusing on the realistic case of pure dephasing in an excitonic qubit, we quantitatively assess the impact of physical constraints on achievable pulse separations, and show that little advantage of high-level decoupling schemes based on concatenated or optimal design may be expected if pulses cannot be applied sufficiently fast. In such constrained scenarios, we demonstrate how simple modifications of repeated periodic-echo protocols can offer significantly improved coherence preservation in realistic parameter regimes. We expect similar conclusions to be relevant to other constrained qubit devices exposed to quantum or classical phase noise.
We theoretically investigate the influence of designed pulse sequences in restoring quantum coherence lost due to background noise in superconducting qubits. We consider both 1/f1/f1/f noise and random telegraph noise and show that the qubit coherence time can be substantially enhanced by carefully engineered pulse sequences. Conversely, the time dependence of qubit coherence under external pulse sequences could be used as a spectroscopic tool for extracting the noise mechanisms in superconducting qubits, i.e., by using Uhrig's pulse sequence [Phys. Rev. Lett. 98, 100504 (2007)], one can obtain information about moments of the spectral density of noise. We also study the effect of pulse sequences on the evolution of the qubit affected by a strongly coupled fluctuator and show that the non-Gaussian features in decoherence are suppressed by the application of pulses.
Uhrig's dynamical decoupling pulse sequence has emerged as a universal and highly promising approach to decoherence suppression. So far, both the theoretical and experimental studies have examined single-qubit decoherence only. This work extends Uhrig's universal dynamical decoupling from one-qubit to two-qubit systems and even to general multilevel quantum systems. In particular, we show that by designing appropriate control Hamiltonians for a two-qubit or a multilevel system, Uhrig's pulse sequence can also preserve a generalized quantum coherence measure to the order of 1+O(TN+1)1+O({T}^{N+1})1+O(TN+1) with only NNN pulses. Our results lead to a very useful scheme for efficiently locking two-qubit entangled states. Future important applications of Uhrig's pulse sequence in preserving the quantum coherence of multilevel quantum systems can also be anticipated.
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.