Resonator-assisted single molecule quantum state detection

The ability to determine the state of a single quantum emitter is essential for quantum information processing. Resonance fluorescence imaging is a convenient and powerful method. During imaging, exciting closed optical transitions ensures that a quantum emitter scatters a large number of photons without leaving a specific ground state, thus making it possible to achieve state-sensitive detection with a high signal-to-noise ratio, even with low photon collection efficiency. This approach has been widely employed in various quantum systems, such as cold atoms Bochmann et al. (2010); Gehr et al. (2010); Fuhrmanek et al. (2011), trapped ions Olmschenk et al. (2007); Myerson et al. (2008), nitrogen vacancy centers Neumann et al. (2010); Robledo et al. (2011) and quantum dots Vamivakas et al. (2010).

Many promising quantum systems are not suited for fluorescent imaging due to their energy structure. For instance, cold molecules have a wide range of applications in quantum chemistry Bohn et al. (2017) and quantum computation DeMille (2002). However, most of the molecules have no real optical cycling transitions due to a myriad of rovibrational levels accessible in a radiative decay process. For specific kinds of molecules, one may find a transition with near-unity Franck-Condon factor for a target ground state, and use multiple lasers to drive an approximately closed transition in a manageable collection of ground and excited states. For instance, SrF Shuman et al. (2010) or CaF Anderegg et al. (2017) molecules can be laser cooled and trapped in a magneto-optical trap.

In general, alternative state detection method is needed for molecules without optically closed transitions. Resonance-enhanced multiphoton ionization is adopted for molecule detection Gabbanini et al. (2000), which ionizes the molecules and detects the subsequent ions. This method is nevertheless completely destructive. Direct absorption imaging of molecules is also reported Wang et al. (2010), when a molecular ensemble has a high optical density.

Cavity quantum electrodynamics opens a new way to implement state detection of a single molecule. Cavity-controlled light-matter interaction enables the manipulation of molecular photon emission properties. When a molecule is located inside a cavity or near a resonator which is tuned to the molecular resonance of interest, the branching ratio of decay into irrelevant states may be greatly suppressed Lev et al. (2008), allowing the interaction with resonator photons for an extended period of time. In addition, considering the molecules directly emit photons into the resonator mode(s), the signal photon collection efficiency may be significantly enhanced compared to the case of emission in freespace.

In this paper, we consider a scheme to detect the quantum state of molecules without a closed transition by utilizing high-finesse Fabry-Perot cavities Hood et al. (2001) or high-Q whispering-gallery mode (WGM) resonators Vernooy et al. (1998); Spillane et al. (2005); Pöllinger et al. (2009); Tien et al. (2011); Chang et al. (2019). Our scheme is inspired by the pioneering experiments for detecting single atoms falling through a microtoroidal resonator Aoki et al. (2006); Shomroni et al. (2014), and probing trapped single atoms inside a mirror cavity Boozer et al. (2006); Khudaverdyan et al. (2009), a fiber-based cavity Gehr et al. (2010); Kato and Aoki (2015), as well as in the vicinity of a photonic crystal cavity Thompson et al. (2013); Goban et al. (2014). In particular, we consider the transmission of a bus waveguide critically coupled to WGMs in a micro-ring resonator [Fig. 1(a)], displaying molecule-induced transparency on resonance due to strong light-molecule interaction. The proposed detection scheme is background-free, as there is no waveguide transmission unless a molecule in the target state couples to the resonator. The maximum scattered photon number is enhanced by the cooperativity parameter $C$ before the molecule is completely pumped away from its initial state. We numerically simulate this open quantum system with a Lindblad master equation and derive analytical solutions in the weak-driving regime. We then extend the single-molecule model to a multi-molecule case considering collective effect. Similar scheme can be realized in monitoring the off-resonant transmissivity in the case of a Febry-Perot cavity (Appendix A). Our scheme can also be applied to single-shot state readout for other quantum emitters Sun and Waks (2016).

For illustration purposes, we take Rb₂ molecule as an example. The relevant internuclear potentials are plotted in Fig. 1(b), in which a single-photon, short-range photoassociation has been utilized for ground state molecule synthesis directly from cold atoms Bellos et al. (2011). Using an optical cavity or a photonic crystal waveguide to radiatively enhance the synthesis efficiency has recently been discussed Perez-Rios et al. (2017); Kampschulte and Denschlag (2018). These recent developments makes Rb₂ a good candidate to demonstrate resonator-enhanced single molecule detection.

The paper is organized as follows. Section II reviews the optical setup of the resonator as well as a simplified energy structure of the Rb₂ molecule. In Sec. III, we derive a formalism to calculate single molecule dynamics with one resonator mode. In Sec. IV, the discussion is extended to the coupling between a single molecule and two resonator modes. In Sec. V, we discuss the multi-molecule dynamics by taking into account collective effects.

We first introduce the optical system, a bus waveguide coupled to an empty micro-resonator as shown in Fig. 1(a), which supports WGMs that circulate either in the clockwise (CW) or the counter-clockwise (CCW) orientation along the resonator. We assume an input field of power $P_{I}$ is injected from one end of the bus waveguide, and analyze the transmission power $P_{T}$ and reflection power $P_{R}$. In the following cases, we assume that back-scattering in the resonator, which couples the CW and CCW modes, occurs at a rate much smaller than the total resonator loss rate $\kappa$, and thus there is negligible mode mixing. Here, $\kappa=\kappa_{e}+\kappa_{i}$, where $\kappa_{i}$ is the intrinsic photon loss rate and $\kappa_{e}$ is the waveguide-resonator coupling rate. Due to the phase matching condition, when the resonator couples light from single end of the bus waveguide shown in Fig. 1(a), only one mode (CW WGM, illustrated as solid line) can be excited and the resonator can be treated like a single mode cavity. Reflection stays at zero ($P_{R}=0$) due to the absence of CCW WGM excitation.

To achieve background-free molecule detection, we consider a critically coupled resonator ($\kappa_{i}=\kappa_{e}$) for zero waveguide transmissivity on resonance ($\Delta_{\text{cl}}=0$). The bus waveguide transmissivity is evaluated by solving the standard Heisenburg-Langevin equation with the single-mode Hamiltonian of an empty resonator

where $\Delta_{\text{cl}}=\omega_{c}-\omega_{l}$ is the detuning between the resonant mode frequency $\omega_{c}$ and the external driving frequency $\omega_{l}$, $\hat{a}^{({\dagger})}$ represents the annihilation (creation) operator of the (CW) resonator mode, excited from the bus waveguide at a rate coefficient $\varepsilon=i\sqrt{2\kappa_{e}\mathcal{I}}$ and $\mathcal{I}=P_{I}/\left(\hbar\omega_{l}\right)$ is the waveguide photon input rate. At steady state, the expectation value of intra-resonator field is found to be $\expectationvalue{\hat{a}}=\varepsilon/\left(i\Delta_{\textrm{{cl}}}+\kappa\right)$. The bus waveguide transmission is the interference between the input field $\sqrt{\mathcal{I}}$ and the out-coupled field from the resonator $i\sqrt{2\kappa_{e}}\expectationvalue{a}$, resulting in a transmissivity Haus (1984)

As shown in Fig. 2(c), at critical coupling the waveguide transmissivity, $T=\absolutevalue{\Delta_{\text{cl}}/(\kappa+i\Delta_{\text{cl}})}^{2}$, drops to zero on resonance. As we discuss in the following, the waveguide transmissivity will be greatly modified when a molecule is present and couples to the resonator. This establishes our scheme to realize background-free molecule detection.

To model the radiative dynamics of a resonator-coupled molecule, we treat it as an effective three-level system, as illustrated in Fig. 1(b). While radiative decay processes can couple a molecular excited state $\ket{e}$ to a collection of (electronic) ground states of different rovibrational energy levels, the molecule-resonator coupling can in principle involve only one designated rovibrational state $\ket{g}$ when the resonator frequency $\omega_{c}$ is aligned with the transition frequency $\omega_{m}$ and the resonator linewidth ($\lesssim O(1)~{}$GHz) is smaller than the relevant rovibrational energy level spacing ($\gtrsim O(10)~{}$GHz) by over an order of magnitude. A third state $\ket{s}$ denotes all other uncoupled states that can accumulate population from spontaneous decay. Given a Franck-Condon factor $f_{\mathrm{FC}}$ between states $\ket{e}$ and $\ket{g}$, the spontaneous decay rate to $\ket{g}$ is $\Gamma_{g}=f_{\mathrm{FC}}\Gamma$ and the decay rate to $\ket{s}$ is $\Gamma_{s}=(1-f_{\mathrm{FC}})\Gamma$, where $\Gamma$ is the total decay rate of the excited state $\ket{e}$.

We denote the coherent coupling rate between the molecule and the resonator mode as $\displaystyle g_{c}=\sqrt{3\Gamma_{g}c^{3}/2V\omega_{m}^{2}}$, where $c$ is the speed of light, $\displaystyle V=\int\epsilon(\mathbf{r})|\mathcal{E}(\mathbf{r})|^{2}d\mathbf{r}/\left|\mathcal{E}\left(\mathbf{r}_{\mathrm{mol}}\right)\right|^{2}$ is the mode volume, $\epsilon(\mathbf{r})$ is the dielectric function, $\mathcal{E}\left(\mathbf{r}\right)$ is the mode field strength, and $\mathbf{r}_{\mathrm{mol}}$ is the molecular position. $g_{c}$ is position dependent due to the mode field intensity variation near the resonator dielectric surface. We assume that molecules are trapped in close proximity of a resonator and hence $g_{c}$ is a constant.

Table 1 lists typical resonator and molecule parameters used in the numerical and analytical calculations throughout this paper. We take Rb₂ molecules coupled to a micro-ring resonator as an example. The relevant internuclear potentials are plotted in Fig. 1(b), where $\ket{g}$ stands for the rovibrational ground state of interest in the $a^{3}\Sigma^{+}_{u}$ triplet potential, and $\ket{e}$ represents an excited state in the $1^{3}\Pi_{g}$ potential (vibrational level $\nu=8$, and angular momentum quantum number $J=1$) Bellos et al. (2011). Due to the finite Frank-Condon overlap between $\ket{g}$ and $\ket{e}$, this transition has been utilized for ground state molecule synthesis directly from cold atoms via a single-photon short-range photoassociation to state $\ket{e}$, followed by spontaneous decay into $\ket{g}$ Bellos et al. (2011). On the other hand, our model optical system is adapted from a recent report on high-Q micro-ring resonators Chang et al. (2019), where we assume that high quality factor $Q>10^{5}$ and large single-photon vacuum Rabi frequency $2g_{c}\sim 2\pi\times 500~{}$MHz can be simultaneously realized to achieve a large coorperativity parameter $C\equiv g_{c}^{2}/\kappa\Gamma$, which is the key parameter to achieve high single-molecule detection sensitivity.

We now analyze the recovery of bus waveguide transmissivity on resonance with the presence of a single molecule as shown in Fig. 2(b). We begin with the first scenario where a molecule couples to a single resonator mode. This applies to the case when a molecule, spin-polarized in a stretched state, couples only to a circularly polarized WGM, and cannot emit photons into the other WGM because of its opposite circular polarization state ¹¹1The WGMs are circularly polarized since they are traveling wave modes with strong evanescent field outside the waveguide. A $\pm\pi/2$ degree out-of-phase axial field component, dictated by the transversality of the Maxwell’s equation and time reversal symmetry, gives the CW and CCW WGMs opposite circular polarization states.. The single-mode light-molecule interaction Hamiltonian $\hat{H}_{1}$ can be written as

where $\Delta_{\text{ml}}=\omega_{m}-\omega_{l}$, $\hat{\sigma}_{-}=\ket{g}\bra{e}$, and $\hat{\sigma}_{+}=\ket{e}\bra{g}$.

Taking into account of the resonator loss and the molecule spontaneous emission, the master equation of the full system is written as

where $\rho$ is the density matrix of the molecule and photon system and the Lindblad operators take the form of $\mathcal{L}[\hat{b}]\rho=\hat{b}\rho\hat{b}^{\dagger}-\frac{1}{2}\hat{b}^{\dagger}\hat{b}\rho-\frac{1}{2}\rho\hat{b}^{\dagger}\hat{b}$ and $\hat{\sigma}_{-}^{\prime}=\ket{s}\bra{e}$. As shown in Table 1, both the coherent coupling rate $g_{c}$ and the total resonator loss rate $\kappa=\kappa_{i}+\kappa_{e}=2\kappa_{e}$ are at least an order of magnitude larger than the molecule spontaneous decay rates $\Gamma_{g(s)}$, thus allowing us to utilize fast resonator-molecule dynamics for state detection before losing the population into the uncoupled states $\ket{s}$.

In the limit of single excitation, the resonator and the molecule form an effective five-level system with states denoted by $\ket{m,n}$ as shown in Fig. 2(a), where $m$ represents the molecular quantum state and $n=0,1$ is the photon number in the WGM. Three coupled states $\ket{e,0}$ and $\ket{g,0(1)}$ on the left of Fig. 2(a) evolve under the cavity QED Hamiltonian $\hat{H_{0}}+\hat{H_{1}}$ that can quickly establish a quasi-steady state under external driving. Population in this subsystem then slowly decays into the uncoupled states illustrated in the right part of Fig. 2(a), evolving as an empty resonator.

Figure 2(b) shows the steady-state transmissivity in an ideal cavity QED system for $\Gamma_{s}=0$. The transmission spectrum is evaluated by substituting the expectation value of the field amplitude $\langle\hat{a}\rangle$ in Eq. (2) with the steady-state solution

Recalling that the WGM frequency $\omega_{c}$ is aligned to the molecular transition frequency $\omega_{m}$, we define detuning $\delta\equiv\Delta_{\text{cl}}\equiv\Delta_{\text{ml}}$. When the cooperativity $C\gg 1$, the transmissivity at zero detuning ($\delta=0$),

nearly recovers to unity. This effect can be understood as the interference of two molecule-photon dressed states that results in a molecule-induced transparency window, similar to an electromagnetically-induced transparency (EIT). The EIT-like effect contrasts the vanishing transmissivity of an empty resonator critically coupled to the bus waveguide as shown in Fig. 2(c). This forms a highly sensitive scheme for quantum state detection similarly found in Aoki et al. (2006).

In the realistic case of $\Gamma_{s}\neq 0$, transmission can only recover for a finite period of time. One expects that the transmission spectrum evolves transiently from an EIT-like curve of Fig. 2(b) to the empty resonator case as shown in Fig. 2(c). Nevertheless, strong resonator coupling suppresses the excited state $\ket{e}$ population, resulting in a much reduced decay rate into the uncoupled states $\ket{s}$ compared to the free-space decay rate $\Gamma_{s}$. Finite transmission through the bus waveguide can thus be collected for a finite time period for molecular state detection.

Using the separation of time scales, we find the analytical solution for the quasi-steady density matrix $\rho^{ss}$ of the system, as detailed in Appendix B, and evaluate the population transfer rate $D$ to the empty resonator. In the weak-driving regime ($\displaystyle\absolutevalue{\epsilon}\ll\kappa,g_{c}^{2}/\kappa$), we find the population of the molecule-resonator dressed state primarily resides in $\ket{g,0}$. The transfer rate $D$ is greatly suppressed due to a small population in $\ket{e,0}$. Based on Eq. (36), We find

where $\rho_{e0,e0}~{}(\rho_{g0,g0})$ is the population in the excited state $\ket{e,0}$ (ground state $\ket{g,0}$), and the superscript $ss$ stands for steady state. $D/\mathcal{I}$ also represents the probability for the dressed state to decay into $\ket{s,0}$.

At zero detuning, we find the decay rate

Comparing $D_{\mathrm{res}}/\mathcal{I}$ with the depumping probability in freespace $\Gamma_{s}/\Gamma=(1-f_{FC})$, the resonator-enhanced $\ket{e}$-$\ket{g}$ transition enjoys a large suppression factor $\sim 1/C$ for depumping into the uncoupled states when $C\gg 1$. As we will show in Eq. (14), this factor suggests that around $\sim C/(1-f_{FC})$ photons may be transmitted through the bus waveguide before the system is converted into an empty resonator and again blocks all resonant input photons.

We validated the slow-transfer model with full numerical calculations Johansson et al. (2013) using the master equation Eq. (4) and the parameters listed in Table 1; see Fig. 3 for an example. The numerical result shows negligible differences from the analytical model in the mean resonator photon number $\langle\hat{a}^{\dagger}\hat{a}\rangle$ [Eq. (40)], as well as the state populations

In the following discussions, we will mainly present our analytical analysis.

Now we derive the transmission spectrum by evaluating the time evolution of the quasi-steady resonator field $\expectationvalue{\hat{a}}=\rho^{ss}_{g1,g0}+\rho^{ss}_{s1,s0}$ as detailed in Eq. (39), where $\rho_{g1,g0}$ and $\rho_{s1,s0}$ are the off-diagonal density matrix elements between states $(\ket{g,1}$, $\ket{g,0})$ and $(\ket{s,1}$, $\ket{s,0})$, respectively. Substituting $\expectationvalue{\hat{a}}$ in Eq. (2), we find

The instantaneous transmission resembles that of a cavity QED system [black curve in Fig. 4(b)] with an EIT window near $\delta=0$ and two absorption dips at $\delta=\pm g$ separated by the vacuum Rabi frequency; $T(\delta)$ eventually evolves to be $\delta^{2}/(\delta^{2}+\kappa^{2})$, the transmissivity of an empty resonator.

At zero detuning, the transmissivity $T(0)$ decays exponentially with increasing input photon number $\mathcal{I}t$ as

which is robust against decay when $C\gg 1$. Figure 4(a) illustrates sample transmission curves at different cooperativity parameters $C$.

As transmission measurement typically involves finite integration time, we calculate the time-averaged transmission spectra

under various time intervals $(0,\tau)$. Figure 4(b) shows how the transmitted signal at various laser detuning $\delta$ evolves with the integration time $\tau$. These spectra demonstrate the transition from an EIT-like behavior in a molecule-coupled resonator to the resonant absorption spectrum in an empty resonator – Two initial transmission dips and a transparency window near $\delta=0$, formed by the destructive interference of two molecule-photon dressed states, gradually fade away to be overtaken by the single resonance of an empty resonator. Apparently the transmission signal at zero detuning or near the two dips corresponding to the dress-state resonances particularly provides sensitive transient signal for the detection of a molecule in the coupled ground state $\ket{g}$.

We consider background-free transmission at zero detuning since it allows us to take the interrogation time $\tau\rightarrow\infty$ and maximize the number of transmitted photons for detection. Figure 5(a) displays the relationship between transmitted photon number $N_{T}(\tau)$ and input photon number $\mathcal{I}\tau$, where

The diagonal dashed line in Fig. 5(a) represents the optimal case of unity transmissivity for $C\rightarrow\infty$ and for a closed transition $f_{FC}=1$. All other cases of finite $C$ and $f_{FC}<1$ fall short of the optimal case and gradually saturates at a maximum photon number

indicating that the molecule eventually decouples from the resonator due to pumping to $\ket{s}$. Figure 5(a) inset shows the approximate linear dependence of $N_{T,\text{max}}$ on the cooperativity $C$ when $C\gg 1$.

On the other hand, for a background-free setup with total single photon counting efficiency $\eta$, the threshold photon number $N_{\mathrm{th}}$ for molecule detection can be made far less than $N_{T,\text{max}}$. It is possible to conduct a nearly non-destructive state measurement, that is, preserving the initial molecular state following state detection.

In Fig. 5(b-c), we calculate the relationship between the transmitted photon number $N_{T}(\tau)$, the depumping probability $P_{s}\approx 1-P_{g}$, and the estimated detection fidelity. We consider approximate Poisson distributions in both the background counts (mean number $\bar{n}_{s}=\mathcal{I}_{\mathrm{dark}}\tau$), when a molecule is uncoupled, and the signal photon counts (mean number $\bar{n}_{g}=\eta N_{T}(\tau)+\bar{n}_{s}$), when a molecule is in state $\ket{g}$. Here, $\mathcal{I}_{\mathrm{dark}}$ is the dark-count rate of a single-photon detector. We define the state detection fidelity for successfully detecting the molecular state as

where $n_{\mathrm{th}}$ is the threshold photon count that maximizes the fidelity. Here, $p_{g}(n)=\sum_{k=n}^{\infty}\bar{n}_{g}^{k}e^{-\bar{n}_{g}}/k!$ and $p_{s}(n)=\sum_{k=0}^{n}\bar{n}_{s}^{k}e^{-\bar{n}_{s}}/k!$. As shown in Fig. 5(c), with typical experiment parameters ($C=50$ and $\eta=0.3$), near-unity fidelity of 95% can be achieved with $\sim 15$% depumping probability and $N_{T}\approx 10$ as in Fig. 5(b).

Lastly, we comment that as large cooperativity $C>1$ guarantees higher transmitted photon counts, Eq. (14) suggests that it is also possible to detect a molecular state using a transition of a small Frank-Condon factor. Figure 6 shows that the transmitted photon number $N_{T,\text{max}}\sim C/2$ even for $f_{FC}\ll 1$. In comparison, the collectable photon number in direct fluorescence imaging is $1/(1-f_{FC})$ without repumping, subject to finite collection efficiency due to limited solid angle span of imaging instrument.

To sum up, in this section we propose a background-free state detection method, using single WGM coupled to a molecule without a closed transition. An alternative scheme using a single-mode Fabry-Perot cavity has also been investigated. To achieve near background-free detection in a cavity, one should instead monitor the transmissivity at the resonance of a molecule-cavity dressed state. A fiber Fabry-Perot cavity, for example, has a record $C=145$ Colombe et al. (2007) that would serve as an excellent candidate for the proposed scheme. Details of the adaptation are described in Appendix A.

In the previous section, we consider single-mode interaction with a spin-polarized molecule. In general, a spin unpolarized molecule can couple to both CW and CCW WGMs. We now consider a general case that a molecule can couple to both modes. We continue to assume negligible back scattering or mode mixing to simplify the discussion. Here, the two modes are degenerate and we model the empty resonator with a two-mode Hamiltonian

where $\hat{a}_{CW}^{({\dagger})}$ and $\hat{a}_{CCW}^{({\dagger})}$ are annihilation (creation) operators for the CW and CCW modes, respectively, and we consider the input field in the bus waveguide excites only the CW mode as in Fig. 1(a).

To illustrate the key signature of molecule-WGM-bus waveguide coupling while keeping the calculation tractable, we continue to use a simple two-level structure to effectively model an unpolarized molecule equally coupled to the CW and CCW modes ²²2While the hyperfine and magnetic sub-level structure should be taken into account in a realistic model, the example given here assumes a simple spherical dipole coupled to the WGMs as an approximation of an unpolarized molecule interacting with the electric field of WGMs.. We write down the two-mode light-molecule interaction Hamiltonian

and assume equal coupling strength with the two modes $g_{CW}=g_{CCW}=g_{c}$. The master equation of the full system is then modified to be

where the two WGMs are assumed to have the same intrinsic loss rates $\kappa_{i}$ and bus waveguide coupling rates $\kappa_{e}$.

In the limit of single excitation, the resonator and the molecule form an effective six-level system shown in Fig. 7(a). State $\ket{g,1_{CW(CCW)}}$ represents the degenerate level with one photon in the CW (CCW) mode and the molecule in $\ket{g}$. The four coupled states on the left of Fig. 7(a) form a cavity QED subsystem with quasi-steady equilibrium, whose population is gradually transferred, via spontaneous decay, to the right part of Fig. 7(a) that evolves like an empty resonator described in Sec. II. The decay rate is similarly described by Eq. (7) except now $g_{c}^{2}$ is replaced by $g^{2}_{CW}+g^{2}_{CCW}=2g_{c}^{2}$, effectively giving a total cooperativity $2C$. We now have

based on Eq. (41) and assuming $\delta=\Delta_{\text{ml}}=\Delta_{\text{cl}}$ and $\kappa=2\kappa_{e}$. The transfer rate at zero detuning is approximately four times slower than that expected in Eq. (8),

due to the increased cooperativity $2C$.

While waveguide transmission is modified with the presence of a single molecule, there is now also reflection in the bus waveguide due to the molecule-excited CCW resonator field which couples to the bus waveguide in the backward direction relative to the input field. As similarly discussed in Sec. III, we evaluate the time-dependent transmissivity and reflectivity using $T^{\prime}=\left|1+i\sqrt{\frac{2\kappa_{e}}{\mathcal{I}}}\langle\hat{a}_{CW}\rangle\right|^{2}$ and $R^{\prime}=\left|i\sqrt{\frac{2\kappa_{e}}{\mathcal{I}}}\langle\hat{a}_{CCW}\rangle\right|^{2}$ with $\expectationvalue{\hat{a}_{CW}}$ and $\expectationvalue{\hat{a}_{CCW}}$ calculated in Eqs. (42) and (43) respectively. We find

where

In Fig. 7(b-c), we calculate the time-averaged transmission and reflection spectra at critical coupling, using the definitions similar to Eq. (12) for $\overline{T}^{\prime}(\delta,\tau)$ and $\overline{R}^{\prime}(\delta,\tau)$. There are now three absorption dips (reflection peaks) found in $\overline{T}^{\prime}(\overline{R}^{\prime})$, resulting from the resonances of three eigenstates within the single excitation subspace of the Hamiltonian $H_{0}^{\prime}+H_{1}^{\prime}$. Compared to Fig. 4(b), the molecule-induced transparency and finite reflectivity near $\delta=0$, albeit with much reduced contrast, continue to provide ideal background-free signal.

At $\delta=0$, we find equal transmissivity and reflectivity,

where the peak values at $t=0$ is reduced to $\approx 25\%$ of the maximum transmission for the single mode case Eq. (11). The reduced total bus waveguide output, $T^{\prime}(0)+R^{\prime}(0)\lesssim 0.5<1$, is due to the excitation of a pure photonic eigenmode that dissipates through the intrinsic resonator loss. Nevertheless, the transfer rate $D^{\prime}_{\textrm{res}}$ in $T^{\prime}(0)\ (R^{\prime}(0))$ is also smaller by four times resulting from the collective coupling strength of two modes. As a result, at zero detuning the integrated transmitted (reflected) photon number $N_{T^{\prime}}(\tau)$ ($N_{R^{\prime}}(\tau)$), defined as in Eq. (13), still leads to the same maximum counts $N_{T^{\prime},\text{max}}=N_{R^{\prime},\text{max}}=N_{T,\text{max}}$ as shown in Figs. 8 and 5(a).

As suggested from the discussions above, the fidelity for molecular state measurement, using either tranmission or reflection signal alone, remains identical to the case of coupling to a single mode as shown in Fig. 5(c). Moreover, simultaneous detection of bus waveguide transmission and reflection can offer superior sensitivity, with one time more signal photons and with non-trivial temporal correlation between the transmitted and reflected photons when one exploits quantum nonlinearity in the molecule-WGM interactions.

In the previous sections, we assume large cooperativity ($C\gg 1$) for single molecule detection. In this section, we discuss the case when more than two molecules in the same state are present in the system. Even with a small cooperativity, one may take advantage of collective effects to achieve state detection with high-fidelity.

We begin with $N$ ground state molecules trapped on a micro-ring resonator, interacting with a single WGM with identical coupling strength $g_{c}$. We consider the resonant case $\delta=0$, and write down the light-molecule interaction Hamiltonian as

where the index $\alpha$ labels individual molecules and $\phi_{\alpha}$ represents a position-dependent phase in the light-molecule coupling since a WGM is a traveling wave.

We incorporate a modified Dicke model to investigate the collective behavior of these resonator-coupled molecules. Let $\displaystyle\hat{J}^{N}_{\pm}=\frac{1}{\sqrt{N}}\sum^{N}_{\alpha=1}\hat{\sigma}^{\alpha}_{\pm}e^{\pm i\phi_{\alpha}}$ be the collective spin lowering and raising operators, we can rewrite the equivalent Hamiltonian for Eq. (24) as

where the molecule-resonator coupling strength is replaced by $\sqrt{N}g_{c}$.

In the limit of single excitation, the resonator-coupled $N$ molecules resemble an effective two-level system with a $N$-molecule ground state $\ket{g_{N}}\equiv\Pi^{N}_{\alpha=1}\ket{g}_{\alpha}$ and an excited state $\ket{e_{N}}\equiv\hat{J}^{N}_{+}\ket{g_{N}}$, in which one molecule gets excited to $\ket{e}$. Spontaneous decay (via emitting single photon into freespace) can either bring the population in $\ket{e_{N}}$ back to $\ket{g_{N}}$ or bring one excited state molecule down to the uncoupled state $\ket{s}$. The full master equation for the resonator-coupled $N$-molecule density matrix $\rho_{N}$ can be expressed as

where the first line of the equation describes the evolution of the $N$-molecule collective states with coupling strength $\sqrt{N}g_{c}$ and decay rate $\Gamma_{g}$, and the second line adds resonator dissipation and single molecular decay into $\ket{s}$.

While full evolution dynamics of Eq. (26) can be evaluated numerically, calculation for large $N$ can be computationally expansive. Here, we develop an analytical approximation in the weak-driving limit. Starting with $N$ molecules in the ground state $\ket{g_{N}}$ weakly excited to $\ket{e_{N}}$ by the resonator mode, the system evolves collectively similar to the single molecule case in Fig. 2(a). Within this $N$-coupled molecule manifold, the system can be described by a simple three level system consisting of $\ket{g_{N},0}$, $\ket{g_{N},1}$ and $\ket{e_{N},0}$ until spontaneous decay into state $\ket{s}$ occurs. If we trace out the molecule that decays into the uncoupled state, not knowing which one did, the system can be described again by a collective state $\ket{g_{N-1}}$ with $N-1$ molecules in the ground state, weakly excited to $\ket{e_{N-1}}$, as detailed in Appendix D. The dynamics can cascade down as prescribed with $N-2,N-3,...,1$ molecule(s) left in the system until all molecules become uncoupled, as illustrated in Fig. 9 (a).

We can calculate the probability $p_{n}$ of having $n$ coupled molecules in the system, using the simple cascade model. This leads to a system of equations,

where $p_{0}$ is for all molecules in $\ket{s}$. The effective transfer rate of the population from $n$ to $n-1$ coupled molecules can be calculated according to Eq. (49). We find

as detailed in Appendix D. For $n\geq 1$, the transfer rate is suppressed by the cooperativity $\sim 1/nC$. We note that Eq. (7) (with $\delta=0$) is the single molecule case of Eq. (28).

We can also derive the bus waveguide transmission by finding the expectation value for the resonator field,

Bus waveguide transmissivity $T_{N}$ for initially $N$ ground state molecules can then be evaluated using Eq. (2).

We have compared the analytical solutions with numerical calculations for mean populations and photon numbers, and found very good agreement for $N=2,3,4$. Equations (27-29) allow us to evaluate the dynamics of waveguide transmission with arbitrarily large number of coupled molecules.

The major advantage for coupling more than one molecules to a resonator is that the collective coupling leads to many more signal photons and higher fidelity without losing significant fraction of ground state molecules to the uncoupled states. To illustrate this, in Fig. 9(b) we calculate the state detection fidelity $\mathcal{F}$ as a function of the ground state molecule loss

It is shown that, under a moderate $C=10$, $N=10$ ground state molecules can be detected with over 99% fidelity with 1% ground state population loss. For even larger $N$, non-destructive state detection with negligible loss can be realized with cooperavity parameter $C<0.1$, as described in Ref. Sawant et al. (2018).

To conclude, we have proposed a background-free state detection scheme for single molecules without an optically closed transition. High-fidelity measurement can be realized in resonators with a sufficiently large cooperativity $C>10$. A possible experiment with cold molecules could begin with an array of cold atoms trapped in optical tweezers Thompson et al. (2013); Kim et al. (2019) or in a lattice of evanescent field traps above the surface of a high-Q micro-ring resonator or a photonic crystal cavity of $C\gtrsim 25$ Chang et al. (2019); Samutpraphoot et al. (2020). Resonator-assisted photoassociation (PA) to a molecular ground state (also with high fidelity) can be performed by simply introducing PA light into the experimental setup Perez-Rios et al. (2017); Kampschulte and Denschlag (2018). Immediately following PA, one can detect the existence of ground state molecules using the proposed scheme with probe photons directly launched into the bus waveguide. This state detection technique could also be employed for atoms Kim et al. (2019) or quantum emitters coupled to a high-Q micro-ring resonator or other whispering-gallery mode resonators.