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arXiv:1705.06883 · Physical Review Applied

Invariant-Based Inverse Engineering of Crane Control Parameters

S. González-Resines, David Guéry-Odelin, Ander Tobalina, Ion Lizuain, E. Torrontegui, Juan Gonzalo Muga

Published Nov 6, 2017• 28 citations
arXivHTMLPDF

Abstract

In physics, similarities can crop up in the most unexpected places. This study, for example, exploits the striking analogy between single ions in microscopic moving traps and the loads moved by mechanical cranes. Though the systems are separated by many orders of magnitude in mass and shuttling distance, results and concepts may be translated from the microscopic domain to the macroscopic, and $v\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}e$ $v\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}a$. ``Shortcuts to adiabaticity'' techniques based on dynamical invariants of motion allow for fast, ultrarobust load control for unknown initial conditions for cranes, while the analysis of energy consumption for cranes can be translated to ion traps.

DOI
10.1103/physrevapplied.8.054008

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In physics, similarities can crop up in the most unexpected places. This study, for example, exploits the striking analogy between single ions in microscopic moving traps and the loads moved by mechanical cranes. Though the systems are separated by many orders of magnitude in mass and shuttling distance, results and concepts may be translated from the microscopic domain to the macroscopic, and v\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}e v\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}a. ``Shortcuts to adiabaticity'' techniques based on dynamical invariants of motion allow for fast, ultrarobust load control for unknown initial conditions for cranes, while the analysis of energy consumption for cranes can be translated to ion traps.
We use the dynamical algebra of a quantum system and its dynamical invariants to inverse engineer feasible Hamiltonians for implementing shortcuts to adiabaticity. These are speeded up processes that end up with the same populations as slow, adiabatic ones. As application examples, we design families of shortcut Hamiltonians that drive two- and three-level systems between initial and final configurations, imposing physically motivated constraints on the terms (generators) allowed in the Hamiltonian.
Pure-state inverse engineering among the schemes of shortcuts to adiabaticity is a widespread utility in applications to quantum computation and quantum simulation. While in principle it can realise the fast control of quantum systems with high fidelity, in practice this fast control is severely hindered by infinite energy gaps and impractical control fields. To circumvent this problem, we propose a scheme of shortcuts to adiabaticity of mixed state based on the dynamical invariant of open quantum system. Our scheme can drives a steady state to a target steady state of the open system by a controlled Liouvillian that possesses the same form as the reference (original) one. We apply this scheme to stimulated Raman adiabatic passage (STIRAP) and find that an almost perfect population transfer can be obtained. The experimental observation with currently available parameters for the nitrogen-vacancy (NV) center in diamond is suggested and discussed.
Different methods have been recently put forward and implemented experimentally to inverse engineer the time-dependent Hamiltonian of a quantum system and accelerate slow adiabatic processes via nonadiabatic shortcuts. In the ``transitionless quantum driving'' proposed by Berry, shortcut Hamiltonians are designed so that the system follows exactly, in an arbitrarily short time, the approximate adiabatic path defined by a reference Hamiltonian. A different approach is based on first designing a Lewis-Riesenfeld invariant to carry the eigenstates of a Hamiltonian from specified initial to final configurations, again in an arbitrary time, and then constructing from the invariant the transient Hamiltonian that connects these boundary configurations. We show that the two approaches, apparently quite different in form and so far in results, are, in fact, strongly related and potentially equivalent, so that the inverse-engineering operations in one of them can be reinterpreted and understood in terms of the concepts and operations of the other one. We study, as explicit examples, expansions of time-dependent harmonic traps and the state preparation of two-level systems.
There is recent interest in determining energy costs of shortcuts to adiabaticity (STA), but different definitions of 'cost' have been used.We demonstrate the importance of taking into account the control system (CS) for a fair assessment of energy flows and consumptions.We model the energy consumption and power to transport an ion by a STA protocol in a multisegmented Paul trap.The ion is driven by an externally controlled, moving harmonic oscillator.Even if no net ion-energy is gained at destination, setting the time-dependent control parameters is a macroscopic operation that costs energy and results in energy dissipation for the short time scales implied by the intrinsically fast STA processes.The potential minimum is displaced by modulating the voltages on control (dc) electrodes.A secondary effect of the modulation, usually ignored as it does not affect the ion dynamics, is the time-dependent energy shift of the potential minimum.The non trivial part of the energy consumption is due to the electromotive forces to set the electrode voltages through the low-pass filters required to preserve the electronic noise from decohering the ion's motion.The results for the macroscopic CS (the Paul trap) are compared to the microscopic power and energy of the ion alone.Similarities are found-and may be used quantitatively to minimize costs-only when the CSdependent energy shift of the harmonic oscillator is included in the ion-energy.