Dynamical normal modes for time-dependent Hamiltonians in two dimensions

Abstract

We present the theory of time-dependent point transformations to find independent dynamical normal modes for 2D systems subjected to time-dependent control in the limit of small oscillations. The condition that determines if the independent modes can indeed be defined is identified, and a geometrical analogy is put forward. The results explain and unify recent work to design fast operations on trapped ions, needed to implement a scalable quantum-information architecture: transport, expansions, and the separation of two ions, two-ion phase gates, as well as the rotation of an anisotropic trap for an ion are treated and shown to be analogous to a mechanical system of two masses connected by springs with time dependent stiffness.

Shortcuts to adiabaticity: Concepts, methods, and applications

“…This technique has been applied to accelerate processes described by two and three level systems such as cotunneling and splitting of two bosons in a double well, see Fig. 5, to generate macroscopically entangled states in a Tonks-Girardeau gas , to design optical waveguide devices (Chung et al, 2017;Liu and Tseng, 2017;Martínez-Garaot et al, 2017), and quantum neural networks (Torrontegui and García-Ripoll, 2019).…”

“…As well, shortcuts have been proposed to produce single photons on demand in an atom-cavity system approximated by three levels (Shi and Wei, 2015). A Toffoli Cavity QED QZD+Invariants Santos and Sarandy (2015) Universal gates N qubits CD driving Chen et al (2015a) Phase Cavity QED QZD+Invariants Liang et al (2015d) Phase Cavity QED QZD+Invariants Liang et al (2015b) CNOT Cavity QED QZD+Invariants Liang et al (2015a) Swap Cavity QED QZD+Invariants Zhang et al (2015a) Non-Abelian geometric Superconducting transmon CD driving Santos et al (2016) N-qubit Four-level system CD driving Song et al (2016d) 1-and 2-qubit holonomic NV centers CD driving Liang et al (2016) Non-Abelian geometric NV centers CD driving Two-qubit phase Two trapped ions Invariants Du et al (2017) Non-Abelian geometric NV centers CD driving Wu et al (2017a) CNOT Cavity QED QZD+Dressed-state scheme Santos (2018) Single-and Two-qubit Two-and Four-level system Inverse engineering Liu et al (2018) Non-Abelian geometric NV centers Invariants Wang et al (2018) Single-qubit Superconducting Xmon qubit CD driving Shen and Su (2018) Two-qubit controlled phase Two Rydberg atoms Invariants Ritland and Rahmani (2018) Majorana Top transmon OCT for noise cancelling Li et al (2018b) 1-Qubit gate&transport Double quantum dot Inverse engineering Yan et al (2019a) Non-Abelian geometric Superconducting Xmon qubit CD driving Lv et al (2019) Non-cyclic geometric Two-level atom CD driving Santos et al (2019) Single qubit Nuclear Magnetic Resonance CD driving Qi and Jing (2019) Single and double-qubit holonomic Rydberg atoms CD driving system of distant nodes in two-dimensional networks (cavities with a Λ-type atom) is approximated by a three-level Λ system in Zhong (2016) and then STA techniques are applied to achieve fast information transfer.…”

Invariant-Based Inverse Engineering of Crane Control Parameters

“…Setting the shortcuts in that case will require considering two harmonic oscillators, by means of a dynamical normal-mode analysis similar to the one done for two ions [57]. (c) Invariant-based inverse-engineering combines well with optimal control theory [60][61][62].…”

“…Invariant-based engineering is the approach best adapted to the peculiarities of trapped ions and has been extensively applied by our group in that context [50][51][52][53][54][55][56][57]. Here, we propose an inverse-engineering method for crane operations based on the invariants of motion of the system.…”

Fast state and trap rotation of a particle in an anisotropic potential

“…S diagonalizes the initial A matrix by the relation A 4 = S T AS and relates old coordinates and momenta in the There is a maximum allowedθ, when one of the normal mode frequencies becomes complexθmax = ω1, see Eq. (9). For a non-rotating trap (θ = 0), these frequencies are simply the axial frequencies ω1,2.…”