Selective far-field addressing of coupled quantum dots in a plasmonic nanocavity

Plasmon–emitter hybrid nanocavity systems exhibit strong plasmon–exciton interactions at the single-emitter level, showing great potential as testbeds and building blocks for quantum optics and informatics. However, reported experiments involve only one addressable emitting site, which limits their relevance for many fundamental questions and devices involving interactions among emitters. Here we open up this critical degree of freedom by demonstrating selective far-field excitation and detection of two coupled quantum dot emitters in a U-shaped gold nanostructure. The gold nanostructure functions as a nanocavity to enhance emitter interactions and a nanoantenna to make the emitters selectively excitable and detectable. When we selectively excite or detect either emitter, we observe photon emission predominantly from the target emitter with up to 132-fold Purcell-enhanced emission rate, indicating individual addressability and strong plasmon–exciton interactions. Our work represents a step towards a broad class of plasmonic devices that will enable faster, more compact optics, communication and computation.

i y y y y y y y y φ  are not rigorously in-phase. Therefore, the phase difference in Supplementary Equation 7 is not exactly 0 and the local field constructive interference at Qj is not optimal.
The combination of the destructive interference at one QD Qi and the constructive interference at the other QD Qj leads to selective excitation of Qj with high excitation selectivity. The condition (excitation polarization parameters θ and ϕ ) for optimal selective excitation of Qj is just the condition for optimal suppression of the excitation of Qi, but not the condition for optimal excitation enhancement for Qj. With the condition for optimal selective excitation, the excitation enhancement for Qj may not be optimal. As the amplitudes of the very weak x-and z-components determine how small ( ) Q E i can be suppressed to, they also determine the best excitation selectivity we can get.
Selective excitation of Q1 and Q2 can be realized in a broad range of wavelengths (by illuminating with wavelength-dependent elliptically polarized light) covering non-resonant and resonant excitations.
Here we present selective excitation of Q2 (excitation suppression of Q1) in a broad range of wavelength.
The source-normalized field amplitude spectra of the x-and z-component electric field at Q1 are shown in Supplementary Fig. 6a. The source-normalized field amplitude spectra of the y-component electric field at Q1 are shown in Supplementary Fig. 6b. In the broad range of wavelength, the y-component is dominant. The phase spectra of the y-component electric field at Q1 is shown in Supplementary Fig. 6c.
Then the polarization parameters ( , ) θ ϕ for optimal excitation suppression of Q1 at different wavelengths can be obtained using the equations We can see from the electric field vectors that the electric fields around the end cap of a GNR are roughly normal to the surface of the GNR. Therefore, at the QDs in the U-shaped gold nanostructure, either the electric field contributed by the y-oriented GNR (G1 for Q1 and G2 for Q2) or the electric field contributed by the x-oriented GNR (G3) should be roughly y-oriented with very weak x-component.
Moreover, we view the U-shaped gold structure as two parts, one composed of G3 that is x-oriented and the other composed of G1 and G2 that are both y-oriented. And we simulate the electric field distributions separately for these two parts as shown in Supplementary Fig. 7. At the locations where the QDs should reside in the U-shaped nanosystem, the x-component electric fields are very weak for both structure parts and for both x-and y-polarized excitations (see Supplementary Fig. 7b,d,f,h).
The second key point is why the phase relation between the local fields at Q1 and Q2 is anti-phase when excited with x-polarized light while it is in-phase when excited with y-polarized light. When excited with x-polarized light, only the x-oriented GNR G3 is excited directly. The plasmonic oscillations in the y-oriented GNRs G1 and G2 are induced by the plasmonic oscillation in G3 through capacitive coupling with the right end and left end of G3, respectively. Therefore, the electric displacement vectors in G1 and G2 are in the opposite direction (as shown in Supplementary Fig. 5e). The electric field at Q1 is contributed by G1 and G3, while the electric field at Q2 is contributed by G2 and G3. Since the electric displacement vectors in G1 and G2 are in the opposite direction, the electric field contributed by G1 at Q1 and the electric field contributed by G2 at Q2 should be in the opposite direction. Moreover, the electric field contributed by G3 at Q1 and the electric field contributed by G3 at Q2 are also in the opposite direction ( Supplementary Fig. 7a). Therefore, the total electric field at Q1 and Q2 are in the opposite direction and the phase of the y-component at Q1 and Q2 are anti-phase as shown in Supplementary Fig. 5d. When excited with y-polarized light, the y-oriented GNRs G1 and G2 are directly excited. Therefore, the electric displacement vectors in G1 and G2 are in the same direction (as shown in Supplementary Fig. 5j). Although G3 is capacitively coupled with G1 and G2 at its two ends respectively, the induced current in G3 by G1 and that by G2 counteracts with each other and therefore the plasmonic oscillation in G3 is very weak as shown in Supplementary Fig. 5j. Therefore, the electric field at Q1 is contributed only by G1 and the electric field at Q2 is contributed only by G2. Since the electric displacement vectors in G1 and G2 are in the same direction, the electric field at Q1 and Q2 should be in the same direction and the phase of the y-component at Q1 and Q2 are in-phase as shown in Supplementary Fig. 5i.

Supplementary Note 3 -Purcell effect for QDs in the plasmonic nanostructure
The analysis of Purcell effect for a QD in a plasmonic nanostructure is a non-trivial problem, because the exciton in a QD is not a simple dipole, but has complicated fine structure levels 1 . Each fine structure level has its decay rate and transition orientation 2 . Therefore, the fluorescence decay or Purcell effect for a QD is influenced by its intrinsic fine structure and its orientation with respect to the plasmonic nanostructure. The Purcell effect determines the energy transfer from the excited QD to the plasmonic mode. The plasmonic mode further determines the polarization of photon radiation.
In the following, we assume that the QD is weakly excited so that the probability of excitation of biexcitons or multiexcitons can be neglected and we only need to consider the decay of monoexcitons.
We also assume that thermalization is much faster than decay dynamics, even with Purcell effect, so that the decay dynamics can be described with an effective decay rate 3 .

Influence of fine structure on Purcell effect
The intrinsic decay rate of a QD is the effective decay rate 3 where the first term is from the dipole-allowed modes, the second term is from the dipole-forbidden, phonon-assisted modes, ρ i ( ρ j ) is the population probability of the exciton state is the decay rate of the exciton state Here 0 0 At room temperature, there is always some dipole-allowed transition state considerably populated, therefore the emission from dipole-forbidden decays is negligible due to their extremely slow decay rate (~1 μs − ) as compared with that of dipole-allowed decays (~1 ns − ) 3 . Therefore, the effective decay rate can be simplified as When the QD is coupled to the plasmonic nanostructure, the decay rate becomes where we can see that the effective Purcell factor is influenced by the intrinsic fine structure. The decay probability from state i is 0 0 from which we can see that Purcell effect can influence the decay probabilities.  Fig. 8a). The decay rate can be expressed as where we have combined the degenerate states since The combination of the transitions from 1L ± (or 1U ± ) correspond to a 2Ddipole in the plane perpendicular to the c-axis of the nanocrystal (Supplementary Fig. 8a) 2,4 . The decay probability from these 2D-dipole transitions is while the decay probability from the linear dipole transition along the c-axis is When QD is coupled to a plasmonic nanostructure, the decay rate of the QD becomes Then the effective Purcell factor that is experimentally measured can be expressed as ρ γ ρ γ ρ γ γ γ ρ γ ρ γ ρ γ from which we can see that the effective Purcell effect is influenced by the intrinsic decay probabilities from the 2D transition dipole and the c-axis transition dipole. For spherical QDs 0 0U p is much smaller than 0 2D p , hence the c-axis is also called 'dark axis' 2 . With the Purcell effect, the decay probability from these 2D-dipole transitions is modified to while the decay probability from the linear dipole transition along the c-axis is modified to From Supplementary Equations 20 and 21, we can see that the 'dark axis' may be made bright by the Purcell effect, as long as the Purcell factor 0U f is much larger than 2D f .

Influence of orientation on Purcell effect
The Purcell factors 2D f and 0U f are influenced by the orientation of the QD in the plasmonic nanostructure, and can be determined as: f , z f and the intrinsic decay probabilities projected to x-, y-and z-axis: where 0 For a QD oriented as shown in Supplementary Fig. 8a, the intrinsic decay probabilities projected to x-, y-and z-axis are ( ) The Purcell factors x f , y f and z f at Q1 and Q2 are numerically calculated and plotted in Supplementary Fig. 8b,c. With the Purcell effects, the decay probabilities projected to x-, y-and z-axis (26)

Simulated and measured Purcell effects
Since 0 0U p is much smaller than 0 2D p , the effective Purcell factor f is maximal when the 'dark axis' is in the xz plane and is minimal when the 'dark axis' is along the y axis. From simulation, we can see that x f and z f are much smaller than y f , thus the maximal effective Purcell factor can be approximately expressed as Then we can get the upper limit of the effective Purcell factor as 1 y 2 f , which is ~93 for Q1 and ~145 for Q2 (see y f at 808nm in Supplementary Fig. 8b). The experimentally measured effective Purcell factor is ~45±3 for Q1, which is significantly below the upper limit implying a non-optimal orientation. The experimentally measured effective Purcell factor is ~132±8 for Q2, which approaches the upper limit implying a near-optimal orientation.
As shown in Supplementary Fig. 9a, the lifetime of Q1 is shorter when both Q1 and Q2 are in the nanosystem (corresponding to the structure measured in Fig. 3 and 4) than when only Q1 is in the nanosystem (corresponding to the structure measured in Fig. 2 of the main text). We attribute this to the change of structure parameters during the process of moving Q2 into the nanosystem. As shown by the simulated Purcell factors in Supplementary Fig. 9b, both the refractive index of the silica-encapsulated Q2 (simply modelled here as a silica sphere) and the decrease of g 2 (gap width between G2 and G3) can cause a red shift of the plasmonic resonance and consequently increase the Purcell factor at the emission wavelength ~808 nm. There is also possibility that the gap width between G1 and G3 is slightly altered since pushing Q2 or G2 during the manipulation process may also move G3 through direct or indirect contact. We stress here that the influence from the existence of Q2 is due to its refractive index, but not the energy transfer. As we will analyze in Supplementary Note 4, the energy transfer rate should be much smaller than the spontaneous emission rate and therefore the existence of energy transfer should not influence the lifetime measurement.
Since the lifetime curves are nearly mono-exponential under selective excitation or selective detection and the minor decay component can be attributed to the other QD due to the finite selectivity (see Supplementary Note 6 for the exponential fitting of the lifetime curves), the decay of each QD can be regarded as mono-exponential. When only Q1 is in the nanosystem, the measured lifetime curve is indeed mono-exponential ( Supplementary Fig. 9a). With strong Purcell effects, the decay dynamics of the QDs remains mono-exponential. The mono-exponential decay behaviours are expected in our measurement, for two reasons. First, the QDs are weakly excited, so the probability of excitation of biexcitons or multiexcitons can be neglected and the measured decay dynamics is of monoexcitons.
Second, at room temperature the decay dynamics is still much slower than thermalization, so the decay dynamics can be well described with an effective decay rate 3 , which is consistence with our analysis  1f) for Q1. This plasmonic mode further determines the far-field emission polarization. Therefore, as long as the Purcell factor for Q1 (Q2) is much larger than 1, the orientation of Q1 (Q2) will not influence its emission polarization. Its emission polarization should be the same as that of a y-oriented 1D dipole emitter at its location. To numerically calculate the far-field polarization, we use a y-oriented 1D dipole at the location of the QD and simulate the near field distribution in a plane slightly below the plasmonic nanostructure and perform a far-field projection routine. The calculated far-field polarizations are in good agreement with the experimental results, as shown in Fig. 2b and Fig. 3b.
It is the distinct locations in the nanosystem that makes Q1 and Q2 couple to distinct plasmonic modes and consequently radiate with entirely different polarization states (roughly orthogonal to each other). To confirm this, we re-assemble the structure by moving the GNRs (Q1 is not moved) so that Q1 locates at the gap between G2 and G3 (which is Q2's location before the re-assembly) as shown by the AFM image in the lower inset in Supplementary Fig. 10 (the tiny bump on G3 is a fragment that sticks during the manipulation). When Q1 is at the gap between G1 and G3 (i.e., before the re-assembly), the polarization angle is ~46° (blue experimental data points and simulated continuous curve in Supplementary Fig. 10 or Fig. 2b). When Q1 is at the gap between G2 and G3 (i.e., after the re-assembly), the polarization angle changes to ~132° (red experimental data points and simulated continuous curve in Supplementary Fig. 10), which is very near the polarization angle of Q2 shown in Fig. 3b (for comparison, also shown with green 'x'-shaped data points in Supplementary Fig. 10).
The GNR farthest from the QD affects the far-field polarization. Before G2 is added, the far-field emission polarization of Q1 is linearly polarized with a polarization angle of ~52° (red experimental data points and simulated continuous curve in Supplementary Fig. 11a), while after G2 is added, the polarization angle rotates to ~46° (blue experimental data points and simulated continuous curve in Supplementary Fig. 11a). The electric field coupled to G2 (comparing field profile in panel c with that in panel b in Supplementary Fig. 11) is roughly anti-phase with the electric field in G1 and thus reduces the y-polarized component in the far-field radiation, which explains the slight rotation of polarization angle.

Supplementary Note 4 -Energy transfer between the QDs
For simplicity, here we model the emitters as point dipoles, one as the donor and the other as the acceptor.
Further more, the dipoles are assumed to have the optimal orientations for energy transfer. For dipoledipole energy transfer at subwavelength distance, the optimal orientation is pointing from donor to acceptor, as shown in the lower two insets in Supplementary Fig. 12a. For energy transfer with the plasmonic nanostructure, the optimal orientation is the y-direction, as shown in the upper inset in  Supplementary Fig. 12b. At the wavelength of ~808 nm (i.e., the emission wavelength of the QDs used in our experiment), the enhancement factor is ~540. With this enhancement, the energy transfer rate between the two dipole emitters (~61 nm apart) coupled with the gold nanostructure will be the same as that between two dipole emitters ~22 nm apart (without the gold nanostructure), as shown by the intersection of the red solid curve and the blue solid curve at ~808 nm in Supplementary Fig. 12a.
In the experiment, the situation is more complex. Most importantly, the QDs has to be modelled as a combination of a 2D dipole and a 1D dipole, and the orientations of the QDs may deviate from the ideal case, which will significantly reduce both the Purcell effect and the enhancement factor of the energy transfer rate. For simplicity, the reduction can be effectively attributed to the orientation deviation of a 1D dipole from the y direction. Suppose that the donor (acceptor) dipole emitter D μ ( A μ ) deviate from the y direction by an angle of D θ ( A θ ). Recall that the Purcell factor while the enhancement factor for the energy transfer rate In the nanosystem, for both ( ) To roughly estimate the enhancement of energy transfer rate, we compare theoretical Purcell factors with the experimentally observed Purcell factors. The theoretical Purcell factor (for the emission wavelength of ~808 nm) is ~188 for the dipole emitter 1 μ and ~290 for the dipole emitter 2 μ ( Supplementary Fig.  12c), while in the experiment we observe a Purcell factor of ~45±3 for Q1 and a Purcell factor of ~132±8 for Q2, which are reduced with a factor of ~4.2 and ~2.2 respectively. Then we can roughly estimate that the enhancement factor of the energy transfer rate is reduced from the theoretical value (~540) by a factor of ~9.2 (the product of the reduction factors ~4.2 and ~2.2) to ~59.
Although the energy transfer rate is expected to be strongly enhanced in the nanosystem, it is still much smaller than the enhanced spontaneous emission rates and therefore, considering the competition between the energy transfer and the spontaneous emission of the donor 6 , the energy transfer efficiency is so low that we can safely neglect the energy transfer in our experiment. For dipole-dipole energy transfer in homogenous free-space, the energy transfer rate ET γ decays rapidly with donor-acceptor distance R and can be expressed as where 0 γ is the intrinsic decay rate of the donor (in the absence of the acceptor) and 0 R is the Förster radius. 0 R is typically in the range of 2-9 nm 7 . Since the energy transfer rate is enhanced to be the same as that between two dipole emitters ~22 nm apart, we can estimate the energy transfer rate to be 6 0 ET 0 9 nm 22 nm 213 At the same time, the spontaneous emission rates of both dipole emitters are also strongly enhanced as shown in Supplementary Fig. 12c Therefore, the energy transfer efficiency i E (i denotes that i μ is the donor), defined as the fraction of energy transferred to the acceptor compared to the total energy released from the decay of the donor is estimated to be lower than

Supplementary Note 5 -Finding optimal polarization for selective excitation and optimal polarizer angle for selective detection
To perform far-field selective excitation and far-field selective detection, the key task is to experimentally find, for each QD, the optimal polarization for excitation suppression and optimal polarizer angle for emission blocking.
When only one QD is in the system, we find the optimal excitation polarization for excitation suppression by successively searching the elliptical polarization parameters ϕ and θ to minimize the emission intensity. First, we set θ to a moderate value 0 θ (e.g., 45  ) and scan ϕ to find the optimal ϕ that minimizes the emission intensity. Then, we set ϕ to this optimal ϕ and scan θ to find the optimal θ that minimizes the emission intensity.  Supplementary Fig. 6d-f) in agreement with those given by direct calculation (solid curves in Supplementary Fig. 6d-f), and slightly better results at wavelengths shorter than 710 nm where the dominance of the y-component degrades.
When both QDs are in the system, if we use the emission intensity as the figure of merit for optimization of the excitation polarization, there is mutual dependence between determination of the optimal polarization for selective excitation and determination of the optimal polarizer angle for selective detection. To find the optimal excitation polarization to suppress either QD, we have to selectively detect the QD to minimize its emission intensity. To find the optimal polarizer angle to selectively detect either QD, we have to selectively excite the QD to measure its emission polarization.  Under this detection condition, we can then find the optimal excitation polarization by successively searching the elliptical polarization parameters ϕ and θ that minimize the detected emission intensity.
First, set θ to a moderate value 0 θ (e.g., 45  ) and scan ϕ to find the optimal ϕ that minimizes the detected emission intensity. Due to finite detection selectivity, the detected emission intensity includes contributions from both Q1 and Q2 and can be expressed as (normalized) The found optimal ϕ is denoted as ( ) 1 ϕ i . Then set ϕ to this optimal value ( ) 1 ϕ i and scan θ to find the optimal θ that minimizes the detected emission intensity (normalized) The found optimal θ is denoted as ( ) , that is, Q2 is suppressed to a certain degree and Q1 is selectively excited with a certain selectivity. Under this excitation condition, we then scan the polarizer angle α to find the polarizer angle that minimizes the detected emission. Due to finite excitation selectivity, the detected emission includes contributions from both Q1 and Q2 and can be expressed as From the found polarizer angle α , we can get the polarization angle ( 1) ψ +  ) doesn't deviate too much from the optimal value ( Q1 90 ψ +  ).
We theoretically demonstrate in Supplementary Fig. 14 the optimization process for the structure simulated in the main text. In Supplementary Fig. 14a Fig. 14a

Supplementary Note 6 -Exponential fitting of fluorescence decay curves
The spontaneous fluorescence decay curves measured when Q1 is selectively excited (blue solid data points in Fig. 4), when Q2 is selectively excited (red solid data points in Fig. 4), and when Q1 and Q2 are equally excited (yellow-green solid data points in Fig. 4), are fitted using the following bi-exponential decay function ( ) ( ) For these three fits, the lifetime parameters 1 τ and 2 τ are shared. The fitting results are as follows. When Q1 is selectively excited (blue solid data points in Fig. 4), the fitting result is When Q2 is selectively excited (red solid data points in Fig. 4), the fitting result is When Q1 and Q2 are equally excited (yellow-green solid data points in Fig. 4), the fitting result is From the fitting results, we can obtain the proportions of the decay components with different lifetimes.
The proportion of the photon counts from the decay component with lifetime 1 τ is 1 1 while the proportion of the photon counts from the decay component with lifetime 2 τ is 2 2 In the decay of the selectively excited Q1 (Q2), the minor ~3.4% (~4.8%) decay component with a lifetime of ~1.88 ns (~6.47 ns) makes the decay slightly bi-exponential. Since the lifetime of the minor decay component is equal to the lifetime of the other QD, we attribute the slight bi-exponential nature to the finite excitation selectivity.
The spontaneous fluorescence decay curves measured when Q1 is selectively detected (blue hollowed data points in Fig. 4) and when Q2 is selectively excited (red hollowed data points in Fig. 4) can also be fitted to slightly bi-exponential decay curves with the lifetimes 1.88 ns and 6.47 ns, as follows.
When Q1 is selectively detected (blue hollowed data points in Fig. 4), the fitting result is When Q2 is selectively excited (red hollowed data points in Fig. 4), the fitting result is When Q1 is selectively detected, ~97.3% of the photon counts comes from the decay with a lifetime of ~6.47 ns, while only ~2.7% comes from the decay with a lifetime of ~1.88ns. When Q2 is selectively detected, ~96.4% comes from the decay with a lifetime of ~1.88 ns, while only ~3.6% comes from the decay with a lifetime of ~6.47 ns. We attribute the slight bi-exponential nature to the finite detection selectivity.
Since the lifetime curves are nearly mono-exponential under selective excitation or selective detection and the minor decay component can be attributed to the other QD due to the finite selectivity, the decay of each QD can be regarded as mono-exponential. When only Q1 is in the nanosystem, the measured lifetime curve is indeed mono-exponential ( Supplementary Fig. 9a).

Supplementary Note 7 -Photon collection efficiency
To measure the excitation cross sections or excitation enhancement factors for QDs, the collection efficiency for QD emission is required. Here, the required collection efficiency is a relative efficiency normalized to the collection efficiency for a horizontally oriented linear dipole. The collection efficiency for QD emission is influenced by three factors: dipole orientation, structure loss and optical transmittance.
Before the QD is coupled to the plasmonic nanostructure, the detection efficiency of QD emission is influenced by the orientation of the QD (approximately a 2D-dipole; see Supplementary Fig. 8a), which can be determined by the measured polarization (Fig. 2b) and the numerical aperture of the collective objective (NA = 0.7) 8 . After the QD is coupled to the plasmonic nanostructure, the radiation pattern is tailored to a linear dipole with horizontal orientation (Fig. 2b and Fig. 3b).
Due to the loss in the plasmonic nanostructure the effective quantum efficiency of the nanosystem is less than 100% 9,10 .The measured large effective Purcell factors show that nearly all the energy from the exciton decay is extracted to the plasmonic mode excited by a y-oriented dipole ( Supplementary Fig.   8d,e). Therefore, the effective quantum efficiency, which is defined as the ratio between the counts of far-field photons and the counts of radiative recombinations of excitons, can be obtained through simulation using a y-oriented dipole. The simulated effective quantum efficiencies for Q1 and Q2 are shown in Supplementary Fig. 8f.
When a linear polarizer is inserted for selective detection, the collection efficiency for the selectively detected QD is further influenced by the transmittance of the polarizer, which can be determined according to the polarizer angle and the measured emission polarization of the selectively detected QD (Fig. 3b).

Structure of the nanosystem
To make the plasmon-mediated interaction between the emitters much faster than dephasing processes in typical solid-state quantum emitters, we modify the plasmonic nanostructure as shown in

Quantum dynamics
In this nanosystem, the plasmon dissipation is much faster than plasmon-emitter interactions, so the coupling between plasmon and emitters is in the weak-coupling regime. We can trace out the plasmonic degrees of freedom and focus on the dynamics of the reduced density matrix ρ for the emitters' subsystem, which is described by the following master equation 11,12

Re
, , (54) Since the transition dipoles of the quantum emitters in the nanosystem are set to the y-direction, i.e., is the y-component electric field at r i when an electric dipole source μ n j y is placed at r j , which is numerically calculated using FDTD simulation.
The simulated ij g and γ ij versus transition wavelength are shown in Supplementary Fig. 15b, where all the rate parameters are in units of the free-space spontaneous decay rate 0 γ , which can be expressed as ( ) At the resonance wavelength 808 nm, the interaction is purely dissipative, with a vanishing 12 g and maximal γ and 12 γ . The extremely enhanced dissipative interaction 12 γ is ~72,000 times faster than the intrinsic decay rate 0 γ of the emitters. For a typical transition dipole moment of 10 Debye, the freespace decay rate 0 γ is ~(16.8 ns) -1 at ~808 nm transition wavelength, and the enhanced dissipative interaction 12 γ reaches ~(0.23 ps) -1 . Such an interaction rate can feasibly overwhelm dephasing processes in typical solid-state quantum emitters [14][15][16][17][18] .
For ideally identical emitters or emitters with finite but small transition energy difference in realistic implementations, as long as the plasmon-enhanced interaction rates are much larger than the dephasing rates, the eigenstates of the singly excited emitter subsystem are the maximally entangled states ( )( Using the basis consisting of the singly excited states ± , the ground state  ). As we demonstrate below, this large decay rate difference leads to spontaneous generation of entanglement between the emitters.

Spontaneous entanglement generation
As an example demonstration of spontaneous entanglement generation, the system is initialized to a singly excited unentangled state 1 2 , e g (a superposition of states Since the system is initialized in a singly excited state and left to decay spontaneously without driving, the dynamics is confined in the reduced basis { } ρ +− = . Around the resonance wavelength 808nm, 12 g is vanishing, therefore ρ +− is real and the concurrence is reduced Supplementary Fig. 15e). From Supplementary Fig. 15e, we can see that both states ± decays from a population probability of 0.5. The state + decays very quickly while the state − decays much more slowly, which quickly leads to imbalance between states ± and thus induces entanglement. At

Entanglement detection
The states ± decay to different plasmonic modes as shown by the electric field profiles in Supplementary Fig. 15f,g. The field profiles are numerically calculated by coherently adding the field profiles from dipole sources 1 μ and 2 μ with phase delay 180° for state + and 0° for state − .
The plasmons from the decay of state + radiate to x-polarized photons with radiation efficiency of ~1.9% (red curve in Supplementary Fig. 15h), while the plasmons from the decay of state − radiate to y-polarized photons with radiation efficiency of ~9.3% (blue curve in Supplementary Fig. 15h).
Therefore, through polarization analysis of the photon radiation, states ± can be distinguished and the state of the system can be analyzed.