Controlling collective spontaneous emission with plasmonic waveguides

We demonstrate a plasmonic route to control the collective spontaneous emission of two-level quantum emitters. Superradiance and subradiance effects are observed over distances comparable to the operating wavelength inside plasmonic nanochannels. These plasmonic waveguides can provide an effective epsilon-near-zero operation in their cut-off frequency and Fabry-Perot resonances at higher frequencies. The related plasmonic resonant modes are found to efficiently enhance the constructive (superradiance) or destructive (subradiance) interference between different quantum emitters located inside the waveguides. By increasing the number of emitters located in the elongated plasmonic channel, the superradiance effect is enhanced at the epsilon-near-zero operation, leading to a strong coherent increase in the collective spontaneous emission rate. In addition, the separation distance between neighboring emitters and their emission wavelengths can be changed to dynamically control the collective emission properties of the plasmonic system. It is envisioned that the dynamic modification between quantum superradiant and subradiant modes will find applications in quantum entanglement of qubits, low-threshold nanolasers and efficient sensors.


Introduction
Since the Purcell effect was proposed [1], the modification of the spontaneous emission rate emitted by quantum emitters matched to a resonant cavity has attracted much attention leading to several intriguing applications, such as efficient nanolaser sources [2], optical communication devices [3], DNA analysis [4], single-photon generation [5] and sensitive optical microscopy [6]. Purcell proved that the spontaneous emission decay rate is not an intrinsic property of the emitter but largely depends on the inhomogeneity of the environment. Therefore, resonant systems, such as nanocavities [7], photonic crystals [8], nanoshells [9], plasmonic waveguides [10] and nanoantennas [11], can be introduced to enhance this decay rate. However, one major limitation towards practical emission enhancement is that it is extremely difficult to boost the total emission of an ensemble of quantum emitters. In this scenario, each emitter usually needs to be accurately placed at a specific location in the resonating system, where large and homogeneous electric field distributions exist. This directly hinders the collective spontaneous emission rate enhancement, especially in the common scenario of a collection of quantum emitters arbitrarily located inside a resonating system. This collective spontaneous emission response, also known as superradiance, can improve the directivity and coherence of the total emitted radiation by an ensemble of quantum emitters. Considerable research efforts have been devoted to the study of superradiance because of its interesting potential applications in quantum communications [12,13], narrow linewidth lasers [14], atom lasers [15] and thermal emitters [16]. The phenomenon of superradiance was originally proposed by Dicke [17], who demonstrated that the radiation intensity emitted by N atoms placed in subwavelength distances was proportional to N 2 instead of the usual N. This phenomenon is based on the constructive interference between emitted waves and has been investigated by using quantum emitters coupled to a variety of photonic environments including microcavities [18,19], metal interfaces [20], plasmonic waveguides [21,22] and left-handed media [23]. However, this effect can only occur when the neighboring quantum emitters are separated by a small fraction of the emitted radiation wavelength [12], which limits its practical applications. In addition, the counterpart mechanism of superradiance is subradiance. It is a destructive interference effect leading to suppressed emission from a collection of active particles, such as atoms [ In recent years, realistic metamaterials with effective epsilon-near-zero (ENZ) permittivity response have generated increased interest, especially due to their peculiar transmission properties that provide, in principle, infinite phase velocity combined with anomalous impedance-matching. The ENZ response has been theoretically predicted [36-39] and experimentally verified [40] using narrow plasmonic waveguides operating at their cut-off wavelength. Uniform phase distribution and large field enhancement is obtained inside the channels of these narrow waveguides. This anomalous quasi-static response is independent of the channel's length and shape and has been used to squeeze and tunnel light In this work, we demonstrate a way to obtain different collective spontaneous emission effects, such as superradiance and subradiance, excited by a collection of quantum emitters placed inside plasmonic channels. ENZ operation is obtained at the cut-off wavelength of these plasmonic waveguides and FP resonances are found at lower wavelengths. Classical electromagnetic calculations are used to compute the Purcell enhancement and radiative efficiency of a single emitter and a pair of two-level quantum emitters embedded inside the plasmonic waveguide at ENZ and FP resonances. The utilized two-level emitters are characterized by the ground state and the excited state. They modeled using the point-dipole approximation, assuming weak excitation (no saturation) and operation in the weak coupling quantum regime [46]. Strong superradiance is obtained at the ENZ wavelength that is independent of the emitters' distance. It can be achieved without the usual constraint of subwavelength distance between emitters and can be obtained even at emitter distances on the order of wavelength. In addition, both superradiant and subradiant modes exist at higher-order FP resonances found in lower wavelengths compared to ENZ. It is demonstrated that these collective responses can be dynamically controlled by changing the emitters' separation distance, location, and frequency of operation. We also consider the collective emission of N emitters uniformly located inside the plasmonic channel at the ENZ wavelength. In this case, the collective decay factor is amplified by almost N times compared to a single emitter's decay rate. This is typical response of systems exhibiting superradiance [29,47]. We also theoretically compute the power time-dependent decay curves for different number of emitters. These results directly correspond to power lifetime measurements that can be obtained in an experimental verification of the proposed plasmonic superradiance effect. Finally, we provide insights on the time-dynamics of the proposed collective emission effects. The proposed quantum plasmonic system is envisioned to have several applications in quantum communication and computing systems on a chip, such as low-threshold nanolasers [35], quantum memories [32], and ultrasensitive optical sensors [48,49].

Optical response of plasmonic waveguides
The geometry of the proposed plasmonic grating unit cell is shown in Fig. 1. A narrow rectangular slit is carved in a silver (Ag) screen, whose permittivity dispersion follows previously derived experimental data [50]. The slit has width w = 200 nm, height t = 40 nm and length l = 500 nm. It is loaded with silica with relative permittivity of= 2.2. This structure was originally introduced before [43] for nonlinear applications and can sustain ENZ and FP resonances. The width w is designed to tailor the cut-off wavelength of the dominant quasi-TE 10 mode propagating inside each slit. At this frequency point, the plasmonic waveguide behaves as an effective ENZ medium and this anomalous impedancematching phenomenon leads to total transmission combined with large field enhancement and uniform phase inside each slit [39]. This effect is independent of the grating's periodicity or channels' length l [43]. In this work, the grating period was chosen to be a = 400 nm and b = 400 nm but similar effects are expected from an isolated plasmonic slit. Since the slits occupy a very small area on the grating surface, the incident waves are mostly reflected at the first interface. However, optical transmission can occur around the cutoff frequency and at higher FP frequencies leading to minimum reflection and maximum transmission. The plasmonic grating is illuminated by a normal incident z-polarized plane wave shown in Fig. 1(a). The computed transmittance is shown in Fig. 2(a) as a function of the incident radiation wavelength. The transmission peak at = 1012 nm corresponds to the cut-off frequency of the dominant quasi-TE 10 mode. The field distribution normalized to the incident wave is homogeneous and enhanced along the channel at this wavelength [ Fig. 2(b)]. As it was expected, the plasmonic waveguide effectively behaves as ENZ material at the cutoff wavelength. For longer wavelengths above the cut-off value, the incident wave is totally 6 reflected by the structure and the transmission is zero. Below the cutoff wavelength, an additional transmission peak appears at = 922 nm corresponding to the first-order FP resonance. The field enhancement distribution at the FP resonance is shown in Fig. 2(b) where a typical standing wave pattern is obtained. The grating is excited by a plane wave impinging at normal incidence but the ENZ operation will be unaffected even at oblique incidence. The ENZ transmittance is 50% due to increased losses at near-infrared (near-IR) coming from the silver waveguide walls. The ENZ peak is slightly lower compared to the FP resonance peak due to the uniform field distribution at the ENZ wavelength, making the optical absorption from the silver walls more effective. The ENZ transmission can be further increased, in case we reduce the waveguide's length, without affecting the ENZ performance.

Spontaneous emission of single emitter in plasmonic waveguides
We compute the spontaneous emission rate excited by a single two-level quantum emitter embedded in the narrow plasmonic waveguide shown in Fig. 1(c). Assuming weak excitation (no saturation) and operation in the weak coupling regime, the emitter can be modeled using the point-dipole approximation [  the permeability of free space and  the relative permeability of the surrounding space.
In plasmonic systems, the total spontaneous emission is decomposed into its radiative (energy transferred into the environment) and non-radiative (associated with system losses) contributions [52] and reflects the radiative emission efficiency [52]. It quantitatively describes how much radiation can escape from the plasmonic system to the surrounding space and is nearly independent of emitter's intrinsic quantum yield [53]. The emitter is assumed to be vertically oriented (z-axis) that guarantees a maximum coupling with both ENZ and FP modes of the plasmonic structure. In free space, a point dipole emitter has LDOS equal to    Fig. 3(a)]. The radiative emission efficiency, quantified by the QY, reaches high values of 0.7 and its distribution is also uniform [ Fig.  3(b)]. On the contrary, at the FP resonance, both emission enhancement and QY are largely dependent on the location of the dipole emitter along the plasmonic channel, as it is shown in Figs. 3(c) and (d), respectively. This is consistent with the electric field standing wave distribution shown before in Fig. 2(b). Additionally, the maximum spontaneous emission rate enhancement is lower at the FP wavelength compared to the ENZ response. Furthermore, the plasmonic waveguide demonstrates a highly directional radiation pattern at the ENZ resonance, which is computed and shown in Fig. 4, assuming that the dipole emitter is vertically oriented inside the nanochannel. The directional radiation pattern is independent of the emitter's position inside the waveguide at the ENZ wavelength. Directional radiation is very important for designing the future integrated optical communication nanodevices.

Collective spontaneous emission of multiple emitters in plasmonic waveguides
Next, we consider the collective spontaneous emission properties and the coherent interactions from a pair of two-level quantum emitters to investigate the effect of superradiance and subradiance. They are placed at the center of the plasmonic waveguide channel along the y-axis and their distance can change [see Fig. 1(d)]. Again the point-dipole approximation is used in these calculations, which is valid for small emitters operating at the weak coupling regime. In addition, the emitters are oscillating in-phase and are made of the same material. This is a typical behavior of molecules inside active bulky media, such as fluorescence materials. We assume that the location of the first emitter i with dipole moment i μ is fixed at position i r (near the channel's edge). The second emitter j with identical dipole moment j μ is placed at position j r , which varies along the y-axis. Both emitters oscillate in phase and are separated by distance d. In this case, the non-local density of states (NLDOS) is computed in order to define the resultant density of states due to interference caused by the second emitter to the first one and inverse. The total decay rate based on the NLDOS is [21]: where i  μ is the complex conjugate of the transition dipole moment of emitter i. Therefore, 12  represents the contribution to the decay rate of emitter 1 at position 1 r due to interference caused by emitter 2 located at position 2 r . Hence, a normalized decay factor is defined to quantify the modification of the collective decay rate due to interference from another emitter: 11 12 22 21 11 22 .
We numerically calculate the normalized total decay factor  . The result is plotted in Fig.   5(a) versus the two emitters' separation distance d at the ENZ (= 1012 nm/ black curve) and FP (= 922 nm/red curve) resonances, respectively. The green dashed line refers to the end of the plasmonic channel. Outside of the slit is assumed to be free space and  naturally converges to one after a small distance from the waveguide's edge. In this case, the second emitter is located outside of the waveguide and the coupling between the emitters due to the plasmonic waveguide modes ceases to exist. Superradiant (subradiant) modes are excited when the normalized decay factor  is larger (smaller) than one due to constructive (destructive) interference between the emitters operating at ENZ and FP resonances. The decay factor  is also calculated when the two emitters are uncoupled and placed in free space [blue curve in Fig. 5(a)]. Note that the normalized total decay factor curves tend to the same limit: This detrimental property severely limits its practical applications. However, subwavelength distance is not needed in order to obtain superradiance when emitters are placed inside plasmonic channels operating at the cut-off (ENZ) wavelength. In addition, the ENZ superradiance emission is directional, as it was shown in Fig. 4. Strong superradiance can also be obtained when the channel's length is increased, which is not going to affect the ENZ response. Therefore, ENZ can extend this interesting effect to regions comparable and even larger to the wavelength. In this case, more emitters can be incorporated inside the nanochannel, leading to much stronger superradiant emission that is strongly directional in space and time [12]. The strong and uniform field enhancement at ENZ resonance [ Fig. 2 is responsible for this effect. Hence, the ENZ mode leads to perfectly coherent interactions between separate emitters. of the emitter's separation distance, consistent with the results in Fig. 5(a). On the contrary, at the FP resonance (= 922 nm), the collective emission properties of the plasmonic system can be dynamically controlled and changed from subradiant state ( 0   ) to free-space state ( 1   ) depending on the emitter's distance. Note that subradiance can only be observed at fixed emitter's distance operating at the FP resonance mode, consistent with the standing wave distribution shown in Fig. 2(b). We also compute the collective emission enhancement from an arbitrary large number of quantum emitters N uniformly distributed in the narrow waveguide. In the case of two quantum emitters placed in the channel operating at the ENZ resonance, the normalized decay factor given by Eq. (5) has been calculated to be 2   (Dicke's perfect superradiance).
Similarly, when N quantum emitters are placed in the plasmonic ENZ system, they will exhibit perfect superradiance and their normalized collective decay factor can reach the maximum value N   , i.e., it will be equal to the total number of emitters. This can be computed by the generalized version of Eq. (5) for N coherently interacting emitters. Note that  is analogous to the total electric field E   . The emitted radiation intensity is proportional to the square of the electric field and, as a result, it will be analogous to N 2 , as it was predicted by Dicke [17]. We numerically calculate this normalized total decay factor  excited by 100 N  emitters at the channel's yz-plane at the ENZ resonance (see Fig. 6). Uniform and strong enhancement of the total collective emission is obtained (  is very close to N inside the nanochannels), which means that all emitters in the nanochannel are involved in the collaborative superradiance effect independent of their positions. Hence, we can pack as many emitters as possible inside the waveguides and we can obtain a giant increase in the total spontaneous emission rate at the ENZ resonance of this plasmonic configuration. The total superradiance of the plasmonic system is only limited by the number of emitters (active molecules) placed inside the nanochannel. Note that the superradiant state excited by 100 N  emitters also has high directionality in the far-field (same radiation pattern as Fig. 4) in the plasmonic ENZ system, which is a basic feature of the coherent superradiance effect [55].

Time-dependent lifetime decays and time-dynamic response
Finally, we calculate the time-dependent lifetime decay curves when one, two and 100 N  emitters are placed inside the plasmonic waveguide nanochannel discussed before. The emitted power from an emitter at point r is proportional to the excitation field, the radiative rate and the total spontaneous emission. It is given from this formula [52,54]: