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Exploring Floquet Systems with Topology and Steady States for Quantum Theorists

Dive into the intriguing world of Floquet-inspired phenomena in quantum systems, exploring topological phases, steady states, Hamiltonian construction, and more. Discover the latest research and advancements in this field with contributions from leading experts. Unravel the complexities of Floquet Hamiltonians, quasi-energy states, and synthetic dimensions to understand the behavior of quantum systems with periodically varying Hamiltonians. Explore the implications of Floquet-driven phenomena on energy pumping, frequency conversion, and topological phases, shedding light on the fascinating intersection of quantum theory and topology. Delve into the mathematical intricacies and cutting-edge simulations that are shaping the future of quantum research.

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Exploring Floquet Systems with Topology and Steady States for Quantum Theorists

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  1. Quantum Floquet Systems: Topology, steady states, and more Funding: Darpa (FENA), NSF, Packard foundation

  2. Hamiltonian construction quantum theorists U V J (taken from: Good night (quantum) construction site)

  3. Hamiltonian construction

  4. Hamiltonian construction

  5. Floquet topological phases? e

  6. Floquet topological phases

  7. Floquet topological phases Netanel Lindner Technion Victor Galitski Maryland Aoki, Oka (2009) Lindner, Galitski, Refael (2010) Kitagawa, Berg, Rudner, Demler (2010)Jiang, Kitagawa,Akhmerov,Alicea,Refael, Pekker, Demler, Cirac, Zoller,Lukin (2011) Grushin, Gomez-Leon, Neupert (2014)

  8. Topological Polaritons Karzig, Bardyn, Lindner, GR (2014) Torsten Karzig Station Q Charles Bardyn Caltech Netanel Lindner Technion

  9. Floquet-inspired new phenomena • Topolaritons: • Anomalous Floquet Topological Anderson Insulator:

  10. Steady states? Driving uncontrolled heating?

  11. Steady states? PRX, 2016 Karthik Seetharam Caltech Charles Bardyn Caltech Netanel Lindner Technion Dehghani, Oka, Mitra (2014)Iadecola, Neupert, Chamon(2015) Genske, Rosch (2015) Bilitewski, Cooper (2014) Chandran, Sondhi (2015) Mark Rudner Copenhagen

  12. Environment Engineering Energy filtered lead System V E Environment Phonon bath

  13. Mathematical interlude: Floquet Hamiltonians • Periodically varying Hamiltonian: with • Hamiltonian Floquet operator: • Energy eigenstates Quasi-energy states:

  14. Band structure folding and the Floquet zone E P

  15. Band structure folding and the Floquet zone E P (1) 1st resonance: Rotating wave approximation.

  16. Band structure folding and the Floquet zone E b b P (1) 1st resonance: Rotating wave approximation. (2) Higher order resonances

  17. Band structure folding and the Floquet zone E 3-photon resonance Floquet Zone b b b b b b P 2-photon resonance 2-photon resonance (1) 1st resonance: Rotating wave approximation. (2) Higher order resonances

  18. Floquet Double Drive: 2d Floquet and topological energy pumping Ivar Martin (Argonne NL) Frederik Nathan (Copenhagen) Bertrand Halperin (Harvard) Gil Refael (Caltech)

  19. Overview • Floquet intro (Not again…) • topological phases 0-d topological drive: Topological Cavity amplifier Quantized pumping And frequency conversion

  20. Floquet wave functions • Simple drive: • Time evolving wave function: Time dependent Schroedinger equation: Synthetic dimension – like a tight binding model See also: Nathan Goldman; Shanhui Fan; Iacoppo Carussoto....

  21. Photon space space vs. real space • Add phase to the drive: • Photon number operator?

  22. Photon space space vs. real space • Add phase to the drive: • Photon number operator? • Photon absorption rate?

  23. Phase vs. Momentum • Why not identify momentum with drive-phase?

  24. Caveats • Drive implies constant momentum change: Can’t access all momentum states. • Dimension of Hilbert space = dim(H0)

  25. Caveats • Drive implies constant momentum change: Can’t access all momentum states. • Dimension of Hilbert space = dim(H0)=2 | π | -π k

  26. Caveats • Drive implies constant momentum change: Can’t access all momentum states. • Dimension of Hilbert space = dim(H0)=2 | π | -π k

  27. Caveats • Drive implies constant momentum change: Can’t access all momentum states. • Dimension of Hilbert space = dim(H0)=2 z • Adiabatic evolution - if y x

  28. Higher dimensional analogy? • 2d and higher dimensional systems also possible: And tight-binding effective Hamiltonian:

  29. Higher dimensional analogy? • 2d and higher dimensional systems also possible:

  30. Commensurate vs incommensurate Incommensurate commensurate Time Quasi-crystal!

  31. Topological synthetic Floquet phases • Use BHZ band structure: • BHZ magic:

  32. Topological synthetic Floquet phases • Use BHZ band structure: • BHZ magic: Topo trivial Berry curvature

  33. Topological synthetic Floquet phases • Use BHZ band structure: • BHZ magic: Topo trivial -1 1 C=1 m/2b C=-1

  34. Semiclassical motion • Berry curvature • BHZ Berry curvature and motion:

  35. Topological synthetic Floquet phases • 2f Floquet BHZ:

  36. Synthetic 2f BHZ • 2f Floquet BHZ: • Semiclassical motion: Quantized energy pumping Quantized

  37. Quantized energy pumping Quantized Quantized energy pumping Energy pumped between lasers:Topological frequency conversion Quantization - if all BZ is covered…

  38. Simulations • Need to calculate: • Parameter regime? Strong driving: Adiabatic evolution: • Initializing

  39. Numerics I: Incommensurate Frequencies • :

  40. Numerics I: Incommensurate Frequencies [strong coupling] • : Fidelity: Work: Power:

  41. Numerics I: Incommensurate Frequencies [weak copling] • : C=1 m/2b C=-1

  42. Numerics II: Commensurate Frequencies • : Periodic Momentum paths Initiate w/ Floquet eigenstates

  43. Numerics II: Commensurate Frequencies [m dependence] • :

  44. Numerics II: Commensurate Frequencies [phase dependence] • :

  45. Quantum – Classical hybrid • Replace one of the modes with a cavity: Much larger Hilbert space…

  46. Quantum – Classical hybrid • Phase diagram: (photons in cavity) Topo Topo Trivial

  47. Quantum – Classical hybrid • Pumping an occupied mode:

  48. Quantum – Classical hybrid • Pumping from zero:

  49. Period tripling • Unexpected stable orbit: Time-crystal?? 

  50. Summary and conclusions • Multiple drives could induce topological effects. • Topological frequency conversion, quantized energy pumping, optical amplifier. • Experimental realizations? Spins, qubits, cavities with rare Earth garnets? • Open questions: Temporal disorder effects. Using dissipation to increase fidelity? Two cavity systems?

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