Floquet engineering the band structure of materials with optimal control theory

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Floquet engineering the band structure of materials with optimal control theory

Alberto Castro, Umberto De Giovannini, Shunsuke A. Sato, Hannes Hübener, and Angel Rubio

Phys. Rev. Research 4, 033213 – Published 19 September 2022

Abstract

We demonstrate that the electronic structure of a material can be deformed into Floquet pseudobands with arbitrarily tailored shapes. We achieve this goal with a combination of quantum optimal control theory and Floquet engineering. The power and versatility of this framework is demonstrated here by utilizing the independent-electron tight-binding description of the $π$ electronic system of graphene. We show several prototype examples focusing on the region around the $K$ (Dirac) point of the Brillouin zone: creation of a gap with opposing flat valence and conduction bands, creation of a gap with opposing concave symmetric valence and conduction bands (which would correspond to a material with an effective negative electron-hole mass), and closure of the gap when departing from a modified graphene model with a nonzero field-free gap. We employ time-periodic drives with several frequency components and polarizations, in contrast to the usual monochromatic fields, and use control theory to find the amplitudes of each component that optimize the shape of the bands as desired. In addition, we use quantum control methods to find realistic switch-on pulses that bring the material into the predefined stationary Floquet band structure, i.e., into a state in which the desired Floquet modes of the target bands are fully occupied, so that they should remain stroboscopically stationary, with long lifetimes, when the weak periodic drives are started. Finally, we note that although we have focused on solid state materials, the technique that we propose could be equally used for the Floquet engineering of ultracold atoms in optical lattices and for other nonequilibrium dynamical and correlated systems.

Received 2 March 2022
Accepted 3 August 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.033213

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Photoinduced effect

Graphene

Bloch-Floquet theorem

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Alberto Castro ^1,2,*, Umberto De Giovannini^3,4,†, Shunsuke A. Sato ^5,4,‡, Hannes Hübener ^4,§, and Angel Rubio^4,6,∥

¹Institute for Biocomputation and Physics of Complex Systems, University of Zaragoza, 50018 Zaragoza, Spain
²ARAID Foundation, 50018 Zaragoza, Spain
³Università degli Studi di Palermo, Dipartimento di Fisica e ChimicaEmilio Segrè, via Archirafi 36, I-90123 Palermo, Italy
⁴Max Planck Institute for the Structure and Dynamics of Matter and Center for Free Electron Laser Science, 22761 Hamburg, Germany
⁵Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan
⁶Center for Computational Quantum Physics (CCQ), The Flatiron Institute, 162 Fifth avenue, New York, New York 10010, USA

^*acastro@bifi.es
^†umberto.de-giovannini@mpsd.mpg.de
^‡ssato@ccs.tsukuba.ac.jp
^§hannes.huebener@mpsd.mpg.de
^∥angel.rubio@mpsd.mpg.de

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Vol. 4, Iss. 3 — September - November 2022

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Figure 1
Schematic diagram of the technique demonstrated in this paper: A graphene sheet is first pumped by a switch-on pulse, that is designed by OCT techniques to prepare the system such that the Floquet bands induced by a periodic multicolor continuous-wave (CW) pulse, that comes later, are populated. This time-periodic driving is also designed by OCT methods, in order to produce Floquet bands with a given predefined shape.
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Figure 2
(a)–(c) Field-free bands (gray) and Floquet pseudobands (blue) of graphene, in the vicinity of the $K$ point. The Floquet pseudobands are computed (a) using a reference circularly polarized field (see text), (b) using fields optimized to create a gap with opposing parallel bands, and (c) using fields optimized to create a gap with opposing concave bands. (d)–(f) Cuts through the $x z$ plane of the previous three-dimensional plots, displaying also the target bands in red and the series of Floquet replicas. These are color-coded: A darker blue color corresponds to a larger Fourier component $〈 u_{k m}^{α} | u_{k m}^{α} 〉$ [see Eqs. (7) and (8)]. (g)–(i) Fourier decomposition (left) and Lissajous plots (right) of the fields corresponding to the pseudobands shown above. FT, Fourier transform.
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Figure 3
Gap closing. (a) Field-free bands (gray) of the modified graphene model in the vicinity of the $K$ point (blue) showing how the gap can be closed. (b) Cut along the $y$ axis of (a). Along with the Floquet bands in the first BZ, the sidebands are also shown. The color intensity of each point is proportional to the value of the corresponding Fourier component. (c) Fourier components and Lissajous figure of the optimized pulse.
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Figure 4
Berry curvature of the valence (left panels) and conduction (right panels) pseudobands. They correspond to (a) and (b) the Floquet pseudobands computed with the reference circularly polarized fields, (c) and (d) the flat pseudobands, and (e) and (f) the negative curvature bands shown in Fig. 2. (a) replicates the results shown in Fig. 3(d) of Ref. [13]. Black dashed circles indicate the area for which the target was defined.
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Figure 5
(a) Optimal switch-on $A_{x}^{so}$ (blue) and $A_{y}^{so}$ (red) pulses, applied during the first 20 periods of duration $T$ , followed by the optimized periodic drivings $A_{x}$ and $A_{y}$ . Lines represent the full pulses $A_{i}^{so} (v^{opt}, t) = S (t) f_{i} (v^{opt}, t) A_{i} (u^{opt}, t)$ , whereas the filled curves are the absolute value of the envelopes $S (t) f_{i} (v^{opt}, t)$ (which become equal to 1 after the switch-on phase). (b) Average population of the lower pseudoenergy Floquet modes: $| 〈 ψ_{k 0} (t) | u_{k}^{0} (t) 〉 |^{2}$ . One can see how, after the switch-on phase, it is approximately constant and equal to 1 since the system has been (almost) transferred to those states.
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