Intro comes later.

• Cavity Quality makes quantum? high Q
• experiment vs theory
• approximation hierarchy: QED→NR-QED→light only / matter-only / semiclassical mix
• approximations beyond the hierarchy (well not really): dipole, few levels, few modes
• coupling hierarchy: perturbative → strong → ultra-strong
• system size:
  • 1 vs many modes
  • small atoms - small molecules - simple solids - nano-particles - bio-molecules - …
• physical categories/fields
  • Plasmonics: Wang, Y., Plummer, E. W. & Kempa, K. Foundations of plasmonics. Adv. Phys. 60, 799–898 (2011)
  • Quantum Optics
    Quantum nature of light
  • Solid State Physics
  • Spectroscopy
  • Quantum Chemistry
  • Molecular Physics
• Quasi-particles: Polaritons, Plasmons, Excitons
• Effects: Long/short range coherence/order, condensation, superfluidity
• Nuclei Effects: Phonon theory, Dissociation, Chemical reactions (proton transfer, structural change of groundstate)
• Photon Interaction

• Condensates
  • Polariton-condensates, see Polaritons
• Plasmonics: Wang, Y., Plummer, E. W. & Kempa, K. Foundations of Plasmonics. Adv. Phys. 60, 799–898 (2011)
  • plasmons are photons that have acquired mass due to there strong interaction with the electron plasma
  • electrons scatter with these particles: QED with non-zero mass photons?
  • excitations of plasmons itself are collective modes
  • screening effect: typically, we need MBPT …? DFT treatment possible? what about groundstates?
• Polaritons:
  • hybrid states between light and matter (electronic, vibronic, collective modes, …)
• LC-circuits, “Circuit QED”
  • Todorov T and Sirtori C 2014 Few-electron ultrastrong light matter coupling in a quantum LC circuit PhysRevX.4.041031 (2014):
    • analyze the transition of the 2 different fundamental regimes of the em-field (possible in a QED-circuit)
      1. optical regime: D and H are fundamental variables, bosonic, Hopfield model, many electrons
      2. electronics regims: H and P are fundamental variables, fermionic, JC-model, few electrons
    • use Dicke-states for the coupled Hamiltonian, which are “adequate for the transition between the 2 regimes”
    • effective bosonization with increasing number of electrons
    • for all electron numbers, there is strong coupling and polaritons! Instead the difference is in the “quality” of the Polariton:
      1. optical regime: only UP and LP for the collective electron mode (electronic transition polariton)
      2. electronical regime: many polaritons from all the states of the few electrons (plasmon polariton)
    • self-interaction term fundamental for the observed effects
• Photon Interaction
  • Firstenberg, O. et al. Attractive photons in a quantum nonlinear medium. Nature 502, 71–75 (2013)
    • “Dispersive coupling” by Rydberg Atoms (use electromagnetically induced transparency)
    • Experimental proof of 2-photon (bound) states (but non-equilibrium)
  • Fermionization of Bosons (1d): Tonks–Girardeau (TG) regime of strongly interacting bosons
  • “Crystallization”
    Chang, D. E. et al. Crystallization of strongly interacting photons in a nonlinear optical fibre. Nature Phys. 4, 884–889 (2008)
    • technique (based on EIT) to create 1d strongly correlated photons that have many particle properties, especially, they could form a lattice and have reduced particle-number fluctuations
  • Theory to describe propagation of interacting photons \\Fleischhauer, M. & Lukin, M. D. Dark-state polaritons in electromagnetically induced transparency. Phys. Rev. Lett. 84, 5094–5097 (2000)
• Electromagnetically induced transparency (EIT) \\Review: Fleischhauer, M., Imamoglu, A. & Marangos, J. P. Electromagnetically induced transparency: Optics in coherent media. Rev. Mod. Phys. 77, 633–675 (2005)
  • “interference between excitation pathways” (at least 3-level system necessary): control laser “dresses” one transition (e.g. 2-3) and thus splits the transition energy in two, so that the probe field that was originally in resonance with another transition (e.g. 1-3) is not interacting anymore → transparency
  • zero absorption of the light with strong dispersion and non-linear effects at the same time (energy)
• Sub-cycle energy exchange between matter and field \\Sommer, A. et al. Attosecond nonlinear polarization and light–matter energy transfer in solids. Nature 534, 86–90 (2016).
  • experimentalist are able to track energy transfer between very short (attosecond) pulses and a matter system
  • they can differentiate between external field and polarization and resolve how energy is pumped and extracted from the system (of course there is also an irreversible part)
  • back reaction of the field to the induced change in matter fundamental
  • more or less only Kerr-effect (2nd order, far form resonance so only refraction index change) \\self-steepening, optical shock-wave formation

• Cavity QED Hamiltonian
  • General procedure to eliminate the $A^2$ term: Vukics A, Grieer T and Domokos P 2014 Elimination of the a-square problem from cavity QED Phys. Rev. Lett. 112 073601
    • dipole approximation for atom-models in general bounded region: separate problem in inner and outer atomic degrees of freedom:
      1. canonical and kinetic momentum are not the same!
      2. $A^2$ term allows for pair-creation/annihilation (whats the problem?)
      3. boundaries introduce atom-dipole self-interaction with its boundaries and influence dipole-dipole interaction between atoms
    • inspired by free space solution (Cohen-Tannoudji: Photons and Atoms, Chap. IV.C) to these “not so nice” properties: introduce new (Polaritonic) coordinates, very similar to length gauge
    • introduction of $P^2$ term, with $P=e_A*r_A*\delta(r-r_A)$: other way to get to the necessity of the r^2 term
    • claim that this approach allows for consistent introduction of few-level approximation WITHOUT taking the dipole self-interaction further into account (really?)
• Dipole Approximation
• $A^2$ term discussion
  • classical field $\rightarrow A^2$ can be eliminated by gauge trafo \\Faisal, FHM & Kamiński, JZ Floquet-Bloch theory of high-harmonic generation in periodic structures . Phys. Rev. A 56, 748–762 (1997)
• Self-interaction term:
  • Small discussion on the issue: Expecially important, if nulcei are taken into account
    Flick, J., Ruggenthaler, M., Appel, H. & Rubio, A. Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry. Proc. Natl Acad. Sci. USA 114, 3026–3034 (2017)
  • Beyond toy models: groundstate does not exist without self-interaction (even in semi-classical limit), correct dipole Hamiltonian only with length gauge?!
    Rokaj, V., Welakuh, D. M., Ruggenthaler, M. & Rubio, A. Light-matter interaction in the long- wavelength limit: no ground-state without dipole self- energy. J. Phys. B Atom. Mol. Opt. Phys. 51, 034005 (2017)
  • Experimental point of view: In USC experiments, they could only fit the experimental data with use of the self-interaction term and beyond RWA
    George J, et al. (2016) Multiple Rabi splittings under ultrastrong vibrational coupling. Phys Rev Lett 117(15):153601
  • Fundamental also in another USC situation: 2d QW of electrons (2 bands) coupled to a 0d-mode (plasmon polaritons)
    Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato,C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, Ultrastrong Light-Matter Coupling Regime with Polariton Dots Phys. Rev. Lett. 105, 196402 (2010)
• Cavity Born-Oppenheimer Approximation (CBOA)
  Flick, J., Ruggenthaler, M., Appel, H. & Rubio, A. Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry. Proc. Natl Acad. Sci. USA 114, 3026–3034 (2017)
  • works for model system well
  • dissociation coordinate “becomes polaritonic”: strong coupled molecule must excite photons for dissociation, fr q(=E)=0 also (uncoupled) anti-binding states are bound!
  • also for “more complicated” nulcei structure, polaritonic effects are visible (Shin–Metiu model)
• (Local) Control Theory
  Flick, J., Ruggenthaler, M., Appel, H. & Rubio, A. Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry. Proc. Natl Acad. Sci. USA 114, 3026–3034 (2017)
  • first ideas of how to control polaritonic chemistry with photonic degress of freedom (extra external current), more proof of principle than anything else …
• QEDFT
  • OEP - Functional
    Pellegrini C, Flick J, Tokatly IV, Appel H, Rubio A (2015) Optimized effective potential for quantum electrodynamical time-dependent density functional theory. Phys Rev Lett 115(9):093001
  • Spontaneous Emission of 2-level system, coupled to 400 modes: Comparison OEP to MF
    Flick, J., Ruggenthaler, M., Appel, H. & Rubio, A. Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry. Proc. Natl Acad. Sci. USA 114, 3026–3034 (2017)
    • OEP captures dipole moment of atom better
    • OEP captures correlation between atom and photons, which results in non-zero field at atom also long after the emission of the photon
• Floquet-(Bloch-)Theory
  • Introduced by Faisal, FHM & Kamiński, JZ: Floquet-Bloch theory of high-harmonic generation in periodic structures. Phys. Rev. A 56, 748–762 (1997)
    • describes non-equilibrium steady-state, generated by driving the (periodic) system with a classical laser. Floquet: Discrete FT of the time-dependent (but entirely periodic) problem
    • analytically soluble model: Kronig-Penney plus 1 frequency laser
    • band-structure is replicated in distance of the photon energy
    • avoided crossings occur where bands would cross (especially visible if field is tuned in resonance with the band-gap): formation of “photon dressed electronic states”
    • induced change of absorption energy: dynamical stark shift
  • pump-probe spectroscopy: interpretation and range of validity of Floquet approximation De Giovannini, U., Hübener, H. & Rubio, A. Monitoring electron-photon dressing in WSe 2 . Nano Lett. 16, 7993–7998 (2016). * dressed states from pump pulse → Floquet-sidebands * measurement process in ARPES with a finite-width probe pulse → performing a time average of the nonequilibrium oscillating electronic structure corresponding to the time integrals in the Floquet analysis