Theoretical framework

We consider the general system depicted in Fig. 1a, in which two optical guiding structures, labelled as the pump and probe waveguides, are connected by a photonic black box. We are interested in the coupling of the emitter to the pump and probe waveguides. In order to measure this, optical power P_in is coupled into transverse mode m of the pump guide, making a field $$ \mathrm{Re}\left\{{\mathcal{E}}_{\mathrm{in}}{\mathbf{u}}_m\left(x,y\right)e^{i\left(kz-\omega t\right)}\right\}$$. Here, u_m(x, y) gives the transverse distribution of the field in mode m, z is the direction of propagation and $$ {\mathcal{E}}_{\mathrm{in}}$$ is the amplitude of the pump light in that mode. In the absence of the emitter, the transmitted pump light in mode m of the probe guide is $$ \mathrm{Re}\left\{t_0{\mathcal{E}}_{\mathrm{in}}{\mathbf{u}}_m\left(x,y\right)e^{i\left(kz-\omega t\right)}\right\}$$, where t₀ is the complex transmission factor.

Fig. 1

Schematic representation of the optical system to be characterised.

a Response of the system without the presence of a quantum emitter. Input light at frequency ω is transmitted (reflected) by the photonic structure with coefficient ∣t₀∣² (∣r₀∣²). b With the presence of a single emitter with ground state $$ ∣g⟩$$ and excited state $$ ∣e⟩$$ separated by energy ℏω₀, the light interferes with the resonance fluorescence of the driven quantum system coupled to the pump and probe waveguides with efficiency β_pump and β_probe, producing Fano transmission (T) and reflection (R) spectra. c Example Markovian structures covered by our general characterisation model, including a continuous waveguide, a resonator in the weak coupling regime and an interrupted waveguide.

Consider an emitter with an upper level $$ ∣e⟩$$ and a lower level $$ ∣g⟩$$ (other levels are sufficiently far from resonance that they can be adiabatically eliminated from the coherent dynamics). When the emitter is put in place, the field has the option of being scattered by the emitter into the probe waveguide, as depicted in Fig. 1b. We make two key assumptions about the dynamics. First, we assume that the local density of electromagnetic states is constant over the spectral linewidth of the emitter²⁹. This Markovian approximation is justified for most single-photon sources where a fast optical response is desirable. Second, we make the semi-classical assumption that the quantum correlations between the pump field and the emitter can be ignored³⁰. With these assumptions, the total output field in the probe waveguide is given by²⁹

Here, Ω is the Rabi frequency, defined as d ⋅ E(r₀)/ℏ, where d is the dipole transition matrix element and the pump field at the site of the emitter is $$ \mathrm{Re}\left\{\mathbf{E}\left({\mathbf{r}}_0\right)e^{-i\omega t}\right\}$$. We choose Ω to be real without loss of generality.

On substituting Eq. (3) and Eq. (4) into Eq. (2), we find

Hence, ignoring a global phase, the field at the output end of the guide is given by

Here, we have introduced ϕ_T, which is the phase difference between the two transmitted fields due to propagation; a further phase shift will come from the lag of the dipole response σ⁻. We have also defined $$ {\beta }_{\mathrm{eff}}=\sqrt{4{\beta }_{\mathrm{pump}}{\beta }_{\mathrm{probe}}}$$. Note that β_pump and β_probe are both between 0 and 1 but β_pumpβ_probe ≤ β_pump(1 − β_pump) ≤ 1/4, so the maximum value is β_eff = 1. It follows that the net transmission power is given by

The density matrix elements are found by solving the optical Bloch equations^19,29, with the result

In an experiment to measure the transmission as a function of frequency, Eq. (9) may be fitted to the spectrum. When S ≪ 1 and the value of Γ₁/(2Γ₂) is known, the fit will yield values for ∣t₀∣, αβ_eff and ϕ_T. However, it is common in a real experiment for the light to be attenuated by the train of auxiliary optics so that the measured powers $$ {\mathcal{P}}_{\mathrm{out}}$$ and $$ {\mathcal{P}}_{\mathrm{in}}$$ have the ratio $$ {\mathcal{P}}_{\mathrm{out}}/{\mathcal{P}}_{\mathrm{in}}=\eta P_{\mathrm{out}}/P_{\mathrm{in}}$$, and the value of η is unknown. For large detuning, the measured transmission $$ {\mathcal{P}}_{\mathrm{out}}/{\mathcal{P}}_{\mathrm{in}}$$ then takes the value η∣t₀∣². On normalising the data to this transmission, we have from Eq. (9)

In this case, the fit yields a value for αβ_eff/∣t₀∣, rather than αβ_eff. One may determine ∣t₀∣ by an auxiliary experiment that compares the device with another that contains no emitter and has ∣t₀∣ = 1. Eq. (10) can be modified in an exactly similar way to give the reflection spectrum as a function of αβ_pump/∣r₀∣, while ∣r₀∣ is determined by comparison with a reflector having ∣r₀∣ = 1. Finally, the value of β_probe can be deduced from the combination of β_eff and β_pump. Of course, β_probe could also be determined from a reflection measurement made on the output side.

For structures where a normal mode decomposition is appropriate, for example, a continuous waveguide or weak cavity as depicted in Fig. 1c, most of the parameters in Eq. (9) and Eq. (10) can be calculated analytically, as we consider further in Section I of the Supplementary Information. Our method becomes essential when dealing with structures that cannot be described by a simple mode decomposition, such as the waveguide gap depicted in Fig. 1c.

Microfluidic integration of single molecules with waveguides

In the early days of single-emitter spectroscopy, it was found that large PAH molecules such as pentacene⁸, terrylene³¹, dibenzanthanthrene (DBATT)³² and dibenzoterrylene (DBT)³³ could be hosted in PAH crystals to form stable quantum emitters in the solid state. In this work, we use DBT-doped anthracene. The molecular structures are shown in Fig. 2a and relevant energy levels of DBT are drawn in Fig. 2b. This well-studied combination has a very weak singlet–triplet inter-system crossing, is highly photostable, has a high probability of radiative decay on the zero-phonon line (ZPL) (shown in blue)^33,34 and has a lifetime-limited resonance width at cryogenic temperatures³⁵.

Fig. 2

Localised growth of DBT-doped anthracene in the vicinity of an interrupted waveguide.

a Molecular structure of anthracene and dibenzoterrylene (DBT). b Jablonski diagram of DBT ground (S₀) and excited (S₁) states excited at Rabi frequency Ω and decaying with rate Γ₁. Triplet states are ignored as the inter-system crossing is very weak³³. c Overview of the grating couplers, the interrupted silicon nitride (SiN) waveguide and the microfluidic channel crossing it. The zoom-in shows details of the intersection between guide and channel. d False-colour optical-microscope image of two devices with the microfluidic channels filled. e Fluorescence excitation spectrum of molecules near the gap in a device at cryogenic temperature. The molecules are excited from the “pump” waveguide and fluorescence is collected by the confocal microscope from the red dot shown in d. f Wavelength distribution of the DBT resonances from the same confocal spot.

The coupling of photons to single PAH molecules has been used in bulk material to demonstrate, for example, a single-molecule optical transistor²³ and few-photon nonlinear optics³⁶, but for applications such as a deterministic photon source, stronger coupling is desirable. A natural way to achieve that is to integrate the emitters into a photonic structure^10,12,13,15, and it is convenient to grow doped organic crystals around the structure by solidifying from a molten mixture^12,37. Normally, the structure is made of inorganic material and the organic molecule couples to an evanescent field. However, the molecule is usually unstable at less than a few hundred nanometres from the inorganic/organic interface³⁸, and therefore it can only be placed in the tail of the evanescent field where the dipolar coupling to the photonic mode is weak. Here, we take a different approach, shown in Fig. 2c, where a silicon nitride waveguide having grating couplers at each end is interrupted by a sub-wavelength gap. After fabricating the waveguide chip, molten anthracene doped with DBT is drawn by capillary forces along a microfluidic channel that cuts across the waveguide and fills the gap, as depicted in Fig. 2c. A numerical simulation, details of which are given in Section II and Supplementary Fig. S1 of the Supplementary Information, shows that the coupling efficiency β_eff for a molecule sitting at the centre of the gap decreases rapidly with the length of the gap. However, with a gap of 400 nm, this can be as high as 30%. A smaller gap can yield higher coupling, but the guide faces on each end of the gap are then close enough to the molecule that they may compromise its optical properties. The coupling can, of course, be much higher with the introduction of a cavity¹⁰.

We began device fabrication with a silicon wafer that had a layer of thermal oxide covered by silicon nitride, and we patterned the interrupted waveguides and grating couplers in the silicon nitride. The microfluidic channels were then fabricated from a sacrificial resist layer on top of which SiO₂ was sputtered. We cleaved the chips to expose the channel entrance at the facets and baked the sample to remove the resist, thereby opening hollow channels. Finally, we filled the channels with DBT-doped crystalline anthracene by controlled heating and subsequent cooling. See “Methods” for details of device fabrication and filling of the capillaries.

To verify that we had stable emitters in the vicinity of the waveguide, we performed fluorescence spectroscopy at a cryogenic temperature under a microscope (see “Methods” for a detailed description of the optical setup). A waveguide chip was filled with DBT-doped anthracene at 10⁻⁴ molar fraction, then cooled in the cryostat to 4.7 K and positioned so that a device having 400-nm gap length and 1-μm channel width (see Fig. 2c) was in focus at the centre of the field of view. Figure 2d shows a false-colour white-light image of the structure. A cw laser was focused onto one of the grating couplers to excite molecules from the “pump” side of the waveguide, and was continuously scanned at low power between 784.5 nm and 785.5 nm to cover the inhomogeneous width of the S_0,0 ↔ S_1,0 transition. The light was then collected from the vicinity of the waveguide gap and sent to a photon counter. An 800-nm long-pass filter removed any scattered laser light together with the ZPL fluorescence, leaving only the red-shifted fluorescence. We plot a slice of the scan in Fig. 2e, which reveals the characteristic Lorentzian resonance peaks of many DBT molecules having a range of resonant frequencies, each being shifted according to its local environment. The histogram of these frequencies shown in Fig. 2f corresponds to a spectral density of 0.2 molecules per GHz at 785 nm, so the individual molecular lines are well resolved. Some of the light is collected from molecules that are well away from the gap but are excited by scattered pump light, and these have poor coupling to the guide. To characterise the strength of the coupling, we therefore use the extinction spectroscopy method developed above, as described next.

Characterisation of the coupling

In this section, we make a practical demonstration of our theory through a transmission measurement, which we compare with Eq. (9). We collected light from the grating on the “probe side” of the waveguide and used a 785 ± 3-nm band-pass filter to remove the red-shifted fluorescence and most of the local phonon sideband³⁹. (We show in Section IV of the Supplementary Information that imperfect filtering has a negligible effect). The grating selectively couples to the x-polarised waveguide mode, which is the mode that we pump. We scanned the pump frequency over the resonance of a single DBT molecule and recorded both the red-shifted fluorescence from the gap and the resonant transmission from the output grating, as indicated in Fig. 2c. The data are plotted in Fig. 3a as open circles in red and blue for the fluorescence and transmission, respectively. A solid line shows the least-squares fit of a Lorentzian to the fluorescence data, which gives us the linewidth, $$ 2{\mathrm{\Gamma }}_2\sqrt{1+S}$$ (full width at half maximum (FWHM)). Knowing that, we then fit Eq. (11) to the transmission data to produce the solid line through the blue data points. For this second fit, we express Eq. (11) as the Fano lineshape

Fig. 3

Coherent characterisation of a coupled single molecule by extinction spectroscopy.

a Experimental data. Red circles: red-shifted fluorescence emerging from the gap, fitted with a Lorentzian curve. Blue circles: transmission at 785 nm observed through the output grating, normalised and fitted according to Eq. (11). b Plots of full width at half maximum (FWHM) linewidth, visibility (V) and asymmetry (q) as a function of the pump power. Error bars signify one standard deviation from fitting the transmission at each power. c, d Computation of the coupling efficiency β_eff and propagation phase difference ϕ_T for a dipole along the x direction, placed in the mirror symmetry plane of the structure. e Normalised transmission spectrum from Eq. (11) expected for lifetime-limited DBT molecules placed inside the channel, at positions 1–3 shown in c and d.

In order to derive β_eff from V₀ and q₀, we measured ∣t₀∣ by comparing the off-resonant transmission of the device with the transmission of a second device, which was identical, except that the waveguide had no gap. We scanned the laser frequency to look for possible cavity resonances in the optical setup, which would have invalidated the method but found only a very weak modulation. This comparison gave ∣t₀∣ = 0.63(6), which differs slightly from the numerically calculated transmission ∣t_sim∣ = 0.81, perhaps because our simulation simplifies the anisotropic refractive index of the anthracene. Having measured ∣t₀∣, the only external parameters needed to deduce a value for β_eff are $$ {\mathrm{\Gamma }}_1^{-1}=\left(4.5\pm 1\right)$$ns⁴⁰ and α = 0.33^35,39, both known from bulk measurements of DBT in anthracene. We have checked (see Supplementary Information Fig. S3d) that Γ₁ is not appreciably altered when the anthracene channel is narrow, and we have shown (see Supplementary Information Fig. S1a) that the radiation rate γ₁ and the branching ratio α are not significantly altered by the electromagnetic response of the waveguide gap. With these two inputs, we find that the β₊ solution gives the unphysical result β_eff > 1, so we conclude that β_eff = β₋∣t₀∣/α = 9(2)%, with the error bar coming roughly equally from the uncertainty in ∣t₀∣ and from the other uncertainties combined. The corresponding solution for the phase difference is ϕ_T = 61(2)° (independent of ∣t₀∣), with the error bar coming primarily from the uncertainties in q₀ and V₀.

It is instructive to compare these results for β_eff and ϕ_T with a numerical simulation (see “Methods”). Figure 3c shows β_eff for a dipole transverse to the guide (along x), placed in the yz plane centred on the guide. (The coupling at the centre of the 400-nm gap is less than the maximum possible 30% because the height and width of the guide are not perfectly optimised). The coupling is the strongest for an emitter placed in the gap, but we note that an emitter outside the gap and close to the guide couples to the evanescent field, as seen by the yellow strip running along the outside of the guide. Figure 3d shows the propagation-phase difference ϕ_T. This phase varies strongly with the position in the gap, in contrast to the behaviour when coupling to a cavity. Also, we find that ϕ_T tends to 90^∘ when the emitter couples to the evanescent field on the side of the guide and far from the gap, as expected for coupling to a continuous waveguide. See Section I of the Supplementary Information for treatments of the continuous waveguide and weak cavity cases. In Fig. 3e, we plot the transmission spectra calculated for weakly pumped, ideally polarised DBT molecules at each of the three positions marked in Fig. 3c, d.

The dashed lines in Fig. 3d show where a dipole lying in the yz plane through the centre of the guide would give the measured value ϕ_T = 61°. If the molecule is in this plane, we expect it to be near the upper contour, for example in the position marked 2, because the lower one is too close to the substrate for photostability. On this line, the calculated coupling efficiency varies in the range 20−21%, which is to be compared with the 9(2)% we have measured. Our molecule has no reason to be aligned along x, so the simulation would be consistent with our measurement if the molecule makes an angle of θ = 49° to the x axis. Of course, there is also no reason for the molecule to sit in the plane x = 0. Looking at the whole surface where ϕ_T = 61°, we find that the simulated coupling varies in the range 11−21%, and conclude therefore that θ is in the range 25°−49°.

Derivation of Eq. (4)

The classical pump field $$ \left\{{\mathbf{E}}_{\mathrm{m}}^f,{\mathbf{H}}_{\mathrm{m}}^f\right\}$$ propagates forward (towards the black box) in transverse mode m at frequency ω and with power P_in. This field leaves the guide and enters the black box, where it induces a dipole moment D in an emitter that radiates the field {E_d, H_d} with power P_d. A fraction β_pump of that radiated power goes back into the pump guide. From the orthogonality of modes⁴⁵ we have

Wanting to relate these fields to D, we note that the dipole at position r₀ produces a current density j_d(r) = − iωDδ(r − r₀). Similarly, the pump field may be viewed as the result of (fictitious) electric and magnetic current densities $$ {\mathbf{j}}_{\mathrm{in}}\left(\mathbf{r}\right)=\mathbf{\delta }\mathbf{n}\times {\mathbf{H}}_{\mathrm{m}}^f$$ and $$ {\mathbf{m}}_{\mathrm{in}}\left(\mathbf{r}\right)=\mathbf{\delta }\mathbf{n}\times {\mathbf{E}}_{\mathrm{m}}^f$$⁴⁵. These lie on a plane surface S far from the black box, whose normal is parallel to the direction of propagation, and δn is a Dirac delta function along the normal. Now we can make use of the reciprocity theorem⁴⁶ to write

Connecting the classical dipole to the quantum emitter, we replace the ratio ∣D ⋅ E(r₀)∣²/P_d by (2ℏΩ)²/(ℏωγ₁)⁴⁶. Both γ₁ and the Rabi frequency Ω are defined in the main text. With this substitution in Eq. (20), we obtain the result given in Eq. (4).

Device fabrication

The waveguides are fabricated from a 200-nm-thick silicon nitride layer on 2 μm of silica on silicon. The waveguide patterns are first written into ma-N 2403 resist by electron beam lithography and transferred into the underlying silicon nitride layer by reactive ion etching with a CHF₃ plasma. We overetch the silicon nitride by 150 nm so that the middle of the waveguide sits 250 nm away from the bottom surface. In this way, the position of maximum coupling is not too close to the bottom surface. The waveguides on the chip have a width of 400 nm and gap lengths ranging from no gap to 400 nm. We terminate the waveguides with gratings based on concentric circles. To avoid reflections, the gratings are designed to couple light at an angle of 10° to the vertical.

To overlay the microfluidic channels, we first spin-coat a 1-μm layer of AZ nLOF 2020 resist that is diluted 4:1 (resist:solvent w/w) with PGMEA. Electron beam lithography exposes the resist along channels that are perpendicular to the waveguides and aligned with the gaps. We then deposit 2 μm of SiO₂ on top of the resist using RF sputtering. Next, the sample is cleaved to expose the resist channels on both facets. Finally, we place the sample in a furnace that is heated to 550 °C in the ambient atmosphere. Under these conditions, we find that the resist is released from the channels without leaving any residue, and we are left with open structures that can be filled with molten DBT-doped anthracene.

Capillary filling

In order to fill the microfluidic channels with doped anthracene, we use growth from the melt by solidification^12,37. We first place a small quantity of DBT-doped anthracene powder (10^–4 mol/mol concentration) on the facets of the chip. The sample is then put on a hotplate in a glove box that is continuously purged with nitrogen. We heat the sample at a rate of 5 °C s⁻¹ and hold the temperature at 210 °C until the channels are visibly filled by the melted material. Finally, we cool the sample at a rate of −5 °C s⁻¹ causing the anthracene to crystallise. This yields long stretches of the capillaries filled by solid anthracene. We check the quality of the DBT molecules in the capillaries using cryogenic fluorescence spectroscopy and we show in Section 3 and Supplementary Fig. S3 of the Supplementary Information that the spectral stability is not appreciably affected by the constrained geometry of the microfluidic channel.

Optical setup

The optical apparatus was a three-beam confocal microscope built around a closed-cycle cryostat (Cryostation, Montana Instruments), as illustrated in Supplementary Fig. S2 of the Supplementary Information. The primary excitation light came from continuously tunable titanium:sapphire laser (SolsTiS, MSquared) that was power-stabilised using an acousto-optic modulator and a proportional integrated-derivative controller (SIM960, SRS). The light was delivered to the apparatus through a single-mode fibre, then collimated with an aspheric lens and polarised before passing through a half-wave plate and a band-pass filter (F1) to produce a linearly polarised beam with adjustable polarisation angle and spectral purity. This entered a 10% transmission (90% reflection) beamsplitter (BS), and the transmitted light was sent to a pair of electronically controlled galvanometer mirrors (GM). Through the use of two lenses in a “4f” configuration (L1, L2), the angular change in the galvanometer mirrors allowed us to adjust the angle of incidence onto an objective lens (LD EC Epiplan-Neofluor ×100, 0.75NA, Zeiss) inside the cryostat without translating across the objective aperture. This in turn caused a focused spot to be raster-scanned across the sample. The back aperture of the objective was overfilled to ensure the minimum spot size of 720-nm full-width half-maximum. The sample was mounted on a 3-axis piezo-controlled translation stage (PS, Attocube), which we used to locate waveguides and bring them into focus. Molecule fluorescence followed the beam path back to the 90:10 BS where the 90% reflected portion passed through a long-pass filter (F2) to remove the excitation laser before being collected in multimode fibre and detected on a silicon avalanche photodiode. By inserting a pellicle BS into the excitation path after the scanning mirrors, we introduced white light (WL) from a lamp onto the sample. This light was then reflected from the sample and off another pellicle BS above the cryostat to an electron-multiplying charge-coupled device (CCD) camera (iXon, Andor), which took wide-field images, such as that shown in Fig. 2d. A second single-mode fibre input (shown within the rectangle labelled “Grating Coupling”) was collimated, polarised, filtered and steered onto a (90:10) beam splitter, before being combined with the main beam path in the “4f” lens setup using a 50:50 beam splitter. The steering mirrors allowed the beam to couple into the pump guide through its grating coupler, giving a typical total coupling efficiency of 8% from fibre to waveguide. Light emerging from the probe guide grating coupler was directed back to a final single-mode fibre (in the rectangle) and thence to the detector that recorded the transmission spectrum.

Finite-difference time-domain simulations

The numerical simulations of the device are performed with three-dimensional finite-difference time-domain (FDTD) analysis using the Meep software package⁴⁷. The structural parameters, as defined in Fig. 2c, are waveguide width = 400 nm, waveguide height = 200 nm, gap length = 400 nm, under-etch = 150 nm, channel width = 1 μm and channel height = 1 μm. We use a mesh size of 16 nm and perfectly matched layers to simulate open boundaries. Anthracene is a biaxial material but for simplicity, we choose to approximate it as isotropic with refractive index n = 1.8.

To compute the transmission through the gap, we use a continuous eigensource to excite the x-polarised mode of the pump waveguide. We determine the power transmitted into the x-polarised mode of the probe waveguide by projecting the field at the output end onto that mode. For coupling efficiency calculations, we use a continuous dipole source placed at a given position in the channel and monitor the total power emitted together with the power coupled into the x-polarised modes of the waveguides.

FDTD simulations also allow us to calculate the phase difference ϕ_T. Using a continuous eigensource to excite the x-polarised mode m of the pump waveguide, we first compute the phase shift of the transmitted light, $$ \mathrm{Arg}\left(t_0\right)$$, which is the phase difference between light in mode m at the entrance of the probe guide and the exit of the pump guide. Mode decomposition is used to isolate the field coupled to mode m of the probe waveguide. For the propagation phase shift of the scattered light, we place an electric dipole at the position of the molecule. The dipole oscillates in phase with the pump field at that position, but the pump field is not turned on. Again we take the difference between the phase of the (dipole) field in mode m at the entrance to the probe guide and that of the pump field (if it were turned on) at the exit of the pump guide. Calling this latter phase shift Δϕ, we have $$ {\varphi }_{\mathrm{T}}=\mathrm{\Delta }\varphi -\mathrm{Arg}\left(t_0\right)$$.