Multidimensional Purcell effect in an ytterbium-doped ring resonator

Due to the dipole-forbidden nature of the 4f–4f transitions of rare-earth ions, their oscillator strengths are very small (∼1 × 10–6). To overcome this limitation, a high-quality-factor (Q) and small-mode-volume (V) (high Q/V) optical cavity can be used to enhance the ion–light interaction in the framework of cavity quantum electrodynamics (QED). In the weak coupling regime of cavity QED, this effect is characterized by the Purcell factor, which quantifies the enhancement of the spontaneous emission (SE) rate. To date, the SE of rare-earth ions has been studied in photonic-crystal cavities12, 13, 14, whispering-gallery mode ring resonators15 and Fabry–Perot cavities16, 17, 18. An erbium-doped Si3N4 ring resonator was also studied in ref. 12, but no cavity QED phenomena were observed due to the low Q/V of the particular ring resonator. In this Letter, we investigate an Yb3+-doped high-Q (∼5 × 106) Si3N4 ring resonator that has an advantage over other types of microcavity because of its all-integrated-optics approach. Because of the structure of the ring resonator, not only is the cavity Purcell effect present, but enhancements from waveguide modes also exist. The former effect is spectrally narrowband and becomes appreciable only in the vicinity of the cavity resonance, while the latter effect is intrinsically broadband19, 20. The long excited-state lifetime of the 4f–4f transition makes thermal depolarization of Yb3+ dipoles important21. This causes the coupling of the ions to the modes in all polarizations, even though the ions are initially resonantly excited in a well-defined polarization. Here, we introduce a model that separates the different contributions to the SE rate into three discrete channels: a one-dimensional (1D) enhancement from the slab waveguide, a two-dimensional (2D) enhancement from the channel waveguide and the three-dimenional (3D) Purcell effect from the cavity22. We refer to these effects as 1D, 2D and 3D Purcell effects. Measurements of the total Purcell factor, depolarization and decoherence as functions of temperature shed light on the different contributions to the enhanced SE rate. The ring resonator (radius of 1.59 mm) consists of a Si3N4 core embedded in amorphous SiO2 cladding23, 24, 25, 26. A schematic, photograph and cross-section of the ring resonator are presented in Fig. 1a–c. The Si3N4 core has a width of 2.8 μm and a thickness of 0.1 μm. Yb ions are implanted into the device with a peak concentration of ∼0.001% (atom numbers). The ions are laterally uniform and are approximately Gaussian distributed in depth with a full-width at half-maximum of 86 nm and a centre 72 nm above the Si3N4 top surface (see Methods). The ring resonator is coupled to two waveguides. A single-mode fibre and three multimode fibres are coupled to the waveguides (see Methods). An energy level diagram of Yb3+ in SiO2 is presented in Fig. 2a. We focus on the optical transition centred at 976 nm, which involves the lowest manifold of the ground state (2F7/2) and the lowest manifold of the excited state (2F5/2) (see Supplementary Fig. 4 for emission spectra). Energy transfer among adjacent ions is the dominant non-radiative decay mechanism for Yb3+ (ref. 27), as illustrated in Fig. 2b. Around 976 nm, the ring resonator supports two transverse magnetic (TM) modes and three transverse electric (TE) modes (Supplementary Fig. 2). Profiles of the fundamental modes denoted by ‘TM0’ and ‘TE0’ are shown in Fig. 1d,e, which correspond to the 2D waveguide modes and cross-sections of the 3D ring resonator modes. The 1D modes are calculated for an infinite Si3N4 slab sandwiched in SiO2 cladding, resulting in two 1D modes, the profiles of which are shown in Fig. 1f,g. The electric fields of the TM (TE) modes are approximately linearly polarized along the y (x) direction. The measured values of Q at 976.0 nm are 4.8 × 106 and 8.3 × 105 for the 3D TM0 and TE0 modes, respectively (Supplementary Fig. 3). A schematic of the set-up used to measure the SE decay rate of Yb3+ is shown in Fig. 2c. The device was mounted in a closed-cycle dilution refrigerator. The output of a narrow-linewidth (∼100 kHz) laser at 976.0 nm was modulated by a mechanical chopper and coupled to fibre 1 through a fibre polarization controller (FPC). The resulting laser pulse has an ‘on’ (‘off’) time of 1 (9) ms and a measured rise and fall time of less than 0.5 μs. The laser frequency was locked to the 3D TM0 mode (see Methods). The FPC was set to maximize the transmission at fibre 2. SE events of the Yb3+ ensemble were counted at the output of fibre 4 with time bins of 0.5 μs through a bandpass filter (λ0 = 976 nm, Δλ = 2 nm). Three SE decay traces measured at 295 K, 5.7 K and 50 mK are shown in Fig. 3. These are not single-exponential functions because of the spatial and spectral distributions of Yb3+ and therefore the variations in the Purcell factors. The fastest decay rate for the trace at 50 mK is clearly greater than those at 295 and 5.7 K. We are interested in the maximum Purcell factor Fm, which is defined as the net enhancement in addition to the free-space SE decay rate k0 such that the maximum SE decay rate is given by k0 + kRFm, with kR being a free-space radiative decay rate. To extract Fm from the traces, we constructed a decay function I(t) with k0 ≡ kR + kET and km ≡ kRFm as two parameters, where kET is an energy transfer rate among ions (see Methods). From fitting the trace at a given temperature with I(t), we obtained the values of k0 and km. Examples at 295 K, 5.7 K and 50 mK are shown in Fig. 3 with k0 = 1.41, 1.30 and 2.00 (±0.01) ms−1 and km = 1.2, 4.6 and 11.7 (±0.2) ms−1, respectively. I(t) is in good agreement with the traces at 295 K, 5.7 K and also with the initial 1 ms of the trace at 50 mK, but it apparently deviates from the trace at 50 mK at longer times leading to an artificially large value of k0 (2.00 ms−1) (see Methods for further discussion). Nevertheless, the extracted value of km is always reliable because the fastest decay occurs within the initial 1 ms, to which part of the trace I(t) yields a very good fit (inset of Fig. 3). The fact that at 5.7 K the value of k0 + km is large and the value of k0 is small compared with those at 295 K causes the crossing of the two traces. Figure 4a shows the values of k0 at different temperatures. k0 decreases with decreasing temperature down to about 10 K due to the decreasing non-radiative decay rate. The model k0 = kR + kET is fitted to the data above 10 K as shown in Fig. 4a, where kET(T) = Bexp(–β/T), as given by ref. 27, with kR, B and β as parameters. The fit results in kR = 1.30 ± 0.01 ms–1, B = 0.13 ± 0.01 ms–1 and β = 51 ± 7 K. The measured Purcell factor Fm = km/kR is shown in Fig. 4b (and inset). Fm is approximately constant at high temperature with a value of ∼0.8 and increases rapidly when T < 50 K, reaching a value of 9.0 at 50 mK. To interpret the results, we introduce a model that includes multidimensional Purcell effects and dipole depolarization. We neglect the Purcell effects from higher-order modes. Fm can be written as a sum of contributions over all the dimensions and modes22: where F1, F2 and F3 are 1D, 2D and 3D Purcell factors for p ≡ TM0 or q ≡ TE0 modes, respectively, fp and fq account for the reduction of the Purcell factor because the ions are located away from the field maximum and 0 ≤ Θ ≤ 1 quantifies the degree to which the ions maintain their initial dipole directions during the lifetime (see Supplementary Section 1 for details of the model). F1 and F2 are independent of temperature, while the expression for F3 contains the homogeneous linewidth of Yb3+ denoted by ΓH and therefore F3 is temperature-dependent. It has been found that ΓH(T) ∝ T1.8 at high temperature due to the coupling to phonons and ΓH(T) ∝ T1.3 at low temperature due to coupling to tunnelling systems28. Here we assume that the crossover occurs at 40 K (following ref. 28). We measure ΓH(T) using photon echo techniques (see Methods and Supplementary Section 4), the results of which are shown in Fig. 5a. Below 80 mK, ΓH(T) depends weakly on temperature, as reported previously in ref. 29. For 80 mK < T < 150 mK, ΓH(T) is well described by a power law of T1.3 with ΓH(1K) = 2.1 ± 0.1 MHz. We also measure Θ using polarization analysis of the resonance fluorescence (see Methods and Supplementary Section 5) and show the results in Fig. 5b. Θ decreases exponentially as a function of temperature with parameter α = 5.1 ± 0.1 K. Because photon echo and lifetime experiments are performed on largely different timescales (microsecond versus millisecond), spectral diffusion has a larger impact on the lifetime measurements, leading to a larger value of ΓH(T) (ref. 30). To accommodate this, ΓH(1K) is treated as the only adjustable parameter in the model. All the other parameters are either obtained through numerical simulations or extracted from measurements. The fit of equation (1) to the data is shown in Fig. 4b (and inset), resulting in ΓH(1K) = 4.8 ± 0.2 MHz. As expected, this value is larger than that measured with photon echo experiments by a factor of 2.3 due to spectral diffusion. Contributions from the 1D, 2D and 3D Purcell effects are shown separately in the figure. The 1D and 2D Purcell factors are constants at the high temperature limit with Θ ≈ 0, but they approach the values for TM0 when T ≪ α (Θ ≈ 1). The 3D Purcell factor is relatively small at high temperature, but it becomes appreciable for T < 50 K where ΓH(T) is comparable to or smaller than the linewidth of the 3D TM0 mode (64 MHz). Ultimately, the value of Fm reaches 9.0 at 50 mK, where the 3D Purcell effect is dominant and is almost entirely determined by the large Q/V of the 3D TM0 mode. Our unique approach of cavity QED has the potential to increase this number through further optimization of the design of the ring resonator. Because of the broadband nature of the 1D and 2D Purcell effects, their temperature dependences are very weak compared to the 3D Purcell effect. This study is therefore best suited to demonstrating the most important 3D Purcell effect and separating it from the combined 1D and 2D effects. In conclusion, we have introduced a novel solid-state cavity QED system based on high-Q Si3N4 ring resonators with implanted rare-earth ions. We have separated the 3D contribution to the Purcell effect from the 1D and 2D contributions and have analysed the temperature dependence. This provides novel insight into the temperature dependence of the decoherence and depolarization of the rare-earth ions. The results indicate that the system has the potential of interfacing single rare-earth ions with single photons on a chip. Yb3+ ions were introduced into the device using ion implantation techniques. Yb2+ ions in the implanter were first selected to have only the isotope 174Yb with zero nuclear spin, and were then implanted with an energy of 360 keV perpendicular to the device with a 200-nm-thick top cladding. After implantation, Rutherford backscattering spectrometry was performed on a reference sample and the data were analysed using NDF31 (Supplementary Fig. 1). The implantation energy was chosen carefully so as not to introduce Yb ions into the Si3N4 core, because elimination of implantation defects in SiO2 and Si3N4 requires different annealing temperatures and their temperature ranges are incompatible with each other32. The device was completed by wafer bonding at 950 °C for 3 h, which also anneals implantation defects in SiO2. The fibre connections to the waveguides were permanent and compatible with cryogenic temperature. A stripped and cleaved optical fibre was first dipped into a small droplet (diameter smaller than 0.5 mm, generated at the tip of a syringe needle) of ultraviolet curing optical adhesive (NOA61, Norland Products). Once the fibre was retracted from the adhesive droplet, a tiny amount of ball-shaped adhesive was attached at the fibre tip. The fibre was then aligned with the waveguide by maximizing the measured resonance transmission and was arranged protruding towards the waveguide facet. Once the adhesive on the fibre tip touched the waveguide, surface tension pulled the adhesive to the coupling point and rendered the coupling area well surrounded by the adhesive. Subsequently, the adhesive was cured by ultraviolet light. The small and axisymmetric adhesive spot at the coupling point ensured that negligible misalignment was induced by thermal contraction of the adhesive itself while cooling to cryogenic temperature. The laser frequency was locked to the top of the fringe of the TM0 mode of the ring resonator during the experiment to compensate for thermal drifts. The laser current was modulated with a small sinusoidal wave at 100 kHz. The signal of the APD was demodulated with a lock-in amplifier to generate an error signal, which was fed back to the laser piezo controller through a proportional-integral controller. The locking was stable, even though the laser light was blocked repetitively by the mechanical chopper. We assumed that the Purcell factor distribution along the Yb3+ sheet ( and z = 0, i = TM0 or TE0) can be approximated as a Gaussian function h(x) at a specific temperature. The position-dependent decay rate k(x) is then readily available as with h(0) = 1. The probability density of Yb3+ decaying with rate k is proportional to the number of ions dN in an infinitesimal interval dk around k: where ρ is a linear doping density of Yb3+. By using a Gaussian function of h(x), we obtain and P(k) = 0 elsewhere, where A is a normalization factor rendering ∫P(k)dk = 1. The excited-state population of Yb3+ is given by W/(k + 2W) according to the steady-state solution of the rate equations, where W is the excitation rate. A decay function I(t) can be built as We assume that W is independent of k and is small compared to k. I(t) can be simplified as which is used to analyse the measured SE decay traces. W increases with decreasing temperature due to the decreasing homogeneous linewidth of Yb3+. At low temperature, the assumption of W ≪ k remains valid for large k values, but it becomes invalid for small k values. This explains the deviation of the decay function from the slow components of the decay trace at 50 mK, resulting in an artificially large value of k0, whereas the fit is precise for the fast components, resulting in a correct value of km. Two laser pulses with widths of 60 and 120 ns were generated with two acousto-optic modulators in series and coupled to the 3D TM1 mode of the ring resonator through fibre 1. Photon echo occurred after the second pulse and was measured through fibre 3 using a SPCM. The echo intensity was maximized by adjusting the laser power while keeping the pulse widths constant. After this optimization, the two pulses ideally performed π/2 and π operations for the ions. The echo intensities were measured with varying delay time between the two pulses at a certain temperature. The data are well fitted with a single-exponential function of delay time. The time constant τ extracted from the fit is related to ΓH(T) as ΓH(T) = 1/4πτ (see Supplementary Section 4 for details). The laser output was modulated with a mechanical chopper and coupled to the 3D TM1 mode of the ring resonator through a beamsplitter and fibre 1. The fluorescence of Yb3+ was coupled back to fibre 1 and the reflective port of the beamsplitter. The polarization of the resonance fluorescence was analysed using a bandpass filter, wave plates, a polarizing beamsplitter and two SPCMs. The wave plates were adjusted when the ring resonator was cooled to 10 mK with fully polarized dipoles of Yb3+ such that the integrated signal of one SPCM denoted C1 was maximized and the integrated signal of the other SPCM denoted C2 was simultaneously minimized. Coupling and detection efficiencies were calibrated at room temperature with fully depolarized dipoles. The efficiency factor η is given by , where and are the values of C1 and C2 at room temperature. The dipole polarization Θ is given by Θ = (ηC1 – C2)/(ηC1 + C2) (see Supplementary Section 5 for details). Download references The authors thank M.P. van Exter for scientific discussions and proofreading of the manuscript, and A.M.J. den Haan, J.J.T. Wagenaar, M. de Wit and T.H. Oosterkamp for operating the dilution refrigerator. This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). This work was supported by NWO VICI grant no. 680-47-604, NSF DMR-0960331, NSF PHY-1206118, DARPA MTO under EPHI contract HR0011-12-C-0006 and the Fund for Scientific Research–Flanders (FWO).