Ensemble-induced strong light-matter coupling of a single quantum emitter

TL;DR

This paper introduces a novel method to achieve strong light-matter coupling of a single quantum emitter by leveraging virtual excitations of a nearby ensemble, enabling enhanced non-linearities and potential quantum applications.

Contribution

The authors propose a new technique using ensemble-induced virtual excitations to strongly couple a single emitter to a cavity, surpassing traditional ensemble coupling schemes.

Findings

Strong coupling of a single emitter achieved via ensemble interactions.

Realistic parameters provided for SiV− defects in diamond.

Potential applications in quantum information and cavity-assisted chemistry.

Abstract

We discuss a technique to strongly couple a single target quantum emitter to a cavity mode, which is enabled by virtual excitations of a nearby mesoscopic ensemble of emitters. A collective coupling of the latter to both the cavity and the target emitter induces strong photon non-linearities in addition to polariton formation, in contrast to common schemes for ensemble strong coupling. We demonstrate that strong coupling at the level of a single emitter can be engineered via coherent and dissipative dipolar interactions with the ensemble, and provide realistic parameters for a possible implementation with SiV$^{-}$ defects in diamond. Our scheme can find applications, amongst others, in quantum information processing or in the field of cavity-assisted quantum chemistry.

Equations

$${\partial }_t\rho =-\mathrm{i}[H_0+H_{\text{JC}}+H_{\text{DD}},\rho ]+\mathcal{L}\rho ,$$

$${\partial }_t{\rho }^{\text{eff}}=-\mathrm{i}[H_0^{\text{eff}}+H_{\mathrm{JC}}^{\mathrm{eff}},{\rho }^{\text{eff}}]+{\mathcal{L}}^{\text{eff}}{\rho }^{\text{eff}}.$$

$$\displaystyle g_{A}^{\text{eff}}=g_{A}-{\rm Re}\big{(}\vec{g}^{T}\mathbf{M}^{-1}\vec{v}\big{)}\,.$$

$${\gamma }_{\pm }=\frac{{\kappa }^{\text{eff}}+{\gamma }_A^{\text{eff}}}{2}\pm \sqrt{\frac{({\kappa }^{\text{eff}}-{\gamma }_A^{\text{eff}}{)}^2}{4}+{\mu }^2}.$$

$$\displaystyle g^{\text{eff}}_{{A}}=g_{{A}}^{(0)}\Big{\{}\cos(ky_{\text{A}})-\frac{{{\gamma}_{{B}}}\Delta_{{B}}g(\vec{r})+{{\gamma}_{{B}}}^{2}f(\vec{r})}{\Delta_{{B}}^{2}+{{\gamma}_{{B}}}^{2}}\cos(ky_{\text{B}})\Big{\}},$$

$$\displaystyle{{\gamma}^{\text{eff}}_{{A}}}={{\gamma}_{{A}}}\Big{\{}1-f(\vec{r})^{2}+\frac{\left[g(\vec{r}){{\gamma}_{{B}}}-f(\vec{r})\Delta_{{B}}\right]^{2}}{\Delta_{{B}}^{2}+{{\gamma}_{{B}}}^{2}}\Big{\}}.$$

Full text

In the strong coupling regime of light-matter interactions, the coherent exchange of energy between matter and light occurs at a rate faster than all dissipative processes Kimble (1998); Raimond et al. (2001). Strong coupling has been actively pursued in atomic Sanchez-Mondragon et al. (1983); Agarwal (1984); Thompson et al. (1992); Brune et al. (1996); Pinkse et al. (2000); Colombe et al. (2007) and molecular Pockrand et al. (1982); Lidzey et al. (1998); Bellessa et al. (2004); Dintinger et al. (2005); Shalabney et al. (2015); Chikkaraddy et al. (2016); Wang et al. (2019); Zhang et al. (2017); Wang et al. (2017) physics, circuit quantum electrodynamics (QED) Blais et al. (2004); Wallraff et al. (2004); Chiorescu et al. (2004); Schoelkopf and Girvin (2008); Gu et al. (2017), and condensed-matter physics Yoshie et al. (2004); Hennessy et al. (2007); Deng et al. (2010); Carusotto and Ciuti (2013), with applications in, e.g., quantum information processing Pellizzari et al. (1995); Turchette et al. (1995); Imamoğlu et al. (1999); Rauschenbeutel et al. (1999); Zheng and Guo (2000); Mabuchi and Doherty (2002); O’Brien et al. (2009), transport Orgiu et al. (2015); Feist and Garcia-Vidal (2015); Schachenmayer et al. (2015); Zhong et al. (2017); Hagenmüller et al. (2017); Rozenman et al. (2018); Paravicini-Bagliani et al. (2019), optomechanics Brennecke et al. (2008); Gröblacher et al. (2009); Hammerer et al. (2009); Xuereb et al. (2012); Restrepo et al. (2014); Benz et al. (2016), and quantum chemistry Hutchison et al. (2012); Thomas et al. (2016); Herrera and Spano (2016); Galego et al. (2016); Flick et al. (2017); Ribeiro et al. (2018); Thomas et al. (2019).

For a single quantum emitter, strong coupling is accompanied by non-linear quantum effects leading to, e.g., the generation of nonclassical states of light used in photon quantum gates Birnbaum et al. (2005); Englund et al. (2007); Lang et al. (2011). Reaching strong coupling, however, is often difficult as it relies on the integration of small volume resonators with large quality factors. For mesoscopic ensembles this requirement is relaxed as the coupling can be collectively enhanced by a factor $\sqrt{N}$ (for $N$-emitters) Dicke (1954); Tavis and Cummings (1968); Thompson et al. (1992). This allows for the realization of collective polariton dynamics Kaluzny et al. (1983); Raizen et al. (1989), however, it does not enhance nonlinear quantum effects for the cavity mode Rempe et al. (1991); Carmichael et al. (1991); Sáez-Blázquez et al. (2018); Trivedi et al. (2019), limiting its use for, e.g., quantum optics and quantum information processing Lukin et al. (2001).

Here, we introduce an alternative approach where an ensemble of two-level quantum emitters with negligible populations modifies the cavity coupling of an individual beneficiary target two-level system. This allows for bringing the single target emitter from the weak to the strong coupling regime, even when the mediating ensemble is only weakly coupled to the cavity. The induced strong coupling is visible in collectively-enhanced single emitter-cavity vacuum Rabi splittings. Remarkably, here the resulting photon non-linearities are also enhanced by collective effects. We identify two distinct mechanisms for reaching strong coupling in the target-cavity system, based on an increase of coherent coupling strength or a modification of dissipation, respectively. In the first case, we provide a possible implementation of our scheme with realistic parameters considering SiV*-* centers in diamond coupled to a microcavity.

Our results can provide a viable path towards the realization of quantum non-linear optics experiments within cavity QED with moderate light-matter coupling strengths or cavity quality factors. In contrast to schemes using active dressing of, e.g., Rydberg states Wade et al. (2016); Motzoi and Mølmer (2018), our setup solely relies on virtual excitations of a nearby ensemble.

The basic scheme is shown in Fig. 1(a). The target emitter $A$ and $N$ ensemble emitters denoted as $B$ are coupled to a cavity mode via a Jaynes-Cummings interaction Jaynes and Cummings (1963) $H_{\text{JC}}=a\left(g_{A}\sigma_{A}^{+}+\sum_{\ell=1}^{N}g_{\ell}\sigma_{\ell}^{+}\right)+{\rm h.c.}$ and to each other via coherent dipole-dipole interactions Lehmberg (1970) $H_{\rm DD}=\sum_{i\neq j}\sqrt{\gamma_{i}\gamma_{j}}g(\vec{r}_{ij})\sigma_{i}^{+}\sigma_{j}^{-}$ (where $i,j=A,1,\dots,N)$. Here, $\sigma_{i}^{\pm}$ denote the raising/lowering operators for the emitters and $g_{i}=g_{i}^{(0)}\cos(ky_{i})$ their position-dependent coupling strength to the cavity mode (wavelength $\lambda$, $k=2\pi/\lambda$, annihilation operator $a$, and cavity mode profile constant along $x$ and $z$). The coherent dipole-dipole coupling strengths depend on the relative positions of the emitters $\vec{r}_{ij}=\vec{r}_{i}-\vec{r}_{j}$ via the anisotropic function $g{(\vec{r}_{ij})}$ (cf. Hagenmüller et al. (2019) for details) and on their decay rate $\gamma_{A}$ and $\gamma_{\ell}\equiv\gamma_{B}$ for all $\ell=1,\cdots,N$. For significant collective enhancement of the $A$-cavity coupling, we require $|\vec{r}_{\ell A}|\lesssim\lambda$, so that the ensemble $B$ is in the near-field of $A$.

The dynamics of the system with density operator $\rho$ is governed by a standard master equation

[TABLE]

where $\hbar\equiv 1$. In a frame rotating at the system $A$’s frequency $\omega_{A}$, $H_{0}=\Delta_{{c}}a^{\dagger}a+\Delta_{B}\sum_{\ell}\sigma_{\ell}^{+}\sigma_{\ell}^{-}$ represents the free Hamiltonian with detunings $\Delta_{c}=\omega_{c}-\omega_{A}$ and $\Delta_{B}=\omega_{B}-\omega_{A}$, where $\omega_{c}$ and $\omega_{B}$ are the bare cavity mode and ensemble emitter frequencies, respectively.

Dissipation is described by $\mathcal{L}\rho$ $=$ $-\kappa\mathcal{D}(a^{\dagger},a)\rho-\sum_{i,j}\sqrt{\gamma_{i}\gamma_{j}}f{(\vec{r}_{ij})}\mathcal{D}(\sigma_{i}^{+},\sigma_{j}^{-})\rho\,$, with superoperator $\mathcal{D}(X,Y)\rho=[X,Y\rho]+[\rho X,Y]$. The first term on the right-hand side governs photon losses at rate $2\kappa$. Diagonal terms ($i=j$) in the second term correspond to standard spontaneous emission of the emitters. Off-diagonal ($i\neq j$) terms give rise to mutual decay into the environment depending on the separation of the emitters via the function $f(\vec{r}_{ij})$, ranging from [math] at large separations to unity at zero separation Hagenmüller et al. (2019). In this near-field limit, they are responsible for the emergence of super- and subradiant modes Gross and Haroche (1982).

We are interested in obtaining an effective dynamics for the target A and the cavity, by adiabatically eliminating Hagenmüller et al. (2019); Zwanzig (1960); Reiter and Sørensen (2012); Schütz et al. (2013) the interacting ensemble of emitters $B$. When the latter is weakly populated, the equations of motion within the sub-system $B$ become linear, $\partial_{t}\langle\sigma_{\ell}^{-}\rangle=-{\rm i}\sum_{m}(\mathbf{M})_{\ell m}\langle\sigma_{m}^{-}\rangle$, with (\mathbf{M})_{\ell m}=(\Delta_{B}-{\rm i}\gamma_{B})\delta_{\ell m}+(1-\delta_{\ell m})\gamma_{B}\big{[}g(\vec{r}_{\ell m})-{\rm i}f(\vec{r}_{\ell m})\big{]} a $N\times N$ matrix Bienaimé et al. (2011); Lesanovsky et al. (2019). Adiabatic elimination of $B$ generally requires that the eigenvalues $\lambda_{\eta}$ of $\mathbf{M}$ are the largest parameters (see Hagenmüller et al. (2019) for details). This ensures that sub-system $B$ evolves fast compared to the reduced system including $A$ and the cavity. The coupling of the latter to $B$ also has to be perturbatively small such that $B$ remains weakly populated.

After elimination of $B$, the effective master equation for the reduced density operator $\rho^{\text{eff}}$ for target $A$ and cavity only reads

[TABLE]

The effective free Hamiltonian $H_{0}^{\text{eff}}=\Delta_{A}^{\text{eff}}\sigma_{A}^{+}\sigma_{A}^{-}+\Delta_{c}^{\text{eff}}a^{\dagger}a$ contains renormalized detunings \Delta_{A}^{\text{eff}}=\omega_{A}^{\text{eff}}-\omega_{A}=-{\rm Re}\big{(}\vec{v}^{T}\mathbf{M}^{-1}\vec{v}\big{)} and \Delta_{c}^{\text{eff}}=\omega_{c}^{\text{eff}}-\omega_{A}=\Delta_{c}-{\rm Re}\big{(}\vec{g}^{T}\mathbf{M}^{-1}\vec{g}\big{)}, which can be generally compensated. Here, $(\vec{g})_{\ell}=g_{\ell}$ and $(\vec{v})_{\ell}=\sqrt{\gamma_{A}\gamma_{B}}[g(\vec{r}_{{\ell}A})-{\rm i}f(\vec{r}_{{\ell}A})]$. The effective $A$–cavity interaction H_{\text{JC}}^{\text{eff}}=g_{A}^{\text{eff}}\big{(}a^{\dagger}\sigma_{A}^{-}+\sigma_{A}^{+}a\big{)} has the modified coherent coupling

[TABLE]

The effective dissipator for the reduced system in Lindblad form reads $\mathcal{L}^{\text{eff}}\rho^{\text{eff}}=-\sum_{\nu=\pm}\gamma_{\nu}\mathcal{D}(L^{\dagger}_{\nu},L_{\nu})\rho^{\rm eff}$ with rates

[TABLE]

The main results of this work follow from Eqs. (3) and (4). We define the system to be strongly coupled when $|g_{A}^{\text{eff}}|$ exceeds the largest rate $\gamma_{+}$. The parameters $\kappa^{\rm eff}=\kappa+{\rm Im}[\vec{g}^{T}\mathbf{M}^{-1}\vec{g}]$ and ${{\gamma}^{\text{eff}}_{{A}}}={{\gamma}_{{A}}}+{\rm Im}[\vec{v}^{T}\mathbf{M}^{-1}\vec{v}]$ in Eq. (4) describe the modification of cavity decay and target emitter linewidth, respectively, due to the interaction with the $B$ ensemble. The rate $\mu={\rm Im}[\vec{g}^{T}\mathbf{M}^{-1}\vec{v}]$ gives rise to mutual decay mechanisms, dissipatively coupling $A$ and the cavity via Lindblad jump operators $L_{+}$ and $L_{-}$(see Hagenmüller et al. (2019)).

Using the mode-decomposition of M in terms of its eigenvectors $\vec{x}_{\eta}$, the effective coupling modification in Eq. (3) reads $\Delta g_{A}\equiv g^{\rm eff}_{A}-g_{A}=-\sum_{\eta}{\rm Re}(\vec{g}^{T}\vec{x}_{\eta}\vec{x}^{T}_{\eta}\vec{v}/\lambda_{\eta})$, which depends both on the mode-overlaps $\vec{g}^{T}\vec{x}_{\eta}$, $\vec{v}^{T}\vec{x}_{\eta}$, and the complex eigenvalues $\lambda_{\eta}$, and can increase with $N$. For example, when the sum $\sum_{\eta}$ is dominated by a nearly homogeneous ($l_{B}\ll|\vec{r}_{\ell A}|\ll\lambda$) coupling to a symmetric mode $\vec{x}_{S}=(1,1,\dots,1)/\sqrt{N}$ of $B$ with eigenvalue $\lambda_{S}$, the enhancement scales as $\sim N$ since both mode-overlaps $\vec{g}^{T}\vec{x}_{S}$, $\vec{v}^{T}\vec{x}_{S}$ scale as $\sim\sqrt{N}$.

In Fig. 1(b) we show the normalized cavity transmission spectrum $\mathcal{T}_{c}$ for the crystalline cubic structure in panel (a), where the symmetric mode $\vec{x}_{S}$ is found to dominate Schütz et al. (2019). Here, $\mathcal{T}_{c}(\omega_{L})$ is computed numerically from Eq. (2) for a weak cavity drive (strength $\phi$) at frequency $\omega_{L}$, and is given by $\mathcal{T}_{c}(\omega_{L})=\langle a^{\dagger}a\rangle(\kappa^{2}/\phi^{2})$ Schütz et al. (2019). We find a collective enhancement of the polariton-splitting that scales approximately $\propto N$, with super-linear corrections for large $N$ due to energy shifts in $\lambda_{S}$. We note that due to the small size of the cube ($l_{B}\ll|\vec{r}_{jA}|\ll\lambda$), the condition $\Delta_{A}^{\rm eff}\simeq\Delta_{c}^{\rm eff}$ is always satisfied. The polariton peaks display a marked asymmetry growing with $N$, which we attribute to the presence of the mutual decay rate $\mu$ Hagenmüller et al. (2019).

Importantly, the collective enhancement leads to strong non-linear effects at the single photon level. Fig. 1(c) shows the equal-time photon-photon correlation function $g^{(2)}(\tau=0)=\langle a^{\dagger}a^{\dagger}aa\rangle/\langle a^{\dagger}a\rangle^{2}$, computed for the same parameters as in panel (b) and using the steady state of Eq. (2) with a weak laser drive Schütz et al. (2019). The figure shows that $g^{(2)}(0)$ becomes $g^{(2)}(0)\lesssim 1$ at frequencies close to the polariton ones, and decreases with $N$. It can reach values $g^{(2)}(0)\ll 1$ for the upper polariton, reflecting the polariton asymmetry in $\mathcal{T}_{c}$ found above. This demonstrates that it is possible to achieve the “photon blockade” regime for a single target quantum emitter solely by increasing the number $N$ of dipoles making up the mediating mesoscopic ensemble.

When $l_{B}\ll|\vec{r}_{\ell A}|$, the collective dipole of $B$ can be approximated by a point dipole of strength $g_{B}^{(0)}=\sqrt{N}g_{A}^{(0)}$ and $\gamma_{B}=N\gamma_{A}$ positioned at ${\vec{r}}_{B}$ (each dipole moment in $B$ is chosen identical to that of $A$). Analytical expressions for the parameters of Eq. (2) can then be obtained in the single-mode limit. In this case the effective coupling strength reads Hagenmüller et al. (2019)

[TABLE]

where ${\vec{r}}\equiv{\vec{r}}_{B}-{\vec{r}}_{A}$. We find an increased effective cavity decay, $\kappa^{\text{eff}}=\kappa+{{\gamma}_{{B}}}g_{{B}}^{2}({\vec{r}}_{\rm B})/(\Delta_{{B}}^{2}+{{\gamma}_{{B}}}^{2})$, as well as a modified linewidth

[TABLE]

The mutual decay term entering Eq. (4) instead reads $\mu=g_{{B}}\sqrt{{{\gamma}_{{A}}}{{\gamma}_{{B}}}}(g(\vec{r}){{\gamma}_{{B}}}-f(\vec{r})\Delta_{{B}})/(\Delta_{{B}}^{2}+{{\gamma}_{{B}}}^{2})$. The equations above show that the parameters in Eq. (2) are generally tuned by a combination of coherent/dispersive Liu et al. (2014) and incoherent/dissipative terms (e.g., $\sim\Delta_{B}g(\vec{r})$ and $\sim{{\gamma}_{{B}}}f(\vec{r})$ in Eq. (5), respectively), and depend strongly on geometrical effects, due to the dipolar nature of the $A$-$B$ interaction. For a given geometry (i.e., positioning of $A$ and $B$, and dipole orientations), $g^{\text{eff}}_{{A}}$ can be significantly increased by modifying the detuning $\Delta_{B}$, an effect that we term coherent inheritance of coupling strength. A competing effect that we dub reduction of linewidth in $A$ can be similarly obtained for ${{\gamma}^{\text{eff}}_{{A}}}$ in Eq. (6) (even in the absence of a cavity, e.g. by putting the third term on the r.h.s. of the equation to zero, and for small separations where $f(\vec{r})^{2}\sim 1$). Such linewidth narrowing effect can be related to the formation of a subradiant mode.

Figs. 2(a) and (c) show contour plots of $|g^{\text{eff}}_{{A}}/\gamma_{+}|$ as a function of the relative distance $r$ between $A$ and $B$, and the dipole-orientation angle $\theta$ [see Fig. 1(a)], for two choices of parameters where coherent inheritance of coupling strength [panel (a)] or reduction of linewidth [panel (c)] are the dominant effects. In the two panels, the bare parameters of Eq. (2) are chosen as $\gamma_{A}<g_{A}^{(0)}<\kappa$ and $\kappa<g_{A}^{(0)}<\gamma_{A}$, respectively. In (a) we further choose a large detuning $|\Delta_{B}|\gg\gamma_{B}$ (with $\gamma_{B}=10^{4}\gamma_{A}$). This implies an effective coherent coupling $g_{A}^{\text{eff}}/g_{A}^{(0)}\approx\cos(ky_{A})-\frac{{{\gamma}_{{B}}}g(\vec{r})}{\Delta_{B}}\cos(ky_{B})$, when the positions of $A$ and $B$ are such that $|g(\vec{r})|>|f(\vec{r})|$. Note that while larger effective couplings can be obtained for smaller $|\Delta_{B}|$, the latter has to be large enough to satisfy the conditions for adiabatic elimination. In (c), instead, $A$ and $B$ are resonant ($\Delta_{B}=0$) and $\gamma_{B}$ is large, such that the effective coherent coupling and the linewidth simplify to $g_{A}^{\text{eff}}/g_{A}^{(0)}\approx\cos(ky_{A})-f(\vec{r})\cos(ky_{B})$ and $\gamma_{A}^{\rm eff}=\gamma_{A}[1-f(\vec{r})^{2}+g(\vec{r})^{2}]$, respectively. Furthermore, we position $A$ such that $g_{A}^{\text{eff}}/g_{A}^{(0)}\approx-f(\vec{r})\cos(ky_{B})$ only stems from the dissipative dipole term $f(\vec{r})$. In both cases, the figure shows that a regime of strong coupling with $|g_{A}^{\text{eff}}/\gamma_{+}|>1$ between the target $A$ and the cavity can be reached (area enclosed by the green line), e.g., in a “head-to-tail” configuration with $\theta\sim 0$ or close to the “magic angle” $\theta\sim 0.3\pi$ in panel (a) and (c), respectively. This is particularly surprising for panel (c), as coherent light-matter coupling is entirely due to dissipative effects.

Fig. 2 confirms strong coupling with a simulation of the time-evolution for the mean populations of $A$, $B$, and the cavity, starting with a single photon. The presence of $B$ induces a clear coherent exchange of energy between the cavity and $A$ (negligible population in $B$), in contrast to the trivially damped behavior for the bare system. The validity of the effective master equation Eq. (2) is confirmed via a comparison to a full simulation of Eq. (1). We note that we have also demonstrated that our scheme is robust against inhomogeneous broadening and moderate positional disorder Schütz et al. (2019).

We now discuss a possible implementation of coherent coupling inheritance with realistic parameters considering SiV*-* centers in diamond coupled to a microcavity of the kind described in Ref. [Najer et al., 2019] [Fig. 3(a)]. Two nanodiamonds separated by a distance $r=60$ nm containing respectively a single ($A$) and an ensemble ($B$) of SiV*-* defects are located above the cavity mirror. The nanodiamond with size $l_{B}$ containing $N$ dipoles $B$ could be a fragment of a bulk diamond that was grown with a high percentage of the centers oriented along the $y$ axis Michl et al. (2014), which coincides with the $\langle 111\rangle$ crystallographic direction 111Note that we are defining this direction as $y$ while defining it as $z$ is more common in the literature..

The energy levels of SiV*-* consist of electronic ground $E_{g}$ and excited $E_{u}$ state doublets with splittings of $\sim 50$ GHz and $\sim 260$ GHz, respectively Hepp et al. (2014). At temperatures below $4$ K and in the presence of a magnetic field $H$ aligned along $z$, the states are Zeeman-split and both $A$ and $B$ species can be prepared in the ground state $\ket{1,\downarrow}$ by spin-flipping optical pumping Evans et al. (2018). The splitting of the $E_{g}$ doublet Meesala et al. (2018) then allows for microwave resolvable addressing to map the dipoles $A$ to the state $\ket{2,\downarrow}$. $A$ and $B$ only couple to the optical transitions $\ket{2}\leftrightarrow\ket{3}$ and $\ket{1}\leftrightarrow\ket{4}$, respectively [Fig. 3(b)] via photons polarized along $z$. A cavity mode with vacuum wavelength $\lambda=737$ nm is resonant with the target $A$ and detuned from $B$ by $\Delta_{B}=2\pi\times 310$ GHz.

The bare couplings $g^{(0)}_{A/B}$ can be estimated by comparing our setup to that of Ref. [Evans et al., 2018], where pairs of SiV*-* were coupled to a diamond photonic crystal cavity with strength $g_{\rm pcc}=2\pi\times 7.3$ GHz and mode volume $V_{\rm pcc}=0.5(\lambda/2.4)^{3}$ ($2.4$ is the refractive index of diamond). In our setup [Fig. 3(a)], we choose a microcavity with mode volume $V_{\rm mc}=1.4\lambda^{3}$ Najer et al. (2019), where the coupling strengths then read $g^{(0)}_{A/B}=g_{\rm pcc}\sqrt{V_{\rm pcc}/V_{\rm mc}}\approx 2\pi\times 1.17$ GHz. In Fig. 3 we model the nanodiamond $B$ as a cube with lattice spacing $d=8$ nm, which provides a density of emitters $\sim 2\times 10^{6}{\rm\mu m}^{-3}$ Bradac et al. (2017). The decay rates are expected to be $\gamma_{A}=\gamma_{B}\sim 2\pi\times 51.5$ MHz Jahnke et al. (2015). We choose $\kappa=2g^{(0)}_{A}$ corresponding to a moderate cavity quality factor $Q\approx 8.7\times 10^{4}$ (see, e.g., Refs. Zhang et al. (2018); Najer et al. (2019); Riedel et al. (2017)). This provides a finesse $F\approx 3\times 10^{4}$ for a cavity of length $L=3\lambda/2\approx 1.1\mu$m. Note that the non-radiative decay of the state $\ket{2}$ with rate $\sim 2\pi\times 13$ MHz Jahnke et al. (2015) can be overcome using a re-pumping scheme.

The effective coupling strength $g^{\rm eff}_{A}$ is displayed as a function of $N$ in Fig. 3(c), showing that the single SiV*-* center $A$ can be brought to strong coupling with the microcavity mode ($g^{\rm eff}_{A}>\gamma_{+}$) due to the presence of the nanodiamond containing the $B$ emitters. Remarkably, this leads to strong non-linear effects $g^{(2)}(0)\approx 0.46$ for $N=1000$ and at a frequency close to that of the upper polariton.

In conclusion, we have investigated different mechanisms of how a single quantum emitter $A$, initially weakly coupled to a cavity mode, can be brought into strong coupling. The effect relies on both coherent and dissipative interactions with a nearby ensemble of identical emitters, and provides a viable path towards the realization of strong coupling at the level of a single SiV*-* center, which has, to the best of our knowledge, never been achieved. In contrast to usual collective strong coupling, our scheme gives rise to collectively-induced strong photon non-linearities and can find important applications in quantum information technologies with Si-V centers. Our work also emphasizes the important role played by optically active environments in cavity-QED. In particular, it is an interesting prospect to explore whether our model could be extended to vibrational strong coupling in molecular ensembles Lather et al. (2019), and to scalable microwave qubits magnetically coupled to a transmission line resonator on the verge of strong coupling Viennot et al. (2015).

Acknowledgements — We are grateful to C. Genet and S. Whitlock for stimulating discussions. Work in Strasbourg was supported by the ANR - “ERA-NET QuantERA” - Projet “RouTe” (ANR-18-QUAN-0005-01), IdEx Unistra - Projet “STEMQuS”, LabEx NIE, and ERC Adv. Grant 788482 “MOLUSC”. G. B. and T. V. acknowledge funding through the Australian Research Council Centre of Excellence for Engineered Quantum Systems EQUS (Project number CE170100009). C. G. acknowledges support from the Max Planck Society and from the German Federal Ministry of Education and Research, co-funded by the European Commission (project RouTe), project number 13N14839 within the research program “Photonik Forschung Deutschland”. This work is supported by IdEx Unistra (project STEMQuS) with funding managed by the French National Research Agency as part of the “Investments for the future program”.