Spontaneous emission in micro- or nanophotonic structures

*Correspondence: ygu@pku.edu.cn 1State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing, China 2Frontiers Science Center for Nano-optoelectronics, Collaborative Innovation Center of Quantum Matter, and Beijing Academy of Quantum Information Sciences, Peking University, Beijing, China Full list of author information is available at the end of the article Abstract


Introduction
Single-photon source in micro-or nanoscale is the basic building block of on-chip quantum information and scalable quantum network. Enhancement of spontaneous emission (SE) in the frame of cavity quantum electrodynamics (CQED), i.e. Purcell effect controlling the spontaneous emission rate through cavity modes, is one of the key principles of realizing single-photon sources [1]. With the trend of on-chip optical quantum devices, realizing single photon emission in the micro-or nanoscale is increasingly important for quantum information process. Local field enhancement or small optical mode volume in micro-or nanostructures brings huge advantages in CQED, quantum information, and light-matter interaction. So far, the enhancement of SE in the micro-or nanoscale has reached great achievements in the areas of whispering gallery microcavities, photonic crystals (PCs), surface plasmon polaritons (SPPs), metamaterials, and so on, which provides abundant choices and methods of controlling and collecting single photons. In this Review, we will mainly focus on the theories, experiments, and up-to-date applications of SE in above areas. These researches demonstrate that Purcell effect has great influence in single-photon source, photonic circuit, and on-chip quantum information.
In the past decades, there were several excellent reviews related to the SE. For examples, Serge Haroche et al. used the concept of cavity quantum electrodynamics to review the SE in 1989 [2], but not any photonic structure was involved. K. J. Vahala pointed out that optical microcavities confine light to small volumes by resonant recirculation which provides an environment for SE [3]. After that, towards to integrated quantum devices, there was a great achievement in this field. H. J. Kimble predicted that quantum nodes in quantum networks can be realized by the optical interactions of single photons and atoms in traditional CQED cavities [4]. In the last ten years, more and more researchers recognized the importance of quantum plasmonics and its applications in quantum information processing [5][6][7]. However, an updated version of reviewing the SE in microor nanophotonic structures is required, including not only the theoretical and practical knowledge accumulated over the past decades, but also the up-to-date applications and development trend of CQED system in the micro-or nanoscale, which is our intention of writing this review.
To quantitatively describe the enhancement of SE, Purcell factor F is defined as F = γ tot /γ 0 , where γ tot is the total SE rate from micro-or nanostructures and γ 0 in vacuum.
According to Purcell theory, F ∝ Q/V with V ∼ L 3 , that is, larger Purcell enhancement requires larger quality factor Q and smaller optical mode size L, which can be provided by various micro-or nanostructures. As shown in Table 1, Q and L for traditional CQED systems are usually less than 10 3 and 0.5 mm, respectively, thus the Purcell factor F is in the order of 10 2 [2,3,8]. With the larger Q (10 4 ∼ 10 9 ) and smaller L (10 ∼ 100 μm), Purcell factors in whispering gallery microcavities can reach to 10 5 [9][10][11]. By creating PC defect cavities, L continues to get smaller to less than 10 μm, but Q is limited in 10 3 ∼ 10 5 due to the transport loss, so Purcell factors are usually less than 10 2 [12][13][14][15]. In SPP structures, the scale of L can even be smaller than 1μm which is often in subwavelength scale and always leads to local field enhancement, but Q is less than 10 2 because of absorbtion loss, thus Purcell factors can be up to 10 4 but with large absorption part [16][17][18][19][20]. Additionally, we also reviewed the photon emission in metamaterials and hybrid structures.
The rest of this Review is organized as follows: to ensure the completeness of the paper, in "A brief review of CQED" section, we briefly introduced the concept and developments of traditional CQED; Then in "The development of spontaneous emission in micro-or nanostructures" section, we summarized the theories, experiments, and applications of the spontaneous emission in various micro-or nanostructures, mainly including whispering gallery microcavities, photonic crystals, plasmon nanostructures, metamaterials, Table 1 Schematic, mode size L, quality factor Q, and Purcell factors F = γ tot /γ 0 for traditional CQED systems [2,3,8], WGMs [9-11], PCs [12][13][14][15], and SPP structures [16][17][18][19][20] and their hybrid structures; Finally, in "Summary and prospects" section, there were the summary and prospects.

A brief review of CQED
Cavity quantum electrodynamics focuses on the interaction between quantized field and quantum emitter at a single-quantum level in the limited space as well as cavities. The study of CQED started from Purcell's discovery in 1946 [1] that the spontaneous emission can be enhanced or inhibited by the surrounding environment. In 1963, Jaynes and Cummings [21,22] set up the interaction between single-mode electromagnetic field and two-level atom, which is now known as Jaynes-Cummings model. In 1983, Goy's group used Rydberg Na atoms to increase the rate of spontaneous emission by about 50 times [23]. Later, Hulet's group achieved suppressed spontaneous emission of Rydberg Cs atoms experimentally [24]. At the end of 1980s, the theory of weak coupling regime of cavity quantum electrodynamics had been in-depth researched [2]. Owing to the improvement of micro-or nanocavity technology [25], strong interactions of single atoms and photons in CQED attracted much attention from quantum entanglement and cold atoms [26-30], which was decades later than weak coupling because of its rigorous conditions. In 1996, Serge Haroche finished the first proof of interaction between single two-level atom and single photon in the cavity [31], and thus became a Nobel laureate in 2012 [32]. Since then, strong and weak coupling regimes have been both key components of CQED [3,4,[33][34][35].
Light trapped in the cavity can only form resonance at some specific eigenfrequencies, i.e., discrete cavity modes, leading to interesting behavior between cavity and emitter [2,[35][36][37][38][39][40][41][42]. Using Jaynes-Cummings model [21,22] with single-mode cavity and two-level emitter [Figs. 1(a-f )], the Hamiltonian and dynamical equations are described as [43,44] where σ † (σ ) is the rising (lowering) operator of the emitter, a † (a) is the creation (annihilation) operator of the mode of nanocavity, and ω a (ω c ) is the frequency of the emitter (nanocavity). This CQED system is governed by three photonic processes [5,7,20,26,44,45], characterized as parameters g, κ, and γ , where g donates the coupling coefficient between emitter and cavity mode, κ describes the dissipative channel from cavity to the thermal reservoir, and γ describes the dissipative channel from emitter to the modes other than nanocavity mode. When talking about weak coupling regime, because g γ , κ, the energy exchanging strength between quantum emitter and cavity is much less than the loss rates of system, so most photons decay into the environment. The SE rate of quantum systems can be controlled by cavity mode and other modes out of cavity, which is so-called Purcell effect. Its emission enhancement is defined as Purcell factor [3,5,7] Fig. 1 Schematic diagram of CQED systems. a A typical CQED system consisting of a single quantum emitter (here is two level) and a cavity with coupling coefficient g, cavity loss κ, and spontaneous emission rate γ [7]. b The probability of excited-state with time evolution and c transmission spectrum for weak (red curve) and strong (blue curve) coupling regimes [5]. d Schematic diagram of Purcell enhancement at weak coupling regime [3]. e Vacuum Rabi oscillation with period energy exchange at strong coupling regime [3]. f Schematic of Rabi splitting energy by dressed state theory [7] which is proportional to quality factors Q and inverse proportional to cavity mode volume V. For any resonant optical cavity, the Q factor is defined as Q = 2π(energy stored)/(energy lost in one cycle of oscillation) [46], thus Q is inversely proportional to κ, which is much important for enhancement of SE in micro-or nanophotonic devices. Also, considering cavities may be open, cavity mode volume V is an effective volume. Micro-or nanophotonic structures with small mode volume and high Q could lead to huge enhancement of SE of quantum emitter due to F ∝ Q/V [5,6]. γ 0 represents the spontaneous emission in vacuum, which can be obtained from Weisskopf-Wigner theory [47]: where μ is the dipole moment of quantum emitter. While for strong coupling regime, i.e., g κ, γ , the energy exchanging strength between quantum emitter and cavity mode is much larger than the loss rate of the system. The coupling coefficient g is described as where E is the electric field corresponding to a single excitation and ε means the environmental permittivity. The coupling coefficient g ∝ Q/ √ V [5,7]. The Rabi splitting caused by strong coupling can be explained by the dressed state theory [48] and in the resonant condition it is If g is much larger than |(κ − γ )/4|, Rabi splitting is close to 2g [ Fig. 1(f )]. Spontaneous emission of quantum emitters of CQED is quite important for realizing single photon sources. Rydberg atoms [22,49,50] and quantum dots [51] are two kinds of technologic realization of single-photon sources owing to the large dipole moments. Also, quantum dots are popularly applied in Purcell experiments with different functionalities caused by various ingredients [52][53][54][55]. For example, InAs quantum dot embedding in GaAs materials can provide broadband spontaneous emission control with high efficiency [56][57][58], while entangled photon pair sources show high brightness and indistinguishability by combining GaAs quantum dots with new broadband photonic nanostructures [59].
The large enhancement of spontaneous emission makes CQED systems elemental in the field of quantum network, which plays an important role in the broad context of quantum information [4]. In the localized quantum nodes, the quantum state is generated, processed and stored in the setting of CQED. And quantum states are transported through quantum channels from site to site with high fidelity. Such quantum networks can be achieved by single photons and atoms provided by CQED systems [4,36,60,61].

The development of spontaneous emission in micro-or nanostructures
According to Purcell's theory, the SE of quantum emitters is controlled by the optical modes of the cavity [1]. To engineer the optical modes in the micro-or nanoscale, people fabricate various micro-or nanostructures, such as nanofiber [62,63], metal-coated nanostructure [64,65], erbium dopants in a cryogenic resonator [66], cold atoms in microcavity [27][28][29][30], and so on. Some of them possess unique nature, like superradiance for atoms [67] and quantum state controlled emission [68]. Nowadays, traditional CQED study starts to combine with micro-or nanophotonics. Here we mainly focus on the spontaneous emission in whispering gallery microcavities, PCs, plasmonic structures, metamaterials, and hybrid photonic structures.

Spontaneous emission in whispering gallery microcavities
Whispering gallery modes (WGMs) are optical modes in which light is confined by continuous total internal reflection in microcavities with circular symmetry [Figs. 2(a, b)] [69][70][71][72]. The study of WGMs started from the research of acoustic wave propagation phenomena in the dome of St Paul's Cathedral in 1910 [73], where one could always hear the whisper of the others at anywhere of the gallery if they all stood near the wall. Since then, electromagnetic WGM microcavities with various shapes were found [74], such as spheres, pillars, disks, and toroids [69]. For a typical 2D WGM microcavity, light propagates in a circular pattern along the cavity wall, so only those modes satisfying mλ = n e L can be self-reinforced, while others are eliminated [Figs. 2(c,d)]. Here, m is an integer angular mode number, λ is the vacuum wavelength, n e is the effective refractive index, and L is the circumference of the microcavity. Some characteristics of WGMs have also been observed, like radiation pressure enhancement effect [75] and structure resonances [76]. The first WGM laser was realized in 1961 [77], and then low-threshold version was developed for integrated optical chip in 1991 [78].
Compared with traditional Fabry-Perot cavities, WGM microcavities have higher quality factor Q, smaller mode volume V, longer photon lifetime, and larger optical density [69,71,[79][80][81][82]. For example, WGMs of silica microsphere provide Q of ∼ 10 9 and small mode volume [79]. Normally, quality factors Q of microcavities are not affected by emitters. However, in cases of emitters with large size, emitters' permittivity makes influence in optical mode of cavities, which leads to the change of quality factors Q [69,83]. According to Eq. (3), higher quality factor Q and smaller mode volume V lead to larger Purcell factor. Therefore, enhanced spontaneous emission can be easily realized in WGM cavities Fig. 2 The basic concepts of WGMs. a Schematic of WGM toroid microcavity [3]. b Schematic of self-reinforced light by total internal reflection [69]. c The typical electric field distribution of WGM in a microsphere [70]. d The radial field distribution of mode shown in (c) [70]. The x-axis in (d) represents radius distance from center, and the unit is μm [10,80]. For the applications in single photon sources [3,[84][85][86][87][88], we concentrate on several typical structures consisting of WGMs and quantum emitters [3,10,11,[89][90][91][92][93].
The first experiment of spontaneous emission of single quantum dot in microdisk was realized in 2000, with the Purcell factor about 6 [10, 84] [Figs. 3(a, b)]. Owing to the strong temperature dependence of single-exciton resonance of the quantum dot, energy from quantum dot to WGMs could be controlled. Same phenomenon was also demonstrated in microsphere cavity [89] [Figs. 3(c, d)]. Spontaneous emission suppression was later demonstrated in similar structures using metal sidewall coatings [90]. In order to measure the spontaneous emission of WGM structures, in 2003, Vahala's group proposed fiber-taper coupling to high-Q microspheres [92]. Using a combination of lithography, dry etching and a selective reflow process, they achieved Purcell factor to 10 5 in on-chip silica toroid-shaped microresonators [ Fig. 3(e)] with ultrahigh Q/V [ Fig. 3(f )] [11,91]. In addition, strong coupling behavior in WGM cavities has also been well developed, where periodical energy exchanging between WGM and emitter leads to Rabi splitting [94][95][96][97].
Single photon source is one of the most important applications of the Purcell effect in microcavities [98]. The pioneering work of single photon source was done in 2000 by A. Kiraz et al. [84]. By using WGM cavity with the zero-phonon line of nitrogen-vacancy (NV) center, Purcell factor of 12 could be reached, with 25% zero-phonon line emission of diamond [88]. Therefore, by enhancing spontaneous emission and collecting single photons, WGM microcavities can realize an efficient integrable single photon source. . c A glass microsphere doped with CdSe nanocrystals in a thin surface shell [89]. d The radiative lifetime of CdSe quantum dots with and without WGM structure [89]. e A high Q/V toroid microcavity on a chip [11]. f Measured Q/V ratio of toroid microcavities by varying principal diameter D [11] Besides single photon sources, spontaneous emission of WGMs has also been applied in single-photon transport [99], quantum many-body simulation [100], and quantum gate [101].

Introduction to PCs
Photonic crystals impose periodic modulation on the refractive index in the scale of wavelengths [Figs. 4(a, b)], allowing Bloch modes for photons similar with electrons in crystal lattices [106]. In the reciprocal space, Bloch modes form photonic band diagrams The basic concepts of PCs. a A schematic diagram of a 2D PC slab [102]. b A schematic diagram of 3D PC [103]. c A photonic band diagram of a 2D PC slab [104]. d The electric field in a L3 cavity by shifting the holes at the cavity edges [12]. e The electric field of a double heterostructure PC waveguide [105] while the frequency range without Bloch modes is called the photonic band gap (PBG) [ Fig. 4(c)]. Thus, undesired radiation can be suppressed in the PBG by the deliberate design of PCs [107,108]. The ability of radiation suppression differs in various PC structures [102,104]. Only the PBG in 3D PCs can be complete [103], where optical modes are prohibited in all wavevectors and polarizations. With easier fabrication for on-chip devices, 2D PCs show higher potentials in applications despite of incomplete band gap.
Defect modes in the PBG [109,110] are commonly utilized as optical microcavities, which can be introduced by removing lattice units [109,111] and changing size or refractive index [112]. These modes possess narrow linewidths due to the suppression of other undesired modes, where electric field is confined in the subwavelength scale [3]. PC microcavities usually possess 10 3 ∼ 10 5 of Q-factors and moderate V (∼ (λ/n) 3 ) [12, [112][113][114][115][116][117][118][119]. The most common design of photonic crystal microcavity is L3 PC microcavity [ Fig. 4(d)], which is produced by removing 3 holes along a row of hexagonal lattices and changing the geometry of several neighbouring holes. Such design avoids the abrupt change at the edge of the microcavity that undesired scattering can be suppressed [12, [120][121][122]. Another scheme of defect modes is to remove the entire row of hexagonal lattices, called W1 defect waveguide [123], which allows light propagation in the line defect. Quality factor Q can be further improved by special modification of PC structures [13], especially in double-heterostructure nanocavity [ Fig. 4(e)] with Q ∼ 6×10 5 [105]. Thus, the PC microcavity with high Q and small V enables the enhancement of spontaneous emission.
PC structures can be fabricated by up-to-date nanotechnologies. Due to the difficulty of the 3D nanofabrication, 2D PCs have occupied the most attention [15,124,125]. To optimize the radiation effect in defect modes, the frequency and spatial position of emitters should be accurately determined. The frequency between cavity and emitters should match, which can be realized by tuning temperature [126], inert gas [127] and electric field [128]. The spatial position of emitters requires positioning technology [5,[129][130][131]. The DNA origami technology provides an effective way with accurate positioning of 10-nm solution [132].
In recent years, some novel PCs with unique optical dispersion relationship have been developed, such as index-near-zero materials [133] and Dirac/Weyl-type PCs [134][135][136][137], which are different from bandgap PCs. These PCs with new characteristics, like long-range dipole-dipole interactions [135,136], may lead to more interesting results of spontaneous emission.
The Purcell enhancement can be realized easily in PC defect cavities. In the defect cavity of square lattices [15], Purcell factor reached 76 and the delay time after the pump in laser response was cut down to a few picoseconds. Thus the early work of single photon sources in PC was conducted in the inverse opal 3D PCs with 5-fold Purcell enhancement [Figs. 5(a, b)] [162]. The Purcell effect also provides possibilities for tailoring polarization by air holes of PC defects [164][165][166]. The single photon emission with linear polarization was achieved in the point defect by designing a PBG only for transverse-electric-like modes [125].
Compared to defect cavities, PC waveguides have the advantage in the high coupling efficiency [163,167,168]. Emitted photons can be coupled into the waveguide mode, especially one-way PC waveguide [Figs. 5(c, d)] supporting unidirectional light emission.  [162]. b Measured decay rates of the excited states in the inverse opal 3D PCs with different lattice parameters [162]. c Illustration of a single photon source based on a one-way PC waveguide [163]. d Effective decay rate with different positions in the structure (c) [163] Also, the 'spin-moment locking' mechanism enables unidirectional transmission [169][170][171]. The enhanced photons emitted from circularly polarized emitter can stimulate the specific transverse spin momentum to achieve the unidirectional transmission [148].
With the increase of Q values of PC microcavities, the interaction between defect modes and the emitters goes into the strong coupling regime i.e., g > κ, γ [12]. In the PC cavities with point defect, Rabi splitting has been observed [126,129,172]. Strongly coupled lightmatter systems provide anharmonic energy structure that allows photon blockade [7], which can be used in various PC microcavities for single photon sources [173][174][175][176][177][178].

Introduction to surface plasmon polaritons (SPPs)
The history of surface plasmon polariton (SPP) can be traced back to 1900s [179][180][181]. Since then, mathematical description of the surface waves had been established and loss phenomena of metallic surfaces had been recorded [182][183][184]. The researches focusing on the trivial surface modes and related phenomenon were reviewed by H. Raether et al [185]. Then the sandwich-like metallic structures with symmetric and asymmetric SPP modes were studied [186,187]. Also, more localized field enhancement nearby various metallic nanoparticles was investigated in the micro-or nanoscale [17, 18, [188][189][190][191][192]. As shown in Fig. 6, two classes of SPPs, i.e., localized SPPs supported by a metallic nanoparticle   [192]. b Electric field contours and c Extinction efficiency of dipole mode of a silver nanosphere with r = 30 nm [192]. d Schematic diagram of SPP wave at a metal-dielectric interface [17]. e The field decaying exponentially with δ m in the metal and δ d in the dielectric for SPP waves [17] The localized SPPs of metallic spheres can be obtained from Mie's theory [188,193,194]. For a small enough metal sphere, i.e., R λ, only dipole resonance is considered and its extinction cross section is described as [193,194] where V 0 = (4π/3)R 3 denotes the particle volume, d is the embedding dielectric permittivity, and m (ω) = m (ω) + i m (ω) is the metal permittivity. The dipole resonance occurs at m = −2 d , which is independent on the radius of metallic sphere. Also, more higher-order resonances can appear when the metallic sphere becomes larger [194]. Besides nanosphere, other nanoparticles such as nanorods have the similar results [195]. When localized SPP happens, most electromagnetic energy concentrates near the metallic nanoparticle [5], namely, it has very small optical mode volume V [ Fig. 6(b)]. Though the quality factor Q is not very high [ Fig. 6(c)], it will enhance the light-matter interaction with the principles of CQED. While for the SPP propagating along the metallic and dielectric surface, its dispersion relation is [17,18,185,186,188] where d and m = m + i m are the dielectric and metallic permittivity respectively. k sp is the complex surface plasmon wavevector, i.e., k sp = k sp + ik sp with This dispersion relation shows that the light cannot couple with the SPP directly because k sp > k 0 for the same frequency ω [17]. The k z in metal or dielectric medium is imaginary, so the field in this perpendicular direction is evanescent [ Fig. 6(e)] as z d and z m respectively [185] For examples, when λ = 600 nm, one obtains for silver [196] z m = 23 nm and z d = 371 nm, and for gold [196] z m = 29 nm and z d = 281 nm. Thus SPP is localized at the interface between metal and dielectric and decays exponentially [189], which causes ultrasmall optical mode volume V and can be applied to the spontaneous emission enhancement. The propagation length of SPP can be found from the imaginary part of wavevector [17], For silver in visible spectrum, δ sp ranges from 10 to 100 μm, and in near-infrared telecom spectrum δ sp can reach 1 mm [17]. This propagation length is enough to support SPP waves in well-designed subwavelength photonic devices [190,197]. Finally, as shown in Fig. 6(d), SPPs only exist for TM polarization [18,191]. All above characteristics of SPP are based on single flat interface between a dielectric and metallic medium, but the same characteristics are also found in multilayer plasmon nanostructures [18,187]. Modern nanofabrication technology especially chemical synthetic technology [198,199], makes it possible to fabricate plasmonic structures. Localized SPPs have been proven by the synthesis of nanoparticles like nanobars, nanorice, and nanorods [198,200,201]. SPP waves in waveguides with various shapes such as nanoscale metallic cylindrical cable [202] and metal-insulator-metal structure [203] have also been studied. Hybrid integration of nanowires makes the propagation length of SPPs enough to function in the subwavelength photonics [197,199,204,205]. Dark plasmons in metallic nanoparticles excited by localized emitter allows for subwavelength guiding of optical energy with almost no radiative losses [206]. The interaction between semiconductor and metallic nanoparticles has been studied, which brings unique nonlinear Fano effect [207,208]. Theory of quantum optics with surface plasmons has also been constructed [209] , which opens the way to the later studies of plasmon CQED and on-chip quantum information.
Additionally, new types of lasers based on SPPs have been developed greatly, named as spasers [210].

Spontaneous emission based on SPP structures
Nowadays, plasmonic devices contribute a lot on strong and weak coupling systems of CQED by providing ultrasmall optical mode volume to enhance light-matter interaction [5-7, 220, 221]. According to localization properties of optical field, we roughly divided the studies of plasmonic CQED into three categories, i.e., localized SPP-based CQED, SPP-based CQED, and gap-plasmon-based CQED. The typical localized SPPbased CQED system [ Fig. 7(a)] is composed of a single emitter like excited molecule or quantum dot, and a metal nanoparticle such as nanorod, nanosphere, or nanocube. The total decay rate γ tot of a single emitter is obtained by considering two channels, one is the radiative loss γ rad emitting to the free space, the other is the nonradiative decay γ abs originating from Ohmic loss inside the metal nanoparticle, i.e., γ tot = γ rad + γ abs . Accordingly, the total, radiative, and nonradiative Purcell factors are obtained by γ tot /γ 0 , γ rad /γ 0 and γ abs /γ 0 , respectively. γ 0 represents the spontaneous emission in vacuum. The total Purcell factor of localized SPP system usually ranges from 10 0 to 10 4 according to the distance between the emitter and plasmon structure [222].
The spontaneous emission of an excited molecule as emitter near a small metal sphere was calculated in the early 1980s [225]. The lifetime of an emitter near metal sphere decreases as emitter approaches the sphere because of nonradiative part γ abs . Also, oscillations in the lifetime occur even at separations of a few wavelengths, which means localized SPP is excited. So far, with the help of advanced microscope technology [ Fig. 7(b)], the enhancement and quenching of emitter fluorescence coupling to noble metal nanoparticle have been investigated [195,223,[226][227][228]. In practical researches, quantum yield q a = γ rad /γ tot is the radio of radiative decay rate and represents the emission probability. The decrease of quantum yield means quenching of spontaneous emission. The local field enhancement of nanoparticle at localized SPP mode leads to an increased excitation rate whereas nonradiative energy transfer to the particle leads to a decrease of the quantum yield, which shows either fluorescence enhancement or quenching [ Fig. 7(c)]. Moreover, the continuous transition from enhancement to quenching was demonstrated with tuning the distance between emitter and nanoparticle [223]. Besides nanospheres, other shapes of particles like nanorods [195], nanorice [198], and nanodisks [229] also provide spontaneous emission enhancement based on localized SPP. To obtain the Purcell factors near the plasmon structures, people have developed various methods, including semiclassical theory [222], Greenfunction [223,230], first-principles [8], and the commercial softwares of FDTD and FEM [20].
In SPPs, evanescent waves decay exponentially perpendicular to the surface, which makes the electromagnetic modes compressed nearby the surface, thus enhances the Fig. 7 Spontaneous emission based on SPP structures. a Schematic diagram of localized SPP-based CQED system [222]. b Sketch of the experimental arrangement with SEM image of a gold nanoparticle [223]. c Fluorescence image of particle-emitter interaction with dip in the center indicating quenching [223]. d Schematic diagram of SPP-based CQED system [224]. e Purcell factor (solid curve) and efficiency of emission into surface plasmons (dashed curve) with different size of wires [224]. f Second-order correlation function g (2) (τ ) of quantum dot fluorescence using SPP-CQED system [224] light-matter interaction. The typical SPP-based CQED system [ Fig. 7(d)] is usually composed of a single emitter and an one-dimension plasmonic nanowire or two-dimension plasmonic nanofilm. In this case, the total decay rates γ tot includes three parts, first is the radiative loss γ rad emitting to the free space, second is effective part γ ev that guided into the SPP through the whole waveguide, third is the Ohmic loss γ abs into the SPP waveguide. Accordingly, the total, radiative, nonradiative, and effective Purcell factors are obtained by γ tot /γ 0 , γ rad /γ 0 , γ abs /γ 0 , and γ ev /γ 0 , respectively. Purcell factor of these SPPbased CQED systems is around 10 0 ∼ 10 1 [224,231,232], which is usually smaller than that of localized SPP-based CQED system.
In 2007, the research showed the generation of single plasmons in silver nanowires coupled to quantum dots [224]. When a single emitter is optically excited in proximity to a silver nanowire, emission photons couple directly to guided surface plasmons in the nanowire, causing the wire's end to light up [ Fig. 7(d)]. The emitter-SPP coupling is stronger for thinner wires, while the out-coupling efficiency of SPP decreases for thinner wires [ Fig. 7(e)]. Thus it's important to choose suitable size of nanowires to keep the balance between the decay rate and collection efficiency. By observing second-order correlation function g (2) (τ ) of quantum dot fluorescence [ Fig. 7(f )], it's demonstrated that the light emission at the nanowire end is a single quantized surface plasmon [231,232]. Considering the 2.5 ∼ 10 folds of Purcell enhancement [ Fig. 7(e)], SPP-based CQED system at weak coupling regime is a good candidate for single-photon source. Meanwhile, the lifetime of exciton is shortened and the broadband nature of coupling is also demonstrated. Furthermore, the SPP waveguides can act as a planar dielectric antenna if multilayer (three layers or more) plasmon structures instead of simple nanowires or nanofilms are used. When placing emitters into their middle layer, the photon collection efficiency at the vertical direction almost approaches to 100% [233,234], which can be used to realize a deterministic single-photon source. Therefore, with the advantage of efficiently collecting single photons, SPP-based CQED systems are quite useful in single-photon source, long-range optical coupling of quantum bits, and so on.
The typical bowtie nanoantenna [ Fig. 8(a)] is composed of two nearby metallic nanotriangles and a single emitter such as a molecule within the gap [16,236]. An enhancement or quenching of the photon emission relies not only on the local-field factor and the quantum yield, but also on the excitation and emission spectra of the emitter [236]. Smaller gap of a metallic bowtie nanoantenna brings stronger local-field factor and less quantum yield. When a bowtie antenna is excited at the localized SPP frequency, light is focused to a hotspot within the gap, leading to a strong enhancement of the SE. Also, the bowtie antenna can function as a low-pass filter for emitter. In the experiment, by using gold bowtie and a single fluorescent molecule with low quantum efficiency, enhancement up to a factor of 1340 is observed [ Fig. 8(b)], showing great potential for high-contrast selection of single nanoemitters [16]. Similarly, other nanoantennas of two metal particles like metal double-spheres [237] or metal double-rods [238] with nanoscale gap, also provide localized gap plasmons which can be applied for large enhancement of the SE.
Large SE enhancement can also occur in the nanogap between two plasmon waveguides, where the emitted photons can couple into guided SPPs. One example is the nonresonant enhancement of spontaneous emission in metal-dielectric-metal (MDM) slab waveguide structures [ Fig. 8(c)] [240,247]. Because of tight confinement of electric fields within the gap, these structures show the strong emission enhancement with factor of 10 1 ∼ 10 2 [ Fig. 8(d)] with almost 100% coupling efficiency to plasmon waveguides. Asymmetric gap waveguide, composed of a silver nanowire and a silver substrate separated by nanoscale dielectric bilayer of Al 2 O 3 embedding the fluorescent dyes Alq 3 [245], is also investigated [ Fig. 8(e)]. This structure with strong field confinement within the . c Schematic of metal-dielectric-metal (MDM) gap plasmon slab structure [240]. d Spontaneous emission enhancement for a MDM structure with different gap sizes [240]. e Schematic diagram of gap-plasmon-based CQED system [245]. f Time-resolved fluorescence measurements for the structure (e) [245]. g Decay rate for the structure (e) with different photonic channels [245]. h Schematic of a silver nanocube situated on a gold film [246]. i Distribution of enhancement of spontaneous emission rate for the structure (h) [246]. j Schematic diagram of silver nanorod-coupled gold nanofilm gap plasmon system with a designed nanofiber [20]. k Purcell factors for different emission paths for the structure (j) with emitter within the nano gap [20] gap can be seen as the nanocavity containing coupled emitters. The time evolution of the emission intensity is also greatly modified on or off resonance to the cavity [ Fig. 8(f )]. As shown in Fig. 8(g), smaller gap brings stronger photon emission. The Purcell factor can even approach to 10 3 when the emitters are placed within the nanogap [245].
Nanocavity consisting of metallic nanoparticles and plasmonic waveguide will be ultrasmall enough to explore more properties of CQED system including both weak and strong coupling regime due to the existence of the gap plasmon. Plenty of researches showed that these nanogaps structures can overcome the challenges of directional emission, room-temperature, broadband operation, high radiative quantum efficiency, and a large spontaneous emission rate [229,246,[248][249][250]. By a film-coupled metal nanocube system with emitters embedded in the dielectric gap [ Fig. 8(h)], SE enhancement can exceed 10 3 [ Fig. 8(i)] with high quantum efficiency (> 50%) and collection efficiency (84%) [246]. This work proves that smaller gap size brings stronger Purcell enhancement with maximum factor ∼ 2 × 10 3 . Moreover, the longitudinal and transverse modes of film-coupled metal nanoparticle system are revealed to collect the emission photons [248]. For collecting and guiding the enhanced photons directly into the on-chip optical circuits in the nanoscale, metallic nanorod-coupled nanofilm structures coupled to a low-loss dielectric nanofiber are designed [ Fig. 8(j)] [20]. Purcell factors of all these channels are shown in Fig. 8(k), with the order of magnitude 10 3 . Based on this design, by tuning the permittivity of the environment via liquid crystal, high-contrast switching of the spontaneous emission can be achieved, companying with high-efficiency extraction of the emitting photons [251]. To overcome high optical loss in the metal, some nanogaps between the dielectric and metallic structures are developed [252,253]. These kind of devices with high quantum efficiency promise an important impact on various fundamental and applied research fields, including photophysics, ultrafast plasmonics, bright single-photon sources, and Raman spectroscopy.

Plasmonic nanolasers & strong coupling
Plasmonic nanolaser is one of the key components for on-chip optical information systems based on SPP combining with simulated emission. Highly localized plasmonic mode allows lasers to work in the subwavelength scale [254][255][256][257][258]. Under the Purcell enhancement, lifetime of emitted photons is changed in well-designed nanocavity, enabling ultrafast responses of nanolasers [259]. In particular, spaser is one type of nanolasers, which can directly generate stimulated emission of quantized surface plasmons [7,259]. The principle of spaser was proposed in 2003 [260], but the large absorption in metal prevented its realization. Until 2009, the first spaser-based nanolaser overcoming loss was demonstrated [261]. Soon the dark emission of spasers was revealed [262]. So far, ultralow-threshold plasmonic lasers under continuous-wave pumping at room temperature have been realized [263][264][265]. Overall, plasmonic nanolasers have great potential in circuits, sensors, superresolution imaging and quantum information, especially chemical and biological nanoscopic sensors [254,259,262,266].
Surface plasmon combining with strong coupling regime of CQED also reaches great achievements. Different from cold-atom studies [26,30], researchers can explore strong plasmon-exciton coupling based on plasmonics at room temperature [19, [267][268][269][270][271][272][273][274], which requires ultrasmall optical mode volume. In practice, it's hard to reach g γ , κ, thus the strong coupling is usually confirmed by observing Rabi splitting of fluorescence spectrum and dynamics of reversible interaction [45,275,276]. Nowadays, strong coupling has been realized in several designs. By using special emitters such as J aggregates, Rhodamine 6G molecules, or other quantum dots, hybrid plasmonic structures enable strong coupling [230,267,268,271,272,274,[277][278][279]. Also, gap plasmon structures provide ultrasmall optical volume for strong coupling regime, such as dimer nanoantennas [269,270] and hybrid nanowires coupling with metallic film [273]. Particularly, when considering evanescent mode from 1D or 2D waveguides [280,281], the enhancement of light-matter interaction benefits from not only gap surface plasmon [19,282], but also compressed optical mode volume by evanescent waves [45,275,276], leading to strong photon-exciton coupling with efficient fluorescence collection and high sensitivity to embedding environment.
Here we mainly introduce the spontaneous emission enhancement in metasurfaces and epsilon-near-zero (ENZ) materials.

Spontaneous emission control with metasurfaces
Metasurface can be seen as a 2D counterpart of metamaterials [297][298][299][300], which was firstly realized by using subwavelength optical antennas in 2011 [ Fig. 9(a) as planar photonic devices [297], polarization conversion [298], radiation manipulation [306,307], and spatiotemporal light control [308]. Recently, metasurface has been demonstrated as a suitable platform for quantum optics [309], including quantum coherence [310,311], quantum interference [305,312], and quantum entanglement [313]. In metasurface, Purcell effect mainly depends on the coupling interaction between emitters and metasurface [314][315][316]. The first research was made by using plasmonic metasurface in 2010, resulting in the Purcell factor of 10 2 [317]. Later, metasurface made up of metal nanoparticles array was fabricated, exhibiting twofold Purcell enhancement in visible spectrum [318]. Purcell effect in metasurface was also realized by Yuan et al. [319] through designing all-dielectric metasurface composed of asymmetric geometric metaatoms. This structure with Purcell enhancement was also proven by other researchers [320,321]. Besides carefully designing metasurface, there are some other ways to enhance the photon emission in metasurface, such as constructing partially reflecting cavity [322] and changing the background refractive index by liquid crystals [323].
The metasurface's ability to modulate spontaneous emission brings novel applications in quantum interference. One fascinating example is anisotropic quantum vacuum [305] Fig. 9 The basic concepts and Purcell enhancement of metasurface. a The SEM image of typical metasurface [301]. The schematic diagrams of controlling the (b) phases [302], (c) frequencies [303], and (d) polarizations [304] of electric field by metasurface. e Metasurface controlling the light polarization for anisotropic quantum vacuum [305]. f Anisotropic Purcell factors of x-and y-polarized emitters in the metasurface (e) [305] through controlling the polarized light reflected by metasurface [ Fig. 9(e)]. Thus the local densities of optical states in x-and y-directions are different, i.e., anisotropic Purcell enhancement occurs [ Fig. 9(f )], and Purcell factors of y-polarized emitters are larger than those of x-polarized emitters. This anisotropic Purcell enhancement can be applied to obtain valley coherence [310], long-lifetime coherence [311], and coherent manipulation [312]. Also, the specific polarization of single photon emission can be generated by other designs of metasurfaces, like bullseye structures metasurface with radial phase distribution [324], and metasurface-based spin-splitting bifocal lens [304].

Spontaneous emission in epsilon-near-zero materials
Epsilon-near-zero (ENZ) material refers to the material with a dielectric constant = 0. The materials with zero-dielectric constant have been experimentally realized in various wavelengths [325][326][327][328][329], such as doped semiconductors in the near-infrared band, topological insulators in the ultraviolet, and polarized materials in the mid-infrared [327,[330][331][332]. If the propagation constant of the waveguide working at the cutoff frequency is zero, its effective propagation wavelength will be infinite, which is equivalent to an ENZ material [333].
In most photonic structures, Purcell factor is very sensitive to the position of the quantum emitter. It always needs an accurate positioning, which is difficult to be realized in the experiments. While in ENZ materials, Purcell factor is not much relative to the emitter's position because of a uniform local density of states [350]. Focused local field in ENZ plates is conductive to the emission of quantum emitter [351,352]. SE of emitters in ENZ materials can be greatly improved by Dicke superradiation effect [353]. The air cavity embedded in the open ENZ materials has two modes of radiation and non-radiation [ Fig. 10(a)], which can enhance or suppress the SE of the emitter in the cavity [ Fig. 10(b)] [354]. For the application to the single-photon source, the directional radiation can be realized within the zero-index metamaterial [Figs. 10(c, d)] [337,338,355].  [354]. b Purcell factors for different amounts of loss (ε ) in the ENZ medium in structure (a) [354]. c Sketch of optical cavity made of zero-index metamaterial [337]. d Purcell factor of directional single photon source in structure (c) [337]

Spontaneous emission in hybrid structures
In the study of Purcell effect, the assembly of hybrid structures from different basic photonic components can achieve many advances superior to those of the individual photonic subunits [5]. In the sections above, we mainly focused on the properties of spontaneous emission in micro-or nanostructures only with single photonic component. However, in the practical CQED applications, hybrid structures composed of multi-photonic-entities are required, such as gap plasmon structures combining WGMs [249] or metamaterials [250]. So far, based on various photonic components, Purcell enhancement can be obtained with novel characteristics like chirality [148] and topological protection [238].
Hybrid structures made up of PCs and plasmonic devices always attract people's attention. In a hybrid photonic structure that consists of a gold dimer placed on the top of a 1D PC cavity, Purcell factors of 0 ∼ 4.9 × 10 3 are available [356], which can be applied as a wonderful switching knob. However, it's a challenge to couple photons from a circularly polarized emitter into photonic structure to simultaneously realize strong spontaneous emission enhancement and unidirectional propagation locked by helicity of optical mode. By combining a photonic crystal and metallic nanoparticle structure [ Fig. 11(a)] to create nanocavities with both strong local-field intensity and high helicity [ Fig. 11(b)] [148], the Purcell factor of circularly polarized photons emitting into the photonic crystal waveguide reaches 148 [ Fig. 11(c)], which is one order of magnitude larger than that without the nanoparticle, and with ≈ 95% of photons propagating unidirectionally along the waveguide [ Fig. 11(c)]. This nanophotonic interface of chiral quantum electrodynamics can be applied in on-chip nonreciprocal quantum light sources, quantum circuits, and scalable quantum networks.
Topological states are some unique optical modes between the optical bands, described by topological invariants in the reciprocal space [357][358][359]. Recently, topological protection has been introduced into the Purcell enhancement with the discovery of mechanism of edge state-led mode coupling under topological protection [238], that localized surface plasmons almost do not have any influence on the edge state, while the edge state greatly changes the local field distribution of surface plasmons [ Fig. 11(e)]. Based on this mechanism, in the well-designed topological photonic structure containing a resonant plasmon nanoantenna [ Fig. 11(d)], an obvious absorption reduction in the spontaneous emission spectra appears due to the near-field deformation around the antenna induced by the edge state [ Figs. 11(f, g)]. Because a plasmon antenna with ultrasmall mode volume provides large Purcell enhancement and simultaneously the photonic crystal guides almost all scattering light into its edge state, the rate of nonscattering single photons reaches more than 10 4 γ 0 [ Fig. 11(g)]. Nonscattering large Purcell enhancement will provide practical use for on-chip quantum light sources, such as single-photon sources and nanolasers.

Summary and prospects
We have reviewed the spontaneous emission enhancement based on the CQED principle in micro-or nanophotonic structures, including whisper gallery microcavities, photonic crystals, surface plasmon metallic nanostructures, metamaterials, and hybrid structures. By carefully designed optical modes in micro-or nanostructures, spontaneous emission enhancement can be obtained with different peculiarities. Whispering gallery microcavities provide high Q, thus large Purcell factors can be obtained easily. PCs require defects and waveguides to create a cavity, which can improve photon collection but Fig. 11 Purcell effect in hybrid structures. a Schematic diagram of coupled structure consisting of PC and silver nanoparticle coupled to a chiral polarized emitter [148]. b Mechanism of spin-locked photonic propagation with chiral emitter [148]. c Purcell factors of structure (a) with high directionality [148]. d Schematic diagram of 1D topological structure including a silver nanoantenna and a quantum emitter [238]. e Mechanism of edge state-led mode coupling [238]. Purcell factors of the nanoantenna (f) without and (g) with topological structure [238] Purcell enhancement is not very large. By using novel PCs like Dirac/Weyl-type lattices, people may find Purcell enhancement with more interesting characteristics. Surface plasmon nanostructures offer large Purcell factors in the nanoscale confined space, but the metallic loss is always a barrier. As for metamaterials, metasurfaces focus on controlling the phase and polarization of emitted photons, while ENZ materials keep the emission enhancement away from influence of emitters' positions. But there are still lots of research gaps that should be filled for Purcell effect in metamaterials. Hybrid photonic structures allow some unique characteristics related to spontaneous emission enhancement superior to single photonic entity. It's worth looking forward to more unpredictable innovations on spontaneous emission enhancement in more hybrid photonic structures. Overall, achievements of spontaneous emission enhancement in micro-or nanophotonic structures can be utilized in the on-chip quantum information process, quantum states controlling, and quantum internet.
It's worth stressing that interdisciplinary between quantum science and micro-or nanophotonics has become the engine of innovation with the state-of-art nanotechnology. Single-photon source based on Purcell enhancement is the basic building block of on-chip quantum information, which can be applied in quantum gates [360], quantum nodes [4], nanolasers [361], and so on. Also, brand-new characteristics from hybrid structures are introduced to single photons especially topological protection [238,[362][363][364][365]. By designing open symmetric cavities with both gain and loss mechanisms [366], one could find amazing additional nature of single photons enhancement at exceptional points in PT-symmetry systems. Besides CQED, more and more quantum areas have been combined with micro-or nanophotonics. For quantum beam splitters [367], if the enhancement of photons exists, it may be easier to observe the Hong-Ou-Mandel interferometry effect, which will provide new way to engineer the quantum states. For quantum holography [368], enhanced entangled photons may lead to the better resolution and less noise. For quantum key distribution [369], it may help lower symbol error rate so that the efficiency of quantum information transport can be improved. Therefore, the subject integration between quantum science and micro-or nanophotonics will be an unstoppable trend.

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Competing interests
The authors declare that they have no competing interests. Tame