Strong coupling spontaneous emission interference near a graphene nanodisk

: In this work, we analyze the spontaneous emission dynamics of a V-type quantum emitter near a graphene nanodisk based on the combination of electromagnetic and quantum dynamical calculations. The presence of the graphene nanodisk gives strong anisotropy to the Pur-cell factors of the quantum emitter, leading to interference effects in spontaneous emission appearing as coupling between the emitter’s upper levels. This effect is further enhanced by the strong light–matter interaction of the quantum emitter with the modiﬁed electromagnetic mode continuum, which induces non-Markovian sponta-neous emission dynamics. We have studied the population dynamics of the quantum emitter at a speciﬁc distance from the center of the graphene nanodisk for various free-space decay widths and different quantum emitter’s initial conditions and have shown weak coupling results appearing with Markovian decay dynamics, obtained for quantum emitters with small free-space decay widths, and population dynamics that exhibits distinctly non-Markovian features, such as prominent decaying Rabi oscillations in the population evolution of the quantum emitter’s excited states and energy exchange between them during the overall population decay into the photonic mode continuum for largest free-space decay widths. Also, for the largest value of the free-space decay width, we obtain signiﬁcant population trapping effects in the excited states of the quantum emitter. Furthermore, we ﬁnd that the population dynamics for spe-ciﬁc light–matter interaction strength conditions between


Introduction
One of the primary objectives in nanophotonics is to establish robust light-matter coupling between quantum emitters (QEs) and nanoscale photonic structures.Such strong light-matter interaction conditions result in the coherent energy exchange between the altered photonic environment due to the presence of the nanoscale structure and the QE, manifested by reversible dynamics of the QE's spontaneous emission.The strong coupling between light and matter can also produce various intriguing phenomena with potentially numerous applications related to chemical reactions, novel technologies at the nanoscale, nonlinear optics, and quantum technology [1], [2].The photonic nanostructures usually explored for achieving reversible spontaneous emission in QEs include foremost plasmonic nanostructures, such as metallic nanoparticles, metal-dielectric interfaces, and metallic or metal-dielectric cavities [3]- [10], while seminal experiments have demonstrated strong light-matter interaction effects in such systems in the frequency regime [11]- [14].
Other proposals for photonic nanostructures achieving strong light-matter with QEs exploit two-dimensional materials, like graphene nanostructures [15], [16], nanoscale disks and layers of hBN [17]- [19], and transition-metal dichalcogenides [20]- [28].In particular, the interaction of light and matter near graphene nanostructures has been studied intensively in recent years [16], [29]- [39], since graphene supports surface plasmons, resulting in interesting optical properties.The surface plasmons modes of graphene can be tuned by multilayer stacking and doping, as well as voltage gating, with frequencies in the near to far infrared part of the electromagnetic spectrum [15].Graphene nanostructures interacting with two-level QEs through light have gathered substantial interest.Pioneering research by Koppens, Chang, and Abajo [15] demonstrated the strong nature of light-matter interaction near a graphene nanodisk, which can be utilized for singlephoton-level quantum optics by positioning a QE close to the nanodisk.Additionally, a QE next to a graphene nanodisk can support strong plasmon-plasmon interaction, manifested by plasmon blockade effects [29] and enhancement of the quadrupolar transitions in QEs [40].Furthermore, high efficiency long-range transfer of energy, which can be also tuned, between two QEs via a graphene nanodisk [30], as well as strong coupling effects in the entanglement dynamics between two QEs [31], has been shown.Also, the interaction between two dipoles and transfer of energy between a QE composed of three levels and a graphene nanoscale disk in a nonlinear photonic crystal have been examined, including the investigation of a strongly driven QE-graphene nanodisk structure with respect to nonlinear optical response [41] and coherent control [34].Recently, the excitation of surface plasmon excitation in a graphene nanodisk by moving a nonexcited two-level QE through the nanodisk has been proposed [37].Also, strong coupling between a QE and a graphene nanoscale disk has been demonstrated by obtaining Rabi oscillations in the excitedstate population dynamics of the QE, as well as trapping of population in the excited state of the QE, and analyzing the non-Markovian features and the quantum speed limit in this structure [42].
Beyond effects in QEs of two-level systems, significant quantum coherence and interference effects are present in QEs with three levels.Specifically, in a QE in V-type configuration, quantum interference effects in spontaneous emission [43] can be created by situating the QE in a photonic environment featuring Purcell effect anisotropy [44].The quantum interference in spontaneous emission is due to the coupling between the two excited states of the Vtype three-level system in an anisotropic vacuum, while no interference occurs in the ordinary vacuum.This effect has been mainly investigated in the weak light-matter regime, adjacent to various nanophotonic structures, such as structures of metamaterials [45]- [47], semiconductor microcavities [48], and plasmonic or polaritonic nanostructures [39], [49], [50].It has also been studied in the strong coupling regime with metallic nanospheres [3] and transition-metal dichalcogenides nanodisks [22], [28].Quantum interference in spontaneous emission results in pronounced effects in coherent nonlinear optics and quantum optics [43], and it can be potentially employed to increase the efficiency of a photovoltaic device [51], among other applications.
In this study, we present a nanoscale graphene-based photonic framework that supports significant quantum interference in a QE's spontaneous emission under conditions of both weak and strong light-matter interactions.In particular, we examine the spontaneous emission of a QE in a V-type configuration in proximity to a graphene nanoscale disk under both weak and strong coupling conditions and show the transition between the weak and strong coupling regimes.Specifically, we combine classical electromagnetic calculations, employed for determining the Purcell enhancement factor of the QE in proximity to the graphene nanoscale disk, with exact quantum dynamical calculations for the probability amplitudes of the excited states of the V-type QE.
The paper is structured as follows: In Section 2, we present the theory used in this work for studying the dynamics of the spontaneous emission of a V-type QE in proximity to a graphene nanoscale disk, as well as the corresponding QE Purcell enhancement factors, obtained by an electromagnetic method based on first-principles.In Section 3, we show and analyze the results on the spontaneous emission dynamics.Finally, we draw conclusions in Section 4.

Theory
In this work, the dynamics of the spontaneous emission of a QE in V-type configuration interacting with a graphene nanodisk is studied; the QE is situated at ⃗ r = (0, 0, z) of a nanodisk-centered originated coordinate system (see Figure 1).The electric dipole moments of the transitions in the QE are ⃗  10 =  ε− and ⃗  20 =  ε+ , where ε± ≡ (ε z ± iε x( y) )∕ √ 2 denote the right-circular (ε + ) and left-circular (ε − ) unit vectors ( is taken to be real).Also,  0 denotes the transition frequency of the two degenerate excited states in the QE in V-type configuration, where the lower state energy is defined as the energy zero-point.In this theoretical work, we follow the macroscopic quantum electrodynamics approach [52], [53].
The system Hamiltonian, after invoking the electric dipole and rotating wave approximations, reads (ℏ = 1 is always in use in the following) where f † (r ′ , ) and f(r ′ , ) denote the electromagnetic field operators for creation and annihilation of bosons, obeying the typical commutation relations [52].We note that the bosons here correspond to polaritons formed between graphene nanodisk plasmonic resonances and the electromagnetic field modes; these are the modes of the photonic environment, which couple to the QE.In Eq. ( 1), the quantum states of a QE in V-type configuration are written as |i; n r ′ , ⟩ = |i⟩ ⊗ |n r ′ , ⟩, where |i⟩(i = 0, 1, 2) (see also Figure 1), and the states of the photonic modes are denoted by |n r ′ , ⟩, with |0  ⟩ ≡ |0 r, ⟩ and |1  ⟩ ≡ |1 r, ⟩ being the vacuum and one photon states of frequency , respectively.Furthermore, the coupling between the QE and photonic states |i; 0  ⟩, (i = 1, 2) and |0; 1  ⟩ is given by where  i0 is the electric dipole transition moment vector between the |i; 0  ⟩, (i = 1, 2), and |0; 1  ⟩ states.In Eq. ( 2),  I (r ′ , ) stands for the imaginary part of the space-and frequency-dependent complex dielectric function of the graphene nanodisk.The important quantity, here, is the classical electromagnetic Green's tensor G(r, r ′ , ).It can be written as G(r, r ′ , ) = G 0 (r, r ′ , ) + G ind (r, r ′ , ), with G 0 (r, r ′ , ) being the homogeneous part of the Green's tensor.The induced part of the Green's tensor is given by G ind (r, r ′ , ); it accounts for the influence of the graphene nanodisk on the photonic environment, thus determining the coupling of the QE with its surrounding continuum of electromagnetic modes.
The planar conductivity, , of the graphene nanodisk, within the random phase approximation [54], determines its optical response as explained in detail in Ref. [30].It is a function of  and T, standing for the chemical potential and the temperature, respectively; it is given by  =  intra +  inter , where the contributions of the intraband,  intra , and interband,  inter , transitions are [55], , (3) respectively.The  intra term models a Drude-type response corrected for impurity scattering, through the term that contains the relaxation time .In this work, as parameters for the in-plane graphene conductivity calculations, we use  = 0.5 eV, T = 300 K, and  = 1 ps.
According to Ref. [30], in order to calculate the induced part of the Green's tensor, the graphene charge density for a nanodisk with radius R is expanded over an appropriate set of functions: , with l and n referring to the angular and radial eigenmodes, respectively, and P (l,0) n denoting the Jacobi polynomials.Since in this work, the QE is located over the graphene nanodisk center, r = (0, 0, z), within the electrostatic regime, the components of the G ind (r, r ′ , ) are given by [30] G ind , (5) and , (6) with Z(z, R) = √ ( z∕R In Eqs. ( 5) and ( 6), the expansion coefficients c l n , for l = 0, 1, describe the dipole source effect on the nanodisk, while l = 1 and l = 0 relate to a QE with a transition dipole moment along x and z, respectively.
The state of the V-type QE is given by It follows that [3] where  = t − t ′ , and In Figure 2, the Purcell factors for a QE with z-oriented (top panel) and x-oriented (bottom panel) transition dipole moments are shown at various distances (d = 5, 10, 15, 20, 30, 50 nm) from a R = 30 nm graphene nanodisk.It is evident that the orientation of the QE transition dipole moment influences the peak locations in the Purcell spectrum, which are associated with plasmonic resonances of the graphene nanodisk, while the distance between the QE and nanodisk has negligible impact on these positions.However, the peak magnitude is influenced by the distance d; as anticipated, the  k (, d), (k = z, x), diminish as d increases.
In the upper two panels of Figure 3, the Purcell factor for a QE with transition dipole moment oriented along the x-axis (upper panel) and along the z-axis (lower panel) are depicted with the QE positioned at d = 10 nm from a graphene nanodisk with different radii R = 7.5, 30, 50 nm.
We see that the peak energies in the Purcell factor spectrum depend on the nanodisk's radius, with a blue shift occurring as the radius enlarges.Additionally, we find that for a greater radius, there is a reduction in the number of peaks and a diminishing of the peak heights compared to nanodisks of smaller radii.In the bottom panel of Figure 3, we show the quantum interference factor We note that the quantum interference factor values can be almost equal to −1 and 1, the values for maximum quantum interference [43], at different energies.This is due to the fact that the plasmonic resonances for zor x-oriented dipole moments occur at different resonance frequencies, which leads to nonoverlapping Purcell enhancement factors for two mutually perpendicular dipole orientations.Before we present the dynamics of the spontaneous emission in the strong coupling regime, we briefly discuss the dynamics in the weak coupling regime, where the Markov approximation can be applied.In such a case, we write where () is the delta function, and Γ j = Γ 0 ( 0 ) j ( 0 , d), with j = z, x.The equations for the probability amplitudes thus become which can also be written as The above two equations clearly demonstrate the importance of the quantum interference factor , which induces the coherent coupling between the upper states.Equations ( 15) and ( 16) can be solved analytically, and for the initial conditions of interest in this work, these solutions are as follows: If initially the QE is in state and if the QE is in an antisymmetric superposition state

Results and discussion
In this section, we show the dynamics of the spontaneous emission of a QE in V-type configuration in proximity to a R = 30 nm graphene nanodisk for several initial states and under various light-matter interaction strength conditions between the QE and the photonic environment.The interaction strength is affected linearly by the vacuum decay width Γ 0 , according to Eq. ( 11).Alternatively, one could affect the coupling strength by changing the distance d between graphene nanodisk and the QE for a fixed vacuum decay width value.We primarily focus on the case with  = −1, i.e., a QE with transition frequency  0 = 0.22379 eV.
In Figure 4, the evolution of the population of a QE in V-type configuration with  0 = 0.22379 eV situated at d = 10 nm above the nanodisk is shown.We show the for the larger Γ 0 .More specifically, the initial population is transferred rapidly back and forth from state |1; 0  ⟩ to state |2; 0  ⟩; also in case of the largest Γ 0 shown, the QE has decayed into the photonic electromagnetic mode continuum, in about 2.5 ps.The fact that the population is transferred between the two excited states of the QE rapidly as the total decay occurs indicates directly that a plasmonic resonance in the graphene nanodisk is on resonance with the QE; in such a case, the photonic environment facilitates the transfer of population efficiently between the QE excited states.For the smaller Γ 0 values, we observe dynamics with minor non-Markovian features and complete decay times over 10 ps (the latter is not shown here).
In Figure 5, the QE population evolution for the same parameters as in Figure 4 for a symmetric and antisymmetric initial state, | S (0)⟩ and | A (0)⟩, respectively, is shown.The dynamics shown indicates that |c 1 (t)| 2 , being always  this dynamics has clear non-Markovian behavior, for all Γ 0 , in particular manifested in the larger Γ 0 cases as decaying Rabi oscillations of the population time evolution.Note that the complete decay times become very different as Γ 0 decreases.More specifically, while for the largest Γ 0 value, the time required for complete decay of the QE population with initial state | S (0)⟩ is about twice larger than the corresponding time in case of the initial state | A (0)⟩, the corresponding time differs greatly as the Γ 0 value becomes smaller, which results to less pronounced non-Markovian dynamics.
In Figure 6, we investigate the population dynamics when the free-space decay width Γ 0 equals 41.357 μeV.
Under such particularly strong light-matter coupling conditions, with initial state |Ψ(0) = |1; 0  ⟩, we observe that the population of states |1; 0  ⟩ and |2; 0  ⟩ decay rapidly into the electromagnetic continuum of modes as modified by the presence of the graphene nanodisk.However, the decay is only partial, since after 300 fs, an oscillatory complete exchange of population between these two states sets in, with no decaying amplitude, resulting in time-averaged trapped population in these states, amounting in total to about 30 % of the initial population of the QE.In the inset of Figure 6, we study the population of states |1; 0  ⟩ and |2; 0  ⟩, when the initial state is |Ψ S (0)⟩ and |Ψ A (0)⟩; in both cases of initial state, we observe trapped population in states |1; 0  ⟩ and |2; 0  ⟩, which in total amounts to about 40 % and 30 % of the initial population, respectively.There is a qualitative difference, though, with the case of initial state |Ψ(0) = |1; 0  ⟩, since now there is no exchange of population between the two excited state of the QE.Evidently, in this case, two noninteracting bound states between the electromagnetic continuum of modes and the states of the QE are formed [6], [42].
We will now present an analytical solution for the case of strong coupling under the approximation of a Lorentzian spectral density.We define The amplitudes a(t) and b(t) obey the equations We approximate J k () = J + () ± J − (), with k = z for the sum and k = x for the difference of the terms in J k (), given by a Lorentzian form, where, Δ ≡  0 −  c denotes the detuning between the transition frequency  0 of the QE and a central mode frequency  c , as in a cavity, and  stands for the spectral width with respect to the given mode.The amplitudes a(t) and b(t) using contour integration and the Laplace transform are given by [42], [56] with β =  − iΔ, q a = √ β2 − 2Γ 0 ( 0 ) z ( 0 , d), and q b = √ β2 − 2Γ 0 ( 0 ) x ( 0 , d).Finally, the analytical solution for c 1 (t) and c 2 (t) are given by In particular, the population of the upper states, in case of an antisymmetric initial state, |Ψ A (0)⟩, are given by while, for a symmetric initial state, |Ψ S (0)⟩, the population of the upper states are given by In the upper and lower panel of Figure 7, the QE population dynamics for a symmetric and an antisymmetric initial state, respectively, computed analytically is presented using Eqs.( 32) and ( 33), after a one-peak Lorentzian function J L () is fitted to J k () when the distance between the QE and a R = 30 nm graphene disk amounts to d = 10 nm [57].The upper panel of Figure 7 corresponds to  0 = 0.37583 eV ( = +1) and the lower panel corresponds to  0 = 0.22379 eV ( = −1).The population evolution obtained analytically for all cases shown is in perfect agreement with the corresponding EMDE-calculated dynamics.Note that in the two cases shown in Figure 7, the QE transition frequency is in each case resonant to a plasmonic mode of the graphene nanodisk; for  = +1, the QE transition frequency is resonant to the lowest-energy peak in the Purcell spectrum for a QE with a transition dipole moment along the z direction, while for  = −1, the corresponding Purcell spectrum peak is for a QE with a x-oriented transition dipole moment.We also note that a very good agreement is also found between the analytical results using the above procedure and the numerical results using the EMDE method for the case presented in Figure 4, where the QE is prepared initially in one of the upper states (not shown here).
We also find that in case of  = −1, the exact |c 1 (t)| 2 = c 2 (t)| 2 dynamics computed with the EMDE methodology for initial state |Ψ A (0)⟩ is generally in poor agreement (not shown here) with the |c A 1 (t)| 2 = |c A 2 (t)| 2 analytic results under strong coupling conditions, while the comparison becomes better as Γ 0 decreases.In analogy, the same fact is true in case of  = +1 when starting with an |Ψ A (0)⟩ initial state (not shown here).We note that in those two cases, the QE transition frequency  0 is not resonant to any peak in the Purcell spectrum that corresponds to a QE transition dipole moment orientation as related to the dynamics according to Eqs. (32) and (33).
Lastly, in Figure 8, we compare the population dynamics |c 1 (t)| 2 and |c 2 (t)| 2 for an initial state |Ψ(0)⟩ = |1; 0  ⟩ at  0 = 0.375831 eV ( = +1) for the smallest Γ 0 considered in this work, which clearly implies dynamics in the Markovian regime, obtained by the EMDE method as well as using Eqs.( 30) and (31); the agreement is perfect between them.It is clear that the population dynamics shows that partial population trapping, actually 50 % of the initial population, occurs within a time scale that a simple exponential decay will be practically completed; this is not a strong coupling effect as discussed above.Here, by taking into account the initial conditions, Eqs. ( 25) and ( 26) imply that at  0 = 0.37583 eV ( = +1), b(t → ∞) = 0 and a(t → ∞) = 1∕ √ 2, which back-transformed to the initial basis leads to |c 1 (t → ∞)| 2 = 1∕4 and |c 2 (t → ∞)| 2 = 1∕4, as observed in Figure 8.
Physically, the rationalization of the effect is based on the fact that under weak coupling conditions, as manifested by the Markovian dynamics in this case, the dynamics of the symmetric state a(t) is affected only by the peak in the Purcell enhancement factors at  = 0.37583 eV; it thus decays totally in the electromagnetic mode continuum as modified by the nanodisk, while the antisymmetric state b(t) at this frequency remains practically nondecaying within the time range considered.

Conclusions
In summary, we have examined the spontaneous emission dynamics of a V-type QE near a graphene nanodisk.The anisotropy of the Purcell factors of the QE due to the presence of the nanodisk leads to interference effects in spontaneous emission, since the two excited states of the QE are coupled.This is further enhanced by the strong light-matter interaction of the QE with the modified electromagnetic mode continuum, which induces non-Markovian spontaneous emission dynamics.
We have analyzed the decay of the QE population when it is 10 nm away from the nanodisk, for various vacuum decay widths of the QE.We found that in the case of larger vacuum decay widths, the QE population decay exhibits distinctly non-Markovian features, such as prominent decaying Rabi oscillations in the population evolution of the QE excited states and energy exchange between them during the overall population decay into the photonic mode continuum.Notably, for the largest vacuum decay widths, we observed significant population trapping effects in the excited states of the QE.In case of smaller free-space decay widths, such effects are absent, and fewer decaying Rabi oscillations occur or even Markovian behavior is found.
Furthermore, we have found that the population dynamics for specific light-matter interaction strength conditions between the QE and the nanodisk depend distinctively on the initial state of the QE, whether it is a single state or a superposition state.This demonstrates the quantum interference effects in the spontaneous emission dynamics of a QE in V-type configuration.Analytical solutions for the population evolution, which show excellent agreement with the exact simulations under resonant conditions between the QE transition energy and plasmonic resonances in the graphene nanodisk, have been also given.We trust that our findings can be beneficial primarily for innovative photonic device development at the nanoscale, as well as quantum technologies, in the near future.

Research funding:
The work of I.T. was funded by an Empirikion Foundation research grant.Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.Conflict of interest: Authors state no conflict of interest.Ethical approval: The conducted research is not related to either human or animals use.Data availability: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Figure 1 :
Figure 1: (color online) Schematic depiction of a V-type QE situated near a graphene nanodisk. 0 denotes the QE transition frequency and R stands for the radius of the nanodisk.

Figure 2 :
Figure 2: (color online) The Purcell enhancement factors for a QE.Top: QE with z-oriented transition dipole moment.Bottom: QE with x-oriented transition dipole moment.Various distances d between the QE and a R = 30 nm graphene nanodisk are considered.

Figure 3 :
Figure 3: (color online) Top: the Purcell enhancement factor for a QE at distance d = 10 nm from a graphene nanodisk of various radii: the QE transition dipole moment is x-oriented (upper) and z-oriented (lower).Bottom: the quantum interference factor (, d) of a QE at distance d = 10 nm from a R = 30 nm graphene nanodisk.

equal to |c 2
(t)| 2 , for a symmetric or an antisymmetric initial state, decays totally into the photonic continuum of modes;
[3]h Γ 0 ( 0 ) being the QE vacuum decay width.We calculate the QE excited-state population dynamics, |c 1 (t)| 2 and |c 2 (t)| 2 , by applying the Effective Mode Differential Equation (EMDE) methodology[3].Note, in Eq. (11),  k (, d) is the Purcell enhancement factor of the vacuum decay rate of a QE at distance z = d above the center of the nanodisk along the k = x, z directions; it is given by