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arXiv:0706.0501

Refocusing of a qubit system coupled to an oscillator

Leonid P. Pryadko, Gregory Quiroz

Published Jun 4, 2007• 0 citations
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Abstract

Refocusing, or dynamical decoupling, is a coherent control technique where the internal dynamics of a quantum system is effectively averaged out by an application of specially designed driving fields. The method has originated in nuclear magnetic resonance, but it was independently discovered in atomic physics as a ``coherent destruction of tunneling''. Present work deals with the analysis of the performance of ``soft'' refocusing pulses and pulse sequences in protecting the coherence of a qubit system coupled to a quantum oscillator.

DOI
10.48550/arxiv.0706.0501

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•Jan 1, 2011•153 citations
Refocusing, or dynamical decoupling, is a coherent control technique where the internal dynamics of a quantum system is effectively averaged out by an application of specially designed driving fields. The method has originated in nuclear magnetic resonance, but it was independently discovered in atomic physics as a coherent destruction of tunneling''. Present work deals with the analysis of the performance of soft'' refocusing pulses and pulse sequences in protecting the coherence of a qubit system coupled to a quantum oscillator.
Dynamical decoupling is a coherent control technique where the intrinsic and extrinsic couplings of a quantum system are effectively averaged out by application of specially designed driving fields (refocusing pulse sequences). This entails pumping energy into the system, which can be especially dangerous when it has sharp spectral features like a cavity mode close to resonance. In this work we show that such an effect can be avoided with properly constructed refocusing sequences. To this end we construct the average Hamiltonian expansion for the system evolution operator associated with a single ``soft'' $\ensuremath{\pi}$ pulse. To second order in the pulse duration, we characterize a symmetric pulse shape by three parameters, two of which can be made zero by shaping. We express the effective Hamiltonians for several pulse sequences in terms of these parameters and use the results to analyze the structure of error operators for a controlled Jaynes-Cummings Hamiltonian. When errors are cancelled to second order, numerical simulations show excellent qubit fidelity with strongly suppressed oscillator heating.
We consider the effect of broadband decoherence on the performance of refocusing sequences, having in mind applications of dynamical decoupling in concatenation with quantum error correcting codes as the first stage of coherence protection. Specifically, we construct cumulant expansions of effective decoherence operators for a qubit driven by a pulse of a generic symmetric shape and for several sequences of $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses. While, in general, the performance of soft pulses in decoupling sequences in the presence of Markovian decoherence is worse than that of the ideal $\ensuremath{\delta}$ pulses, it can be substantially improved by shaping.
Over the past decade we have seen an explosion of demonstrations of quantum coherence in atomic, optical, and condensed matter systems. These developments have placed a new emphasis on the production of robust and optimal quantum control techniques in the presence of environmental noise. We discuss the use of dynamical decoupling as a form of open-loop quantum control capable of suppressing the effects of dephasing in quantum coherent systems. We introduce the concept of dynamical decoupling pulse-sequence construction as a filter-design problem, making connections with filter design from control theory and electrical engineering in the analysis of pulse-sequence performance for the preservation of the phase degree of freedom in a quantum superposition. A detailed mathematical description of how dephasing and the suppression of dephasing can be reduced to a linear control problem is provided, and used as motivation and context for studies of the filtration properties of various dynamical decoupling sequences. Our work then takes this practical perspective in addressing both "standard" sequences derived from nuclear magnetic resonance and novel optimized sequences developed in the context of quantum information. Additionally, we review new techniques for the numerical construction of optimized pulse sequences using the filter-design perspective. We show how the filter-design perspective permits concise comparisons of the relative capabilities of these sequences and reveals the physics underlying their functionality. The use of this new analytical framework allows us to derive new insights into the performance of these sequences and reveals important limiting issues, such as the effect of digital clocking on optimized sequence performance.
Coherent-optical-control schemes exploit the coherence of laser pulses to change the phases of interfering dynamical pathways and manipulate dynamical processes. These active control methods are closely related to dynamical decoupling techniques, popularized in the field of quantum information. Inspired by nuclear magnetic resonance spectroscopy, dynamical decoupling methods apply sequences of unitary operations to modify the interference phenomena responsible for the system dynamics thus also belonging to the general class of coherent-control techniques. This article reviews related developments in the fields of coherent optical control and dynamical decoupling, emphasizing the control of tunneling and decoherence in general model systems. Considering recent experimental breakthroughs in the demonstration of active control of a variety of systems, we anticipate that the reviewed coherent-control scenarios and dynamical-decoupling methods should raise significant experimental interest.
Over the past decade we have seen an explosion of demonstrations of quantum coherence in atomic, optical, and condensed matter systems.These developments have placed a new emphasis on the production of robust and optimal quantum control techniques in the presence of environmental noise.We discuss the use of dynamical decoupling as a form of open-loop quantum control capable of suppressing the effects of dephasing in quantum coherent systems.We introduce the concept of dynamical decoupling pulse-sequence construction as a filter-design problem, making connections with filter design from control theory and electrical engineering in the analysis of pulse-sequence performance for the preservation of the phase degree of freedom in a quantum superposition.A detailed mathematical description of how dephasing and the suppression of dephasing can be reduced to a linear control problem is provided, and used as motivation and context for studies of the filtration properties of various dynamical decoupling sequences.Our work then takes this practical perspective in addressing both "standard" sequences derived from nuclear magnetic resonance and novel optimized sequences developed in the context of quantum information.Additionally, we review new techniques for the numerical construction of optimized pulse sequences using the filter-design perspective.We show how the filter-design perspective permits concise comparisons of the relative capabilities of these sequences and reveals the physics underlying their functionality.The use of this new analytical framework allows us to derive new insights into the performance of these sequences and reveals important limiting issues, such as the effect of digital clocking on optimized sequence performance.