Optimally robust shortcuts to population inversion in two-level quantum systems

Abstract

We examine the stability versus different types of perturbations of recently proposed shortcuts to adiabaticity to speed up the population inversion of a two-level quantum system. We find the optimally robust processes by using invariant-based engineering of the Hamiltonian. Amplitude noise and systematic errors require different optimal protocols. IntroductionManipulating the state of a quantum system with time-dependent interacting fields is a fundamental operation in atomic and molecular physics, with applications such as lasercontrolled chemical reactions, metrology, interferometry, nuclear magnetic resonance or quantum information processing [1][2][3][4][5]. For two-level systems, there are several approaches proposed for attaining a complete population transfer, for example, π pulses, composite pulses, adiabatic passage and its variants. In general, the π pulses may be fast but highly sensitive to variations in the pulse area, and to inhomogeneities in the sample [1]. Used first in nuclear magnetic resonance [6], composite pulses provide an alternative to the single π pulse, with some successful applications [7,8], but still need accurate control of pulse phase and intensity. A robust option is, in principle, adiabatic (slow) passage, which is however prone to decoherence because of the effect of noise over the long times required. A compromise is to use speeded-up 'shortcuts to adiabaticity', which may be broadly defined as the processes that lead to the same final populations as the adiabatic approach in a shorter time.Several methods to find shortcuts to adiabaticity have been put forward [9-19] for twoand three-level atomic systems. The transitionless or counter-diabatic (CD) control protocols proposed by Demirplak and Rice [9] and Berry [10] start from a reference time-dependent Hamiltonian H 0 and provide an extra interaction that cancels the diabatic couplings. This results in an exact following of the adiabatic dynamics of the reference Hamiltonian, in principle in an arbitrarily short time. They have been applied, for example, to speed up the rapid adiabatic passage (RAP) for an Allen-Eberly scheme [11]. Modified by a unitary transformation [17], New Journal of Physics 14 (2012) 093040 (http://www.njp.org/)

Fast shuttling of a trapped ion in the presence of noise

“…While it has been used to design protocols for fast ion shuttling [15][16][17][18][19], it has also been employed in the field of internal-state control to find protocols which are resilient to noise [20,21]. In this section, we provide a brief review of this method for the ion-shuttling scenario.…”

Robust coherent superposition of states using quasiadiabatic inverse engineering

“…We substitute Eqs. (2) or (3) directly into the Schrödinger equation and obtain the following auxiliary differential equations [20,21,23,24,25]:…”

“…However, the process does not always lead to the exact target state. On the other hand, the invariant-based inverse engineering STA [20,21] allows one to freely design the system evolution from the desired initial state to the final state. In this paper, the invariant-based inverse engineering and quasiadiabatic condition are combined for the first time for the production of coherent superposition of states.…”

Shortcut to adiabaticity in a Stern-Gerlach apparatus

“…More precisely, we design a suitable magnetic field profile by using the reverse engineering approach based on the Lewis-Riesenfeld invariants [7][8][9]. These methods enable one to guarantee a transitionless evolution faster than the time scale imposed by the adiabatic regime.…”