Revisiting Quantum Optics with Surface Plasmons ... - ACS Publications

Perspective pubs.acs.org/journal/apchd5

Revisiting Quantum Optics with Surface Plasmons and Plasmonic Resonators Francois Marquier, Christophe Sauvan, and Jean-Jacques Greffet* Laboratoire Charles Fabry, Institut d’Optique Graduate School, CNRS, Université Paris-Saclay, Palaiseau, France ABSTRACT: Surface plasmon polaritons can be used to confine fields at the nanoscale and are now one of the working horses of nanophotonics. This perspective deals with recent studies aiming at doing quantum optics experiments with surface plasmons. A first class of studies deals with one or two single plasmons and aims at observing wave-particle duality, squeezing, and coalescence of plasmons. A second class of studies deals with cavity quantum electrodynamics with localized plasmons in both the weak coupling regime and the strong coupling regime. KEYWORDS: surface plasmons, quantum optics, nanoantenna, strong coupling, Purcell effect

Q

uantum plasmonics has recently emerged as a new field in the exploration of light-matter interactions. There are essentially two motivations to study surface plasmons polaritons (SPPs) in the quantum regime. One of them is to investigate plasmonics when either field or matter is quantized. This raises the question of the validity limits of a description based on permittivities and classical fields. The second motivation is to explore the modification of light−matter interactions at the nanoscale and observe truly quantum phenomena. A major motivation to explore the nanoscale

cavity quantum electrodynamics with unprecedented large electromagnetic fields. We briefly summarize the key concepts and discuss in details very recent results in the field of plasmonic antennas with a particular discussion of the quenching issue and how it can be overcome. We finally present recent experimental evidence of strong coupling. In the concluding section, we address some other fields where new results are expected such as electrical excitation of surface plasmons, nonlinear quantum plasmonics and super-radiance.

regime is the fact that the electric field scales as

SINGLE PLASMON EXCITATION AND DETECTION In order to perform quantum optics with surface plasmons, a source of single plasmon is needed. Different ways to excite a single quantum of energy of surface or localized plasmons have been recently demonstrated. The simplest approach consists in using a single photon source. Illuminating a metallic structure acting as a plasmon launcher with single photons has been successfully explored both theoretically4 and experimentally.5−10 It is also possible to directly excite a plasmon with a quantum emitter. The first excitation of localized plasmons by single molecules were performed in the context of nanoantennas designed to enhance the fluorescence.11,12 The first direct excitation of a single propagating surface plasmon has been observed in 2007 coupling the fluorescence of a single colloidal CdSe quantum dot to a silver nanowire (see Figure 1A).13 Statistics on light scattered by the end of the nanowire showed an antibunching, thereby demonstrating the single particle behavior of the SPP along the wire. Soon after, the wave behavior of a single SPP excited by a diamond NV center was demonstrated using the same kind of apparatus and focusing on fringes appearing in the emitted spectrum (see

ℏω ϵ 0V

■

when the

field is confined in a volume V. This scaling entails a very large light−matter coupling and the possibility of nonlinear effects with very few photons. In this perspective, we limit the scope of the discussion to quantized electromagnetic fields while using a classical description of metals. In other words, we deal with quantum optics with surface plasmons. Regarding the effects of a quantum description of electrons in metals, the reader is referred to a recent review.1 Two excellent reviews summarized a few years ago the progress in quantum optics with plasmons.2,3 In this Perspective, we focus on very recent achievements in quantum plasmonics which had been on the agenda of the community for many years: observation of truly quantum effects with propagating surface plasmons such as surface plasmon coalescence, coupling of quantum emitters with plasmonic antennas with Purcell factors larger than 100 and quantum efficiencies above 50%, observation of the strong coupling regime at the single emitter level. The first part of this Perspective focuses on experiments aiming at exploring quantum optics with surface plasmons. These experiments raise the question of the decoherence that we address in the section Decoherence. The section Quantum Emitters Coupled to Plasmonic Resonators is devoted to cavity quantum electrodynamics with plasmonic nanoresonators. Plasmonics provides the possibility of confining electromagnetic fields at deep subwavelength scales, opening new opportunities to study © 2017 American Chemical Society

Received: Revised: Accepted: Published: 2091

May 12, 2017 August 22, 2017 August 25, 2017 August 25, 2017 DOI: 10.1021/acsphotonics.7b00475 ACS Photonics 2017, 4, 2091−2101

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Figure 1. Examples of quantum optics experiments with surface plasmons. (A) Single plasmon excitation by a quantum emitter in the vicinity of a silver nanowire. The light scattered at the end of the nanowire shows an antibunching behavior demonstrating the single particle level. Reproduced with permission from ref 13. Copyright 2007 Springer. (B) Wave-particle duality demonstrated on a metallic nanowire. The wave behavior is linked to the fringes that are observed in the spectrum scattered from both ends of the nanowire. Reproduced with permission from ref 14. Copyright 2009 Springer. (C) Preservation of polarization entanglement between two photons when one has been in a plasmonic state when transmitted through a metallic transmission grating. Reproduced with permission from ref 27. Copyright 2002 Springer. (D) Two-particle interferences on a plasmonic beam splitter. The dip in the coincidence rate demonstrates that guided surface plasmons behave as bosons, as photons in the seminal Hong-OuMandel experiment. Reproduced with permission from ref 9. Copyright 2009 Springer. (E) Two-plasmon interference experiment with a lossy beamsplitter. Due to the presence of losses, SPP anticoalescence can be observed. Reproduced with permission from ref 28 Copyright 2017 Springer.

Figure 1B).14 For further experiments, the precise location of the emitter around the nanoantenna is crucial to excite efficiently the resonant plasmonic mode: the deterministic control of the emitter position becomes thus an important goal. A few groups succeeded to deterministically control the position of the quantum emitter near a metallic nanowire using an AFM tip to position the NV center15 or near plane interfaces using a single NV center located at the apex of a scanning tip.16 In the latter, single SPP statistics are observed through leakage microscopy. New developments in photolithography17 allow now the deterministic positioning of the quantum emitters near metallic structures exciting single SPPs.18−20 Other methods appeared recently, using DNA to fix fluorescent molecules on binding sites,21 electrostatic behaviors of materials22 or the fact that electromagnetic resonances could play the role of an optical trap for a single nanodiamond particle containing a single NV center.23 In most experiments, surface plasmons are detected after being converted into photons by scattering, diffraction or leakage in a higher refractive index medium. The quantum of energy associated with a SPP can however be also directly detected. A superconducting structure located at the end of a plasmonic guide (stripe or wire) is used to reach the quantum sensitivity. A first demonstration has been published in 2009, detecting directly the current excited by plasmons arrival,24 although the single particle level of the signal was not directly proved. A year later, the single-energy-quantum level was

reached.25 In that case, a bias was applied to the superconducting wire in order to create a current close to the critical current. The arrival of a single SPP is thus able to change the state from superconducting to normal, so that a resistance appears giving rise to a voltage pulse between the electrodes. These setups confirm that local detection of single surface plasmon is possible without converting the surface wave into a propagating photon. This surface plasmon detection technique has been used to prove the coalescence of two identical surface plasmons on a beam splitter.5,8 A recent paper highlights the theoretical possibility to electrically detect single graphene plasmons.26

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REVISITING QUANTUM OPTICS WITH SURFACE PLASMONS Using the new capabilities to excite and detect single surface plasmons, several fundamental quantum optics experiments have been revisited. Wave-particle duality has been probed for single SPPs guided by a silver nanowire.14 It has also been observed with single surface plasmons propagating along a flat interface using a plasmonic beam splitter to measure the intensity correlation.10 A fundamental quantum interference experiment is the twoparticle interference on a beam splitter known as Hong-OuMandel experiment.29 In such an experiment, two identical bosonic particles impinging on a balanced beam splitter are expected to be detected both on the same output of the beam 2092

DOI: 10.1021/acsphotonics.7b00475 ACS Photonics 2017, 4, 2091−2101

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coherence triggered by processes entangling the quantum state with its environment. A surface plasmon is a polaritonic excitation with both an electromagnetic and an electronic component. It is well-known that many quantum effects in electronic transport can only be seen at very low temperature and on length scales smaller than the coherence length. This suggests that the electronic part of the plasmon polariton could be detrimental for the observation of quantum effects in plasmonics. However, as pointed out in the previous section, many quantum optics experiments have been successfully reproduced with surface plasmons. We compare hereafter the case of plasmons at optical frequencies to low temperature mesoscopic systems to explain the origin of the different behaviors. Let us first remind the origin of decoherence for electrons in low temperature mesoscopic systems such as charge qubits. Here, decoherence refers to the loss of the phase of the electronic wave function due to electron−electron interaction. The main mechanism is due to a random electric field generated by Johnson-Nyquist noise. This random field acts on the electrons and induces pure dephasing.46 As a consequence, electron−electron interaction in these systems is detrimental for the observation of quantum effects. When dealing with a plasmon at optical frequencies, the situation is very different because the energy scale of the elementary excitations involved are different. First, the plasmon is not a single electron excitation but a polarization wave in the metal. Its decoherence can be characterized by a decay time T2 given by 1/T2 = 1/T2* + 1/2T1, where the decay time T1 corresponds to absorption processes and the time T2* corresponds to pure dephasing processes. These quantities have been measured for both extended and localized plasmons.47,48 It turns out that the experiments show no evidence of pure dephasing mechanisms. The remaining contribution to decoherence is thus absorption. The typical decay time T1 is on the order of 5−10 fs. This decay channel corresponds to the absorption of a plasmon and the generation of a hot electron. The question is, therefore, what is the effect of pure absorption processes on the quantum properties of quantum optics experiments? Let us consider now the effect of absorption on correlation measurements that are the only experiments showing truly quantum effects. The simplest experiment is the measurement of the intensity correlation at the two outputs of a beam splitter illuminated by a single plasmon state. Absorption affects the measurements by reducing the number of counts per second on a detector. However, if a single plasmon is detected, that particular plasmon was obviously not absorbed. In other words, any detection of a single plasmon is a postselection process of nonabsorbed plasmons so that absorption merely reduces the counting rate. We now consider the remote control of a plasmon entangled with a photon. In such an experiment, the measurement involves measuring the coincidence between a photon and a heralded plasmon. Here again, the coincidence events detected are a post selection of the events with no absorption of both the plasmon and the photon. Similarly, when performing a Hong Ou Mandel experiment with plasmons, intensity correlation measurements are performed at the output of a beam splitter illuminated by two plasmons. In practice, it amounts to count the number of coincidences on the two detectors. Only the nonabsorbed photons contribute to the coincidence detection events. In summary, for these three types of experiments, the interaction of a plasmon with

splitter. This is the so-called coalescence effect. This pure quantum interference has been demonstrated very recently in an integrated-optics configuration with plasmonic waveguides (see Figure 1D).5−9 It is of importance to note that the coalescence effect is not the only possible scenario for surface plasmons. It has been recently experimentally demonstrated that losses associated with a plasmonic beam splitter can be used as a new degree of freedom to reveal new two-particle quantum interferences displaying anticoalescence of surface plasmons (see Figure 1E).28 It has also been shown that a nonlinear absorption process could occur onto the plasmonic beamsplitter.28 The probability that only one plasmon survives after the beamsplitter is indeed zero, both incident plasmons surviving or being absorbed together. In fact, even if the latter experiment has been conducted for the first time with surface plasmons, the result is not related to the nature of the bosonic particle involved in the interference and could be observed also with photons for instance.30−33 Beyond interferences, entanglement experiments have been performed. It has been shown first that two photons remain entangled when one of the electromagnetic excitation has experienced a plasmonic state through a metallic periodic array of holes (see Figure 1C).27,34,35 In 2015, conservation of path entanglement was demonstrated in a fibered Mach−Zehnder interferometer where photons were converted into surface plasmons in a dielectric-loaded SPP waveguide on each path.36 The possibility to entangle a single photon and a single SPP led recently to the remote preparation of a single-plasmon state.37 One of the major advantages of quantum optics is to manipulate nonclassical states that display unusual behaviors, particularly in the case of noise reduction. In some configurations, it is for instance possible to reduce the experimental noise below the shot-noise limit when exploiting squeezed states, as it has been theoretically predicted in the early eighties.38 It raises the question to what extent such strategies can be applied to quantum plasmonics. It is indeed known that linear losses can degrade squeezed states,39,40 so that one can wonder if such states can really be mediated by plasmonic modes. In 2009, Huck and co-workers41 showed that squeezed photonic states converted into long-range surface plasmons and then back to photons conserve their quantum properties. The modification of the squeezed state is similar to the modification induced by a beam splitter with an equivalent transmission factor and is only degraded by vacuum noise. This is of interest to improve plasmonic sensing. Using squeezed states, a 5 dB sensitivity improvement has been recently demonstrated in a Kretschmann configuration, exciting surface plasmons on a gold thin film.42,43 Another possibility is to take advantage of quantum correlations to remove part of the noise in the measurement. Frequency-entangled photons have thus led to an improvement of the signal-to-noise ratio in a goldnanoparticles spectroscopy experiment when using correlations to cancel the background optical noise as well as the detectors electronic noise.44 These experiments pave the way toward quantum plasmonic sensing, taking advantage, despite the losses, of the possibility to go both beyond the shot-noise limit and below the diffraction limit, as it has been recently highlighted by Lee and co-workers.45

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DECOHERENCE When envisioning to perform quantum optics experiments with surface plasmons, the question of decoherence is critical. Here we denote as decoherence the destruction of quantum 2093


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electrons in a metal is not detrimental when performing quantum measurements at the single plasmon level apart from the obvious reduction of the detection probability due to losses. Another possible process inducing decoherence is the inelastic collision of a plasmon (or photon) with an electron. This is called electronic Raman effect. An incident photon is reflected with a Raman shift and part of the excitation remains in the metal. Such a process leaves an imprint of the photon in the metal thereby generating an entanglement with the environment. The same process can exist a priori for a plasmon. However, the electronic Raman scattering cross section is extremely small so that this phenomenon is negligible. This low efficiency is attributed to the efficient screening of the charge density fluctuations by the conducting electrons. It is only recently that these very inefficient inelastic processes have been observed.49

Coupling Constant and Mode Volume. We first introduce the matrix element of the coupling Hamiltonian in the Jaynes-Cummings formalism, Wif = d·Em(re) = ℏg, where d is the transition dipole moment of a two-level system and Em(re) is the electric field of the resonator mode at the emitter position. The coupling constant of the interaction g appears as the inverse of the characteristic time scale of energy transfer between the cavity mode and the emitter. By denoting γ the radiative decay rate of the emitter into the continuum of vacuum modes in the absence of cavity, we see that the cavity modifies the emitter dynamics provided that g > γ. There are two possibilities to enhance the coupling factor. The first one consists in using emitters with a large dipole moment (i.e., a large oscillator strength); this is the case for J-aggregates or quantum dots for instance. The second way to obtain a large coupling constant g is to enhance the electric field of the cavity

QUANTUM EMITTERS COUPLED TO PLASMONIC RESONATORS Spontaneous emission is not an intrinsic property of matter, it can be tailored by modifying the emitter environment. Coupling a single emitter to a resonant system is of particular interest and constitutes the basis for cavity quantum electrodynamics (CQED). The generic term “cavity” refers to a resonant system that has taken different forms over the years. In its pioneering work, Purcell proposed to couple a radio frequency emitter to a resonant electrical circuit.50 Since the 1980s, a series of experiments have been performed on atoms in microwave51 and optical cavities.52,53 The progress in microand nanotechnologies observed since the early 1990s has then allowed major achievements with solid-state emitters coupled to photonic microcavities.54−58 More recently, light emitters such as fluorescent molecules and quantum dots have been coupled to plasmonic systems.11,12,59,60 In what follows, we summarize recent advances in the field of quantum emitters coupled to plasmonic nanoresonators. We first introduce a few figures of merit that are usually used in CQED to characterize the interaction between a single emitter and an optical resonator. Our objective is not to detail the theoretical formalism underlying CQED, but to provide the reader with the keys to understand the main peculiarities of plasmonic nanoresonators for tailoring spontaneous emission compared to other solid-state systems: extremely small volumes, small quality factors, and presence of quenching. We also define two regimes of spontaneous emission in a cavity. The weak coupling regime (also known as the bad cavity limit) corresponds to the enhancement or inhibition of the emission rate with a possible modification of the angular radiation pattern. In the strong coupling regime, spontaneous emission becomes a reversible process and Rabi oscillations may be observed. We then review recent theoretical and experimental achievements with plasmonic nanoresonators working in both regimes. Recent advances include, in particular, the observation of very fast spontaneous emission with a good radiative efficiency, an important milestone for optoelectronics applications, and the demonstration of strong coupling between a single emitter and a plasmonic nanoresonator. We introduce here the main figures of merit that are usually defined in CQED to discriminate between different spontaneous emission regimes. The reader interested by a more detailed description of the theoretical formalism underlying CQED is referred to refs 61 and 62 and to a recent review article for the case of plasmons.63

mode. The latter is proportional to

■

ℏω 2V ϵ 0 n 2

, with V the mode

volume and n the refractive index at the emitter position. Therefore, confining the electromagnetic energy ℏω in a small volume V enhances the interaction. The volume V of the resonator mode is a second fundamental figure of merit. For optical microcavities, the mode volume is usually defined as the volume that would be occupied by an electromagnetic field containing the same energy as the cavity mode, but with a constant energy density equal to the maximum energy density at an antinode of the mode field,62 V = ∫ ud3r/max(u), with u the modal energy density. This definition relies on a perturbation approach in which energy dissipation is introduced by simply broadening the eigenmodes of an ideal conservative system. However, if this trick is accurate for resonators with a small leakage, it becomes largely unfounded for nanoresonators for which energy dissipation cannot be neglected.64 As a consequence, the usual definition of the mode volume is inappropriate for plasmonic nanoresonators.65−67 The concept of mode volume is nevertheless fully applicable to plasmonics provided that a proper definition be used. For deriving a correct expression of the mode volume, one needs to use the natural (quasinormal) modes of open, absorbing and dispersive resonators. This is a nontrivial theoretical issue that had been studied in the 90s in the simple case of 1D geometries.68,69 Recently, general results have been established for arbitrary geometries and materials.70 Because of energy dissipation, the phase of the field has to be taken into account in the mode volume; this gives rise to non-Lorentzian, Fanolike spectral behaviors.70 Other definitions of the mode volume valid for nonmagnetic materials have also been proposed.71−73 Note that the mode volume definition proposed in ref 71 has been recently debated in refs 74 and 75. Strong and Weak Coupling Regimes. We now introduce the resonator damping that is related to radiative leakage and absorption. This decoherence channel is introduced in the formalism through the resonator full width at half-maximum κ and the corresponding quality factor Q = ω/κ. If the mode volume characterizes the spatial confinement of the electromagnetic field, κ gives a measure of the spectral confinement. In the limit where the emitter line width is much smaller than the resonator line width, two regimes can be considered. If 2g > κ, the photon can be reabsorbed by the emitter before leaving the cavity. The process can be repeated so that spontaneous emission becomes reversible and Rabi oscillations may be observed: the excitation is periodically exchanged between the emitter and the resonator. This so-called strong coupling 2094


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regime is very interesting as it allows controlling the excitation state of the emitter using a cavity if the cavity can be tuned on a time scale shorter than 1/g. The second regime of interaction is characterized by the condition κ > 2g. The lifetime of the photon in the cavity is shorter than the coupling time between the resonant mode and the emitter so that the photon escapes the cavity before being reabsorbed by the emitter. Any cavity excitation is quickly lost into either a propagating photon or heat if absorption is present and there are no oscillations. This regime is usually referred to as the bad cavity limit, the weak coupling regime, the Purcell regime or the antenna regime. The rationale for the name “weak coupling” is that the frequency of the emitter and the cavity are only slightly modified. By contrast, in the strong coupling regime, the coupling of the emitter and the cavity leads to new hybrid eigenstates with significant frequency shifts. Controlling the interaction regime amounts to control the parameter 2g/κ. By introducing the

γres γ

ℏe 2

where re =

e2 4πϵ0mc 2

re 2πλ

(1)

c 3Nλ 2 κ 2π V

λ3 V

is the classical radius of the electron. This

> 2.710−5.

Purcell Factor and Cooperativity. The bad cavity limit allows inhibiting or on the contrary enhancing the spontaneous emission rate of the emitter. The enhancement has an upper bound, the so-called Purcell factor, that can be reached if the transition frequency, the position and the polarization of the emitter are matched with those of the resonator mode. The Purcell factor is defined as the ratio between the decay rate in the cavity mode and the decay rate in the absence of cavity. It can be written either as a function of the coupling constant g or as a function of the volume V, as initially proposed by E. M. Purcell.50 As the resonator is spectrally broad compared to the emitter transition, we can use Fermi golden rule to derive the decay rate in the presence of the cavity: γres =

2π |Wif |2 D(ω) ℏ2

Fp =

(2)

γ

=

4g 2 κγ

(3)

We now recover the original form introduced by Purcell by inserting the explicit forms of the interaction energy ℏg = d

ℏω 2V ϵ 0 n 2

and the spontaneous decay rate of the emitter

in a homogeneous medium with refractive index n, γ =

3N λ 3 ω

4g 2 γ(κ + γ + 2γ *)

(5)

It is seen that the Purcell factor is obtained by replacing in the usual form, the cavity decay rate κ by an effective decay rate κ + γ + 2γ*. Plasmonic Resonators. Now that we have introduced important figures of merit for characterizing spontaneous emission in a cavity, let us discuss the main peculiarities of plasmonic nanoresonators compared to other solid-state systems. A first specificity of plasmonic systems is the fact that the quality factor Q is of the order of 5 to 10 because of the absorption in metals at optical frequencies. This feature makes plasmonic resonators broadband and therefore robust against many potential problems such as spectral diffusion or inhomogeneous broadening. However, a low quality factor is detrimental to reach the strong coupling regime. Fortunately, a second characteristic of plasmonic nanoresonators is to offer

where D(ω)=2/(πκ) is the value of the density of states on resonance assuming that the resonator mode has a Lorentzian profile with width κ. Using the notation Wif = ℏg, the decay rate of the resonator is thus given by γres = 4g2/κ. By normalizing this decay rate by the decay rate γ we find the Purcell factor FP: γres

(4)

= 2 V κ = NFp = 4CN . When considering the case N 4π = 1, there is obviously no collective behavior, but the cooperativity C1 appears to be equal to the Purcell factor apart from a numerical factor 4. Thus, exp(−4C1) is the transmission factor of a cavity containing one two-level system on resonance and 4C1 is the optical thickness. Figures of Merit in the Presence of Dephasing. In the above discussion, we have not considered pure dephasing processes of the two-level system. However, when dealing with molecules in condensed matter at ambiant temperature or quantum dots, pure dephasing processes modify significantly the dynamics of the system. We introduce the pure dephasing rate γ* = 1/T2* so that the total spectral width of the emitter is given by γ⊥ = γ∥/2 + γ*. Using a master equation, it can be shown that the figures of merit are modified by the presence of this dephasing term.77,78 The condition to observe Rabi oscillations requires that the Rabi splitting 2g be larger than the full width at half-maximum of the peaks given by κ + γ + 2γ*. In the bad cavity (weak coupling) regime, the acceleration of the spontaneous decay rate on resonance is given by a modified Purcell factor77,78

shows that the strong coupling regime can be obtained by increasing the quality factor Q, the oscillator strength f, and by decreasing the mode volume V. By setting λ = 600 nm, we find that the strong coupling regime condition is given by

Q f

3 4g 2 3 ⎛⎜ λ ⎞⎟ Q = 2⎝ ⎠ κγ 4π n V

If one assumes that the decay rate in all other possible channels (i.e., different from the cavity mode) is approximately the same as the decay rate in a homogeneous medium, then the total decay rate Γ in the presence of the cavity is simply given by Γ = γ + γres = γ(1+FP). Note that the measurement of the decay rate acceleration Γ allows identifying g using eq 3 without defining an effective mode volume. It is noteworthy that the dimensionless number 4g2/κγ is sometimes called cooperativity. It is used to characterize the absorption of a photon by an ensemble of N atoms in a cavity. The name was first introduced for the quantity CN = Ng2/κγ in the study of the nonlinear transmission by an ensemble of N molecules in a Fabry−Perot cavity with volume V.76 The absorption coefficient 1/labs is given by ρσ, where ρ = N/V and σ is the absorption cross section. The absorption cross section of a two-level system at resonance is a universal quantity given by 3λ2/2π. A photon spends a time 1/κ on average in the cavity before leaving the cavity so that it travels over a distance L = c/ κ in the cavity. If we now compare this distance and the absorption length, we find the optical thickness

oscillator strength f of the emitter, d 2 = 2mω f , where e and m are the electron charge and mass, we obtain 2g λ3 =Q f κ V

=

nd 2ω3 : 3πℏϵ0c 3

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extremely small volumes. For the smallest optical microcavities such as micropillars or photonic crystal cavities, the mode volume is of the order of λ3.55 Moving to plasmonics allows for a dramatic shrinking of the mode volume down to values of the order of λ3/104.70 This is a clear benefit of working in the nearfield regime with plasmonic nanoresonators as compared to usual optical cavities: for the same emitter, the coupling constant g can be several orders of magnitude larger. Finally, a last specificity of plasmonic systems is the presence of quenching. An emitter in its excited state can decay by emitting a photon that is coupled neither to the resonator mode nor to the radiation continuum but that is absorbed locally in the nearby metal. Such a nonradiative decay channel can have a very large rate (>100γ) if the emitter is located at a distance of a few nanometers from a metal surface. Quenching competes against the desired decay into the cavity mode and it appears as an additional decoherence channel that does not exist in conventional optical microcavities. This is a clear limitation of plasmonics that needs to be overcome to realize plasmonic light sources with a good radiative efficiency or working in the strong coupling regime. Recent works have demonstrated that the presence of quenching is not a fundamental limitation; its impact can be drastically reduced in carefully engineered plasmonic nanoresonators.

Figure 2. Radiative efficiency ηr of different plasmonic systems as a function of their total decay rate enhancement Γ/γ. The dashed black curve shows emission at λ = 650 nm close to a planar air/gold surface. The three solid blue curves correspond to gold nanospheres of different radii (R1 = 30 nm, R2 = 40 nm, and R3 = 50 nm) in air. The spheres are resonant at λ = 575, 600, and 650 nm. The distance between the emitter and the metal is varied along the curves between 1 and 40 nm. Red circles represent recent experimental results82−85 and green squares show theoretical data.86,87 The different geometries used in these works are schematically illustrated by the insets; red dots show the emitter position, blue shapes correspond to dielectric materials, while yellow (gray) shapes correspond to gold (silver).

WEAK COUPLING REGIME: TOWARD FAST AND EFFICIENT SPONTANEOUS EMISSION Light emitters in weak coupling regime with a resonator experience a modification of their spontaneous decay rate due to the so-called Purcell effect.50 Of particular interest is the case where the decay rate is enhanced compared to the same emitter in vacuum because the local density of states (LDOS) of the resonator is much larger than that of a homogeneous medium. The total population decay rate Γ in the presence of the resonator (inverse of the emitter lifetime) is not the only figure of merit that has to be considered to characterize the weak coupling between a quantum emitter and a plasmonic resonator. Indeed, because of the presence of absorption, only a fraction of the total decay rate corresponds to photon emission in the far-field, the remaining part corresponding to absorption in the metal. Therefore, one usually defines the radiative and the nonradiative decay rates Γr and Γnr, with Γ = Γr + Γnr, and the associated radiative and nonradiative efficiencies, ηr = Γr/Γ and ηnr = Γnr/Γ = 1 − ηr. The radiative efficiency ηr is also sometimes referred to as the external quantum efficiency (extrinsic to the emitter), in contrast to the internal quantum efficiency, which accounts for possible nonradiative processes that are intrinsic to the emitter (e.g., internal relaxation in vibrational degrees of freedom in a molecule). It is well-known that a light emitter located at a few nanometers away from a metal surface is quenched: an excited state can decay extremely fast but the decay is dominated by nonradiative processes in the metal.79 In other words, the total decay rate is extremely large but the radiative efficiency is close to zero. The classical picture associated with this effect is the ohmic losses in the metal due to the very large electric field decaying as 1/d3 generated by a dipole in the near-field at a distance d. This phenomenon is illustrated in Figure 2 that displays the radiative efficiency ηr of different plasmonic systems as a function of their total decay rate enhancement Γ. The dashed black curve corresponds to emission at λ = 650 nm close to a planar gold surface; the distance between the

emitter and the metal is varied along the curve in the range 1− 40 nm. For an emitter located at 20 nm from the gold surface, the radiative efficiency is 20% and the SE is 4× faster than in vacuum. If the distance to the metal is decreased, the SE enhancement can be increased by several orders of magnitude but this is at the cost of a drastic decrease of the radiative efficiency: for distances below 5 nm, the decay rate enhancement can be larger than 1000 but with a negligible radiative efficiency smaller than 1%. Early works in plasmonics have demonstrated that, even if the numbers can be slightly improved, the general trend seems to be the same for an emitter located close to a plasmonic nanoantenna.11,12,80,81 The three solid blue curves in Figure 2 show the radiative efficiency of an emitter close to a gold nanosphere for different radii, R1 = 30 nm, R2 = 40 nm, and R3 = 50 nm. The nanospheres are resonant at λ = 575, 600, and 650 nm. Replacing the metal plane by a resonant sphere increases the radiative efficiency for decay rate enhancement smaller than 100. However, to obtain extremely large decay rate enhancements (Γ > 500γ), one needs to approach the emitter at only a few nanometers from the metal surface. At these distances quenching is still the dominant process and the radiative efficiency remains limited to at most 10%. As a consequence, it was thought for a long time that spontaneous emission in plasmonic nanoresonators was plagued by quenching and that it was not possible to benefit both from an extremely large decay rate (Γ > 500γ) and a large radiative efficiency (ηr > 0.5). Recent theoretical and experimental works have demonstrated that quenching is in fact not a fundamental limitation of plasmonic systems. It has been understood that the contribution to the LDOS of resonators generating highly confined fields can produce an LDOS, which can be locally as large or larger than the contribution of the non radiative modes. If this highly confined field pertains to a mode with a large radiative decay rate, then the efficiency of the resonator can be good. Several works have shown that well-chosen nano-

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spontaneous decays. Such a SE regime is promising for the realization of light-emitting devices with modulation speeds larger than usual semiconductor lasers.96

resonator geometries can lead to extremely large SE enhancements with good radiative efficiencies.82−89 In 2012, Russel and co-workers measured decay rate enhancements of 1800 from organic dye tris(8-hydroxyquinoline) aluminum (Alq3) embedded in a thin layer of Al2O3 between a silver substrate and a silver nanowire.82 A radiative efficiency of 54% was inferred both from experimental data and numerical calculations. More recently, other experimental works have demonstrated large decay rate enhancements in the range 100−1000 and a radiative efficiency larger than 50%.83−85,88,89 In ref 85, the value of the radiative efficiency has been inferred both from experimental data and numerical calculations. In refs 83, 84, 88, and 89, the radiative efficiency has been calculated. The figures of merit reported in refs 82−85 are shown by the red circles in Figure 2. A variety of light emitters have been used in these different works: fluorescent molecules,82,83,85 III−V semiconductor nanorods,84 or colloidal quantum dots.88,89 Many of these works were based on metal−insulator−metal (MIM) structures also called nanogap devices. A common feature of these works is the location of the light emitter, which is in all cases embedded in a thin dielectric layer sandwiched between two metallic structures. The latter are a metal film and a metal nanowire,82 a metal film, and a metal nanocube,83,88,89 a gold-nanosphere dimer,85 and a gold-nanowire dimer.84 The experimental evidence that these emitting devices are not limited by quenching has been explained by studying in details the process of light emission in a two-dimensional (2D) nanogap.90,91 These theoretical works have shown that the dominant decay channel for a light emitter in a 2D nanogap is the emission into the so-called gap-plasmon mode supported by these structures,92−94 even for vanishing gap thicknesses. The corresponding decay rate has the same variation with the gap thickness d as the quenching; they both vary as 1/d3.91 As a consequence, the ratio between both decay rates is approximately constant with the gap thickness. In addition, the authors of ref 91 have calculated that this ratio is largely favorable to the emission of gap-plasmons, which in turn participate to build the resonator mode. These counterintuitive results explain why quenching is not the dominant decay channel in MIM emitting devices, even for gap thicknesses of a few nanometers. It is important to note that the possibility to overcome quenching is not limited to MIM nanogaps. Numerical calculations have shown that large SE enhancements and good radiative efficiencies can also be obtained with other plasmonic nanoresonators. SE enhancements of the order of 5000 and radiative efficiencies of 70% have been calculated for gold nanocones lying over a high-index substrate.86 These results are marked by a green square in Figure 2. Finally, let us emphasize that experimental demonstrations of extremely large SE enhancements of the order of 1000 are remarkable for two reasons. First, until now, such large SE enhancement factors had never been achieved experimentally with dielectric microcavities. To our knowledge, the largest SE enhancement measured for a quantum dot in a photonic crystal microcavity is 75.95 Ultrafast SE is thus a specificity of plasmonic nanoresonators as compared to dielectric microcavities. Second, if the possibility to drastically enhance SE rates with metallic structures was known since a long time, it was thought that the radiative efficiency was plagued by quenching, that is, nonradiative energy transfer to the nearby metal. These results evidence that quenching can be overcome and they open the way toward integrated light sources with ultrafast

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STRONG COUPLING REGIME The strong coupling regime between an emitter and a resonator (be it photonic or plasmonic) is a remarkable effect which produces hybrid excitations of the cavity and the emitter, namely, between a macroscopic object and a single quantum object. This effect is easier to observe with ensembles of N emitters as the interaction is enhanced by the factor N. It can be observed at the single emitter level showing that the energy spectrum of a single emitter can be modified by designing an appropriate cavity. If the coupling to the cavity can be controlled, this paves the way to the manipulation of light− matter interaction at the individual quantum system scale with possible applications to quantum information. The development of the cavity quantum electrodynamics started with the seminal paper by Purcell.50 It took almost 50 years to observe strong coupling52,53 at the single atom level. We have seen that strong coupling is favored by a large oscillator strength. Hence, quantum dots are good candidates to observe the effect. Using microcavities with quality factors on the order of 10000 and quantum dots with oscillator strengths on the order of 100, the Rabi splitting has been observed in micropillars,56 photonic crystal cavities,57 and microdisks.97 The observation of strong coupling between a single emitter and a plasmonic resonator is a formidable challenge as the decay time of plasmons is on the order of tens of fs so that the Rabi oscillation needs to be on a few fs scale. However, it was predicted theoretically that it could be observed.98−104 Two recent experiments have reported the observation of strong coupling at the level of a few emitters. A vacuum Rabi splitting with silver bow-tie antennas with a gap of about 20 nm has been observed using CdSe/ZnS quantum dots.105 The vacuum Rabi splitting has also been observed with methylene blue molecules whose oscillator strength is on the order of 0.5. This rather low value was compensated by using very small gaps on the order of 1 nm formed between a gold surface and a 40 nm gold nanosphere.106 The dipole moment of the molecule was maintained aligned normal to the surfaces by encapsulating the molecule in macrocyclic cucurbit[7] uril molecules which are pumpkin-shaped molecules with hollow internal volumes. By depositing a monolayer of these molecules on a gold mirror, and filling them with methylene-blue molecules, it has been possible to achieve 1 nm gaps. On both cases, the frequency splitting between both hybrid eigenstates of the strongly coupled system was observed by measuring the scattering spectrum. Figure 3 summarizes the landscape of the experiments conducted with atoms, molecules and quantum dots on one hand, with Fabry−Perot cavities, semiconductors cavities and plasmonic resonators on the other hand. Inspired by eq 5, we plot log(2g/γ) along the vertical axis and log[2g/(κ + γ + 2γ*)] along the horizontal axis so that a constant Purcell factor appears as a line with slope −1 since log(2g/γ) = logFp − log[2g/(κ + γ + γ*]. The left part of the graph corresponds to the bad cavity or antenna regime, whereas the Rabi oscillation regime corresponding to (2g > κ + γ + γ*) is in the right part of the graph. The quantity 2g/γ is given by

3 4π

3/2

λ4 1 n Vre f 4

, which

depends on the mode volume but also on the wavelength, the 2097


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50%.107 This is expected to have many applications. One of them is the development of light emitting diodes that could be modulated beyond 1 GHz.96 Another important potential application is in the field of inelastic tunneling light emission. It has been known for 40 years that a tunnel junction can emit light due to inelastic tunneling of an electron.108 This effect can be observed with a scanning tunneling microscope if the bias voltage is on the order of 2 V. The radiative efficiency of this process is very low and typically on the order of one photon for 104 or 105 electrons. The electrons lose their energy through nonradiative energy transfer to electrons. By accelerating the spontaneous emission rate into the cavity mode while keeping the plasmonic antenna radiative efficiency above 50%, it should be possible to increase significantly the efficiency of the emission process. Recent results include the observation of 2 orders of magnitude enhancement of the signal as well as shaping of the spectrum using a plasmonic antenna109 and the observation of the modulation of light emission at 1 GHz by inelastic tunneling.11 One of the most recent achievements is the demonstration of plasmonic resonators operating in the strong coupling regime. An intrinsic property of strong coupling with a two-level system is the importance of the nonlinearity due to the saturation of the emitter. Remarkably, this nonlinear effect can be observed in the very few photon regime. Indeed, for a two-level system, the onset of saturation is expected for a critical photon number in the cavity given by γ(γ + 2γ*)/8g2,77 where γ* is the pure dephasing decay rate. It follows that even with less than a single photon in the plasmonic resonator on average, the system can be in the nonlinear regime if g is large enough. The origin of this nonlinearity is the saturation of the two-level system when the Rabi splitting is on the order of the width γ. These effects have been observed with microcavities53 and micropillars.110 This regime has not yet been reported with plasmonic cavities. Another opportunity for quantum plasmonics is superradiance. In a nutshell, super-radiance is a collective process resulting from constructive interferences of the fields emitted by different emitters resulting in a faster spontaneous emission process. The classical super-radiance effect is simply the result of interferences between N classical monochromatic emitters leading to an increase by a factor N of the emitted power. Remarkably, this classical picture is still valid111 when dealing with N atoms or N molecules in the single excitation regime, namely, only one emitter is excited among N. We stress that this low excitation regime differs from the regime described by Dicke, where N noninteracting atoms are excited and a burst of radiation is emitted, a regime also called superfluorescence.112 The effect has been observed with self-assembled quantum dots.113 While the perspective of increasing the spontaneous emission rate by a factor N, which can be very large is very attractive, there are many issues to be addressed. The interferences can be canceled by dephasing due to either inhomogeneous broadening of the emitters spectrum or to pure dephasing produced by the interaction with the environment.114 A second effect that needs to be included in the analysis is the presence of interactions between emitters, an effect which is enhanced by plasmonic resonators.115−119 Another key issue for practical applications is to find a way to selectively excite the few superradiant modes among the many subradiant modes. The interplay between Purcell effect, cavity enhanced interaction, super-radiance, and dephasing is still an open question to a large extent.

Figure 3. Comparison of plasmonic resonators, semiconductors cavities and Fabry−Perot cavities. The system is in the Rabi oscillation regime when the abcissa is larger than 1 and in the bad cavity limit when it is smaller than 0.1. The vertical axis is 2g/γ. It depends essentially on the volume confinement but also on the dielectric constant and the oscillator strength. Plasmonic resonators are on the upper part of the graph with 2g/γ > 104, semiconductor cavities are at the bottom with 2g/γ on the order of 102 and Fp ≈ 1000, the smallest Fabry−Perot cavity (upper right corner) has 2g/γ > 105 and Fp > 106.

oscillator strength, and the refractive index, which takes large values for semiconductors. It is seen that plasmonic resonators corresponding to very small mode volumes and therefore to very large values of g are in the upper part of the graph. We also note that the critical photon number providing the onset of saturation is given by γ2/8g277 in the absence of dephasing. It is seen in the figure that this number is much smaller than one for all experiments. The observation of a nonlinear behavior with few photons has been reported with cavities but not yet with plasmonic resonators. Depending on the resonator decay rate, plasmonic resonators can be either in the bad cavity regime or in the strong coupling regime. We have plotted for comparison, the position of previous experiments with semiconductor cavities (photonic crystal cavity, micropillar, microdisk) and with cold atoms in Fabry−Perot cavities. Note that the smallest Fabry−Perot cavity53 reaches a very large value of 2g/γ despite a lack of nanophotonic confinement owing to the small value of the factor n4f with atoms in vacuum.

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OUTLOOK All the plasmonic components of a quantum optics experiment are now available. It is possible to generate locally a single plasmon, it is also possible to guide and control its propagation and to detect directly the surface plasmons. The realization of fully integrated plasmonic quantum circuits is now entering the engineering domain. One of the key issues to be addressed will be the efficiency of the circuits by further reducing propagation losses and insertion losses. We have stressed in this perspective that one of the key achievements in the field of light emission assisted by plasmonic antennas is to overcome quenching to a large extent even though quantum emitters are located at distances on the order of a few nanometers. Very large Purcell factors can now be obtained while the radiative efficiency remains above 2098


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AUTHOR INFORMATION

Corresponding Author

*E-mail: jean-jacques.greff[email protected]. Phone: +33 (0) 1 64 53 31 86. ORCID

Jean-Jacques Greffet: 0000-0002-4048-2150 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Alain Aspect, Antoine Browaeys, Daniel Esteve, and Philippe Grangier for helpful discussions. This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d’Avenir Program (Labex NanoSaclay, Reference: ANR-10LABX-0035). J.J.G. thanks the Institut Universitaire de France and the chair Safran-IOGS for supporting this research.

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