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arXiv:0709.0588 · Physical Review A

Optimization of short coherent control pulses

Stefano Pasini, T. Fischer, P. Karbach, Götz Silvester Uhrig

Published Mar 7, 2008• 52 citations
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Abstract

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.

DOI
10.1103/physreva.77.032315

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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 \ensuremathπ\ensuremath{\pi}\ensuremathπ and \ensuremathπ2\frac{\ensuremath{\pi}}{2}2\ensuremathπ​ pulses.
Numerically we simulate the effect of optimized coherent control pulses with a finite duration on a qubit in a bath of spins. The pulses of finite duration are compared with ideal instantaneous pulses. In particular, we show that properly designed short pulses can approximate ideal instantaneous pulses up to a certain order in the shortness of the pulse. We provide examples of such pulses, quantify the discrepancy from the ideal case, and compare their effect for various ranges of the coupling constants.
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.
We generalize the problem of the coherent control of small quantum systems to the case where the quantum bit (qubit) is subject to a fully general rotation. Following the ideas developed in Pasini et al (2008 Phys. Rev. A 77, 032315), the systematic expansion in the shortness of the pulse is extended to the case where the pulse acts on the qubit as a general rotation around an axis of rotation varying in time. The leading and the next-leading corrections are computed. For certain pulses we prove that the general rotation does not improve on the simpler rotation with fixed axis.
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.
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.