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Strong Coupling and Quantum Molecular Plasmonics

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Strong Coupling and Quantum Molecular Plasmonics Strong coupling and molecular plasmonics Javier Aizpurua http://cfm.ehu.es/nanophotonics Center for Materials Physics, CSIC-UPV/EHU and Donostia International Physics Center - DIPC Donostia-San Sebastián, Basque Country AMOLF Nanophotonics School June 17-21, 2019, Science Park, Amsterdam, The Netherlands Organic molecules & light Excited molecule (=exciton on molecule) Photon Photon Adding mirrors to this interaction The photon can come back! Optical cavities to enhance light-matter interaction Optical mirrors Dielectric resonator Veff Q ∼ 1/κ Photonic crystals Plasmonic cavity Veff Q∼1/κ Coupling of plasmons and molecular excitations Plasmon-Exciton Coupling Plasmon-Vibration Coupling Absorption, Scattering, IR Absorption, Fluorescence Veff Raman Scattering Extreme plasmonic cavities Top-down Bottom-up STM ultra high-vacuum Wet Chemistry Low temperature Self-assembled monolayers (Hefei, China) (Cambridge, UK) “More interacting system” One cannot distinguish whether we have a photon or an excited molecule. The eigenstates are “hybrid” states, part molecule and part photon: “polaritons” Strong coupling! Experiments: organic molecules Organic molecules: Microcavity: D. G. Lidzey et al., Nature 395, 53 (1998) • Large dipole moments • High densities collective enhancement • Rabi splitting can be >1 eV (~30% of transition energy) • Room temperature! Surface plasmons: J. Bellessa et al., Single molecule with gap plasmon: Phys. Rev. Lett. 93, 036404 (2004) Chikkaraddy et al., Nature 535, 127 (2016) from lecture by V. Shalaev Plasmonic cavities as optical resonators λ V Ultra-compact (V∼sub-nm), Ultrafast (1/γa∼ fs), Room Temperature (g∼eV) Effective mode volume V/ volume mode Effective Chikkarady et al. Nature 7, 535 (2016) Plasmonic Nanocavities and Quantum Emiters: QEs: quantum dots, diamond vacancies, dye molecules, J-aggregates, TMDs… Shegai, Pelton, Sandhogar, Bozhevolnyi, Lukin, Sanvitto, Liedl, Acuña, Raschke, Mikkelsen, Hecht, Baumberg… Strong Coupling Classical treatment A molecule as an harmonic oscillator Molecule (or other emitter) In frequency domain The nanoresonator Scattering hν + E + + - - - Extinction Absorption We assume that the nanophotonic resonance is a simple Lorentzian Or equivalently The plasmonic cavity For a small metallic particle or radius R surrounded by vacuum giving Or equivalently Other plasmonic resonators follow similar expressions A key plasmonic nanoresonator: the plasmonic cavity Sphere - plane Sphere - sphere dipole – mirror dipole near-field interaction dipole – dipole near-field interaction High field enhancement in the gap due to resonant near-field coupling • local light sources • enhanced Raman signals (detection of single molecule Raman signals) • nonlinear effects Dipolar sphere-sphere near-field interaction Scattered field: 3 ε i −1 Polarizability of spheres: α = 4πa i i E ∝α E ε i + 2 sca eff in Effective polarizability of interacting dipoles (dipole approximation): α1 α1 2a 2a z r z Ein r Ein α 2 α 2 α1 α 2 α1 α 2 α1+ α 2 + 3 α1+ α 2 − 3 α = πr α = 2πr eff α α eff α α 1− 1 2 1− 1 2 4π 2r 6 16π 2r 6 Resonance shift effects – two resonant particles 2 10 ω p ε =ε = − 7.5 2 metal spheres: 1 2 1 2 ε′ (drude term) ω + iγω 5 γ = 2.0 2.5 ω 0.2 0.40.6 0.8 1 1.2 1.4 -2.5 α -5 ε ′′ 1 2a -7.5 -10 z = 0 z E r z = 0.3a in αeff (a.u.)80 60 z = 3a α 2 40 z α α 1 2 20 α1+ α 2 + 3 α = πr eff α α ω 1− 1 2 0.2 0.4 0.6 0.8 1 1.2 1.4 π 2 6 4 r Arg()αeff ()rad 3 2.5 2 dipolar approximation predicts 1.5 • resonance shifts 1 ω ω p ∝α 0.5 • field enhancement ( E sca eff) Ein ω 0.2 0.4 0.6 0.8 1 1.21.4 Modes energy shift due to coupling Sphere Coupled dimer Sphere + ω1 _ + + a a + _ _ + l=2 _ l=1 + _ + Prodan et al, Science 302, 419 (2003) _ l Nordlander et al, ωωlp= Nanoletters 4, 899 (2004) 21l + − ω1 Schmeits and Dambly, Phys. Rev. B 44, 12706 (1991) Coupled modes in a metallic dimer Schmeits and Dambly, Phys. Rev. B 44, 12706 (1991) The gap plasmon: a distorted dipole -Q Q - - + + Isolated rod: dipole - + - - + + ~ L -Q Q -Q Q Weak coupling: - - + + - + + - + -- + two dipoles - - + + S - - + + ~ L ~ L -Q Q -Q Q - - Strong coupling: ++ - + + - + -- + distorted dipoles - - ++ S - + + < L < L Spectral complexity: higher-order modes Gold spherical dimers (a=60nm) Closely located spheres - + - + - + - + Romero, et al. • The dipole peak redshifts when closer Optics Express • New multipolar peaks appear. 14, 9988 (2006) Gap antenna: localization and field-enhancement Hot-sites Single antenna Gap-antenna Plasmonic cavity-molecule coupling Molecule cavity It is useful to write these equations using the notation g defined to be the same as in the quantum formulation!!! One obtains: Plasmonic cavity-molecule coupling Molecule cavity Briefly back to the time domain This is equivalent to an harmonic oscillator system k1 kg k2 m1 m2 Dynamics of two coupled mechanical oscillators Dynamics of two coupled mechanical oscillators Tuning of the frequency of the mechanical resonator frequency Losses Resonant k1 k1 Without external illumination Molecule cavity ≈Eigenvalue problem ( )( ) = 0 Without external illumination Molecule cavity ≈Eigenvalue problem Eigenvalues Molecule cavity Energy of the modes Losses Zero detuning Molecule cavity Energy of the modes Losses Two different situations Square root is imaginary Weak coupling Energy of the modes does not change, only the losses (decay) are modified Regimes of light-matter coupling Weak coupling regime Modified EM mode density modify radiative decay Zero detuning Molecule cavity Energy of the modes Losses Two different situations Square root is imaginary Weak coupling Energy of the modes does not change, only the losses (decay) are modified Square root is real Strong coupling Two new modes at new energies -> polaritons Regimes of light-matter coupling Strong coupling regime Interaction faster than decay Hybrid light-matter states (polaritons / dressed states) vacuum Rabi oscillations (coherent energy exchange) Zero detuning Molecule cavity Energy of the modes Losses Two different situations Square root is imaginary Weak coupling Energy of the modes does not change, only the losses (decay) are modified Square root is real Strong coupling Two new modes at new energies -> polaritons Note: The equation is the mathematical condition to obtain splitting of the modes. However, for practical purposes, it might be better to impose that the splitting/coupling is sufficiently larger than the sum of the losses (see Törmä and Barnes, Rep. Prog. Phys. 78 013901 (2015)) Strong coupling in frequency: anticrossing Plasmon population The energies never cross Notice also the evolution of the linewidth J. Bellesa et al, PRB 80, 033303 (2009) Strong coupling in time: Rabi Oscillations Energy at zero detuning P. Vasa et al, Nature Photonics 7, 128–132 (2013) The oscillations can be seen as the beating between the two different frequencies The Rabi frequency is thus Progressive emergence of strong coupling Extinction ) ) 1100 nm g=0.025 g=0.05 g=0.1 g=0.15 ( κ κ κ κ wavelength 900 Cavity ) ) 1100 nm g=0.2 g=0.25 g=0.3 g=0.35 ( κ κ κ κ wavelength 900 Cavity ) ) 1100 nm g=0.4 g=0.5 g=0.75 g=1 ( κ κ κ κ wavelength 900 Cavity 900 1100 900 1100 900 1100 900 1100 Excitation wavelength(nm) Excitation wavelength(nm) Excitation wavelength(nm) Excitation wavelength(nm) Time domain is (can be) the key to identify Strong Coupling Regimes of light-matter coupling Weak coupling regime Modified EM mode density modify radiative decay Weak coupling regime: Purcell Factor The cavity acts as a new decay channel for the emitter Molecule Cavity Decay of the molecule into the cavity Decomposing the molecular losses into spontaneous decay and other losses Purcell Factor The Purcell Factor does not depend on the emitter ( and has the same dependence) Weak coupling regime: Fano resonances ‘A’ complex Lamb shift Purcell Factor Weak coupling regime: Fano resonances Lamb shift Purcell Factor Example of application of this classical description: Controlling single molecule coupling in a plasmonic cavity Experiment Theory Dnorm Y. Zhang et al. Nature Communications. 8, 15225 (2017) Strong Coupling Quantum treatment Hamiltonian Plasmon Molecule Plasmon Molecule Coupling illumination illumination Coupling between one (or more) localized single cavity mode and a single photon emitter The losses are introduced via Lindblad operators and the density matrix Density Matrix Master Equation Incoherent Losses Strong coupling Focusing on the coupling and ignoring losses Looking for eigenvalues and eigenvectors of this lossless Hamiltonian In matrix form Strong coupling Considering n=1 And making the (non-rigorous) transformation We recover Estimation of the parameters in the Hamiltonian for the single plasmonic resonator With effective volume So that The volume can be calculated in a similar manner as for dielectric cavities, but • Different contributions of electric and magnetic energy • (R. Esteban et al., New J. of Physics 16, 013052, 2014) • Need to subtract the radiative radiation (A. F. Koenderink, Optics Letter 35, 4208, 2010) Estimation of coupling strengths for a typical single plasmonic resonance Laser-molecule coupling Cavity-molecule coupling Molecule Cavity The key why we use plasmons Laser-cavity coupling Quantum versus classical The Jaynes-Cummings Ladder Uncoupled Coupled system Uncoupled molecule cavity Ground State The state-dependent splitting introduces: • Non-Linearities • Non -classical correlations Acknowledgements -Rubén Esteban - Francisco J. García Vidal Annex: Some thoughts There is significant discussion in the literature about when it is justified to model the plasmon as a single mode as presented here. This approach assumes that a single Lorentzian-like mode dominates the response. This can be valid to a large degree for well chosen systems, but not always. Approaches to overcome this limitations include • Quasi-normal modes (groups of Hughes, Lalanne, Giessen… ) • Formulations based on the Green’s function (for example, H.
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