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Physical Review A

Arbitrarily accurate narrowband composite pulse sequences

Nikolay V. Vitanov

Published Dec 14, 2011• 35 citations
Publisher

Abstract

Narrowband composite pulse sequences containing an arbitrary number $N$ of identical pulses are presented. The composite phases are given by a very simple analytic formula and the transition probability is merely ${\mathrm{sin}}^{2N}(A/2)$, where $A$ is the pulse area. These narrowband sequences can be made accurate to any order with respect to variations in $A$ for sufficiently many constituent pulses, i.e., excitation can be suppressed below any desired value for any pulse area but $\ensuremath{\pi}$.

DOI
10.1103/physreva.84.065404

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Arbitrarily accurate composite pulse sequences (4 pages)

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Narrowband composite pulse sequences containing an arbitrary number NNN of identical pulses are presented. The composite phases are given by a very simple analytic formula and the transition probability is merely sin2N(A/2){\mathrm{sin}}^{2N}(A/2)sin2N(A/2), where AAA is the pulse area. These narrowband sequences can be made accurate to any order with respect to variations in AAA for sufficiently many constituent pulses, i.e., excitation can be suppressed below any desired value for any pulse area but \ensuremathπ\ensuremath{\pi}\ensuremathπ.
We present three classes of symmetric broadband composite pulse sequences. The composite phases are given by analytic formulas (rational fractions of $\ensuremath{\pi}$) valid for any number of constituent pulses. The transition probability is expressed by simple analytic formulas and the order of pulse area error compensation grows linearly with the number of pulses. Therefore, any desired compensation order can be produced by an appropriate composite sequence; in this sense, they are arbitrarily accurate. These composite pulses perform equally well as or better than previously published ones. Moreover, the current sequences are more flexible as they allow total pulse areas of arbitrary integer multiples of $\ensuremath{\pi}$.
Composite pulse sequences, which produce arbitrary predefined rotations of a two-state system at an angle $\ensuremath{\theta}$ on the Bloch sphere, are presented. The composite sequences can contain arbitrarily many pulses and can compensate experimental errors in the pulse amplitude and duration to any desired order. A special attention is devoted to two classes of $\ensuremath{\pi}/2$ sequences---symmetric and asymmetric---the phases of which are given by simple formulas in terms of rational multiples of $\ensuremath{\pi}$ for any number of constituent pulses. This allows one to construct arbitrarily accurate $\ensuremath{\pi}/2$ composite rotations. These $\ensuremath{\pi}/2$ composite sequences are used to construct three classes of arbitrarily long composite $\ensuremath{\theta}$ sequences by pairing two $\ensuremath{\pi}/2$ composite sequences, one of which is shifted by a phase $\ensuremath{\pi}\ensuremath{-}\ensuremath{\theta}$ with respect to the other one.
Composite pulses---sequences of pulses with well-defined relative phases---are an efficient, robust, and flexible technique for coherent control of quantum systems. Composite sequences can compensate for a variety of experimental errors in the driving field (e.g., in the pulse amplitude, duration, detuning, chirp, etc.) or in the quantum system and its environment (e.g., inhomogeneous broadening, stray electric or magnetic fields, unwanted couplings, etc.). The control parameters are the relative phases between the constituent pulses in the composite sequence, an accurate control over which is required in all composite sequences reported hitherto. In this paper, we introduce two types of composite pulse sequences which, in addition to error compensation in the basic experimental parameters, compensate for systematic errors in the composite phases. In the first type of such composite sequences, which compensate for pulse area errors, relative phase errors of over 10% can be tolerated with reasonably short sequences while maintaining the fidelity above the 99.99% quantum computing benchmark. In the second type of composite sequences, which compensate for simultaneous pulse area and detuning errors, relative phase errors of over 5% can be compensated.
Systematic errors in quantum operations can be the dominating source of imperfection in achieving control over quantum systems. This problem, which has been well studied in nuclear magnetic resonance, can be addressed by replacing single operations with composite sequences of pulsed operations, which cause errors to cancel by symmetry. Remarkably, this can be achieved without knowledge of the amount of error ϵϵϵ. Independent of the initial state of the system, current techniques allow the error to be reduced to O(ϵ3)O({ϵ}^{3})O(ϵ3). Here, we extend the composite pulse technique to cancel errors to O(ϵn)O({ϵ}^{n})O(ϵn), for arbitrary nnn.