Probing the spectral density of the surface electromagnetic fields through scattering of waveguide photons

The spectral density of the metal-surface electromagnetic fields will be strongly modified in the presence of a closely-spaced quantum emitter. In this work, we propose a feasible way to probe the changes of the spectral density through the scattering of the waveguide photon incident on the quantum emitter. The variances of the lineshape in the transmission spectra indicate the coherent interaction between the emitter and the pseudomode resulting from all the surface electromagnetic modes. We further investigate the quantum coherence between the emitter and the pseudomode of the metal-dielectric interface.


Results
We consider a two level QE positioned close to a two-dimensional metal-dielectric interface as depicted in Fig. 1(a). The QE is coupled electromagnetically to the SP modes on the metal surface. The Hamiltonian of this QE-metal film system can then be written within the rotating-wave approximation 17  where † a k ( ) a k is the creation (annihilation) operator of the k-mode surface-EM-filed, ω k is the frequency of the k-mode surface-EM-field, and σ + σ ( ) − is the raising (lowering) operator for the QE. Here, g k describes the coupling strength between the QE and the k-mode surface-EM-field, and ħ is set to be unity throughout the paper.
A recent research 15 studied the spectra density which comprises information about the density of the surface EM-fields, and also the QE-field coupling. The results reveal that when the separation z [ 0 , as seen in Fig. 1

(a)]
Scientific RepoRts | 6:21673 | DOI: 10.1038/srep21673 between the QE and the metal surface decreases to a small distance (≤ 10 nm), the spectra density ω ( ) J k can change from smooth to the Lorentzian distribution, where, γ 0 is the spontaneous decay rate of the QE into free space. In this work, we consider the silver thin film with thickness h= 5 nm to be the metal surface. Therefore, ω = 9 p eV (for silver) is the plasma frequency, and ω 0 denotes the energy spacing of the QE. Here, c is the speed of light, γ = .
0 07 p eV (for silver) denotes the field-damping rate of the surface-EM-field into Ohmic losses, and ω = .
3 48 c eV (for silver thin film) is the main peak of the Lorentzian function.
As a result, the Lorentzian spectra density of the system allows us to map the system to the excited state of QE coherently coupled with a strength g eff to a pesudomode 15,18 with a Markovian dissipation γ ( ) eff as depicted in Fig. 1(b). The dynamical evolution of the QE is govern by the master equation 15 , H a a g a a eff e ff p p eff p p 0 describing the energy of the QE, the pseudomode ( ) ω ħ eff and the coherent interaction between them with strength g eff . Here † a p ( ) a p is the creation (annihilation) operator of the pseudomode. In Eq. (3), ρ ρ = ( ) t is the density matrix of the total system. By taking the notation ( )   A single waveguide-photon incident on the quantum emiiter enables us to detect the coherent interaction between the quantum emitter and the metal surface. In this work, we consider silver thin film with thickness h = 5 nm to be the metal surface. We have assumed the silver film is embedded into a dielectric material with ε = 1 d . Here z 0 is the separation between the quantum emitter and the silver surface, and the permittivity of the silver film with the plasma frequency ω = 9 p eV, ε = .
,∞ 5 7 m , and the Ohmic losses γ = . 0 07 p eV. (b) An illustration of the model. When the separation z 0 is small enough, the spectra density of the system becomes Lorentzian. The system can then be regarded as a quantum emitter coherently coupled to a pseudomode with dissipative Ohmic losses. (c) The ratio of the effective coupling strength g eff and the decay rate into Ohmic losses γ p as a function of the separation z 0 .
According to Eq. (3), the population shows reversible dynamics between the QE and the pseudomode which represents all the electromagnetic modes of the metal surface. Since the pseudomode is originated from the Lorentzian spectra density [Eq. (2)] of the system, its energy ω eff is exactly the main peak ω c . The effective loss γ eff of the pesudomode corresponding to the width of the Lorentzian function is the Ohmic loss γ p , while the coherent coupling strength g eff can be given by 15 , the dependence on ω 0 cancels, the effective coherent coupling strength then depends mainly on the separation z 0 . In Fig. 1(c), we plot the coupling g eff as a function of the separation, and as can be seen, it decreases with growing separation z 0 , which coincides with its near-field nature 1 . Notice that when the separation z 0 increases, the Lorentzian spectra becomes flatter, and the coherent interaction fades out accordingly 15 .
Experimentally, to observe the reversible dynamics of the system, one needs to perform the time-resolved measurements to the population evolution of the QE. However, the reversible dynamics inevitably suffers the dissipations such as the Ohmic losses. It might not be easy to observe the dynamics in practice.
Here, we propose a feasible way to detect the existence of the coherent interaction between the QE and the pseudomode through scattering of the waveguide-photon fields. We consider a waveguide coupled to the QE as shown in Fig. 1(a). A single waveguide-photon injected from the left is coherently scattered by the QE. After performing the rotating-wave approximation 17 , the total Hamiltonian of the system with the additional probing waveguide-photon fields becomes, a is the creation (annihilation) operator of the η-mode waveguide-photon, ω η is the frequency of the waveguide-photon, and η g describes the coupling strength between the QE and the η-mode waveguide-photon, which leads to the decay of the QE into the waveguide. Note that, here the QE is coupled to both the waveguide-photon-fields and the surface-EM-fields. However, since the surface-EM-field decays exponentially with distance from the surface 1,2 , the interaction between the surface-EM-fields and the waveguide is extremely small (see detailed discussion in the experimental realization part), and cannot affect the coupling between the emitter and the waveguide. The coupling strength η g can therefore be treated as a perturbation to the original QE-metal film system and can be further assumed to be frequency-independent, this assumption is equivalent to the Markov approximation 19,20 . Because we are only interested in the way the system behaves in experiments, it is sufficient to include the dissipative channels in Eq. (3) via introducing the non-Hermitian term γ − / i 2 eff in the total Hamiltonian and the "quantum jump" 21 term can be neglected.
The scattering eigenstate state of the above combined system can be written as 5,22 : where −, | 0 0 p describes that the QE is in the ground state with no excitation in the pseudomode and the photon field state, while ξ ξ ( ) p is the probability amplitude that the QE (pseudomode) absorbs the excitation. We also assume that the field is incident from the left of the waveguide, φ ( ) Here, t and r are the transmission and reflection amplitude, respectively, and θ ( ) x is the unit step function. The total Hamiltonian [Eq. into SPPs. This rate can be included in the QE Hamiltonian with an additional non-Hermitian term γ − / i 2 sp . Here the SPP modes, not a pseudomode, plays the role of a dissipative Markovian environment. In the QE-SP system, due to the relative strong coupling, the rate γ sp can be large compared with γ 0 and γ p , we set 1,5,25 γ γ γ = ( + ) 10 sp p 0 in this work. As seen in Fig. 2(a), the profile of the transmission coefficient T (red-dashed curve, with ω = 2 0 eV, = z 4 0 nm) is just the dissipative single particle transmission lineshape 5,12 . However, when the spectral density of the surface-EM-fields is Lorentzian, the QE-pseudomode coherent coupling g eff exists (thus, eff eV) significantly changes to two-dip lineshape. We therefore can precisely detect the existence of the coherent interaction between the QE and the pesudomode.
The green-dotted-dashed curve (with ω = 2 0 eV, = z 8 0 nm, and = g eff 0.14 eV) in Fig. 2(a) shows that for larger separation z 0 , g eff decreases as shown in Fig. 1(c), leading to a remarkably different profile comparing with the black-solid curve. In order to compare the profiles of T with different energy spacings of the QE, we also plot the blue-dotted curve (with ω = 3 0 eV, = z 4 0 nm, and = . g 0 39 eff eV). As can be seen, it has a different zero point of T from the case of ω = 2 0 eV. These results indicate that through the scattering of the waveguide-photon, we can detect not only the existence of the coherent interaction, but the varies of the coupling, and even the effect of different energy spacings of the QE from the transmission spectra. In plotting Fig. 2(a), we assume the coupling between the waveguide-photon and the QE is = η g 50 meV. However, this coupling η g in practice varies with the QE-waveguide separation 26 , we therefore plot the transmission coefficient T with different η g in Fig. 2(b). As shown, although the lineshape becomes sharper when reducing the strength of η g , the profile can still be distinguished. This means that even with small η g , we can still detect the existence of the coherent interaction g eff .

Discussion
Since the pseudomode has the coherent interaction with the QE, it will be interesting to study the quantum coherence between the QE and the pseudomode. We calculate the concurrence 27 which quantifies the degree of the bipartite entanglement and also shows the behavior of the quantum coherence. In our scattering approach, after tracing out the microwave-photon fields, the reduced density matrix of the bipartite state is a pure state, and the concurrence simply takes the form =  Fig. 3(a), we plot the concurrence as black-solid (red-dashed) curve for parameters adopted from the black-solid (green-dotted-dashed) curve in Fig. 2(a). As can be seen in Fig. 3(a), the concurrence is non-zero positive, indicating that there exists quantum coherence between the QE and the pseudomode. One can also see that there is a dip in the concurrence lineshape, and when g eff decreases, the dip shrinks. In order to see more insights about the behavior of the concurrence, in Fig. 3(b) we plot the black-solid curve in (a) again without the dissipations, meanwhile, we also plot the normalized ξ (red-dashed) and ξ | | p (blue-dotted). The dips occurs when the energy of the incident waveguide-photon ω ( ) η is resonant with the energy of the pseudomode ω ( ) c . This is because in our scattering approach, the QE state and the pseudomode are within a different subspace from the field states. When the excitation is in the QE, it transfers to the pseudomode via the coherent interaction, the excitation therefore completely transfers to the pseudomode when ω ω = η c . This also explains why the dip shrinks when the coherent coupling g eff becomes smaller. For the experimental realizations, high QE-waveguide coupling strength can be achieved with dielectric waveguides such as photonic crystal waveguides 28 and dielectric slot waveguides 29 . However, in order not to seriously affect the spatial structure of the surface-EM-fields, we suggest to utilize the plasmonic nanowire 30 as the probing waveguide, and the II-VI colloidal quantum dots (e.g., CdSe/ZnS quantum dots with exciton energy around 2-2.5 eV) placed close to a silver thin-film to form the QE-metal film system. Due to the nature that being a near field, the intensity of the surface-EM fields decays exponentially away from the surface 1,2 . In this work, we set the QE-waveguide coupling η g to be a few tens of meV, which is small compared with the QE-metal film coupling g eff (∼ a few hundreds of meV). Figure 1(c) and Eq. (4) gave us a rough estimation that to achieve this magnitude of the coupling strength, the plasmonics metal-nanowire waveguide is about 30 nm away from the QE. Given the physical size of the II-VI colloidal quantum dot (∼ 10 nm) and the QE-metal film separation (∼ 5 nm), the plasmonic waveguide is actually about 50 nm away from the metal film. With this separation, the intensity of the surface-EM fields decays to 6 orders smaller 15 . It therefore can only very slightly affect the plasmonic nanowire waveguide, and similarly, the fields of the waveguide can only slightly affect the metal thin film. This is the reason we treat the waveguide fields and the surface-EM fields as independent fields, and QE-waveguide coupling η g as a perturbation to the original QE-metal film system. The plasmonic metal-nanowire therefore plays a role probing the reversible dynamics in the QE-metal film system.
Summarizing, we propose that through the scattering of the waveguide-photon incident on a quantum emitter, one could precisely measure the coherent interaction leading to reversible dynamics in population between the quantum emitter and the dielectric-metal interface. The behavior of the quantum coherence between the quantum emitter and the metal surface has been also studied by calculating the bipartite entanglement.