Cavity antiresonance spectroscopy of dipole coupled subradiant arrays
David Plankensteiner
Christian Sommer
Helmut Ritsch
Claudiu Genes
An array of $N$ closely spaced dipole coupled quantum emitters exhibits super- and subradiance with characteristic tailorable spatial radiation patterns. Optimizing their geometry and distance with respect to the spatial profile of a near resonant optical cavity mode allows to increase the ratio between light scattering into the cavity mode and free space by several orders of magnitude. This leads to a distinct nonlinear particle number scaling of the relative strength of coherent light-matter interactions versus decay. In particular, for subradiant states the collective cooperativity increases much faster than the typical linear $\propto N$ scaling of independent emitters. This extraordinary collective enhancement is manifested both in the intensity and phase profile of the sharp collective emitter antiresonances detectable at the cavity output port via transmission spectroscopy.
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\begin{document}

\title{Cavity antiresonance spectroscopy of dipole coupled subradiant arrays}
\author{David Plankensteiner}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria}
\author{Christian Sommer}
\affiliation{Max Planck Institute for the Science of Light, Staudtstra{\ss}e 2,
D-91058 Erlangen, Germany}
\author{Helmut Ritsch}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria}
\author{Claudiu Genes}
\affiliation{Max Planck Institute for the Science of Light, Staudtstra{\ss}e 2,
D-91058 Erlangen, Germany}
\date{\today}

\begin{abstract}
An array of $N$ closely spaced dipole coupled quantum emitters exhibits super- and subradiance with characteristic tailorable spatial radiation patterns. Optimizing their geometry and distance with respect to the spatial profile of a near resonant optical cavity mode allows to increase the ratio between light scattering into the cavity mode and free space by several orders of magnitude. This leads to a distinct nonlinear particle number scaling of the relative strength of coherent light-matter interactions versus decay. In particular, for subradiant states the collective cooperativity increases much faster than the typical linear $\propto N$ scaling of independent emitters. This extraordinary collective enhancement is manifested both in the intensity and phase profile of the sharp collective emitter antiresonances detectable at the cavity output port via transmission spectroscopy.
\end{abstract}

\pacs{42.50.Ar, 42.50.Lc, 42.72.-g}

\maketitle

The confinement of atoms and photons in small volumes with very low loss has been a renowned success~\cite{haroche1989cavity,walther2006cavity,kimble1998strong} as it allows for tests of light-matter interactions where the quantum nature of both comes into play. In a cavity quantum electrodynamics (CQED) setup the photon-emitter interaction strength $g \propto  \mu \mathcal{E}$ for a particle with dipole moment $\mu$ is strongly enhanced by decreasing the field mode volume and thus increasing the local field per photon $\mathcal{E}$. In a standard Fabry-P\'{e}rot cavity geometry this is achieved by closely surrounding the emitter with two high-reflectivity mirrors. The atom-photon interaction time is then enhanced by a factor roughly proportional to the cavity finesse characterizing the number of round trips a photon can make before escaping to the environment at a rate $\kappa$. At the single quantum emitter level, this has facilitated experimental progress towards strong coupling allowing the study of single photon nonlinear effects, such as the photon blockade regime~\cite{birnbaum2005photon}, of vacuum Rabi splittings and other tests of fundamental quantum optics effects~\cite{yoshie2004vacuum,casanova2010deep}.\\ %[\textcolor{red}{Could we expand a bit? also add Rempe's paper here.}].
\indent As a characteristic quantity of coupling strength, the single emitter cooperativity $C=g^2/(\kappa \gamma)$ (where $\gamma$ is the rate of spontaneous decay into modes outside the cavity) is a well established figure of merit for strong light-matter interactions when $C\gg1$. Since for a single emitter the dipole moment matrix element $\mu$ of the coupled transition appears both in  $g\propto \mu$ and $\gamma \propto \mu^2$, the cooperativity $C$ is merely a geometric factor independent of $\mu$ \cite{vuletic2001three}. This means that cavity design (increasing the finesse and decreasing the transverse mode area) is the central aspect for reaching high single emitter cooperativity. In an alternative approach, in the case of a lossy cavity where $\kappa$ is large, one often tries to reach a large effective cooperativity by coupling $N$ emitters simultaneously to a cavity mode such that the cooperativity scales like $C\ts{eff} \propto C N$. An implicit important assumption is that the emitter-cavity coupling increases proportionally to $N$ while the free space emission stays constant (independent emitter approach). In reality, especially for small particle-particle separations, the coupling to the vacuum modes is inherently \textit{collective}, leading to the occurrence of superradiantly decaying ($>\gamma$) and subradiantly decaying states ($<\gamma$). Such behavior has recently attracted interest in 1D and 2D sub-wavelength spaced atomic arrays used in topological quantum optics, high extinction media or photon storage~\cite{bettles2015cooperative, bettles2016cooperative, bettles2016enhanced, shahmoon2016cooperativity,perczel2017topological,asenjo2017exponential}.  \\
\begin{figure}[b]
\includegraphics[width=0.9\columnwidth]{fig1.pdf}
\caption{\emph{System setup}. Optical cavity (supporting different transverse modes) coupled to a rigid array of dipole-dipole interacting quantum emitters. Light-matter interaction creates an antiresonance dip and a fast phase switch around the emitter resonance in the cavity transmission spectrum as shown on the right for a single emitter with $g=\kappa/10=2\gamma$.}
\label{fig1}
\end{figure}
In this paper we introduce an alternative, improved path, towards reaching a high cooperativity based on collective dissipative effects. The mechanism involves the separate optimization of the coherent coupling of the emitters to the cavity mode and of the incoherent emitter-vacuum coupling. For a configuration of $N$ closely spaced emitters (separation less than the transition wavelength $\lambda\ts{e}$), the coupling to free space vacuum modes can be strongly suppressed~\cite{dicke1954coherence,haroche1982superradiance,plankensteiner2015selective}. At the same time, a periodic arrangement of emitters in a rigid geometry (for example implanted inside a solid-state matrix), transversely placed inside a single cavity mode can lead to optimized collective coupling~\cite{zoubi2008bright}. The upshot is that $C\ts{eff}$ scales strongly in a nonlinear fashion with $N$ as the effective collective free space decay rate $\gamma\ts{eff}$ can be dramatically suppressed.\\
\indent Apart from a mathematical proof of principle, we propose an example for the practical implementation of phase imprinting using higher order transverse cavity modes leading to the preferential excitation of subradiant collective states. The effect is directly observable by homodyne detection at the cavity output, displayed both in amplitude and phase antiresonant behavior~\cite{sames2014antiresonance,rice1996cavity}. As opposed to the strong coupling regime exploited in~\cite{sames2014antiresonance}, this paper considers the bad cavity regime $\kappa\gg g\sqrt{N}$ where typically one expects modest antiresonance phase shifts. Due to the collectively increased cooperativity, very narrow antiresonances occur accompanied by extremely fast and large phase shift switches rendering such a system perfect for high resolution spectroscopy.

\section{Model}
Let us consider an ordered ensemble of quantum emitters modelled as two-level systems with ground state $\ket{g}_i$ and an excited state $\ket{e}_i$ (split by frequency $\omega\ts{e}$) located at $\textbf{r}_i$ (for $i=1,...,N$) (see Fig.~\ref{fig1}). The levels are connected by individual Pauli raising and lowering operators $ \sigma_i^\pm$ with $\sigma_i^x=\sigma_i^+ + \sigma_i^-$, $\sigma_i^y=-i(\sigma_i^+-\sigma_i^-) $ and $\sigma_i^z= \sigma_i^+ \sigma_i^- - \sigma_i^-\sigma_i^+$. The emitters are embedded in a static 2D support, transversely placed in the center plane of a single higher order transverse electromagnetic (TEM) mode at frequency $\omega_\text{c}$  (see~\fref{fig1}). At position $(\textbf{r},z)$ along the cavity axis $z$ the electric field operator is proportional to $a\cos(kz) f(\textbf{r}) \boldsymbol{\epsilon}_y$, where $a$ is the annihilation operator of the cavity mode, $k=\omega\ts{c}/c$, $f(\textbf{r})$ is the transverse spatial mode profile and $\boldsymbol{\epsilon}_y$ denotes linear polarization in the $y$-direction. The cavity is laser driven at frequency $\omega\ts{l}$ with power $P$ through one mirror. In a frame rotating at $\omega\ts{l}$ (and $\hbar=1$) the dynamics of the mode of interest is described by $H_\text{c} = \Delta\ts{c}a^\dag a + i\eta\left(a^\dag - a\right)$, where $\Delta\ts{c}=\omega\ts{c} - \omega\ts{l}$ and $\eta=\sqrt{2P\kappa/\omega\ts{l}}$. Cavity damping with decay rate $\kappa$ occurs via the collapse operator $a$.\\
\indent At dense spacing ($|\textbf{r}_i - \textbf{r}_{i+1}| < \lambda\ts{e}$) one has to account for the direct emitter-emitter interactions via the induced transition dipole moments $\boldsymbol{\mu}_i$. The collective dynamics is governed by the free Hamiltonian $H\ts{e} = \Delta_\text{e}\sum_i\sigma_i^+\sigma_i^- $ and the collective part $H\ts{dip}=\sum_{i, j : i\neq j}\Omega_{ij}\sigma_i^+\sigma_j^-$, where $\Delta\ts{e} = \omega\ts{e} - \omega\ts{l}$ and $\Omega_{ij}$ is the strength of the coherent dipole-dipole interaction between emitters $i$ and $j$ (see Appendix). Moreover the incoherent collective dynamics leads to mutual decay rates $\gamma_{ij}$ that can be accounted for in a master equation approach for the total system density matrix $\rho$ via the Lindblad superoperator $\mathcal{L}\ts{e} [\rho] = \sum_{i,j}\gamma_{ij}\left(2\sigma_i^-\rho\sigma_j^+ - \sigma_i^+\sigma_j^-\rho - \rho\sigma_i^+\sigma_j^-\right)$~\cite{lehmberg1970radiation} or equivalently, via the Heisenberg Langevin equations (HLE)~\cite{gardiner2004quantum} (see Appendix).\\
\indent In the single cavity mode limit, the interaction is described by the Tavis-Cummings Hamiltonian $H\ts{int} = \sum_i g_i\left(a^\dag\sigma_i^- + a\sigma_i^+\right)$, where the coupling strength $g_i$ of an emitter at position $(\textbf{r}_i, z_i)$ is proportional to $\cos(kz_i) f(x_i,y_i) \boldsymbol{\epsilon}_y\cdot \boldsymbol{\mu}$.

\section{Single emitter antiresonance}
We consider a reference system with a single emitter in the low excitation limit $\langle\sigma^z_i\rangle\approx -1$ where a linear coupled set of Langevin equations can be derived. For a resonant interaction (i.e. $\Delta=\Delta\ts{c}=\Delta\ts{e}$) this leads to the following mean field equations:
\begin{align}
\braket{\dot{a}} &= -(\kappa +i\Delta)\braket{a} + \eta -ig\braket{\sigma^-},
\label{single_HE1}
\\
\braket{\dot{\sigma}^-} &= -(\gamma+i\Delta)\braket{\sigma^-} - i g \braket{a}.
\label{single_HE2}
\end{align}
These equations exhibit the phenomenon of atomic antiresonances \cite{alsing1992suppression,sames2014antiresonance} where the resonantly driven atomic dipole oscillates in a way to counteract the cavity drive and leads to a minimum of transmission~\cite{zippilli2004suppression}. We analyze its dependence $\gamma$ studying the steady-state output field at the right mirror $\braket{A_\text{out}} = \sqrt{\kappa}\braket{a}$, which reads
\begin{align}
\braket{A_\text{out}} &= \frac{\eta\sqrt{\kappa}}{i\Delta+\kappa+g^2/(i\Delta + \gamma)}.
\end{align}
The complex amplitude transmission coefficient is  $t = \sqrt{\kappa}\braket{A_\text{out}}/\eta$ and the relative phase shift caused by the emitter is $ \phi-\phi\ts{c} $, where $\phi=\text{Arg}(t)$ and $\phi\ts{c}=-\arctan\left(\Delta/\kappa\right)$ is the phase shift of the bare cavity. Scanning the laser frequency ($\Delta$) we find that the coherent transmitted intensity through the cavity contains an antiresonance dip around $\Delta=0$ with a corresponding jump in the phase shift (see \fref{fig1}). Fitting the antiresonance with a Lorentzian (see Appendix), we find a depth of $1-T(\Delta=0)=C(C+2)/(C+1)^2$, and a width that can be approximated by $\gamma(C + 1)=g^2/\kappa + \gamma$ (for a regime where both $g,\gamma\ll\kappa$). An almost vanishing transmission is then a signature of reaching a regime of strong cooperativity ($C\gg 1$).

\section{Collective antiresonance of emitter arrays}
As $C$ is independent of $\mu$, particles with larger dipole moments will only broaden the antiresonance. For emitter arrays this is however no longer valid: one can distinguish between the 1D and 3D radiative properties of the ensemble. For 3D \textit{collective} subradiant resonances of an array (where emission outside the cavity mode is minimized), we can then optimize the coupling to the cavity mode which results in extremely sharp and deep antiresonances accompanied by a large phase change within a narrow frequency range. The immediate upshot of such a regime is an enhanced cooperativity which is the ideal limit for high resolution spectroscopy.

The average over the HLEs for many emitters can be cast in a convenient vector form
\begin{align}
\dot{\braket{a}} &= -i\Delta_\text{c}\braket{a}+\eta - i\textbf{G}^\intercal\braket{\boldsymbol{\sigma}} - \kappa \braket{a},
\label{HE1}
\\
\dot{\braket{\boldsymbol{\sigma}}} &= -i \Delta\ts{e} \braket{\boldsymbol{\sigma}} - i \boldsymbol{\Omega}\braket{\boldsymbol{\sigma}} - i \textbf{G}\braket{a} - \boldsymbol{\Gamma}\braket{\boldsymbol{\sigma}},
\label{HE2}
\end{align}
where now $\boldsymbol{\sigma}$ and $\textbf{G}$ are vectors with entries $\sigma_i^-$ and $g_i$. The matrices $\boldsymbol{\Omega}$ and $\boldsymbol{\Gamma}$ have the elements $\Omega_{ij}$ and $\gamma_{ij}$. In steady-state we find
\begin{align}
\braket{A_\text{out}}  &= \frac{\eta\sqrt{\kappa}}{i\Delta\ts{c}+\kappa + \textbf{G}^\intercal\textbf{G}/[i\Delta_\text{eff}(\Delta\ts{e}) + \gamma_\text{eff}(\Delta\ts{e})]},
\end{align}
where the effective $\Delta\ts{e}$-dependent collective energy shifts and linewidths are derived from the matrix $\textbf{M}(\Delta\ts{e})= i\Delta\ts{e}\mathbbm{1}+i\boldsymbol{\Omega}+\boldsymbol{\Gamma}$ as real and imaginary parts
\begin{align}
\label{delta_eff}
\Delta_\text{eff}(\Delta\ts{e}) &= \Im\left\{\frac{\textbf{G}^\intercal\textbf{G}}{\textbf{G}^\intercal\textbf{M}^{-1}(\Delta\ts{e})\textbf{G}}\right\},
\\
\gamma_\text{eff}(\Delta\ts{e}) &= \Re\left\{\frac{\textbf{G}^\intercal\textbf{G}}{\textbf{G}^\intercal \textbf{M}^{-1}(\Delta\ts{e})\textbf{G}}\right\}.
\label{gamma_eff}
\end{align}
In analogy to the single emitter case we can define an effective $N$-emitter cooperativity by
\begin{align}
\label{CN_def}
C\ts{eff}(\Delta\ts{e}) = \frac{\textbf{G}^\intercal\textbf{G}}{\kappa\gamma_\text{eff}(\Delta\ts{e})}.
\end{align}
This equation provides a main message of the paper, as it shows that the numerator and denominator no longer share the same dependency on $\mu$. As $\gamma_\text{eff}$ is not a natural constant of the ensemble, but strongly dependent on the relative positioning and phase of individual emitters, one can reach subradiant states with $\gamma_\text{eff}\ll\gamma$. By proper design of the cavity transverse field amplitude profile, the numerator can at the same time be maximized, resulting in a scaling up of $C\ts{eff}$ well above the independent emitter case $N g^2/(\kappa \gamma)$.

\section{Two emitters}
Let us elucidate the mechanism in the two emitter case, with adjustable separation $d=|\textbf{r}_1-\textbf{r}_2|$. We distinguish two fundamentally different cases: i) uniform coupling $\textbf{G}=(g,~g)^\intercal$ and ii) opposite coupling $\textbf{G}=(g,-g)^\intercal$, resulting in $\textbf{G}^\intercal\textbf{G}=2g^2$ for both cases. The matrix of interactions can be diagonalized with eigenvalues $i(\Delta\ts{e}\pm\Omega_{12})+(\gamma\pm\gamma_{12})$, signaling the presence of collective super- and subradiant states ($\gamma \pm \gamma_{12}$) shifted with $\pm \Omega_{12}$ from the emitter resonance $\omega\ts{e}$.

In the extreme case where $d \ll \lambda_e$, the mutual decay approaches $\gamma_{12}\to\gamma$ and the effective cooperativity reaches i) $C\ts{eff}\to C=g^2/(\kappa\gamma)$ and ii) $C\ts{eff}\to\infty$, respectively. Accounting for the shifts induced by the coherent part of the dipole-dipole interaction, i.e. tuning the cavity such that $\Delta\ts{c}=\Delta\ts{eff}(\Delta\ts{e})$, we find that the depth and the width of each antiresonance is $C\ts{eff}(C\ts{eff}+2)/(C\ts{eff}+1)^2$ and $\gamma\ts{eff}(C\ts{eff} + 1)$, respectively. For $d\to 0$ we hence have i) an antiresonance that is just as deep as for the single emitter but twice as wide (superradiance), and ii) an antiresonance that has a depth of 1 and is sharp with a width of $2g^2/\kappa$ (subradiance). While the width of the antiresonance is still limited by $g$, the phase switch bandwidth is independent of $g$. This is a direct measure of the subradiance as the slope of the phase switch in this limit is $1/\gamma\ts{eff}$ (see Appendix). The result is reminiscent of the one in~\cite{sames2014antiresonance}, however in a very different and less stringent regime where only weak coupling is required and where usually moderate phase shifts are expected; in contrast, for $\gamma\ts{eff}\to 0$, the phase even exhibits a $\pi$ phase change within an extremely narrow frequency range since in this regime $\lim_{\Delta\to 0^\pm}\left(\phi - \phi\ts{c}\right) = \pm\pi/2$.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig2.pdf}
\caption{\emph{Targeting antiresonances}. (a), (c), (e) show the cavity amplitude transmission and (b), (d) and (f) the corresponding phase for a scan of the laser frequency. The upper row corresponds to $\omega\ts{c}=\omega\ts{e}$ and asymmetric coupling $g_i=(-1)^i g$; the middle row illustrates frequency selection as $\omega\ts{c}\simeq\omega\ts{m=N}$. Finally, we also match the symmetry to the state corresponding to the subradiant antiresonance (see Appendix). The parameters are $N=10$, $g=\kappa/50$, $\gamma=\kappa/40$ with a chain separation of $d=0.08 \lambda\ts{e}$.}
\label{fig2}
\end{figure}

\section{Addressing collective subradiant states}
The above results can be generalized to large $N$. Analytical considerations can be made under a nearest neighbor approximation for $H\ts{dip}$ in the single-excitation regime, very well justified at small inter-particle distances and weak driving. A simple diagonalization of $H\ts{dip}$ gives rise to an $N$-band problem with energies $\omega\ts{m} = \omega\ts{e}+2\Omega \cos[m\pi/(N+1)]$ for $m$ running from $1$ to $N$. The Lindblad term then shows a ranking of levels from superradiant ($m=1$) to very subradiant ($m=N$) for $d\ll\lambda\ts{e}/2$~\cite{plankensteiner2015selective}. Moreover, the eigenvectors $|m\rangle=\sqrt{2/(N+1)}\sin[mj\pi/(N+1)]\ket{j}$ have a specific geometry with almost full symmetry ($m=1$) to almost full asymmetry ($m=N$). The two distinct cases involving uniform $\textbf{G}=(g,~g,...)^\intercal$ and opposite couplings $\textbf{G}=(g,-g,...)^\intercal$ then almost perfectly address these fully symmetric $|m=1\rangle$ and asymmetric $|m=N\rangle$ states.\\
\indent Illustrated in~\fref{fig2}a,b is a scan of the collective resonances with $g_i=(-1)^i g$ and $\omega\ts{c}=\omega\ts{e}$. Both the dip and phase show an off-resonant selection of collective subradiant states. We then selectively target a given state by fitting the cavity resonance to its energy as shown in~\fref{fig2}c,d. To achieve this, we focus around the state $|m=N\rangle$ with energy $\omega\ts{m=N}$ and we recalculate the state's energy by imposing $\Delta\ts {eff} (\delta)=0$ after which we set $\omega\ts{c}=\omega\ts{e}-\delta$ (then $\delta$ corresponds to a value close to $\omega\ts{m=N}-\omega\ts{e}$). This insures the resonance condition between the cavity and the targeted collective emitter state is fulfilled. We finally compare the results to an ideal procedure where the components of $\textbf{G}$ are chosen such that they match the geometry of the target state~\fref{fig2}e,f.\\
\indent The characteristics of the antiresonances can be quantified by $C\ts{eff}$ (see~\fref{fig3}). As above, we assume the asymmetric cavity field profile with $\textbf{G}=(g,-g,...)^\intercal$ and make a reference plot $C\ts{opt}$ as a function of $d$. The optimal $C\ts{opt}$ is obtained from Eq.~\eqref{CN_def} by substituting the decay rate with the minimal eigenvalue of $\boldsymbol{\Gamma}$~\cite{ostermann2014protected}. In reality, owing to imperfect phase matching to the most subradiant state as well as to the inherent level shifts brought on by the dipole-dipole interaction, the effective gain is more modest. Nevertheless, as suggested by the blue (solid) curve in~\fref{fig3}, for $d<0.5\lambda\ts{e}$ the enhancement is considerably larger than in the non-interacting quantum emitters case.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig3.pdf}
\caption{\emph{Scaling of cooperativity}. $C\ts{eff}$ for $N=10$ as function of $d$. The effective cooperativity (blue, solid) is compared to an idealized case of perfect subradiance (yellow, dashed). We used $\Delta\ts{c}=\Delta\ts{eff}=0$, $g=\kappa/30$ and $\gamma=\kappa/40$.}
\label{fig3}
\end{figure}

\section{Subradiance using transverse phase gradients}
While in practice individual phase imprinting on the sub-wavelength scale is not a trivial task, we present an implementation using 1D or 2D ensembles transversely placed in the center of a cavity, in the focal point of a higher order TEM mode. In the plane of the emitters, the field profile of a Gaussian-Hermite mode of order $m,n$ is $f\left(x,y\right)= A_{mn} H_m\left(\sqrt{2}x/w\right)H_n\left(\sqrt{2}y/w\right)e^{-(x^2 + y^2)/w^2}$. Here $H_n(x)$ is the $n$-th Hermite polynomial, $w$ is the beam waist and $A_{mn}=\sqrt{2/(\pi 2^{(m+n)}m!n!)}$. Higher order TEM modes exhibit multiple extrema of opposite signs in the transverse profile. For a sufficiently small $w$ (of the order of $\lambda\ts{e}$), adjacent extrema can be closely spaced (for a TEM$\ts{m0}$ mode around $w/\sqrt{m}$) resulting in the desired coupling asymmetry. We illustrate the phase imprinting mechanism for a chain illuminated by a transverse TEM$\ts{m0}$ with increasing $m$ in~\fref{fig4}a. While for small $m$ the cooperativity decreases (owing to a decrease in $|\mathbf{G}|$), at higher $m$ the alternating field phases are partially addressing asymmetric collective states of high robustness resulting in a considerably enhanced cooperativity. The very sharp cavity response for a fixed mode $m=21$ is shown in~\fref{fig4}b,c in comparison to the modest results expected for a TEM$_{00}$ illumination. We have moreover numerically investigated 2D geometries as well and found for example in the case of a $3\times3$ square array with $d= 0.2\lambda\ts{e}=w$ and $g=\kappa/20=2\gamma$ (as depicted in~\fref{fig1}) an enhancement of cooperativity from the bare value $Ng^2/\gamma = 0.9$ to $C\ts{eff}\approx 80.1$.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig4.pdf}
\caption{\emph{Targeting subradiance via transverse mode driving}. (a) Cooperativity as a function of $m$ for $N=10$ and under illumination with TEM$\ts{m0}$ mode. The inset shows a decrease in cavity coupling $|\mathbf{G}|$. (b,c) Comparison of antiresonance signatures for TEM$\ts{00}$ addressing (red, dashed) vs. TEM$_{m0}$ addressing (blue, solid) with $m=21$, corresponding to the point selected in (a). The parameters are $d=0.2\lambda\ts{e}$, $w=\lambda\ts{e}$, $g=\kappa/30$, $\gamma=\kappa/40$ and $\Delta\ts{eff}=0$ at $\Delta\ts{c}=0$.}
\label{fig4}
\end{figure}

\section{Conclusions}
We have shown that tailoring the \textit{collective} dissipative dynamics of $N$ dipole coupled emitters leads to high effective cooperativity even in the regime $Ng^2/(\kappa\gamma)\ll1$. The immediate consequence is the occurrence of a narrow antiresonance dip with fast spectral phase switching without the need of strong individual coupling as in Ref.~\cite{sames2014antiresonance}. As it applies to narrow atomic transitions it hints towards applications for precision spectroscopy and in quantum network characterization. Further applications in hybrid optomechanics~\cite{juan2016near,dantan2014hybrid} where emitter-mediated phonon-photon interactions can greatly benefit from sharp resonances~\cite{dantan2014hybrid} can be envisaged. As opposed to using a lossy cavity field as an engineered bath leading to superradiance as in Ref.~\cite{mascarenhas2013cooperativity}, we only considered the naturally occurring environment provided by the free space radiation modes. We also mention that the regime treated here is perturbative, i.e. the emitters do not modify the bare mode functions of the cavity mode. Very interesting phenomena shall occur when the collective effects can strongly modify the scattering rate into the cavity mode as in Refs.~\cite{bettles2015cooperative, bettles2016cooperative, bettles2016enhanced, shahmoon2016cooperativity} leading to collective atomic reflectivity close to unity and opening up a novel regime of CQED. Similar considerations can be used to analyze metamaterial arrays, where classical analogues of subradiant states are also experimentally seen~\cite{jenkins2016many}.

\begin{acknowledgments}
We acknowledge financial support by the Austrian Science Fund (FWF) within the Innsbruck doctoral school  DK-ALM: W1259-N27 (D.~P.) and project number P29318-N27 (H.~R) as well as from the Aarhus Universitets Forskningsfond and the Max Planck Society (C.~G). We also acknowledge useful discussions with Robert Bettles. Graphs are done with the open source library Matplotlib~\cite{hunter2007matplotlib}.
\end{acknowledgments}
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  {Kleppner}}, }\bibfield  {title} {\enquote {\bibinfo {title} {Cavity
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   {journal} {Physics Today} }\textbf {\bibinfo {volume} {42}}, \bibinfo
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  {\enquote {\bibinfo {title} {Cavity {Q}uantum {E}lectrodynamics},} }\href
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  {Nature} }\textbf {\bibinfo {volume} {436}}, \bibinfo {pages} {87--90}
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\end{thebibliography}%

\pagebreak

\begin{appendix}

\begin{center}
\large{\textbf{APPENDIX}}
\end{center}

\renewcommand{\theequation}{A.\arabic{equation}}
\renewcommand\thefigure{A.\arabic{figure}}
\setcounter{equation}{0}
\setcounter{figure}{0}

\subsection{Appendix 1: Vacuum mediated coherent and incoherent dynamics}
For a collection of emitters at positions $\textbf{r}_i$ the collective spontaneous emission rates and the coherent dipole-dipole interaction strengths are \cite{lehmberg1970radiation}
\begin{align}
\gamma_{ij} &= \frac{3\gamma}{2}F(\textbf{k}\ts{e}\cdot\textbf{r}_{ij}),
\\
\Omega_{ij} &= -\frac{3\gamma}{4}G(\textbf{k}\ts{e}\cdot\textbf{r}_{ij}),
\end{align}
respectively. Here, $\gamma$ is the single emitter free space decay rate, $\textbf{k}\ts{e}$ is the transition wave vector and $\textbf{r}_{ij}:=\textbf{r}_i-\textbf{r}_j$. The functions $F$ and $G$ are defined as
\begin{align}
F(x) &= \left(1-\cos^2\theta\right)\frac{\sin x}{x} +
\notag \\
&+ \left(1-3\cos^2\theta\right)\left(\frac{\cos x}{x^2}-\frac{\sin x}{x^3}\right),
\\
G(x) &= -\left(1-\cos^2\theta\right)\frac{\cos x}{x} +
\notag \\
&+ \left(1-3\cos^2\theta\right)\left(\frac{\sin x}{x^2} + \frac{\sin x}{x^3}\right),
\end{align}
where $\theta$ is the angle drawn by the induced dipoles $\boldsymbol{\mu}$ and the separation vector $\textbf{r}_{ij}$. Note, that for all computations in the paper we assumed the dipoles to be oriented along the y-direction.

\subsection{Appendix 2: Cavity input-output relations}
Consider a cavity with two mirrors $A$ and $B$. We drive the cavity through the mirror $B$ and measure the output at the opposite mirror $A$. Furthermore, we assume that both mirrors have identical losses of $\kappa/2$. The input-output relations of the total input and output operators (i.e. the input white noise on top of the classical input) for both mirrors are
\begin{align}
B_\text{in} + B_\text{out} &= \sqrt{\kappa}a,
\\
A_\text{in} + A_\text{out} &= \sqrt{\kappa}a.
\end{align}
Taking the classical average of the above equations, and assuming the drive through $B$ at amplitude $\braket{B_\text{in}}=\eta/\sqrt{\kappa}$,
\begin{align}
\frac{\eta}{\sqrt{\kappa}} + \braket{B_\text{out}} &= \sqrt{\kappa}\braket{a},
\\
\braket{A_\text{out}} &= \sqrt{\kappa}\braket{a}.
\label{A_out_relation}
\end{align}
Hence, we find that the output at port $A$ reflects the cavity field according to \eqref{A_out_relation}.

\subsection{Appendix 3: Emitter input noise in the HLE}
Writing the input noise in the HLEs for the emitter ensemble is not straightforward as the decay is not diagonal. We first note that for a diagonal Lindblad term describing the damping of a system via multiple damping channels with operators $c_j$ and rates $\nu_j$, the HLE~\cite{gardiner2004quantum} of an arbitrary system operator $A$ is
\begin{align} \label{diagonal_HLE}
\dot{A} &= i[H,A] - \sum_j[A, c_j^\dag]\left(\frac{\nu_j}{2}c + \sqrt{\nu_j}c_\text{in}\right) +
\notag \\
&+ \sum_j\left(\frac{\nu_j}{2}c^\dag + \sqrt{\nu_j}c_\text{in}^\dag\right)[A,c],
\end{align}
where $c_\text{in}$ is the uncorrelated white input noise $\braket{c_\text{in}(t)c_\text{in}^\dag(t')}=\delta(t-t')$.

For correlated emitters, it is possible to diagonalize the Lindblad term such that each decay channel is described by a damping operator which is a linear combination of all $\sigma_j^-$. We may perform the transformation back to the original non-diagonal form in order to find the input noise terms. To this end, let \textbf{T} be the real and orthogonal ($\textbf{T}^{-1}=\textbf{T}^\text{T}$) matrix which diagonalizes the matrix $\boldsymbol{\Gamma}$ as
\begin{align}
\text{diag}\left(\lambda_1,...,\lambda_N\right) &= \textbf{T}^{-1}\boldsymbol{\Gamma}\textbf{T},
\end{align}
where $\lambda_j$ is the $j$th eigenvalue of the decay matrix. Defining a set of damping operators
\begin{align}\label{Pi_op}
\Pi_j^\pm &:= \sum_k \left(\textbf{T}^{-1}\right)_{jk}\sigma_k^\pm,
\end{align}
we may write \cite{ostermann2014protected}
\begin{align}
\mathcal{L}_\text{e}[\rho] &= \sum_i\frac{\lambda_i}{2}\left(2\Pi_i^-\rho\Pi_i^+ - \Pi_i^+\Pi_i^-\rho - \rho\Pi_i^+\Pi_i^-\right).
\end{align}
Obviously, this Lindblad term is diagonal and hence the HLE may be cast into the form given by \eqref{diagonal_HLE}. The input noise terms of the emitter operators $\sigma_{i,\text{in}}^\pm$ follow the transformation rules given by \eqref{Pi_op}. Transforming the HLE for any emitter operator $A$ back into the non-diagonal form gives the usual terms for the deterministic parts. For the noise terms, however, we have
\begin{align}
\sum_j[A,\Pi_j^+]\sqrt{\lambda_j}\Pi_{j,\text{in}}^- &= \sum_j[A,\sigma_j^+]\xi_j(t),
\\
\sum_j[A,\Pi_j^-]\sqrt{\lambda_j}\Pi_{j,\text{in}}^+ &= \sum_j[A,\sigma_j^-]\xi_j^\dag(t),
\end{align}
where we have implicitly defined our correlated emitter noise terms $\xi_j$ as
\begin{align}
\xi_j(t) &:= \sum_{k,l} \textbf{T}_{jk}\sqrt{\lambda_k}(\textbf{T}^{-1})_{kl}\sigma_{l,\text{in}}^-.
\end{align}
Hence, the HLE for any emitter operator $A$ is
\begin{align}
\dot{A} &= i[H,A] - \sum_{ij}[A,\sigma_i^+]\left(\frac{\gamma_{ij}}{2}\sigma_j^- + \xi_i(t)\right) +
\notag \\
&+ \sum_{ij} \left(\frac{\gamma_{ij}}{2}\sigma_j^+ + \xi_i^\dag(t)\right)[A,\sigma_i^-],
\end{align}
with the spatially correlated white noise $\xi_i$. From the definition of the noise it is straightforward to show that
\begin{align}
\braket{\xi_i(t)\xi_j^\dag(t')} &= \gamma_{ij}\delta(t-t').
\end{align}


\subsection{Appendix 4: Lorentzian shape of the antiresonance}
We verify the Lorentzian profile of the antiresonance for a single emitter interacting with a single cavity mode by fitting
\begin{align}
B(\Delta)=\left|\frac{\kappa}{i\Delta+\kappa}\right|^2-\left|\frac{\kappa}{i\Delta+\kappa+g^2/(i\Delta + \gamma)}\right|^2,
\end{align}
with a Lorentzian
\begin{align}
\left|\frac{\beta s}{i\Delta+\beta}\right|^2,
\end{align}
where $\beta$ gives the linewidth while $s$ gives the height. We compute immediately
\begin{align}
s=1-\frac{1}{(1+C)^2},
\end{align}
and from
\begin{align}
\beta &= \sqrt{\frac{2B(0)}{\frac{\partial^2 B}{\partial\Delta^2}(0)}}
\end{align}
we obtain
\begin{align}
\beta = \sqrt{\frac{\kappa^2(g^2+\kappa \gamma)^2(g^2+2\kappa \gamma)}{g^6+4g^4 \kappa \gamma + 2 \kappa^3 \gamma (\kappa^2 +\kappa \gamma +2 \gamma^2)+g^2(\kappa^4 +6 \kappa^2 \gamma^2)}}.
\end{align}
In the limit that both $\gamma,g\ll\kappa$ when only keeping terms of $\mathcal{O}(\kappa^4)$ and higher, this reduces to $\beta\approx\gamma(1+C)$.

The results obtained above are valid for two emitters with the replacements $\gamma\to\gamma\ts{eff}$, $C\to C\ts{eff}$ and the detuning $\Delta\ts{c}=\Delta\ts{eff}=\Delta$.


\subsection{Appendix 5: Characterizing the phase of the field}

In order to describe the phase analytically, consider that for the single emitter transmission coefficient with $\Delta\ts{c}=\Delta\ts{e}=\Delta$, we have
\begin{align}
\frac{\Im\{t\}}{\Re\{t\}} &= \frac{\Delta\left(g^2 - \Delta^2 - \gamma^2\right)}{\kappa\left(\gamma^2 + \Delta^2\right) + \gamma g^2}.
\end{align}
The phase of the field is then given by the arctangent of the above expression. For sufficiently small phase shifts ($C\ll 1$), the phase is well approximated by the argument of the arctangent, such that
\begin{align} \label{phi_app}
\frac{\partial}{\partial\Delta}\left(\phi - \phi\ts{c}\right) &\approx \frac{C\gamma\left(\kappa + \gamma\right)\left(\gamma^2\left(1 + C\right) - \Delta^2\right)}{\kappa\left(\gamma^2 \left(1 + C\right) + \Delta^2\right)^2}.
\end{align}
Finding the roots of this expression yields the detuning at which the phase shift of the emitter is maximal (positive $\Delta$) or minimal (negative $\Delta$), respectively,
\begin{align}
\Delta &= \pm\gamma\sqrt{1 + C}.
\end{align}
We note, that for large cooperativity the approximation of the phase does no longer hold, however, given that $g,\gamma\ll\kappa$ the position of the maximum and minimum are well approximated by the above detuning even if $C\gg 1$, such that
\begin{align}
\max_\Delta\left[\phi - \phi\ts{c}\right] &\approx \max_\Delta\left[\phi\right] &\approx \arctan\left(\frac{C}{2\sqrt{1 + C}}\right).
\end{align}
Here we wrote down the maximum of the phase and note that the minimum only differs from this expression with a negative sign. Now, depending on the sign of $\Delta$, taking the limit of $\gamma\to 0$, the phase either goes to $\pi/2$ or $-\pi/2$, while $\Delta\to0$, i.e.
\begin{align}
\lim_{\Delta\to0^\pm} \phi = \pm\frac{\pi}{2}.
\end{align}
This means that the phase exhibits a jump of magnitude $\pi$ when crossing the resonance, which can also be seen when computing the slope. To this end, consider the fact that close to the resonance where $|\Delta|\ll\kappa$, the phase is small such that the approximation in Eq.~\eqref{phi_app} is valid again. In the limit of $|\Delta|\ll\gamma,g\ll\kappa$ but large cooperativity $g^2\gg \kappa\gamma$ and hence $g\gg\gamma$, this expression becomes
\begin{align}
\phi - \phi\ts{c} &\approx \phi \approx \frac{\Delta}{\gamma}.
\end{align}
We conclude that the slope at the resonance (where the phase changes sign) is proportional to $1/\gamma$ and hence diverges for $\gamma\to 0$, which is in agreement with the previous result where the phase shift is maximal or minimal at $\Delta\to 0$ depending on whether one takes the left-sided or right-sided limit.

Finally, we note that the above discussion is also applicable to two emitters when replacing $\gamma \rightarrow \gamma\ts{eff}$, $C \rightarrow C\ts{eff}$, and setting $\Delta\ts{c}=\Delta\ts{eff}=\Delta$.


\subsection{Appendix 6: Matching frequency and symmetry of antiresonances}

For \fref{fig2}(e),(f) we set
\begin{align}
\textbf{G}\approx &(0.72, -1.44,  2.03, -2.46,  2.68,
\notag \\
&-2.68,  2.46, -2.03,  1.44, -0.72)^\intercal \times 10^{-2} \kappa,
\end{align}
which corresponds to the coefficients of the eigenvector of $\boldsymbol{\Omega}$ that had the largest overlap with the eigenvector of $\boldsymbol{\Gamma}$ corresponding to the smallest eigenvalue and therefore the smallest decay rate \cite{ostermann2014protected}. The frequency was matched by numerically solving $\Delta\ts{eff}(\delta)=0$, which lead to $\delta\approx 0.234\kappa$, and setting $\Delta\ts{c}=\Delta\ts{e}-\delta$.

\subsection{Appendix 7: Comparison to exact numerics}

Let us now comment on the accuracy of the linearization used to obtain the form of eqs. \eqref{single_HE1},\eqref{single_HE2} and \eqref{HE1},\eqref{HE2}. From \fref{figA1}, it is clear that as long as we keep the driving strength $\eta$ weak enough, the requirements for the low-excitation limit are fulfilled rendering the analytics exact (cf. red dots and dark blue line in \fref{figA1}). If the driving becomes too strong, the excitation of the emitters is no longer negligible resulting in a discrepancy between the full numerics and the analytics (cf. dashed, light blue line in \fref{figA1}). Nevertheless, this does not change the results qualitatively, i.e. there still is a subradiant antiresonance.

\begin{figure}[ht]
\includegraphics[width=.8\columnwidth]{figA1.pdf}
\caption{\emph{Comparison of analytics with exact numerics}. We compare the full numerics, i.e. without using $\braket{\sigma_i^z}\approx -1$, to the analytical result in an extremely subradiant regime for a 4 emitter chain with opposite coupling and without dipole-dipole shifts ($\Omega_{ij}=0$). The inset is zoomed in on the antiresonance to emphasize the agreement. The parameters are $d=0.02\lambda\ts{e}$, $g=\kappa/20=2\gamma$ and $\Delta\ts{e}=0$.}
\label{figA1}
\end{figure}

\end{appendix}

\end{document} 
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