BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Michael Weinstein (Columbia University) DTSTART:20230202T163000Z DTEND:20230202T173000Z DTSTAMP:20260423T040035Z UID:SIAM-PDE/31 DESCRIPTION:Title: Effective gaps for time-periodic Hamiltonians modeling Floquet materials \nby Michael Weinstein (Columbia University) as part of Seminar In the Analysis and Methods of PDE (SIAM PDE)\n\n\nAbstract\nFeatured Article: "Effective gaps in continuous Floquet Hamiltonians"\, SIAM J Math Analys is 54 #1 2022. \n\nFeatured Article Authors: Amir Sagiv and Michael Wein stein\n\n\nFloquet media are a type of material\, in which time-periodic f orcing is applied to alter the material’s energy transport properties. E xamples in classical and quantum physics include (i) applying a time-oscil lating electric field to a graphene sheet to influence quantum electronic transport\, and (ii) replacing a periodic array of standard optical wavegu ides by an array of helically coiled waveguides to influence the flow of l ight propagation in the array. The ability to modify energy transport via time modulation is recognized as having great potential for applications t o information transfer\, information processing\, and other applications. \n\nWe discuss Floquet materials governed by a class of parametrically for ced Schrödinger equations\, which arises in the above examples: \ni ψt = Hε (t) ψ\, Hε(t) = H0 + ε W ( ε t \, -i ∇ ). (1)\n\ nThe operator H0 = - Δ + V ( x ) models a graphene-like static material \; the potential V has the symmetries of a honeycomb tiling of the plane. The family of self-adjoint operators T ↦ W ( T\, -i ∇ ) is assumed to be Tper – periodic. \n\nWhile the energy transport of the unforced syste m ( W ≡ 0 ) is governed by the band structure (Floquet-Bloch spectral th eory) of H0 \, that of the forced system is characterized by the monodromy operator\, the unitary mapping ψ0 ↦ ψ ( ε-1 Tper ) \, and its associ ated quasi-energy spectrum. Gaps in the band spectrum and quasi-energy spe ctrum\, respectively\, of static and Floquet materials\, play an important role in their transport properties. \n\nThe evolution under Hε (t) of ba nd-limited Dirac wave-packets (a natural model of physical excitations) is well-approximated on very large time scales by an effective time-periodic Dirac equation\, which has a gap in its quasi-energy spectrum. But little is known about the nature of the quasi-energy spectrum of (1)\, and it is believed that no such quasi-energy gap occurs. \n\nWe explain how to tran sfer quasi-energy gap information about the effective Dirac dynamics to co nclusions about the full Schrödinger dynamics (1). In particular\, we sho w that (1) has an effective quasi-energy gap\, an interval of quasi-energi es whose corresponding modes have only a very small spectral projection on to the subspace of band-limited Dirac wave-packets. The notion of effectiv e quasi-energy gap is a physically relevant relaxation of the strict notio n of quasi-energy spectral gap. Physically\, if a system is tuned to drive or measure at momenta and energies near the Dirac point of H0 \, then the resulting modes in the effective quasi-energy gap will only be weakly exc ited and detected. \n\nThis lecture focuses on joint work with Amir Sagiv (Columbia University) appearing in the article: Effective gaps in continuo us Floquet Hamiltonians\, SIMA 54 #1 2022.\n LOCATION:https://researchseminars.org/talk/SIAM-PDE/31/ END:VEVENT END:VCALENDAR