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Journal of Physics B Atomic Molecular and Optical Physics

Short coherent control pulse with small random errors in its direction

Zhi-Kun Su, Shaoji Jiang

Published Jul 2, 2012• 0 citations
Publisher

Abstract

The finite-amplitude short coherent control pulse with small random errors in its direction is considered.We derive the conditions for it to approximate an ideal δ-shaped pulse up to an error in the second order of pulse duration.Whether the symmetric and asymmetric pulse shapes would fulfill these conditions is also analyzed.The result shows that, under one of these conditions, all symmetric pulse shapes can fulfill this condition but not for all asymmetric ones.

DOI
10.1088/0953-4075/45/14/145501

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The finite-amplitude short coherent control pulse with small random errors in its direction is considered.We derive the conditions for it to approximate an ideal δ-shaped pulse up to an error in the second order of pulse duration.Whether the symmetric and asymmetric pulse shapes would fulfill these conditions is also analyzed.The result shows that, under one of these conditions, all symmetric pulse shapes can fulfill this condition but not for all asymmetric ones.
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.
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.
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.
For a generic dynamical decoupling sequence employing a single-axis control, we study its efficiency in the presence of small errors in the direction of the controlling pulses. In the case that the corresponding ideal dynamical decoupling sequence produces sufficiently good results, the impact of the errors is found to scale as $\ensuremath{\xi}{}^{2}$, with negligible first-order effect, where $\ensuremath{\xi}$ is the dispersion of the random errors. This analytical prediction is numerically tested in a model in which the environment is modeled by one qubit coupled to a quantum kicked rotator in chaotic motion. In this model, with periodic pulses applied to the qubit in the environment, it is found numerically that periodic bang-bang control may outperform Uhrig dynamical decoupling.
Very short optical pulses, traveling in sharp-line resonant absorbers without change of shape, are investigated with a new approximation technique. Instead of employing the long-pulse or slowly varying envelope assumption, we have developed a method of approximation based on a series expansion in powers of a small parameter which is related to the electric field strength. The principal advantages of the method are its ability to treat very short pulses, and the simplicity with which it can be extended, both from lower to higher levels of approximation, and from one type of steady-state pulse to another. In this paper we use the approximation method to study very short pulses which require retention of the second-derivative terms in the Maxwell envelope and phase equations. Corrections to the well-known McCall-Hahn results are easily found to first order in the small parameter. As expected, the corrections begin to be important only for picosecond and shorter pulses.