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12074 Revmodphys.91.045001.Pdf UCC Library and UCC researchers have made this item openly available. Please let us know how this has helped you. Thanks! Title Shortcuts to adiabaticity: Concepts, methods, and applications Author(s) Guéry-Odelin, D.; Ruschhaupt, Andreas; Kiely, Anthony; Torrontegui, E.; Martínez-Garaot, S.; Muga, Juan Gonzalo Publication date 2019-10-24 Original citation Guéry-Odelin,D., Ruschhaupt, A., Kiely, A., Torrontegui, E. Martínez- Garaot, S. and Muga, J.G. (2019) 'Shortcuts to adiabaticity: Concepts, methods, and applications', Reviews Of Modern Physics, 91(4), 045001, (54 pp). doi: 10.1103/RevModPhys.91.045001 Type of publication Article (peer-reviewed) Link to publisher's https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045001 version http://dx.doi.org/10.1103/RevModPhys.91.045001 Access to the full text of the published version may require a subscription. Rights © 2019 American Physical Society Item downloaded http://hdl.handle.net/10468/9796 from Downloaded on 2021-09-23T21:08:23Z REVIEWS OF MODERN PHYSICS, VOLUME 91, OCTOBER–DECEMBER 2019 Shortcuts to adiabaticity: Concepts, methods, and applications D. Gu´ery-Odelin Laboratoire de Collisions Agr´egats R´eactivit´e, CNRS UMR 5589, IRSAMC, Universit´e de Toulouse (UPS), 118 Route de Narbonne, 31062 Toulouse CEDEX 4, France A. Ruschhaupt and A. Kiely Department of Physics, University College Cork, Cork, Ireland E. Torrontegui Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain S. Martínez-Garaot and J. G. Muga Departamento de Química Física, UPV/EHU, Apdo. 644, 48080 Bilbao, Spain (published 24 October 2019) Shortcuts to adiabaticity (STA) are fast routes to the final results of slow, adiabatic changes of the controlling parameters of a system. The shortcuts are designed by a set of analytical and numerical methods suitable for different systems and conditions. A motivation to apply STA methods to quantum systems is to manipulate them on timescales shorter than decoherence times. Thus shortcuts to adiabaticity have become instrumental in preparing and driving internal and motional states in atomic, molecular, and solid-state physics. Applications range from information transfer and processing based on gates or analog paradigms to interferometry and metrology. The multiplicity of STA paths for the controlling parameters may be used to enhance robustness versus noise and perturbations or to optimize relevant variables. Since adiabaticity is a widespread phenomenon, STA methods also extended beyond the quantum world to optical devices, classical mechanical systems, and statistical physics. Shortcuts to adiabaticity combine well with other concepts and techniques, in particular, with optimal control theory, and pose fundamental scientific and engineering questions such as finding speed limits, quantifying the third law, or determining process energy costs and efficiencies. Concepts, methods, and applications of shortcuts to adiabaticity are reviewed and promising prospects are outlined, as well as open questions and challenges ahead. DOI: 10.1103/RevModPhys.91.045001 CONTENTS 2. Examples of invariant-based inverse engineering 13 a. Two-level system 13 I. Introduction 2 b. Lewis-Leach family 14 A. Overview of shortcuts to adiabaticity 2 3. Scaling laws 15 B. Motivation and scope of this review 3 4. Connection with Lax pairs 16 II. Methods 4 D. Variational methods 16 A. Overview of inverse engineering approaches 4 E. Fast forward 17 1. Quantum transport 5 1. The original formalism 17 2. Spin manipulation 5 2. Streamlined fast-forward approach 18 3. Beyond mean values 5 3. Generalizations and terminology 19 B. Counterdiabatic driving 6 F. FAQUAD and related approaches 19 1. Superadiabatic iterations 8 G. Optimal control and shortcuts to adiabaticity 20 2. Beyond the basic formalism 9 H. Robustness 21 “ ” a. Physical unitary transformations 9 1. Error sensitivity and its optimization using b. Schemes that focus on one state 10 perturbation theory 21 c. Effective counterdiabatic field 10 a. Illustrative example: Control of a two-level d. Dressed-states approach 10 system 21 e. Variational approach 11 b. Optimization using perturbation theory f. Counterdiabatic Born-Oppenheimer dynamics 12 in other settings 22 g. Constant CD-term approximation 12 2. Other approaches 23 C. Invariants and scaling laws 12 I. Three-level systems 23 1. Lewis-Riesenfeld invariants 12 0034-6861=2019=91(4)=045001(54) 045001-1 © 2019 American Physical Society D. Gu´ery-Odelin et al.: Shortcuts to adiabaticity: Concepts, methods, … 1. Apply counterdiabatic shortcuts to the full three-level Λ system 23 2. Applying invariant-based inverse engineering shortcuts to the full three-level Λ system 24 3. Using STA techniques after mapping to a two-level system 24 J. Motional states mapped into a discrete system 24 III. Applications in Quantum Science and Technology 24 A. Trapped ions 25 1. Dynamical normal modes 25 2. Ion transport 27 3. Other operations 28 B. Double wells 29 FIG. 1. A turtle on wheels is a good metaphor for shortcuts to C. Cavity quantum electrodynamics 29 adiabaticity. The image is inspired by the artist work of Andree D. Superconducting circuits 30 Richmond. E. Spin-orbit coupling 31 F. Nitrogen-vacancy centers 31 G. Many-body and spin-chain models 32 illustrates the concept pictorially by comparing diabatic, H. Metrology 34 adiabatic, and STA paths in a discrete system. IV. The Energy Cost of STA, Engines, and the Third Law 34 The shortcuts rely on specific time dependences of the A. Energy costs 34 control parameters and/or on the addition of auxiliary time- B. Engines and refrigerators 37 dependent couplings or interactions with respect to some C. Third law 38 reference Hamiltonian or, more generally, a Liouvillian or V. Open Quantum Systems 38 transition-rate matrix. STA methods were first applied in A. Concept of adiabaticity for open systems 38 simple quantum systems: two- and three-level systems or a B. Engineering the environment 39 particle in a time-dependent harmonic oscillator. They have C. Mitigating the effect of environment 39 since come to encompass a much broader domain since slow D. Non-Hermitian Hamiltonians 40 processes are quite common as a simple way to prepare the VI. Optical Devices 40 state of a system or to change conditions avoiding excitations VII. Extension to Classical and Statistical Physics 42 A. Counterdiabatic methods in classical mechanics 42 in a wide spectrum of areas from atomic, molecular, and B. Mechanical engineering 43 optical physics, solid state, or chemistry to classical mechani- 1. Cranes 43 cal systems and engineering. In parallel to such a large scope, 2. Robustness issues 44 different methods have been developed and applied. C. STA for isolated dilute gases 44 This description of shortcuts needs some caveats and 1. Boltzmann equation 44 clarifications. In many quantum mechanical applications, 2. Extension to Navier-Stokes equation 45 the final state in an STA process reproduces the set of D. Shortcuts for systems in contact with a thermostat 45 adiabatic probabilities to find the system in the eigenstates 1. The overdamped regime 45 of the final Hamiltonian. The mapping of probabilities from 2. Connection with free energy and irreversible work 46 3. Extensions 46 VIII. Outlook and Open Questions 46 List of Symbols and Abbreviations 47 Acknowledgments 47 Appendix A: Example of Lie Transform 47 Appendix B: Counterdiabatic Hamiltonian for a Two-level Hamiltonian with Complex Coupling 48 References 48 I. INTRODUCTION A. Overview of shortcuts to adiabaticity “ ” Shortcuts to adiabaticity (STA) are fast routes to the final FIG. 2. Schematic example of adiabatic, diabatic, and STA results of slow, adiabatic changes of the controlling param- processes. The system is initially in the third level. In the eters of a system. Adiabatic processes are here broadly defined adiabatic path (short-dashed red) the system evolves always as those for which the slow changes of the controls leave some along the third level. In a diabatic evolution (long-dashed blue) dynamical properties invariant, the “adiabatic invariants,” the system gets excited by jumping across avoided crossings. such as the quantum number in quantum systems or phase- Along an STA path (green dots) the system does not always travel space areas in classical systems. Figure 1 informally portrays along the third level but it arrives at the third level in a shorter the central idea of shortcuts to adiabaticity. Figure 2 also time (the time dimension is not explicitly shown in the figure). Rev. Mod. Phys., Vol. 91, No. 4, October–December 2019 045001-2 D. Gu´ery-Odelin et al.: Shortcuts to adiabaticity: Concepts, methods, … initial to final settings, without final excitation but allowing the design of operations explicitly intended to achieve it (see for excitations en route, holds for any possible initial state in an example in Sec. III.A), as well as indirectly, since a some state-independent STA protocols and simple enough mitigated decoherence reduces the need for highly demand- systems. These state-independent shortcuts are ideal for ing, qubit-consuming error-correction codes. maximal robustness and to reduce dependence on temper- • The second main paradigm of quantum information ature, so as to avoid costly and time-consuming cooling to the processing is adiabatic computing or quantum annealing, ground state. However, except for idealized models, state which may be accelerated or made possible by changing independence holds only approximately and for a domain of initial and/or final Hamiltonians, modifying the interpolation parameters, for example, within the harmonic approximation path between initial and final Hamiltonians, or adding in a trapped ion. In other applications state independence is auxiliary terms to the transient Hamiltonian. For adiabatic not really necessary or it may be difficult or impossible to quantum computing and other applications with complex achieve as in chaotic systems. Yet shortcuts may still be found systems, progress is being made to find effective STA without for a chosen specific state, typically the ground state or for a using difficult-to-find information on spectra and eigenstates.
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