Fast expansions and compressions of trapped-ion chains

Abstract

We investigate the dynamics under diabatic expansions/compressions of linear ion chains. Combining a dynamical normal-mode harmonic approximation with the invariant-based inverseengineering technique, we design protocols that minimize the final motional excitation of the ions. This can substantially reduce the transition time between high and low trap-frequency operations, potentially contributing to the development of scalable quantum information processing.

Dynamical normal modes for time-dependent Hamiltonians in two dimensions

“…Let us consider two singly-charged, positive ions in a linear trap at lab-frame coordinates q 1 and q 2 , coupled via Coulomb interaction and trapped in an external, possibly time dependent, harmonic potential [9,10]. The external potential can be translated [9] and, in addition, expanded or compressed [10].…”

“…While this is the form that has been used to speed up several operations on trapped ions [8][9][10][11][12][13][14], for the discussion of the separability of these systems in Sec. III it is enough to examineH and we shall omit the momentum shift transformation.…”

“…In particular we analyze if independent dynamical normal modes can be defined by point transformations. This is a key issue to design and engineer fast and robust protocols for operations needed to develop scalable quantum information processing: ion transport [8,9,12], trap expansions or compressions [10], ion splitting [11], phase gates [14], or rotations [13].…”

“…Independent dynamical normal modes ("dynamical" because their definition depends on time due to the changing external control) are very useful, not only to describe the motion in a simple way, but also to inverse-engineer the dynamical operations. Specifically, for operations on one ion in a two-dimensional (2D) potential or on two ions interacting in a one-dimensional (1D) line [8][9][10][11][12][13][14], we noticed that uncoupled dynamical normal modes cannot always be defined. Each case was analyzed separately but a generic understanding of the conditions that determine the coupling/uncoupling of normal-mode coordinates was missing.…”

“…In this paper we study the possibility to define independent dynamical normal modes in systems described by two dimensional, timedependent Hamiltonians. While the question is interesting per se and relevant for a broad span of externally controllable physical systems near equilibrium, our main motivation has been the need to understand and possibly improve on recent work to inverse-engineering fast and robust operations to drive the motion of trapped ions [8][9][10][11][12][13][14]. Trapped ions constitute the most developed physical platform to implement quantum information processing.…”