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arXiv:0906.2070 · Physical Review A

Optimized pulses for the perturbative decoupling of a spin and a decoherence bath

Stefano Pasini, P. Karbach, Carsten Raas, Götz Silvester Uhrig

Published Aug 21, 2009• 20 citations
arXivHTMLPDF

Abstract

In the framework of nuclear magnetic resonance, we consider the general problem of the coherent control of a spin coupled to a bath by means of composite or continuous pulses of duration ${\ensuremath{\tau}}_{\text{p}}$. We show explicitly that it is possible to design the pulse in order to achieve a decoupling of the spin from the bath up to the third order in ${\ensuremath{\tau}}_{\text{p}}$. The evolution of the system is separated in the evolution of the spin under the action of the pulse and of the bath times correction terms. We derive the correction terms for a general time dependent axis of rotation and for a general coupling between the spin and the environment. The resulting corrections can be made to vanish by an appropriate design of the pulse. For $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses, we demonstrate explicitly that pulses exist which annihilate the first- and the second-order corrections even if the bath is fully quantum mechanical, i.e., it displays internal dynamics. Such pulses will also be useful for quantum information processing.

DOI
10.1103/physreva.80.022328

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In the framework of nuclear magnetic resonance, we consider the general problem of the coherent control of a spin coupled to a bath by means of composite or continuous pulses of duration \ensuremathτp{\ensuremath{\tau}}_{\text{p}}\ensuremathτp​. We show explicitly that it is possible to design the pulse in order to achieve a decoupling of the spin from the bath up to the third order in \ensuremathτp{\ensuremath{\tau}}_{\text{p}}\ensuremathτp​. The evolution of the system is separated in the evolution of the spin under the action of the pulse and of the bath times correction terms. We derive the correction terms for a general time dependent axis of rotation and for a general coupling between the spin and the environment. The resulting corrections can be made to vanish by an appropriate design of the pulse. For \ensuremathπ\ensuremath{\pi}\ensuremathπ and \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 pulses, we demonstrate explicitly that pulses exist which annihilate the first- and the second-order corrections even if the bath is fully quantum mechanical, i.e., it displays internal dynamics. Such pulses will also be useful for quantum information processing.
We consider the coherent control of a quantum bit by the use of short pulses with finite duration ${\ensuremath{\tau}}_{\mathrm{p}}$. By shaping the pulse, we perturbatively decouple the dynamics of the bath from the dynamics of the quantum bit during the pulse. Such shaped pulses provide single quantum bit gates robust against decoherence which are useful for quantum-information processing. We extend previous results in two ways: (i) we treat frequency-modulated pulses and (ii) we pass from time-independent baths to analytically time-dependent baths. First- and second-order solutions for $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses are presented. They are useful in experiments where amplitude modulation is difficult to realize.
The coherent control of a small quantum system is considered. For a two-level system coupled to an arbitrary bath we consider a pulse of finite duration. We derive the leading and the next-leading order corrections to the evolution operator due to the noncommutation of the pulse and the bath Hamiltonian. The conditions are computed that make the leading corrections vanish. The pulse shapes optimized in this way are given for $\ensuremath{\pi}$ and $\frac{\ensuremath{\pi}}{2}$ pulses.
Reliable long-time storage of arbitrary quantum states is a key element of quantum information processing. In order to dynamically decouple a spin or quantum bit from a dephasing environment by non-instantaneous pulses, we introduce an optimized sequence of N control π pulses that are realistic in the sense that they have a finite duration and a finite amplitude. We show that optimized dynamical decoupling is still applicable and that higher-order decoupling can be reached if shaped pulses are implemented. The sequence suppresses decoherence up to the order , with T being the total duration of the sequence and τ mx the maximum length of the pulses. The exponent depends on the shape of the pulse. Based on existing experiments, a concrete setup for the verification of the properties of the advocated sequence is proposed.
Dynamical decoupling (DD) is a powerful method for controlling arbitrary open quantum systems. In quantum spin control, DD generally involves a sequence of timed spin flips (π rotations) arranged to either average out or selectively enhance coupling to the environment. Experimentally, errors in the spin flips are inevitably introduced, motivating efforts to optimize error-robust DD. Here we invert this paradigm: by introducing particular control "errors" in standard DD, namely, a small constant deviation from perfect π rotations (pulse adjustments), we show we obtain protocols that retain the advantages of DD while introducing the capabilities of quantum state readout and polarization transfer. We exploit this nuclear quantum state selectivity on an ensemble of nitrogen-vacancy centers in diamond to efficiently polarize the ^{13}C quantum bath. The underlying physical mechanism is generic and paves the way to systematic engineering of pulse-adjusted protocols with nuclear state selectivity for quantum control applications.
We consider pulses of finite duration for coherent control in the presence of classical noise. We derive the corrections to ideal, instantaneous pulses for the case of general decoherence (spin-spin relaxation and spin-lattice relaxation) up to and including the third order in the duration τ_p of the pulses. For pure dephasing (spin-spin relaxation only), we design πand π/2 pulses with amplitude and/or frequency modulation which resemble the ideal ones up to and including the second order in τ_p. For completely general decoherence including spin-lattice relaxation the corrections are computed up to and including the second order in τ_p as well. Frequency modulated pulses are determined which resemble the ideal ones. They are used to design a low-amplitude replacement for XY8 cycles. In comparison with pulses designed to compensate quantum noise less conditions have to be fulfilled. Consequently, we find that the classical pulses can be weaker and simpler than the corresponding pulses in the quantum case.
High-fidelity control of quantum systems is crucial for quantum information processing, but is often limited by perturbations from the environment and imperfections in the applied control fields. Here, we investigate the combination of dynamical decoupling (DD) and robust optimal control (ROC) to address this problem. In this combination, ROC is employed to find robust shaped pulses, wherein the directional derivatives of the controlled dynamics with respect to control errors are reduced to a desired order. Then, we incorporate ROC pulses into DD sequences, achieving a remarkable improvement of robustness against multiple error channels. We demonstrate this method in the example of manipulating nuclear spin bath via an electron spin in the nitrogen-vacancy center system. Simulation results indicate that ROC-based DD sequences outperform the state-of-the-art robust DD sequences. Our work has implications for robust quantum control on near-term noisy quantum devices.