Metal Hybrid

Perspective pubs.acs.org/journal/apchd5

Strong Light−Matter Interaction in Quantum Emitter/Metal Hybrid Nanostructures Parinda Vasa*,† and Christoph Lienau*,‡ †

Department of Physics, Indian Institute of Technology Bombay, 400076 Mumbai, India Institut für Physik and Center of Interface Science, Carl von Ossietzky Universität Oldenburg, D-26111 Oldenburg, Germany

‡

ABSTRACT: Surface plasmon polaritons (SPPs) are spatially confined electromagnetic field modes at a metal-dielectric interface capable of generating intense near-field optical forces on ultrafast time scales. Within the field of photonics, SPPs carry significant potential for guiding and manipulating light on the nanoscale. The intense SPP fields substantially enhance light−matter interactions with quantum emitters (QEs). Thus, hybrid systems comprised of SPP resonators and various types of QEs constitute key components of the modern photonics applications. Recent advances in nanotechnology have enabled fabrication of high quality QE/ metal hybrid nanostructures, in which several aspects of light−matter interactions, including those in the quantum regime have been demonstrated and extensively studied. The present Perspective explores the central phenomenon in light−matter interaction in emitter/metal hybrid nanostructures, namely, the strong dipole coupling between QEs and SPPs, particularly between excitons (Xs) and SPPs. We provide a concise description of the relevant background physics and discuss the dynamics of the coupled QE−SPP modes. We also review the extent to which the strong QE−SPP coupling has been enhanced to reach the challenging but fascinating fundamental quantum mechanical limit of a single QE coupled to a single SPP mode. Studies have demonstrated that these remarkable hybrid nanostructures supporting single QE−SPP coupled mode can potentially open up diverse exciting possibilities like single-molecule sensing, nanoscale light sources, single-photon emitters, and all-optical transistors. KEYWORDS: surface plasmon polaritons, light−matter interaction, weak dipole coupling, strong dipole coupling, quantum emitters, metal nanostructures fields on nanoscale but cannot easily function as a source or a switch. The nonlinear effects of purely metallic nanostructures are rather weak and very high intensities of the order of TW/ cm2 are usually needed to generate an appreciable nonlinear response.20 On the other hand, quantum emitters (QEs) like atoms, molecules and excitons (Xs) in semiconductors have intrinsically large nonlinearities and optical gain but are highly localized in nature, unable to transport information.12,21−28 This functionality of excitonic systems is being extensively used in electronics and active photonics technologies to realize current optoelectronic devices.29 However, their applications are currently limited by either the slower speed of purely electronic circuits or relatively large mode size of the photons.11 The ability of SPPs to guide light4,5 and localizing it to nanoscale dimensions, as in the hot spot of a plasmonic nanoresonator,30,31 in the gap between two nanoparticles32,33 or at the tip of an antenna34 could result in a strong field enhancement.4−6 Thus, enabling efficient intensification of

S

urface plasmon polaritons (SPPs) are hybrid modes of light waves coupled to free electron oscillations in a metal that can be laterally confined below the diffraction limit.1−3 Metals, owing to their large negative dielectric constant exhibit unique optical properties allowing SPP field localization on the scale of the skin depth (Figure 1), which remains submicron throughout much of the visible and near-infrared spectral range. The spatial extent of the SPP fields is dictated by the geometry of the metallic nanostructures rather than by the wavelength of the light.4−6 By choosing appropriate geometries, passive plasmonic systems such as a gap,5 metallic nanowires,7 or chains of metallic nanoparticles,8 can be formed. These can act as novel SPP waveguides, which allow for guiding light over distances in one- or two-dimensional nanostructures,9,10 much smaller than those that can be achieved in diffraction limited photonic structures made of purely dielectric waveguides and resonators.11,12 The ultimate localization length of SPP fields is limited by the nonlocal response of the metal electrons and the onset of quantum tunneling to ∼1 nm.9,10,13−15 In spite of these advantages of being able to control light on the scale similar to present day electronic devices, the applications of plasmonics are limited in modern photonic devices due to high Ohmic losses16−18 and inability of SPPs to add functionalities like light emission and control. Like photons, SPPs are also bosonic particles,19 which can rapidly transport electromagnetic © 2017 American Chemical Society

Special Issue: Strong Coupling of Molecules to Cavities Received: Revised: Accepted: Published: 2

June 20, 2017 September 14, 2017 October 9, 2017 October 9, 2017 DOI: 10.1021/acsphotonics.7b00650 ACS Photonics 2018, 5, 2−23

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reported coupling strengths of as high as ∼650 meV in metallic nanostructures covered with photochromic molecules coated metal nanostructures (Figure 2D),52 and even higher coupling strengths have been reached in other systems.93,94 Therefore, it is possible to observe a strong coupling between light and matter in this class of hybrid structures, even at room temperature without any need for a closed cavity. One of the central goals for these systems is to control light via light by efficiently creating nonclassical hybridized states of light and matter.41,95−98 Such a control offers advantages of better than THz speed52,93,94 and nanoscale integration3−6 not attainable by other existing technologies like electronics, active photonics, or plasmonics. Fundamentally different types of all-optical devices like efficient and directional single-photon sources,31,64,69−71 quantum gates and single-photon transistors,97−100 based on coupled emitter−SPP modes have been already envisaged. Another related unexplored facet is the ability of strong coupling to modify the electromagnetic environment of an emitter, and in turn the potential energy landscapes.101−108 Such a modification may lead to controlling chemical reactions and interactions in atomic condensates.109−111 Coupling to SPPs may also extend the energy transfer range between two emitters, which otherwise may be restricted to just a few nm.112 So far, there is only a limited number of experimental studies on the role played by the local plasmonic environment or the local plasmonic density of states on such resonant energy transfers,113 and these studies are limited to the weak coupling regime. Theoretical studies62,114,115 suggest that it will be of interest to explore how this resonant energy transfer is modified as the coupling is increased into the strong coupling regime. Thus, studying how quantum emitters and SPPs interact may lead to key advances in multiple directions and will certainly unravel several new phenomena. The progress in this field is also being driven by the rapid advances in nanoscience and nanotechnology. Recent developments in fabrication techniques to create very high-quality QE/metal hybrid nanostructures have prepared the stage well for exploring these fascinating quantum phenomena. As with most emerging research areas, there are still many difficult questions that must be answered. For example, it will be important to establish whether sufficiently high field enhancement, reasonable coherence times and control of the optical response are achievable in this class of hybrid nanostructures. Also, we will need to learn how to design structures with optimized properties, which is challenging due to the complex optical properties of these nanoscale entities. So far, we have provided a brief overview of plasmonics with a focus on aspects that are important in the context of strong emitter−SPP coupling. A number of excellent more extensive texts on plasmonics and its applications are available in the literature.116−127 In this Perspective, we broadly discuss topics related to the strong dipole light−matter interaction in QE/ metal hybrid nanostructures (Figure 2).41,52,53,59,62,64,91 We will focus on just five specific topics: (i) QE−SPP interactions in the weak coupling regime; (ii) QE−SPP interactions in the strong coupling regime; (iii) ultrafast hybrid mode dynamics; (iv) theoretical modeling, and (v) emerging applications of strongly coupled hybrid nanostructures. Many other topics, such as active plasmonic devices, molecular plasmonics for biology and nanomedicine,128,129 plasmonics for photovoltaics and solar cells,118,124−126 and SPPs in magneto-optical materials,130−132 are not included. Also, we will not review

Figure 1. Light−matter interaction in a quantum emitter (QE)/metal hybrid nanostructure. The surface plasmon polariton field (E⃗ SPP) is confined to the metal−dielectric interface. The plasmonic resonator (|SPP⟩) and one or several quantum emitters (|QE⟩), for example, excitons are coupled to the vacuum continuum (|V⟩) via dephasing rates γQE and γSPP, respectively. The dipole interaction between the QE and the SPP vacuum field results in a periodic exchange of energy at twice the frequency known as the vacuum Rabi frequency, ΩR = μESPP. Here, μ represents the excitonic transition dipole moment. Two 2 γQE

γ2

, it is known as the coupling regimes are identified: if Ω2R ≤ 2 + SPP 2 weak coupling or Purcell regime, whereas higher Ω2R corresponds to the strong coupling regime. This coupling regime is characterized by the formation of coupled modes, lower (|LP⟩) and upper polaritons (| UP⟩). In realistic situation, radiative and nonradiative damping as well as the pure dephasing processes play an important role for the loss of phase coherence between QEs and SPPs. Accordingly, the damping rate here refers to the total dephasing.

light−matter interaction with QEs, switching with a very small number of photons and overall reduction of the size of a photonics device to match that of an electronic device. During the past decade, there has been an upsurge of new interest in the field dealing with the emission and all-optical control of SPPs.35−39 The focus has been an active plasmonic system, schematically depicted in Figure 1, in which a plasmonic nanostructure is coupled to one or several QEs.40−69 As summarized in Figure 2, the light−matter interaction in such nanostructures is substantially modified by the coupling of transition dipole moments of QEs to local SPP fields.40,41 Since plasmonic fields can be confined to deep subwavelength dimensions,4−10 the resulting local SPP fields and their vacuum fluctuations can be very strong (Figure 2A).41 These vacuum field fluctuations will largely affect the optical response of the QE,40−69 now enabling even a strong coupling between a single QE and a single SPP mode at room temperature (Figure 2C).59 Such hybrid systems connect two fundamentally different quantum worlds of electromagnetic fields (high transportability and low nonlinearity) and QEs (low transportability and high nonlinearity) and open up new opportunities for exploring intriguing physical effects like Purcell enhancement,31,63−71 surface-enhanced Raman scattering,72−76 vacuum Rabi splitting,40−61 optical Stark effect,77−79 polariton lasing,80,81 and condensation.82−84 Lasing on nanoscale in hybrid semiconductor/metal structures has already been demonstrated.81,85−92 Recently, Schwartz et al. have 3

DOI: 10.1021/acsphotonics.7b00650 ACS Photonics 2018, 5, 2−23

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Figure 2. (A) Dipole coupling between a QE and SPP can strongly affect the light−matter interaction in the hybrid nanostructures.41 In this Perspective, we broadly discuss topics related to the (B) change in dynamics arising from the weak QE−SPP coupling;64 (C, D) QE−SPP interactions in the strong coupling regime including interaction with a single QE59 and ultrastrong coupling;52 (E) ultrafast dynamics of the hybridized modes in the strong coupling regime;53 (F) some aspects of theoretical modeling,62 and (G) emerging applications of coupled hybrid nanostructures.91 Images reprinted with permission from the references as indicated: (A) Copyright 2013 Nature Publishing Group, (B) Copyright 2016 American Chemical Society, (C) Copyright 2016 Nature Publishing Group, (D) Copyright 2011 American Physical Society, (E) Copyright 2013 Nature Publishing Group, (F) Copyright 2011 American Physical Society, (G) Copyright 2017 Nature Publishing Group.

continuum is of crucial importance for the nonlinear optical properties of hybrid systems, discussed in the later part of this article. According to the Wigner-Weisskopf model, the exponential relaxation of the emitter can be written as an integral over the interaction between the emitter mode and each of the vacuum modes as given in eq 1

techniques to fabricate and characterize the hybrid nanostructures because extensive texts and several comprehensive review articles already exist on these subjects.4,5,116,133 Even for the topics discussed here, there are, by necessity, some omissions. Strong coupling in ensembles, for example, itself is a vast topic and all the reports will not be included. We have made selections based on novelty and relevance to other topics discussed here. For all topics, we will provide a brief introduction, describe a few representative efforts as well as state-of-the-art experiments and discuss challenges before outlining the future directions.

C ̇ (t ) ∝

∫ μ2 ω3dω ∫ e−i(ω−ω )(t−t′)C(t′)dt′ ∝ −ΓC(t ) 0

(1)

Here, Γ is the damping rate, μ is the transition dipole moment, and C(t) represents the excited state occupation amplitude dynamics.135,136 Due to the averaging over the continuum of vacuum states, the phase memory is lost and the spontaneous emission is an irreversible process. The energy once released to the environment cannot flow back to the emitter in a coherent fashion.135,136 If this emitter is now placed within a lossless resonator having the same characteristic resonance frequency ω0, the situation can again be described as a two-level system coupled to an oscillator. Now, with the resonator mode acting as a second, undamped oscillator. The coupling between the emitter and the electric field of the resonator mode, including its vacuum fluctuations, as described by the Jaynes-Cummings model results in single photon being emitted into the resonator mode and absorbed by the emitter.134−139 Based on this model, the probabilities of finding an emitter in the excited state (e) and the ground state (g) are described as

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BASIC ASPECTS OF THE QUANTUM EMITTER DYNAMICS An emitter, for example, an atom or an exciton making a transition from an excited state to a ground state emits a photon with a characteristic frequency ω0 into the continuum of the vacuum radiation modes. In free space, that is, in the absence of an external resonator, this irreversible process is the most fundamental light−matter interaction occurring due to the coupling of an emitter to the electromagnetic vacuum field.134−138 The idea of spontaneous emission goes back to Albert Einstein when he studied Planck’s blackbody spectrum using the principle of detailed balance. Consequently, the rate of spontaneous emission is still known as the “Einstein A coefficient”. Victor Weisskopf presented a method for analyzing this interesting problem of spontaneous emission together with Eugene Wigner based on the interaction between a single emitter and a continuum of vacuum modes in the environment. The problem is analogous to the interaction between a quantum-mechanical two-level system, the emitter, and a continuum of quantum-mechanical harmonic oscillator modes. The difference between the Fermionic character of the two-level system and the bosonic character of the

Pe(t ) = cos2(Ω R n + 1 t ) 2

Pg(t ) = sin (Ω R n + 1 t )

and (2)

Thus, if the resonator mode is initially in nth eigenstate (a state of precise photon number n), then there are times [corresponding to cos2(Ω R n + 1 t ) = 1] when the emitter 4


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Table 1. Weak and Strong Coupling Regimes of Light−Matter Interaction in QE/Metal Hybrid Nanostructures

information. This reversible exchange of the energy is a characteristic of the interaction with a single resonator mode. In contrast, in free-space spontaneous emission, although there exists a finite probability to return to the original excited state for an individual mode, the summation over the continuum of modes results into totally destructive interference among the probability amplitudes. For a single emitter coupled to a single resonator mode, the Rabi frequency increases with the number

will be found in the excited state by absorbing a photon with certainty. Similarly, there are instances when the emitter will be in the ground state by emitting a photon into the resonator mode. For n = 0, this periodic exchange of energy between the emitter and the resonator mode occurs at twice the frequency known as the vacuum Rabi frequency given by ΩR = μE, where E is the strength of the vacuum electric field fluctuations within the resonator.140,141 As in this case, there is only one resonator mode involved, the energy can flow back and forth between the emitter and the resonator mode without any loss of the phase

of photons present in the mode as Ω R n + 1 . These states 5


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with n > 0 are known as the higher rungs of Jaynes-Cummings ladder.134−139 As the Rabi frequency scales linearly with field, the field localization within a smaller volume as generated by SPP modes enhances the interaction and increases ΩR. If N emitters are present in a resonator supporting multiple coherent modes, the system essentially behaves as a system of coupled classical oscillators,142,143 as shown by Michael Tavis and Frederick Cummings.144 Now, the coupling between emitters and SPPs results in a normal mode splitting with an effective Rabi frequency that is given by ΩR = √N√MμE, where M represents the number of field modes. In this limit, the coupling strength no longer increases with the occupation of the SPP mode (n). This fundamental light−matter interaction between emitters and field modes is reversible as along as the system is completely isolated from the surrounding. In reality, like the emitter, the resonator also has a finite photon lifetime, governed by its line width (or the number of modes), which will limit the periodic exchange of energy between the two oscillators by allowing it to flow irreversibly into the vacuum continuum. The emitter and the resonator are said to be weakly coupled if ΩR is lower compared to the rate at which the energy is transferred to the continuum by either the emitter or the resonator, that is the line widths of the individual uncoupled oscillators. If ΩR is higher or comparable to the line widths, than the emitter and the resonator are said to be strongly coupled. Apart from the dephasing induced by radiative damping of QEs and SPPs, other mechanisms such as pure dephasing processes reflecting energy fluctuations due the coupling of the QE to the environment, nonradiative energy relaxation processes, scattering processes or inhomogeneous broadening in ensembles may also contribute to the spectral width.16,40,47,50 For the system to be in the strong coupling regime, ΩR has to be larger compared to the total dephasing rate. In particular, for resonant QE and SPP modes, the strong coupling regime is γ2

important to briefly introduce some of the challenges that are associated with reaching the strong coupling regime in hybrid plasmonic nanostructures. As depicted, in Figure 1, the dominant QE-field interaction in these hybrid structures comes from the coupling of a single or an ensemble of QE to a locally enhanced SPP field at a metal−dielectric interface.40−69 Since SPP fields are short-range evanescent fields, this interaction can only be strong if the separation between QE and interface is small, typically few tens of nm. Both the QE and the plasmonic resonator are radiatively coupled to the continuum of vacuum states via their individual dephasing rates γQE and γSPP, respectively. While typical radiative lifetimes of QE are comparatively long (in the ps− ns range),50,64,147 plasmonic resonators usually exhibit relatively short radiative damping times in the 10−100 fs range16,148 and, thus, broad line widths compared to microcavities and photonic crystals. In presence of such an enhanced damping, the strong coupling regime is not easily attainable. Nevertheless, it is still observed in plasmonic systems coupled to Xs due to the intense field localization as well as large dipole moment. Thus, high local vacuum field amplitude and comparatively large excitonic transition dipole moments of excitonic QEs are essential.56−60 The coupling strength can be further increased by using high concentrations of the emitters42−55 and by enhancing the field strength, for example, by mode volume reduction.31,58−61,70,71

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EMITTER−SPP INTERACTION IN THE WEAK COUPLING REGIME The linear optical properties of a QE/metal hybrid nanostructure exhibiting dipolar coupling between an ensemble of QEs and SPP fields are well explained as those of two coupled damped harmonic oscillators representing the QEs and SPP system, respectively. For a hybrid structure comprising an ensemble of emitters and supporting multiple SPP modes, the dipole coupling is given by ΩR = ∫ μeff(r)ESPP(r)d3r. Here, μeff ∝ √N denotes an effective dipole moment density. ESPP(r) gives the average strength of the local vacuum electric field fluctuations of the SPP modes at a position r.53,79,147 Weak coupling results when the dissipation or the dephasing of the individual oscillator overwhelms these coherent Rabi dynamics. The free-space population dissipation rates are proportional to the spectral density of modes per unit volume or the free-space vacuum field strength at the resonance frequency, in absence of other nonradiative relaxation channels. In proximity of a plasmonic structure, there is an enhancement of the vacuum field near the SPP resonance frequency and the mode density can significantly exceed that in the free-space. Like in the case of photons, the vacuum SPP field at the resonance frequency

γ2

+ SPP reached if Ω 2R > QE ,40 that is, if the Rabi splitting 2 2 exceeds the sum of the dephasing rates of QE and SPP mode. The primary characteristic features of these two coupling regimes are summarized in Table 1. The weak coupling regime, also known as the Purcell regime is identified by substantial changes in emitter dynamics but there is only a weak shift in the resonance frequency.11,40,143 In this regime, there is an enhancement of the emitter relaxation rate within a resonator by a factor known as Purcell factor given by

3 4π 2

3

() λ0 n0

Q , V0

where

n0 is the refractive index, Q is the resonator quality factor, and V0 is the mode volume.143,145,146 The Purcell factor reflects the increase in the density of states of a resonator compared to the free space at the resonance wavelength. In the strong coupling regime, however, a quasi-static weak optical probing (with field strength much weaker compared to ESPP) of the dispersion relation of the QE/metal hybrid structure in frequency domain, reveals two distinct resonances (Figure 1).11,40,140−143 The spectra reveal an upper polariton (UP, higher energy) and lower polariton (LP, lower energy) resonance, separated by normal mode splitting (ΩNMS = 2ΩR). These UP and LP resonances correspond to newly formed hybridized QE−SPP modes, which have partly matter and partly field character. The physical effects associated with these two coupling regimes in hybrid nanostructures comprising excitonic QEs and SPPs will be discussed in the following sections. For this, it is

ωSPP associated with each mode is given by

ℏωSPP 2ε0εdVSPP

, where

VSPP is the SPP mode volume and ε0εd is the permittivity of the dielectric medium. A proper definition of the mode volume, for lossy, plasmonic modes is a topic of current research.149−151 For a planar, lossy plasmonic structure, the mode volume can be defined as VSPP = LxLyLz, with LxLy representing the in-plane coherence area of the plasmonic mode.67 The SPP field confinement length of such a localized mode, Lz, along the outof-plane direction leads to the electric field enhancement by a factor

εm εd

, where εm is the dielectric constant of the metal.1,2

Thus, an emitter having a transition frequency matching with the SPP resonance frequency will experience an enhancement 6


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Figure 3. Diverse hybrid nanostructures exhibiting weak QE−SPP interaction. Plasmonic resonators with reduced cavity volume like metal (A) YagiUda antenna,12 (B) antenna with corrugations,69 (C) tip,155 (D) quadrumer,156 (E) plasmonic patch-antenna,64 and (F) nanoparticles attached with biological molecules157 are used to enhance the directional spontaneous emission or Raman scattering signals. (G) Localized SPP field can also be used to manipulate cold atoms in tailored dipole potentials.110 (H, I) Configurations of a SPASER: plasmonic counterpart of a laser,85,86,88,89,163 in which the conventional resonator is replaced by a plasmonic nanostructure. Due to the reduced SPP mode volume, as compared to photons, a SPASER can be much smaller in size. Images are reprinted with permission from the references as indicated: (A) Copyright 2010 American Association for Advancement of Science, (B) Copyright 2011 American Chemical Society, (C) Copyright 2013 Nature Publishing Group, (D) Copyright 2014 Nature Publishing Group, (E) Copyright 2016 American Chemical Society, (F) Copyright 2008 American Physical Society, (G) Copyright 2011 Nature Publishing Group, (H) Copyright 2005 American Physical Society, (I) Copyright 2009 Nature Publishing Group, and (J) Copyright 2009 Nature Publishing Group.

dots (QDs) or quantum wells (QWs) having narrower line widths exhibit large enhancement compared to dye molecules. However, for semiconductor nanostructures with intrinsically high luminescence efficiency, close to unity, the plasmonic resonators are mainly used to achieve directional emission from quantum emitters. Due to the broad line width of the plasmonic resonator, it may not only enhance the relaxation rate from the resonant QE but also enhance the emission from nonresonant electronic states, which can effectively even result in a reduction in the quantum yield. This feature makes it relevant to search for plasmonic resonators with reduced line width, for example, by designing plasmonic resonators with dark modes or by coupling hybrid QE−plasmonic systems to low quality factor dielectric microcavities.154,158 Plasmonic antennas are broadband, strongly scattering nanostructures acting as a coupling link between the propagating and the localized near-fields.31,70,71 They are the counterparts of conventional radio and microwave antennas but operate in the visible regime and provide an effective mechanism to couple light in and out of nanoscale assemblies or even single QE. Many obstacles stand in the way of turning a single QE into a bright, fast, single-photon source. QEs are point like dipoles, emitting isotropically in the far-field. Due to their small size compared to the light wavelength, efficient, directional in- and out-coupling is a challenging task. Also, the emitter decay rates are typically slow (nanosecond to several picosecond time scales), fixed through their electronic structure. Hence, irrespective of the type of QEs, ranging from single atom to QD, near-unity efficiency, on-demand single-photon source emitting in a desired collection channel

to its spontaneous decay rate by the Purcell factor. More interestingly, as the enhancement comes about from coupling to the vacuum field fluctuations of localized SPP resonator mode, the spontaneous emission is also directional, which otherwise is not achievable by simply increasing the emitter density and/or the excitation density. Applications of Purcell enhancement or Purcell effect are based on this dual benefit: enhancement of spontaneous emission rate by increasing the local mode density and directional out-coupling via the resonator modes. For the spectral regions, which do not overlap with the SPP resonances, the mode density could in principle be even below that in freespace, effectively suppressing the spontaneous emission. This might occur near the destructive interference dips of Fano resonances,148,152−154 or in dark plasmonic modes that are very weakly coupled to the far-field radiation. Purcell enhancements of spontaneous emission processes have been extensively investigated in microcavities and in photonic crystals.11,143 They have also been reported in plasmonics structures for various types of QEs: atoms, molecules, dye molecules and semiconductor nanostructures.65−71 Figure 3 provides an overview of the diverse configurations involving localized as well as propagating SPPs used to explore the weak coupling regime. For designing plasmonic resonators to observe Purcell effects, a small cavity volume as in case of metallic nanoantenna (Figures 3A,B),12,69 tips (Figure 3C),76,155 and nanostructures (Figure 3D− F),64,156,157 is important because the enhancement driven by improving the resonator quality factor is limited by the spectral width of the emitter. Also, QEs like semiconductor quantum 7


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∼λ2/400 was also observed. More recently, a tunable SPASER in the near-IR range with reduction in threshold pumping has also been demonstrated in a liquid-dye-gold-disc-array hybrid nanostructure.90 These demonstrations of subwavelength plasmonic laser action at visible frequencies suggests new sources that may produce coherent light far below the diffraction limit81,85−91 for novel applications including those in life sciences (Figure 2G).91 The Purcell factor in such hybrid nanostructures is ∼10. They are anticipated to have significant impact on the fields of active photonic circuits, biosensing and, possibly even quantum information technology. This substantial enhancement in spontaneous emission rate is a result of strong mode confinement and therefore is intrinsic to SPASERs. Yet, it does not directly affect the lasing threshold since both stimulated and spontaneous emission rate for a given mode are enhanced by the same factor.164 However, these initial studies confirm the ability of SPASERs to downscale the physical size of devices, as well as their optical modes. Yet, this size reduction comes at the price of substantial Ohmic losses in the metal and thus a strong damping of coherent charge oscillations in the metal. These losses have a significant influence on the SPASER characteristics, in particular, their lasing threshold and line width.164 More research is needed to show whether plasmonic nanolasers can significantly advance optoelectronics by reconciling the length scales of electronics and optics. Another physical effect that strongly relies on SPP field enhancement and weak QE−SPP coupling is that of surface enhanced Raman scattering (SERS).72−76,155,156 The exact mechanism of SERS is still a matter of investigation. The existing theories are based on local field enhancement and the formation of charge transfer complexes. Like the Purcell effect, SERS also is driven by dual benefits of plasmonic structures. First, the SPP field enhancement magnifies the intensity of incident light and the smaller mode volume ensures better coupling with the Raman scatterer, increasing the Raman scattering signal. This signal is then further amplified by the improved out-coupling mechanism, yielding much stronger Raman signal. Raman enhancements by factors as large as 1010−1011 have been reported.73,75 SERS may now be considered as a well-established technique, and a wide variety of metal nanostructures supporting SPPs have been proposed or utilized. To further improve the spatial resolution of SERS, a metal tip (Figure 3C) is generally employed as a plasmonic resonator.76,155 The small tip radius (500.71 Considering that their relatively broad resonances result in low Q ≤ 100, the large Purcell factors are due to the small mode volume. Broad resonances, however, have an advantage of large emission tunability and acceptance bandwidth. It is remarkable that within a short span of few years, 2−3 orders of magnitude enhancement in Purcell factors (Figure 3E) of plasmonic nanoantennas has been achieved.64 While near-unity efficiency single photon emission has been achieved with dielectric structures,25 the efficiency of quantum emitters coupled to metallic structures has so far been limited to significantly lower values.64,162 Strongly confined SPP fields also offer promising scenarios for investigating cold atoms in tailored dipole potentials, which could enable a single atom and a single plasmonic interaction (Figure 3G).62,99,109−111 To this end, generation of Bose− Einstein condensates above plasmonic micro- and submicrometre structures has been reported.110 It would be interesting to investigate the influence of more complex, spatially regularly or irregularly shaped electromagnetic fields for trapping atoms. Such potentials could lead to the realization of nanoscale devices for matter-wave optics. The changes in the relaxation dynamics induced by the weak emitter−SPP coupling can also be exploited to achieve stimulated emission of SPPs. A SPASER (surface plasmon amplification by stimulated emission of radiation), as proposed by David Bergman and Mark Stockman is the plasmonic counterpart of a laser,85,86 in which the resonator is replaced by a plasmonic nanostructure. Due to the reduced mode volume of SPP modes as compared to photons, a SPASER can be much smaller in size.85−91 A SPASER-based nanolaser can act as a source of coherent radiative as well as nonradiatve energy. Due to inherently short, few-femtosecond-SPP lifetime, ultrafast SPASERs have also been proposed. Earlier attempts to demonstrate SPASER action relied on overcoming losses in an optically pumped dye film deposited on a silver film in attenuated total refection (ATR) technique (KretschmannRaether prism-coupling; Figure 3H).163 Due to the weaker confinement and the lower gain coefficient of the dye, such hybrid structures showed only a marginal enhancement of the reflectivity on optical pumping. Later, amplified spontaneous emission at a relatively lower threshold pumping has been reported in gold nanoparticles coated by a gain medium (Figure 3I)88 and in CdSe nanowires placed close to a silver film (Figure 3J).89 In case of the latter, a reduction in mode area to 8


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Figure 4. Overview of various emitter/metal hybrid nanostructures exhibiting strong X−SPP coupling. (A) Hybrid nanostructures comprising CdSe nanoparticles and silver film,51 (B) GaAs quantum well (QW) and gold nanogroove-array,47 (C) Like metal nanostructures, graphene can also support SPPs, which can strongly interact with phonons in the surrounding dielectric (SiO2).168 (D−F) J-aggregate or dye-coated grating, nanoantenna array, and hole array, respectively.112,147,169 In (A−C), normal mode splitting is ∼10−60 meV. Due to a larger dipole moment of Xs in molecular aggregates and organic semiconductors compared to inorganic QW or QDs, (D−F) exhibit much larger normal mode splitting (∼100− 450 meV). (G−J) Hybrid nanostructures in which coupling of SPP modes with few excitons or molecules have been observed.56−59 Similar configurations can be optimized to achieve a quantum strong coupling regime of a single emitter coupled to SPP mode at room temperature. Images reprinted with permission from the references as indicated: (A) Copyright 2010 American Chemical Society, (B) Copyright 2008 American Physical Society, (C) Copyright 2014 American Chemical Society, (D) Copyright 2014 American Chemical Society, (E) Copyright 2014 American Physical Society, (F) Copyright 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, and (G) Copyright 2013 American Chemical Society, (H) Copyright 2015 American Physical Society, (I) Copyright 2016 Nature Publishing Group, and (J) Copyright 2016 Nature Publishing Group.

Neglecting damping effects, the energies of these mixed states, which are coherent superpositions of the individual QE (e.g., exciton/X) and SPP states are given by ω UP/LP =

directional emission, nanolasing and SERS. Various other areas like SPP-enhanced light-harvesting, photocatalysis, and photodissociation also benefit from weak coupling, but are not discussed here. Even though advantages and applications of weak emitter−SPP coupling have been well established, a microscopic understanding and theoretical modeling of several physical effects, for example, SERS, are yet to be developed. Also, the fabrication of high quality plasmonic resonators relies on advanced lithography techniques, which can provide sub-10 nm resolution.116,165 Therefore, these techniques so far remain relatively expensive and difficult to implement. A comprehensive theoretical description and the invention of easier and more versatile fabrication techniques would lead to many interesting applications of weak QE−SPP coupling in a wide range of topics.

ωQE + ωSPP

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EMITTER−SPP INTERACTION IN THE STRONG COUPLING REGIME As discussed earlier, the strong coupling regime is reached if the Rabi frequency exceeds the sum of all dephasing rates of the γ2

1

± 2 (ωQE − ωSPP)2 + 4Ω 2R .40,50,53,147 If the optical probe frequency is maintained at the original uncoupled SPP resonance, introduction of an emitter in the proximity of the resonator drastically increases the probe transmission (or reflection).11,97 Thus, strong coupling leads to new absorption bands and, thus, a decrease in probe reflectivity at the UP/LP resonances. Hence, the absorption at the uncoupled resonance energy decreases, and correspondingly the probe reflectivity increases.11,40,43,48,166 In a state-of-the-art hybrid nanostructure supporting strong coupling between a single QE and a single SPP mode,56−60 the resulting extreme sensitivity of the hybrid optical response to the QE’s presence could be used for implementing quantum information processing. However, due to the strong radiative damping of SPPs, and other dephasing mechanisms present in QEs, reaching this fundamental limit is challenging and has only been achieved very recently in quantum dots,58 dye molecules,59 and J-aggregates.60 The role of dephasing in the strong coupling regime is discussed in the following section. So far, most experimental studies have demonstrated strong coupling between an ensemble of QEs and a large number of SPP modes.42−55 Even though this configuration representing the “classical” limit of a normal mode splitting143 is substantially different from the quantum mechanical limit of a single QE-mode coupling, it offers opportunities to explore several fascinating phenomena. Since Xs in molecular aggregates and in organic and inorganic 2

γ2

+ SPP optical excitations, Ω 2R > QE .40 In this regime, a QE can 2 2 coherently and periodically exchange energy with a SPP mode for an appreciable time. This interaction is mediated by vacuum field fluctuations and results in the formation of hybrid exciton/ SPP states even in the absence of any external light field. Weak optical probing of such a strongly coupled system thus reveals two distinct resonances, UP and LP corresponding to hybridized emitter−SPP polariton states (Figures 1 and 2). 9


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theoretical description for the resulting hybrid modes needs to be developed. This regime has been predicted to give rise to intriguing phenomena such as photon blockade, nonclassical or squeezed state generation, emission of correlated photons, superradiant phase transitions, and ultraefficient light emission, all accompanied by a breakdown of the standard master equation. Novel ultrastrong coupling based devices and circuits are expected to play an important role in quantum information processing. Recently, ultrastrong coupling has been observed in superconducting circuits in the GHz regime,181 in inter subband transitions in semiconductor structures in the THz regime,178−180 or in metal−dielectric microcavities in the midIR.182 Also, Rabi splitting energies of more than 1 eV have recently been found in semiconductor microcavities.183 Such high values of the Rabi splitting have interesting, yet unexplored consequences for the coherent dynamics of these coupled system. They imply that the radiative energy is coherently and periodically exchanged between the molecules and SPP field on an extremely fast sub-10 fs time scale, which might find several interesting applications, for example, in ultrafast switching.52 In most of these experiments, the interaction investigated is between SPPs and an electronic transition, but it could also be realized for a specific vibrational transition,184 for instance, to modify the reactivity of a chemical bond.104−107 Thus, it can be seen as being analogous to a catalyst,125,128 which changes the reaction rate by modifying the potential energy landscape. Such a major reorganization of the energy levels induced by SPP vacuum fields can result in distinct modifications of physical and chemical properties of the molecule. Recently, altered chemical landscapes by strongly coupling the governing reaction pathways to the vacuum field of a microcavity have been reported for the same SPI molecule.107 The results obtained indicate that the strongly coupled emitter−SPP system is characterized by a much longer interaction range, orders of magnitude larger than the size of the individual molecule.112 Since the interaction length between two molecules in free-space is only a few nanometers, it may not be so easy to rationalize how the collective interaction among such a large number of molecules affects the energy levels of an individual molecule. Again, the origin lies in the enormous enhancement of the Rabi coupling, scaling with √N, when coherently coupling a large number of molecule to a SPP mode. Such a coupling leads to a delocalization of electronic wave functions over many molecules and, thus, to significant modifications of their energy landscapes.104−108,112,185 In spite of the tremendous progress in this field, many aspects of these long-range interactions in collective systems are still not clear. In particular, the effects of dipolar couplings on multidimensional potential energy surfaces governing the reactivity of larger molecules are largely unknown. It is also expected that other properties of molecules, like ionization potential and the electronic affinity can also be modified by the coupling to vacuum fields.169 The vacuum field tailored physical and chemical material properties thus open up largely unexplored area with high scientific and technological potential, warranting comprehensive investigations. The studies discussed here have been performed on ensembles of emitters in close proximity of an extended plasmonic structure supporting several modes (Figure 4A− F).42−55 The large number of emitters and modes is essential for attaining strong or ultrastrong coupling regimes. Another fundamental challenge in this regard is the interaction between a single emitter and a single SPP mode. This ultimate regime of

semiconductor nanostructures are characterized by large transition dipole moments, strong QE−SPP coupling is generally studied in semiconductor/metal hybrid nanostructures.42−55,167 Accordingly, in several topics discussed in the following sections, QE is represented by an exciton (X). Strong X−SPP interaction has been reported for various types of semiconductor/metal hybrid nanostructures: QDs (Figure 4A,I),51,58 QW (Figure 4B),47 graphene (Figure 4C),168 and organic molecules (Figure 4D− H,J).56,57,59,112,147,169 Like metal nanostructures, graphene nanostructures also support SPPs in mid-infrared spectral range,170,171 which can strongly couple to the optical phonons of the surrounding dielectric material (Figure 4C).168 The earlier reports of strong coupling for SPP systems were reported in a J-aggregate/metal nanostructure.167 In Jaggregated dye molecules, the excitonic states of closely packed dye monomers with parallely aligned transition dipole moments are coherently coupled, resulting in a delocalization of the excitonic wave function across a large number of monomer units.172,173 This exciton delocalization leads to a pronounced red shift as well as narrowing of the excitonic absorption resonance, which are correlated with a substantial increase of the excitonic transition dipole moment exceeding that of the monomer exciton by the square root of the number Ncoh of the coherently coupled monomer excitons. These large transition dipole moments results in a rapid superradiant exciton decay,174,175 and make J-aggregates particularly interesting for exploring strong coupling to plasmonic nanoresonators. Accordingly, normal mode splittings on the scale of 100−450 meV have been observed in J-aggregate/metal and molecule/ metal hybrid nanostructures (Figure 4D,E).48,112,147 Interestingly, strong coupling is also observed in dye molecules with a broad absorption band, such as Rhodamine 6G, if sufficiently high concentrations are used.49 Inorganic semiconductor hybrid nanostructures exhibit relatively lower splitting ∼10−60 meV47,176 but are more photostable compared to organic molecules and can be tuned electronically.177 Initially, the studies were based on hybrid structures comprising Kretschmann-Raether configuration.42,43,49 Later, periodically modulated surfaces such as gratings, arrays of nanogrooves, -slits, and -dots supporting propagating SPPs,44,52,53,112 were used as they offer higher field confinement and are more promising for device applications. Chemically synthesized nanovoids, nanorods, and nanospheres supporting localized SPPs have also been used.33,46,48,55 Some of the organic semiconductors and photochromic dye molecules, like spiropyran (SPI), have unusually large transition dipole moments and can reach exceptionally high Rabi-splitting values, ∼10× higher than in inorganic semiconductor/metal hybrid nanostructures.52,169 Schwartz et al. have demonstrated a normal mode splitting of ∼650 meV,52 the highest value in an QE/metal hybrid nanostructure in a metal nanohole array coated with a polymer layer containing photochromic SPI molecules (Figure 4F).169 They have observed a splitting that is nearly 30% of the molecular transition energy (2.2 eV), suggesting that such systems could potentially go into the regime of ultrastrong coupling. The ultrastrong light−matter interaction is an emerging field, studying materials in which the coherent dipole interaction energy approaches the transition energy of the emitter.52,93,94,178−181 In this regime, the rotating wave approximation breaks down and the optical response is expected to deviate from the well-established quantum electrodynamical Jaynes-Cummings model.134−139 A new 10


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plasmonic wavguide could create efficient two-qubit entanglement. Two effects are important here: First, the radiationless resonant transfer of electromagnetic energy (Forster transfer) between the two QEs. If this is sufficiently strong to overcome the losses of the dissipative waveguide, this will lead to the formation of two new hybrid states (|+⟩ and |−⟩ in Figure 6B).62 Second, a fraction of the radiation that is spontaneously emitted by one QE is coupled into the waveguide and may be reabsorbed by the other QE. This second type of interaction results in a modification of the radiative decay rates of the new hybrid states, which may either by enhanced (super-radiance) or decreased with respect to those of the uncoupled QE.187 These cooperative emission phenomena,188−192 first studied by Robert Dicke,193 are expected if the spontaneous emission of QE is efficiently coupled into waveguides with sufficiently small losses. First signatures of such cooperative emission phenomena have been seen for J-aggregate QE coupled to plasmonic nanoslit resonators.147 So far, however, no experimental evidence for long-distance entanglement has been provided. Potentially even more intriguing are the pronounced modifications of the transport properties of an ensemble of excitons that is strongly coupled to quasi-one-dimensional electromagnetic (either photonic or plasmonic) mode that have been predicted theoretically by Feist et al.194 Also, these phenomena deserve detailed experimental investigation. For this, it would most definitely be helpful to further enhance the strength of QE−plasmon coupling, for example, by reducing the mode volume and damping of the plasmonic mode and by enhancing the strength of the dipole coupling. To this end, ultrastrong coupling between X and localized SPP in a nanoshell has been predicted and enhancement of the coupling strength by approaching sharp silver tips has been suggested.61,191,195 Though configurations like bow-tie antenna (Figure 4I)58 do not show very directional out-coupling, the narrow gap generates very small mode volume and high field confinement, giving rise to large Rabi splitting. While light localization into such small volumes can, in principle, be achieved in the gap between two metallic nanoparticles,9,32,45,56 preventing rapid energy loss is much more challenging and, so far, remained unresolved problems. Radiative damping can be prevented by confining light in nonradiative, optically dark modes.148,152,153,196,197 Preventing rapid nonradiative mode damping requires to minimize the fraction of the mode energy that is stored inside the metal. Potentially, this may be achieved by ingenious engineering of ultrasmall hybrid metal-dielectric nanocavities or by discovering metallic materials with reduced losses at room temperature.18 More work along these directions is certainly needed to overcome the rapid SPP energy loss. Attempts are also being made to make use of an atomic Bose− Einstein condensate to reach this fundamental limit.109,110 Such atomic systems have the advantage of very long coherence times and are composed of relatively pure two-level systems without competing many-body interactions. The recent observation of strong coupling between a single molecular exciton and single plasmonic mode at room temperature,58−60 indeed makes it likely to expect that such challenging experiments may be performed in the near future. Most importantly, these preliminary results on hybrid structures comprising few emitters suggest that it will be highly interesting to explore their transient optical nonlinearities.50,53,79,105 The small number of Xs that is needed to reach strong coupling directly means that an excitation by only a few photons will greatly change the optical spectra.

a vacuum Rabi splitting clearly goes beyond that of two strongly coupled classical oscillators. The single emitter−SPP system is intrinsically highly quantum mechanical in nature and offers opportunities to study, test and implement quantum optical phenomena and theory on comparatively long length scales. The implementation of innovative coherent control schemes,186 could help to explore those phenomena. The observation of vacuum field induced effects like higher rungs of Jaynes-Cummings ladder, revival of Rabi dynamics, electromagnetically induced transparency, creation of extremely longlived dark states that are totally isolated from the environment are some of the characteristic phenomena, which can potentially be investigated in this regime. The biggest hurdle here is to overcome strong SPP radiative damping. Innovative configurations of bow-tie antenna and gap modes between two metal tips have been proposed to sufficiently enhance the SPP vacuum field to boost ΩR from μeV to several meV range. To this end, much effort has been devoted in the past few years to the study of the coupling between one or few quantum emitters and localized surface plasmons (LSP) of single metallic nanoantennas (Figure 4G−J) at room temperature.56−60 Initially, giant (normal mode) Rabi splittings of up to 400 meV have been deduced by Schlather et al. from linear optical spectra for dye-coated gold dimers (Figure 4G),56 but it soon became clear that the interpretation of dips in such spectra requires great care. Due to the large damping of the plasmonic modes, a more detailed line shape analysis is needed to reliably deduce the coupling strength. More recently, convincing evidence for normal mode splitting has been seen in linear extinction and photoluminescence spectra of dye-coated gold nanorod ensembles, single gold nanoprisms (Figure 4H),57 and bow-tie antenna (Figure 4I).58 It is estimated that about 70 excitons are coupled to hot spots on the surface of the nanoprism, whereas in the case of bow-tie antenna, the number approached a single exciton. Very recently, Chikkaraddy et al. and Liu et al. have provided evidence for reaching the fundamental regime of strong single-molecule−SPP coupling in a nanosphere-patch antenna (Figure 4J)59 and silver nanorod,60 hybrid nanostructure, respectively, at room temperature. The strong coupling regime is established based on the observation of a distinct anticrossing in the extinction spectra of hybrid nanostructure that statistically have just one molecule. These linear optical experiments measure the intensity of electric fields that are scattered from the nanoantenna. More information might be gained by directly probing scattering at the electric field level.147 Such measurements could reveal both the amplitude and phase of the scattered fields and thus would allow to fully reconstruct the linear response function of the hybrid antenna. Important differences between the optical response of such single QE hybrid structures and multi-QEemitter systems are expected to be seen in nonlinear experiments, and in particular in coherent, time-resolved spectroscopic observations. Such experiments could uncover the rich physics of the Jaynes-Cummings model by studying single QEs coupled to plasmonic nanoresonators. Several recent theoretical studies explore plasmon-induced entanglement between distant excitons and pronounced modifications of excitonic transport due to strong X−SPP coupling.40 In particular, the recent theoretical studies by the group of Garcia-Vidal have also explored plasmon-induced entanglement between distant QEs.62,114,187 The work shows that the strong coupling of two QEs, spatially separated by several wavelengths, to a single quasi-one-dimensional 11


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ultrafast pump−probe studies, directly measuring the femtosecond lifetimes of the polariton modes.147 The observations suggest that the optical properties of the hybrid systems are largely influenced by the interplay between the coherent as well as the incoherent energy exchange processes between Xs and SPPs.147 These results are somewhat different from what is known from, for example, semiconductor microcavities,200−206 where usually the radiative decay rates of the cavity mode are much longer and pure dephasing and inhomogenous broadening effects are more pronounced. They show that, in hybrid X−SPP systems, radiative damping phenomena have a profound influence not only on the energetics of the coupled system but in particular also on the dynamics of the polariton modes. Despite the very different nature of both elementary excitations, tailoring of radiative damping phenomena may be an important means to control the polariton dynamics. The consequences of these collective damping phenomena are most evident if both subsystems are resonant and have equal radiative damping rates (ΓQE = ΓSPP). In this case, one of the two polariton modes is transformed into a perfectly dark mode and this dark hybrid mode is thus completely decoupled from the environment. It thus leads to a total suppression of spontaneous emission or to a population trapping. Such a dark mode is also characterized by an extremely long coherence time. This is a situation that is analogous to the electromagnetically induced transparency206 but entirely induced by vacuum field fluctuations. Thus, the incoherent coupling provides a possibility to control the polariton dynamics, which is potentially of interest for tailoring quantum interference phenomena in active plasmonics. A further topic that warrants comprehensive investigation is the collective behavior of the QE−SPP polaritons and the role of the disorder in extended emitter/metal hybrid nanostructures. Several experiments investigating the interplay between strong coupling and disorder in semiconductor hybrid nanostructures now indicate that the resulting hybrid exhibits long-range phase coherence of strongly coupled exciton/ plasmon polarition modes.50,53,112,207,208 Such long-range phase coherence of strongly coupled polaritons in molecular aggregate/metal hybrid systems has been demonstrated by Aberra Guebrou et al.208 This experiment shows that the inplane coherence length of the hybrid modes is on the order of few micrometers even though the wave functions of the individual emitters are localized on an extremely small, fewnanometer scale. The dependence of the normal mode splitting on the concentration of emitters, the sample area and the active layer thickness in disordered materials like molecular aggregates and dye molecules are indicative of such collective phenomena.44,49 The coherence of spatially isolated emitters induced by hybridization with SPPs has never been directly evidenced. However, a series of interference experiments probing spatial coherence of the transmitted and emitted light using Young’s double-slit type interferometric measurements have been reported.112,207,208 Studies of spatial coherence in dye/metal and molecular aggregate/metal systems near the transition from the weak to the strong coupling regime indicate direct relationship between coherence length and polariton line shape. These experiments as well as other theoretical results show that a high degree of spatial coherence is attainable for emitter−emitter distances of much more than one optical wavelength are achievable in state-of-the art plasmonic structures. Values of the polariton coherence length of about 1−5 μm are estimated in molecular aggregate/metal hybrid

Preliminary studies of those nonlinearities have been carried out on the ensembles of emitters to demonstrate an optical Stark effect and X−SPP polariton condensation at room temperature. Attempts are now being aimed toward realizing polariton scattering and lasing. Further experimental and theoretical work is certainly needed to fully understand and exploit this complex light−matter interaction.

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QE−SPP POLARITON RESPONSE AND DYNAMICS As we have seen, a QE (frequently an X) placed in a plasmonic resonator is radiatively coupled both to the environment and to the modes of the resonator. We have already discussed the resulting coherent exchange of photon energy due to their resonant interaction. This interaction, being coherent, preserves the phase of the optical polarizations of the material.40,134−139 It leads to the formation of hybridized QE−SPP polaritons and has been widely studied for propagating as well as localized SPPs by observing anticrossing in the polariton dispersion relations. Importantly, this coherent interaction not only affects the energetics but also dynamics of the coupled system.188−190,193,198 For zero QE−SPP detuning (on-resonance condition), the polaritons may be expected to have population relaxation rates that are equal to the average of the relaxation rates of the uncoupled QE, ΓQE, and SPP mode, ΓSPP.147 The coupling between the two resonances can, however, also largely affect the radiative decay of the coupled system.147,189 Either the QE or the SPP may spontaneously emit a photon into the surrounding, which is then reabsorbed by the other emitter. This results in an incoherent exchange of photon energy between the two systems, in contrast to the dipolar Rabi coupling, which is driven by the coherent polarizations of either system. Being due to a spontaneous emission process, this incoherent interaction is also mediated by vacuum field fluctuations and exists even in the absence of any external excitations. Its magnitude is proportional to ΓQEΓSPP and it mainly affects the radiative damping of the coupled modes. The resulting cooperative damping phenomena, known as sub- and super-radiance,192,193 are found to play a dominant role in the optical properties of many strongly coupled systems such as trapped ions, molecular aggregates, QDs, QWs, and SPPs.148,188−190,193,196,198 Both aspects of the dipole coupling are conceptually well understood in the framework of a phenomenological coupled oscillator model,47,147 briefly discussed in the following section in the context of X−SPP coupling. Though emitter−SPP polaritons have attracted considerable interest due to their possible applications and are, accordingly, studied widely, very little is experimentally known about this incoherent interaction channel. At present, the effects of strong coupling on the radiative damping of such hybrid systems have remained essentially unexplored. Quite recently, a few experimental studies have investigated the relation between strong coupling and the radiative damping of the X−SPP polaritons in J-aggregate/metal hybrid nanostructures.50,53,79,105,147,199 A series of experiments, both in the spectral and temporal domains, have revealed distinctly different damping rates for the upper and lower X−SPP polaritons under resonant excitation conditions. By analyzing the line shape of angle-resolved reflectivity spectra of a Jaggregated dye film coupled to a plasmonic nanoslit grating, we concluded that the lifetime of the LP mode exceeds that of the UP mode due the formation of sub- and super-radiant polariton modes. This difference in lifetime has been confirmed by 12


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The fundamental distinguishing feature of the strong coupling, the periodic ultrafast energy transfer between emitters and SPPs could be a crucial microscopic mechanism for an efficient and all-optical control of light on the nanoscale. Vasa et al. have presented the first real-time observation of ultrafast Rabi oscillations in a molecular aggregate/metal nanostructure, verifying the coherent energy transfer directly in the time domain.53 We have investigated the transient evolution of the polariton response under resonant as well as off-resonant pumping.50,53,79,214 As depicted in Figure 5A, resonant optical

nanostructures comprising J-aggregated, dye-coated, nanoslit arrays.53,112 The estimates are based on an average X−SPP lifetime of 30 fs and an average X−SPP polariton group velocity of one-third that of the speed of light in vacuum. Broadband spectral interferometry experiments performed to analyze the X−SPP polariton response at the field level have been reported, together with a detailed line shape analysis in ref 147. This study indicates that even in a disordered J-aggregated dye sample, X−SPP polaritons that result from the strong coupling of ensemble of excitons to the SPP mode of a gold nanoslit grating are predominantly homogeneously broadened at room temperature. The results suggest that the coupling of the inplane SPP wave function leads to a polariton mode that is delocalized over many exciton sites. The delocalized wave function effectively averages over the disordered potential, and this may reduce the inhomogeneous contribution in the polariton line widths and lead to an increased exciton conductance in the hybrid structure.193,194 Such a transition from inhomogeneous to homogeneously broadened line shape has also been observed in semiconductor quantum well microcavities and has been shown to be related to the motional narrowing or the averaging performed by the cavity mode.209−211 Further investigations using near-field techniques and two-dimensional spectroscopy are likely to provide much more direct insight into the interplay between the long-range polariton coherence and disorder in these active plasmonic structures. So far, we have discussed the polariton response in the linear regime. For applications, however, the nonlinear response is more important as it can add functionalities like ultrafast and all-optical switching or polariton lasing to the otherwise linear response of SPPs. Due to the fast polariton response times and short Rabi oscillation period, both in 10−50 fs time scale, investigations of nonlinear polariton response are challenging. Early experiments have probed the changes in the polariton dispersion relation induced by ultrafast optical pumping on a subpicosecond time scale.50,79,106,107,212 In these experiments, the nonresonant ultrafast strong optical pump mainly saturated the exciton density in the active layer, in turn reducing the Rabi frequency. The transient Rabi frequency, proportional to the square root of the number of Xs in the ground state is given by Ω R (t ) = Ω 0R N0(t ) − N1(t ) , where Ω0R is the Rabi frequency in the absence of the pump pulse and N0 and N1 represent the exciton population fraction in the ground and excited state, respectively.50,79,212 Controlled and reversible switching from strong (Ω0R = 55 meV) to weak (ΩR(t) = 0 meV) coupling on a subpicosecod time scale has been directly evidenced by mapping the transient dispersion relations of the coupled excitations. The expected faster polariton relaxation dynamics compared to uncoupled X dynamics has also been reported.50,79,212,213 In contrast, Schwartz et al. have reported longer relaxation times for LP mode in J-aggregate/metallic planar cavity hybrid structure under nonresonant excitation,199 which could be related to motional narrowing, 209−211 incoherent population transfer from other states, or dark mode formation.202 Further experiments are needed to investigate the role of various mechanisms in determining the polariton dynamics. All these experiments have been carried out using nonresonant pump pulses or an incoherent excitation. For a more direct analysis of the nonlinear polariton response and polariton damping, experiments under resonant excitation conditions are needed.

Figure 5. (A) On-resonance simultaneous excitation of UP and LP modes by an ultrafast, broadband pump pulse initiates a periodic energy transfer between X and SPP, resulting in out-of-phase oscillations in their number density. The time evolution of the differential reflectivity (ΔR/R) at the LP resonance clearly reveals a periodic change in SPP population on sub-40 fs time scale associated with the real-time observation of Rabi oscillations.53 (B) Red-shifted, off-resonant pump pulse transiently shifts the polariton levels. The pump induced optical Stark effect generates large, fully coherent transient changes in probe reflectivity on ultrafast time scale.79 (C) Schematic of an ultrafast two-dimensional (2D) map corresponding to the coupled X−SPP system. 2D spectroscopy permits real-time mapping of density matrix elements.218−224 Image reprinted with permission from the references as indicated: (A) Copyright 2013 Nature Publishing Group.

pumping leads to oscillatory modulations of the reflectivity of the coupled structure with a period of about 20−30 fs. These modulations directly reflect the Rabi oscillations in the SPP population due to the coherent transfer of energy between SPP excitations of the metallic grating and excitons in the coupled Jaggretate.53 The 10 fs resonant pump laser preferentially excites SPP. This impulsive nonequilibrium excitation induces the population oscillations between SPPs and Xs that are the timedomain signature of the strong coupling. The measurements not only unambiguously establish strong Rabi coupling but also provide important new insight into the optical nonlinearites of these hybrid systems. Surprisingly, the dominant nonlinearity governing the nonlinear response is again found to be the transient reduction in normal mode splitting induced by the saturation of Xs. Under more intense pumping, a coherent manipulation of the coupling energy by controlling the exciton 13


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density on ∼10 fs time scale was also demonstrated. These results can be visualized as an important step toward coherent, all-optical ultrafast plasmonic circuits and devices. Other manifestations of fundamental, coherent nonlinear light−matter interactions like the optical Stark effect (OSE) also has been explored in the strongly coupled molecular aggregates/metal hybrid nanostructures.77−79 The OSE involves transient shifts of energy levels in the presence of a nonresonant light field (Figure 5B).79 The strong X−SPP interaction is found to significantly affect the OSEs. Importantly, it has been shown that the near-resonant coherent excitation of the polariton modes, for example, the LP modes, not only dresses the LP mode, but, by virtue of strong coupling, also transiently Stark shifts the UP mode, even though it is greatly off-resonantly (>200 meV) detuned from the pump. This offers a very interesting approach for coherently controlling and enhancing OSEs in strongly coupled systems by transiently dressing the Rabi frequency.79 It has been predicted that largely enhanced OSE induced shift of up to ∼100 meV could be attained in ultrastrong coupling regime, making hybrid nanostructures interesting, not only for applications but also for exploring the underlying physics. So far, all ultrafast experiments have been performed on large ensembles of Xs and with comparatively high pulse energies in nanojoule regime. Highly efficient ultrafast coherent switching with femtojoule pulse energies may be reached for reduced sample area ≤1 μm2, even without considering plasmonic field enhancement. Even lower switching powers down to a few photons only may in principle be reached in tailor-designed hybrid nanoantennas that contain few quantum emitters and optimize the coupling of far field light into an ultrasmall antenna gap.56−60 Such ultrafast plasmonic switches and/or transistors would constitute entirely new nanophotonic devices that could create fundamentally different opportunities for alloptical communication, computation and, in principle, even quantum information processing. Such devices might operate at power levels and switching efficiencies that to date can only be reached with electronic transistors. At the same time, they might operate at extremely high operation speeds in the Terahertz regime, in principle, only limited by the femtosecond lifetime of coupled hybrid mode. These hybrid exciton/ plasmon devices might therefore combine favorable properties of electronic and all-optical devices and would truly merge plasmonics and quantum electronics. The experiments discussed here suggest that the dominant physical mechanism for the nonlinear polariton response that should be considered when designing such structures is the transient reduction in normal mode splitting induced by the coherently or incoherently generated X population. Yet, it is known from other material systems, specifically inorganic semiconductor microcavities with larger Bohr radii, that other physical mechanisms such as pump-induced Coulomb interactions, changes in the X dephasing rate, alterations of the disorder potential, and polariton thermalization may also contribute to the transient reduction in the normal mode splitting.215−217 Careful intensity dependent studies of the polariton dispersion may provide important additional insight into the effects of such interactions on the polariton nonlinearity. Such studies could also shed light on the other nonlinear interactions like polariton lasing,80,81 condensation,82−84 and scattering.204 It would be interesting to explore the polariton nonlinearity in the fundamental quantum limit of single X and single SPP coupling. The demonstration and ultrafast control of higher rungs of

Jaynes-Cummings ladder and optically induced transparency are examples of nonlinear effects, which could govern the nonlinear response in this regime. We have seen that the dominant signature of the correlated dynamics in a strongly coupled X−SPP hybrid system is a periodic and fully coherent exchange of energy between excitons and plasmons. This energy exchange, however, is difficult to observe in ensemble studies due to two primary challenges: (i) The rapid dephasing of coherent X−SPP polarization on a few-tens of femtosecond times requires advanced ultrafast spectroscopy techniques to probe these dynamics in the time domain. (ii) The presence of other spectral broadening mechanisms, in particular an inhomogeneous broadening,147 of the excitonic transitions, masks the Rabi splitting. An extremely powerful, emerging experimental technique that can address both of these challenges is ultrafast two-dimensional (2D) spectroscopy,218,219 schematically shown in Figure 5C. It allows tracking of spectral changes in the time domain and on a time scale of currently somewhat less than 10 fs; it also offers a possibility to separate contributions from incoherent (inhomogeneous broadening) processes from quantum coherence and permits measurement and imaging of the nondiagonal coupling elements of the Hamiltonian of a coupled system.220−222 Essentially, it is akin to real-time imaging of the density matrix of a hybrid system. So far, there are no reports of two-dimensional spectroscopy or even higher dimensional spectroscopy experiments,223,224 in strongly coupled X−SPP systems, which can track the coherent energy transfer into optically dark states. With the rapid development of 2D ultrafast spectroscopy, it is now expected that such experiments will provide new and detailed insight into to the complex correlations and energy transfer processes in these hybrid nanostructures.

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THEORETICAL MODELING TO UNDERSTAND THE FORMATION AND DYNAMICS OF STRONGLY COUPLED POLARITON MODES Presently, most theoretical descriptions of the optical response of emitter−SPP coupling rely on classical models describing the response of the emitter as a damped harmonic oscillator,40,47,48,147 and that of the metal by a dielectric function given by Drude’s model and its variants.1,2,13 In this classical approach, the interface between the different materials is assumed to be abrupt. The different types of relaxation and dephasing processes enter into the dielectric function of each material. Changes in the dielectric function due to electron tunneling across the interface are not considered. These models can describe the formation of two polariton modes in the strong coupling regime separated by a normal mode splitting. The linear optical response of the composite system may then be calculated using a transfer matrix model or by finite element simulations of Maxwell’s equations.1,2,225 This approach, however, makes it difficult to rationalize the effects of radiative and nonradiative relaxation or dephasing processes on the dielectric functions of the coupled materials. It is easier and physically more transparent to include these processes in quantum mechanical models for the optical response.14,15,226−229 Due to the complexity in treating larger open quantum systems, existing studies are generally based on semiclassical approach and phenomenological models.166,230 Here, the QE, generally an exciton is treated as quantum mechanical entity represented by a two-level system (|X⟩) governed by the Schroedinger equation, whereas the SPP field 14


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(|SPP⟩) is still a classical electromagnetic field. The damping or the coupling to the vacuum states (|SPP⟩) is included phenomenologically. The optical response is obtained either by solving Bloch equations or using the Green’s function approach.231 A model based on thermodynamical approach has also been proposed.232 All these models can satisfactorily describe the linear optical response of several systems but have not yet been used to describe the quantum dynamics of strongly coupled polariton modes in the linear and nonlinear excitation regime. Recently, a phenomenological density matrix formalism53,213 and a macroscopic quantum electrodynamic approach have been introduced,233 which can bridge this gap. Also, theoretical work aimed at providing a quantum description of the plasmonic field is underway.234 Here, we briefly discuss the framework of a phenomenological model based on coupled oscillators to describe the linear response of a strongly coupled X−SPP system.147,235 It is a straightforward method to incorporate some of the quantum mechanical effects within a semiclassical approach and can be easily extended to more advanced Bloch equations formalism. Owing to its simplicity and wide scope, it has been successfully used by several groups to explain their experimental results.40,47,48,147 In the strong coupling regime, that is, for a Rabi frequency ΩR exceeding the relevant relaxation and dephasing rates, the linear optical spectra of the hybrid semiconductor/metal nanostructure are governed by the mixed X−SPP modes formed by the dipole coupling between Xs and SPP fields. In the basis {|10⟩, |01⟩} of two-particle states (X and SPP), we can cast the nonhermitian Hamiltonian in the form of a 2 × 2 matrix ⎡⎛ ω ⎛ Γ′ ΩR ⎞ ΓX‐SPP ⎞⎤ X ⎟ − i⎜⎜ X ⎟⎟⎥ M = ℏ⎢⎜⎜ ⎢⎣⎝ Ω* ω ⎟⎠ ⎥ Γ Γ ⎝ X‐SPP SPP ⎠⎦ R SPP

the respective modes. In this model, the coupling constants and the damping rates have been introduced phenomenologically and their values are obtained by fitting experimental data. Though pure dephasing has been included, the inhomogeneous broadening has not been accounted for. It can be added by convoluting χ(ω) with the inhomogeneous distribution of polarition mode frequencies. Since in coupled X−SPP systems, radiative damping is usually quite pronounced, inhomogeneous broadening may not have a strong effect and eq 4 may satisfactorily describe the polariton optical response. All three, classical, semiclassical, and fully quantum, mechanical approaches lead to the same prediction for the occurrence of hybridized modes and the normal mode splitting in linear optical spectra. In the linear regime, the difference mainly lies in providing appropriate microscopic explanations for the various types of dephasing mechanisms. Yet, these models become fundamentally different as soon as it comes to the description of a nonlinear optical response. By using the Liouville equation,53,79,230 the quantum mechanical model can easily be generalized to fully account for the time evolution of the density matrix of the coupled system. Microscopic models for the various relaxation and dephasing processes can be incorporated through Lindblad equations.136,198 Such models can quantitatively account for yet unobserved phenomena such as higher rungs of the Jaynes-Cummings ladder or revivals of Rabi oscillations. In combination with a quantitative modeling of the plasmonic resonator using finite difference time domain simulations of Maxwell’s equations, such a quantum-dynamical treatment would provide a comprehensive theoretical framework offering microscopic understanding of a wide range of phenomena in active plasmonic systems.

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EMERGING APPLICATIONS AND DEVICES BASED ON STRONGLY COUPLED HYBRID NANOSTRUCTURES Optical nonlinearities play an important role in modern photonic functionalities such as generation and detection of light, creation of ultrashort pulses and implementations of ultrafast all-optical switching or signal processing. As we have seen, plasmonic nanostructures can be used to efficiently tailor electromagnetic interactions. Thus, QE−SPP hybrid excitations have the potential to combine the enhanced optical nonlinearities of QEs with the favorable light manipulation properties of SPPs.36−39,85 The development of suitable QE/ metal hybrid structures presents a key step towards the realization of novel active plasmonic devices, combining the operational bandwidth of photonics with the size scalability of electronics. The term “active plasmonics” was introduced by Krasavin et al. in 2004,35 and since then, a number of other techniques have been demonstrated to optically enhance, modulate, and control the propagation of guided SPPs by incorporating a gain medium or an active component.236,237 In such modulators, the control or the modulation speed is generally limited by the stimulated emission rate of the active medium. Since QE−SPP interactions typically enhance the relaxation rate, coupled QE−SPP systems provide a wider choice of active materials and faster modulation speed. Another important application of strongly coupled emitter− SPP systems lies in developing fast and efficient all-optical switches.238−242 Out of several variants of active plasmonic switches that have been demonstrated so far, the ultrafast coherent nonlinearity of X−SPP polaritons induced by transient Rabi oscillations provides the record switching

(3)

Here, ΓX′ = ΓX + Γnr X denotes the sum of the radiative and nonradiative decay rates of the X state. The off-diagonal elements of the first term in the Hamiltonian represent the reversible coherent X−SPP Rabi coupling whereas those in the second term represent the irreversible incoherent coupling. The complex eigenfrequencies ω̃ UP,LP of the coupled upper (UP) and lower (LP) exciton−SPP modes are then given by the eigenvalues of M as ω̃ UP,LP = 1 (ωX + ωSPP − i(ΓX′ + ΓSPP) ± A) 2

with A = Δ2 + 4CD and Δ = ωX − ωSPP − i(Γ′X − ΓSPP). Here C = ΩR − iΓX‑SPP and D = Ω*R − iΓX‑SPP. In the absence of ω +ω damping the expression reduces to ωUP/LP = X SPP ± 2

(ωX − ωSPP)2 + 4Ω 2R . The UP and LP wave functions are then given as normalized eigenvectors of M: |UP,LP⟩ = ((Δ ± A)|10⟩ + 2D|01⟩)/BUP,LP with a normalization constant BUP,LP = |Δ ± A|2 + 4 |D|2 . Using first-order perturbation theory, the linear susceptibility of the hybrid system is 1 χ (ω) = ε0ℏ

⎛ ⎞ |μi |2 |μi |2 ⎜ ⎟⎟ + ∑⎜ ω̃ − iγi* − ω ωĩ * + iγi* + ω ⎠ i ⎝ i (4)

where the sum extends over the UP and LP oscillators, 1 μUP,LP = B ((Δ ± A)Ω X + 2DΩSPP) denotes the effective UP,LP

dipole moment and γi* = 1/T2,i * are the pure dephasing (excluding dephasing due to the population relaxation) rates of 15


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Figure 6. Active plasmonic device structures for (A) Single-photon transistor scheme based on the dipolar coupling between a QD and SPP modes of a metallic nanowire.97 The gate photon (red pulse) switches the QD from the exciton ground state |g⟩ to a metastable state |s⟩ that is decoupled from SPPs. The input pulse at the X−SPP polariton frequency is then preferentially transmitted instead of being reflected. (B) Two-qbits coupled by a SPP waveguide. The strong coupling maintains phase correlation over a long distance preserving the entanglement.62 Images reprinted with permission from the references as indicated: (A) Copyright 2007 Nature Publishing Group and (B) Copyright 2011 American Physical Society.

speed, by far exceeding 1 THz.52,53 A highly attractive feature of such plasmonic switches is the potentially very low switching energy per bit. Plasmonic field localization can greatly increase the dipolar interaction between the quantum emitters and the plasmonic fields, offering, in principle, switching energies of just a few photons. Reaching switching energies of a few femtojoule would make all-optical switches competitive with electronic switches. Reductions in mode volume,56−60 longer dephasing time, and advanced tailoring of emitter−metal interactions in hybrid nanostructures will be important for the future development of such low-power all-optical and ultrafast switches and all-optically tunable active plasmonic devices. Several designs for such highly efficient plasmonic switches have recently been proposed. An interesting approach has been introduced by Chang et al.97 They consider a single quantum emitter, for example, a semiconductor QD with exciton ground state |g⟩ and excited state |e⟩, that is strongly coupled to a continuum of SPP modes of a metallic nanowire (Figure 6A). The tight confinement of the plasmonic modes of the wire results in a large Purcell enhancement of the spontaneous emission rate. For sufficiently large enhancement factors, a weak plasmonic field, propagating along the wire, will mainly be back-reflected if the QD is in its ground state. This geometry can now be transformed into a single photon transistor by adding an external, classical control field (a gate pulse), which drives the emitter from |g⟩ to a suitable metastable state |s⟩, decoupled from the SPPs. Hence, the emitter can no longer interact with the SPP field and the incoming pulse is transmitted. In principle, only a single photon is needed to switch the emitter from |g⟩ to |s⟩ and back. Similar transistor schemes may also be envisioned in the regime of vacuum Rabi splitting. Here, the Jaynes-Cummings model predicts that the Rabi frequency scales as n + 1 and thus increases substantially with the number of SPPs in the plamonic resonator. This opens up a yet unexplored possibility for ultrafast single-plasmon switching by exploiting the strong coupling in the quantum regime. Such active quantum-plasmonic devices are not restricted to plasmonic transistors or modulators. Dipolar interactions between QEs and SPPs can also be used to create longdistance entanglement between two resonant quantum bits. Gonzalez-Tudela et al.62 have theoretically shown that SPPs in realistic one-dimensional waveguides,243,244 are excellent candidates to act as mediators for achieving large values of

entanglement between two distant qubits (Figure 6B). Even though a number of proof of principle experiments strongly support the feasibility of such quantum plasmonic devices, significant challenges like sufficiently large Rabi frequencies at room temperature and long coherence times need to be addressed before key quantum phenomena such as higher rungs of the Jaynes-Cummings ladder, long-distance entanglement, or ultrafast single-plasmon switches and transistors will be demonstrated in such hybrid systems.

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FUTURE DIRECTIONS We have reviewed five broad areas concerning dipole interaction between QEs and SPPs in hybrid nanostructures. Impressive results have been achieved in all of these areas. Substantial progress seems possible with continued improvements in nanofabrication techniques, ultrafast spectroscopy and theoretical support. QE/metal hybrid nanostructures offer exciting opportunities to investigate novel manifestations of vacuum field induced light−matter interactions in many-body systems at room temperature. Such effects were traditionally studied only in very pure systems like atoms and molecules. Plasmonic systems offer ultimate field confinement in a broad spectral range and thus great enhancements in Purcell factors and dipolar coupling strengths. This offers exciting prospects for engineering Rabi coupling in solid-state nanosystems, yet comes at the expense of considerable radiative and nonradiative losses. Future research will have to explore strategies how to tailor the loss channels. Recently introduced concepts such as the reduction of nonradiative losses by collective strong coupling of QEs,233 or by coupling hybrid plasmonic−QE systems to low quality factor dielectric microcavities,154,158 might present interesting avenues toward this far-fetched goal. Future research will have to exploit these basic properties in novel types of active plasmonic devices, among which ultrafast plasmonic modulators, switches, transistor and lasers appear particularly interesting and promising. Several other key areas in quantum−plasmonics are yet to be explored. The nonequilibrium quantum optical response of hybrid structures and quantum plasmonic devices in the strong coupling are upcoming research fields and will greatly benefit from higher field enhancement and low-loss designs. Two other areas which deserve special note are those of active plasmonic structures for photovoltaic and chemical or biosensing applications. Light-harvesting structures for photo16


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voltaic devices115,118−121 and sensors245−247 that use field enhancement, evanescent field coupling, and alterations of potential energy landscapes have already been developed. Incorporating structures exhibiting high Purcell enhancement, like nanoantennas as transducers, offer the potential to greatly enhance their sensitivity and efficiency. The broad research interest and technological impact that plasmonic structures like nanoparticles, waveguides, and nanoantennas have had during the past years suggests that many other intriguing applications for active plasmonics devices will emerge in the very near future.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Parinda Vasa: 0000-0002-3182-0736 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to all our group and department members for stimulating discussions. Financial support by the Indian Institute of Technology Bombay, Department of Science and Technology, Council of Scientific and Industrial Research and Board of Research in Nuclear Sciences, Government of India, Deutsche Forschungsgemeinschaft [SPP1391, SPP1839 (Tailored Disorder) and SPP 1840 (Qutif)], the Korea Foundation for the International Cooperation of Science and Technology (Global Research Laboratory Project, K20815000003), and the German-Israeli Foundation (GIF Grant No. 1256) is gratefully acknowledged.

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