Abstract
There has been an increasing interest in alloptical analogues of electromagnetically induced transparency and Autler–Townes splitting. Despite the differences in their underlying physics, both electromagnetically induced transparency and Autler–Townes splitting are quantified by a transparency window in the absorption or transmission spectrum, which often leads to a confusion about its origin. While the transparency window in electromagnetically induced transparency is a result of Fano interference among different transition pathways, in Autler–Townes splitting it is the result of strong fielddriven interactions leading to the splitting of energy levels. Being able to tell objectively whether an observed transparency window is because of electromagnetically induced transparency or Autler–Townes splitting is crucial for applications and for clarifying the physics involved. Here we demonstrate the pathways leading to electromagnetically induced transparency, Fano resonances and Autler–Townes splitting in coupled whisperinggallerymode resonators. Moreover, we report the application of the Akaike Information Criterion discerning between alloptical analogues of electromagnetically induced transparency and Autler–Townes splitting and clarifying the transition between them.
Introduction
Coherent interactions of light with multilevel atoms can dramatically modify their optical response via quantum interferences between various excitation pathways, or via strongcouplingfieldinduced energy level splitting. The process known as electromagnetically induced transparency^{1,2,3,4} (EIT) is the result of Fano interferences^{5,6} that require coupling of a discrete transition to a continuum. EIT creates a narrow transparency window by eliminating a resonant absorption. EIT has a rich variety of applications such as ultraslow light propagation^{7}, light storage^{8,9}, dissipationfree light transmission and nonlinear optics with weak light. Autler–Townes splitting^{10} (ATS) involves fieldinduced splitting of energy levels and is not associated with interference effects; yet it creates a transparency window because of the doublet structure in the absorption profile. It has been used for measuring transition dipole moments^{11} and quantum control of spin–orbit interactions^{12}.
Coherent processes leading to EIT and ATS have been studied in: atomic gases^{7,13}, atomic and molecular systems^{14}, solidstate systems^{15}, superconductors^{16,17}, plasmonics^{18}, metamaterials^{19}, optomechanics^{20,21}, electronics^{22}, photonic crystals^{23} and whisperinggallerymode microresonators (WGMRs)^{24,25,26,27,28,29,30,31,32,33}. Systems in which EIT and ATS have been studied are listed in Fig. 1. Using the identifications listed in Fig. 1 and Table 1, it can be easily shown that equations and relations describing the EIT and ATS in one of these systems are equivalent to those in the others. The existence of EIT and ATS in plasmonics, metamaterials, photonic crystals and WGMRs is critical for onchip control of light at room temperature. EIT and ATS in these systems do not suffer from experimental complexities that are common in solid state and atomic media (for example, a lowtemperature environment, the need for stable lasers matching the atomic transitions or propagationscaling limitations because of controlfield absorption).
WGMRs have been a fruitful platform to study various aspects of classical and alloptical analogues of EIT and ATS. Fano resonances and EIT have been observed in a silica microsphere^{28}, a polydimethylsiloxanecoated silica microtoroid with two WGMs^{29}, two directly coupled silica microspheres^{25,30}, two indirectly coupled silicon microrings^{26}, and indirectly coupled microdisk and microtoroid^{31}. In these EIT implementations, two frequencydegenerate WGMs of high and lowquality factors (Q) are coupled, and destructive interference of the optical pathways cancels the absorption leading to a narrow transmission peak. If a frequency detuning is introduced, the transmission spectra show sharp asymmetric Fano resonances. ATS has been observed in directly coupled silica microspheres^{25}, directly coupled silica microtoroids^{34}, hybrid systems formed by directly coupling polydimethylsiloxanecoated silica microtoroids with silica microtoroids or microspheres^{35} and directly coupled polyethylene and quartz disks in the THz domain^{36}. In these systems, ATS originates from the lifting of the frequency degeneracy of the eigenmodes, hence their splitting into two resonances because of strong interresonator coupling. The spectral region between the split modes corresponds to a transparency window. A scattererinduced coupling between the frequencydegenerate clockwise and counterclockwise travelling modes of a WGMR can also lead to modesplitting^{37,38,39}. This has been used to detect and measure nanoscale objects with single particle resolution^{37} and to directly measure the Purcell factor^{40}.
The effects of ATS in the absorption profile resemble that of EIT, in that both of the processes display a transparency window, that is, a reduction in the absorption profile. This similarity has led to much confusion^{16,41,42} and many discussions^{17,43,44,45,46} on how to discriminate between EIT and ATS just by looking at the experimentally obtained absorption/transmission spectra, without prior knowledge on the system. The sharpness of the dip in the absorption and the imaginary part of the susceptibility (or the peak in the transmission) has been used as an intuitive and informal criterion to judge whether EIT takes place or not. However, such a test is very subjective. For example, a peak in the transmission spectrum is in general much sharper than the dip in the absorption spectrum of the same system. Therefore, one has to first decide whether to focus on the sharpness of the transmission or the sharpness of susceptibility/absorption. Moreover, systemspecific parameters affect the output spectra. Namely, relaxation rates, coupling strengths and cleanliness of the samples differ among different systems and determine the sharpness of the spectrum. For example, these parameters in superconducting systems are at least one to two ordersofmagnitude less than their atomic counterparts; hence, the dip in the susceptibility of superconducting systems is lesser sharp than that of alkaline atoms^{47}. Therefore, objective and systemindependent methods and tests are needed to make claims of EIT. Since identifying whether an experiment involves EIT or ATS is important for applications, objective methods to discriminate between them have been sought.
Many studies have been carried out to identify the conditions to observe EIT or ATS^{43,44,45,46,47}. Recently Anisimov et al.^{17} have proposed to use the Akaike Information Criterion (AIC)^{48} as an objective test to discern EIT from ATS in experimentally obtained absorption or transmission spectra, and to identify the spectra from which one cannot derive a conclusive result on whether EIT or ATS has played a role. They have successfully applied this test to an experiment with a onedimensional (1D) superconducting transmission line coupled to a flux qubit, concluding that the reported data^{16} do not support the claim of EIT. In a recent study, Giner et al.^{49} demonstrated the suitability of the AIC criterion to evaluate EIT and ATS in experiments with cold caesium atoms. Oishi et al.^{50} suggested that the transient response of a system can be used to discern EIT from ATS. The transient response exhibits a sharp spike when the system is prepared for EIT; however, when it is prepared for ATS, an oscillatory signal attributed to coherent energy exchange between the resonators is observed. Note that AIC discriminates EIT from ATS in the frequency domain, whereas the transient response method discriminates them in the time domain.
Up to this date, there was no study to discern EIT from ATS in coupled optical WGMRs. Here we systematically investigate Fano resonances, EIT and ATS in coupled microtoroids, identify the transition from EIT to ATS (and vice versa) and use AIC to discern EIT from ATS from experimentally obtained transmission spectra. Our results show the suitability of the AIC for discriminating EIT from ATS in systems of coupled WGMRs.
Results
Experimental setup
Our system consists of two directly coupled silica microtoroidal WGM resonators μR1 and μR2, with μR1 coupled to a fibre taper (Fig. 2). We fabricated the silica microtoroids at the edges of two separate silicon wafers, such that when the wafers were brought closer to each other, the microtoroids began exchanging energy. The wafers were placed on separate nanopositioning systems so that the distance between the microtoroids was finely tuned to control the coupling strength κ between them. The coupling strength κ decreases exponentially with increasing distance. The probe light in the 1,550nm band from a narrow linewidth tunable laser was coupled into a WGM of μR1 via the fibre taper. The same fibre taper was also used to couple out the light from the WGM. The output light was then sent to a photodetector connected to an oscilloscope, to obtain the transmission spectra as the wavelength of the input light was linearly scanned. Fibrebased polarization controllers were used to set the polarization of the input light for maximal coupling into WGMs. A thermoelectric cooler was placed under one of the wafers so that resonance frequency of the WGM of interest in a microtoroid could be tuned via the thermooptic effect, to control the frequency detuning of the chosen WGMs in the two microtoroids. A tuning range of 8 GHz was achieved. The microtoroids supported many WGMs in the same band but with different quality factors Q, which is the signature of the amount of loss or dissipation (the lower the loss, the higher the Q and the narrower the linewidth of the resonance mode). This allowed us to investigate the effects of Q of the selected modes on the Fano, EIT and ATS processes by choosing WGM pairs with different Qcontrasts. In addition to the ability of choosing different WGM pairs, our setup allowed us to investigate Fano, EIT and ATS processes and the transitions among them by steering the system independently via the coupling strength or the frequency detuning between the selected WGMs. In our experiment, we selected three different sets of WGM pairs with the intrinsic quality factors (Q_{μR1} and Q_{μR2}) of (1.91 × 10^{5}, 7.26 × 10^{7}), (1.63 × 10^{6}, 1.54 × 10^{6}) and (1.78 × 10^{6}, 4.67 × 10^{6}). Note that the intrinsic Q includes all the losses (for example, material, radiation and scattering) except the coupling losses. Since the probe light is input at the μR1 side with a fibre taper, the Q of the μR1 is smaller than the above intrinsic Qvalues because of the additional coupling losses (that is, μR1 has more loss than μR2).
Analogy between coupled resonators and threelevel atoms
Here we will elucidate the analogy between atomic and photonic coherence effects leading to EIT and ATS. Using coupledmode theory, we find the equations of motion for the complex intracavity field amplitudes A_{1} and A_{2} in the steady state as
where and γ_{2} denote the total losses in μR1 and μR2, respectively, is the intrinsic loss of μR1 and γ_{c} is the coupling loss between the fibre taper and μR1, δ_{1}=ω−ω_{1} and δ_{2}=ω−ω_{2} denote the detuning between the frequency ω of the probe light field A_{p} and the resonance frequencies ω_{1} and ω_{2} of the WGMs, and κ is the coupling strength between the WGMs. In the EIT and ATS experiments, we set ω_{1}=ω_{2}=ω_{0} via the thermooptic effect by thermally tuning the frequency of one of the WGMs to be equal to the frequency of the other. Consequently, for the degenerate frequencies ω_{1} and ω_{2} we have Δ=ω_{2}−ω_{1}=0, and in the rotated frame (ω_{0}→0) we have δ_{1}=δ_{2}=ω. In addition, note that in the system depicted in Figs 1b and 2 the input and output ports are at the side of μR1; hence, the output field is given as where the intracavity field A_{1} can be written as with
where we used α_{k}=iω_{k}+γ_{k}/2, with k=1, 2. This solution χ has a form similar to the response of an EIT medium (threelevel atom) to a probe field. Then, we can write the normalized transmission T=A_{out}/A_{p}^{2} as
where χ_{i} is the imaginary part of χ. Since and we can rewrite the transmission as
Thus, it is sufficient to analyse the behaviour of χ_{i} to understand the conditions leading to EIT or ATS (Supplementary Note 1). This is similar to considering the imaginary part of the susceptibility that determines the absorption of a probe in an atomic system. This analogy between the atomic media and the coupled WGMRs can be extended to other systems by using the analogy map given in Fig. 1 and Table 1.
The eigenfrequencies of this coupled system can be found from the denominator of equation (3) and are given as ω_{±}=(−iα_{1}−iα_{2}±β)/2, with β^{2}=4κ^{2}−(α_{1}−α_{2})^{2}. This reveals a transition at the threshold coupling strength 2κ_{T}=α_{1}−α_{2}=(γ_{1}−γ_{2})/2, where we have used the fact that in our system γ_{1}>γ_{2}, as stated in the previous section. We define the regimes where κ<κ_{T} and κ>κ_{T} as the weak and strongdriving regimes, respectively, and κ=κ_{T} as the transition point. Using the eigenfrequencies, we can rewrite the expression in equation (3) as
where χ_{±}=∓(ω_{±}+iα_{2})/β=−1/2±iξ/β satisfying χ_{+}+χ_{−}=−1 and ξ=(γ_{1}−γ_{2})/4.
In the weakdriving regime quantified by κ<κ_{T}, β is imaginary, that is β=iβ_{i} and Re(β)=β_{r}=0. This leads to real χ_{±} (that is, Im(χ_{±})=χ_{±i}=0) with Re(χ_{±})=χ_{±r}=−1/2±ξ/β, and imaginary eigenfrequencies with Re(ω_{±})=ω_{±r}=0 and Im(ω_{±})=ω_{±i}=−ζ±β/2, where ζ=(γ_{1}+γ_{2})/4. Thus, the supermodes have the same resonance frequencies and are located at the centre of the frequency axis but have different linewidths quantified by their imaginary parts. The imaginary parts of χ are then given by
which consists of two Lorentzians centred at ω=0 with different signs (that is, the first term in the equation is negative, whereas the second term is positive). The transmission in this regime becomes
where all the parameters whose values cannot be determined precisely are incorporated into the coefficients C_{k} and Γ_{k} that can be used as free parameters to perform curvefitting to experimentally obtained transmission spectra. Clearly, the opposite signs of the Lorentzians lead to a destructive interference that results in a transmission profile exhibiting a transparency window similar to that of EIT.
In the strongdriving regime, where κ>>κ_{T} is satisfied, β=2κ is real (that is, β_{i}=0 and β_{r}=2κ) implying ω_{±}=−iζ±κ, that is the resonances are located at frequencies ±κ with a spectral distance of 2κ between them. The resonance linewidths are quantified by Im(ω±)=−ζ. Approximating χ_{±} as χ_{±}=−1/2 we find the imaginary part of χ as
which implies that χ_{i} is the sum of two samesign Lorentzians centred at ±κ. The transmission in this regime is then given by
where C, Γ and δ are the free parameters that can be used in curvefitting to experimentally obtained transmission spectra. Clearly, the transmission in this strongdriving regime presents a symmetric doublet spectra and the observed transparency is because of the contribution of two Lorentzians.
In the intermediatedriving regime^{51} quantified by κ>κ_{T}, β=β_{r} is real (that is, β_{i}=0). This leads to complex eigenfrequencies ω_{±}=(−iγ_{1}−iγ_{2}±2β_{r})/4 and complex χ_{±}=−1/2±i(γ_{1}−γ_{2})/4β_{r}. Thus, the supermodes have different resonance frequencies located at ±β_{r}/2 but have the same linewidths quantified by their imaginary parts Im(ω_{±})=ω_{±i}=(−γ_{1}−γ_{2})/4. Consequently, we have
and
where ε=β_{r}/2. The expression in the second bracket of equation (12) is the sum of two Lorentzians, similar to the expression obtained for the strongdriving regime in equation (10), implying the contribution of ATS. The expression in the first bracket corresponds to the interference term and can be controlled by choosing the loss of the coupled modes. For example, choosing two modes satisfying γ_{1}=γ_{2} will lead to C_{1}=0, and hence the expression T_{EIT/ATS} will become the same as T_{ATS}. This implies that to observe ATS, the linewidths (that is, Q) of the coupled WGMs should be very close to each other as will be demonstrated in the experiments discussed below.
The theoretical shapes and more detailed discussions of these operating regimes are given in Supplementary Note 1 and Supplementary Figs 1–3.
Fano resonances and EIT in experiments with coupled WGMRs
EIT is a result of strong Fano interferences and takes place when a highQ WGM of one microresonator is directly coupled to a lowQ WGM of a second microresonator with zerodetuning in their resonance frequencies. In order to demonstrate this, we chose a lowQ mode in μR1 (Q_{μR1}=1.91 × 10^{5}) and a highQ mode in μR2 (Q_{μR2}=7.26 × 10^{7}). We then set the distance between the resonators such that the coupling strength was smaller than the loss contrast of the resonators. At this point, we continuously tuned the frequency of the highQ mode in μR2 such that it approached to the frequency of the lowQ mode in μR1. As the frequencydetuning between the modes gradually decreased, the modes became spectrally overlapped. Consequently, we first observed an asymmetric Fano lineshape with the peak located closer to the lowerdetuning side (Fig. 3a, upper panels), and then a transparency window appeared at zerodetuning ω_{2}−ω_{1}=0 (Fig. 3a, middle panel). The linewidth of the transparency window was 5 MHz. The asymmetry of Fano resonances decreased as we approached to zerodetuning. As the frequency of the highQ mode was further increased, detuning started to increase again leading to the emergence of Fano lineshapes whose peaks were also located closer to zerodetuning (Fig. 3a, lower panels). Finally, when the frequency was increased such that there was no overlap between the modes, Fano lineshapes were lost and we observed two independent Lorentzian lineshapes corresponding to the two modes in μR1 and μR2.
Next, we studied the effect of coupling on the transparency window by first setting the frequency detuning of the low and highQ modes to zero, and then tuning the distance between the microtoroids (Fig. 3b). Note that the coupling strength here corresponds to the strength of the control field in atomic systems. We observed that as the coupling strength was increased (that is, the distance between the microtoroids decreased) the transmission at the transparency window increased from 0.63 to 0.98 (Fig. 3c). During this process, the linewidth of the transparency window increased from 3 to 43 MHz (Fig. 3c).
Experimental demonstration of ATS in coupled WGMRs
Contrary to EIT, ATS is not a result of Fano interferences but requires a strongcoupling between two WGMs of similar Q. In order to demonstrate this, we chose the mode with Q_{μR1}=1.63 × 10^{6} in μR1 and the mode Q_{μR2}=1.54 × 10^{6} in μR2. We first tuned the resonance frequencies of the modes to have ω_{1}=ω_{2} (that is, zerodetuning) when the microtoroids were sufficiently away from each other so that they could not exchange energies (that is, no coupling). At this stage, the transmission showed single resonance with Lorentzian lineshape (Fig. 4a, lowest panel). As we started to bring the resonators closer to each other (that is, increased coupling strength), the single resonance split into two resonances whose spectral distance (that is, mode splitting) increased with increasing coupling strength. For large coupling strengths, the transmission spectra were wellfitted by two Lorentzian resonances.
Next, we chose two detuned resonance lines and set the coupling to strongcoupling regime (that is, resonators are very close to each other). We observed that the split modes in the transmission were not symmetric (Fig. 4b, upper panel), and they had different transmission dips. This can be attributed to the unequal distribution of the supermodes in the two resonators. As we tuned the spectral distance between the WGMs by increasing the frequency of the mode in μR2, the split modes started to approach each other (that is, decreasing modesplitting) and the difference between their transmission dips decreased. At zerodetuning the resonances became symmetric, that is they are Lorentzian with the same linewidths and transmission dips (Fig. 4b, middle panel). Here the supermodes are equally distributed between the resonators. When the frequency of the mode in μR2 was further increased beyond the zerodetuning point, the modes repelled each other leading to an avoided crossing during which they interchanged their linewidths and transmission dips (Fig. 4b, lower panels).
Discerning EIT from ATS using AIC
As we have discussed in the previous sections, in the theoretical model and the experimental observations, EIT and ATS both display a transparency window in the transmission spectrum of coupled microresonators (that is, similar to a threelevel quantum system). However, EIT is because of Fano interference and hence requires coupling between a highQ mode (that is, discrete system) and a lowQ mode (that is, continuum) in the weakdriving regime, while ATS is because of strongcoupling induced splitting of resonance modes and requires the interaction between modes with similar Q in the strongdriving regime. Thus, depending on the relative Q’s of the interacting modes and their coupling strength, the system of coupled microresonators can operate in either the EIT or the ATS regimes. Discerning whether a transparency window in the transmission spectrum of the system of coupled resonators is the signature of EIT or ATS without a priori information on the Qs and coupling strength between the modes is crucial.
Here we performed experiments under various conditions of our coupledresonator system, obtained transmission spectra and used the AIC proposed to discriminate between EIT and ATS in atomic systems to discern EIT or ATS. The AIC provides a method to select the best model from a set of models based on the Kullback–Leibler (K–L) distance between the model and the truth. The K–L distance quantifies the amount of information lost when approximating the truth. Thus, a good model is the one that minimizes the information loss and hence the K–L distance. Then AIC quantifies the amount of information lost when the model λ_{i} with k_{i} unknown parameters out of N models is used to fit the data x=x_{1}, x_{2}, ..., x_{n} obtained in the measurements, and is given as I_{i}=−2logL_{i}+2k_{i}, where L_{i} is the maximum likelihood for the candidate model λ_{i} and 2k_{i} accounts for the penalty for the number of parameters used in the fitting. Then, the relative likelihood of the model λ_{i} is given by the Akaike weight . In the case of leastsquares and the presence of technical noise in the experiments, a fitness test using perpoint (mean) AIC weight , where Ī_{i}=I_{i}/n, can be applied. In our study involving only two models (N=2), we can rewrite w_{i} and as
with w_{EIT}+w_{ATS}=1 and .
In our experiments, we acquired 10,000 data points (n=10,000) to form a transmission spectrum at each setting of coupling strength and frequencydetuning, and we repeated the measurements to obtain 10 transmission spectra at each setting, to take into account the fluctuations and uncertainty in the measurements. We used T_{EIT} and T_{ATS} given in equation (8) and equation (10), respectively, to fit the transmission spectra obtained in the experiments. Then we used the AIC tests proposed by Anisimov et al.^{17} by calculating (w_{EIT}, w_{ATS}) and to determine which of the models (EIT or ATS) is the most likely for the experimental observation.
In Fig. 5, we present typical curves of (w_{EIT}, w_{ATS}) and obtained at three different experimental settings, corresponding to three different regimes of operation, as the coupling strength was increased: EITdominated (Fig. 5a), ATSdominated (Fig. 5b) and EITtoATS transition regime (Fig. 5c). The models assigned using AIC to the experimental data agree very well with the requirements to observe EIT or ATS.
In the first case (Fig. 5a), the WGMs in the resonators were chosen such that their decay rates, quantified by Q_{μR1} (that is, γ_{1}) and Q_{μR2} (that is, γ_{2}), significantly differed from each other (that is, Q_{μR2}/Q_{μR1}~400). We calculated κ_{T}=γ_{1}−γ_{2}/4=312.8 MHz, which was larger than the coupling strengths used in the experiments (κ<100 MHz). Starting from (that is, both the EIT and the ATS models are equally likely) for very weak coupling strength (0≤κ≤15 MHz≪κ_{T}), the likelihood of EIT model increased as the coupling strength was increased up to 100 MHz. Thus, we conclude that in this setting, which corresponded to weakdriving regime (κ<κ_{T}), data obtained in the experiments favour the EIT model, as revealed by .
In the second case (Fig. 5b), the decay rates of the coupled WGMs were very similar to each other (that is, Q_{μR2}/Q_{μR1} ~0.95), and we estimated the critical coupling strength as κ_{T}=16.2 MHz, which was smaller than the coupling strengths considered κ>60 MHz. Therefore, as demonstrated in the model, this falls in the strongdriving regime (κ>>κ_{T}), where ATS is expected. Indeed, in this experimental setting, starting from , the likelihood of the ATS model increased as the coupling strength was increased up to 400 MHz. Thus, the data obtained in the experiments favours the ATS model as revealed by .
The third case (Fig. 5c) revealed an interesting phenomenon: transition from an EITdominated regime to an ATSdominated regime through an inconclusive regime, where both EIT and ATS are equally likely. The decay rates of the chosen WGMs were similar (that is, Q_{μR2}/Q_{μR1} ~2.6); larger than that of the setting of Fig. 5b but much smaller than that of the setting in Fig. 5a. We estimated the critical coupling strength as κ_{T}=29.5 MHz. In this case, the EIT model first dominated when the coupling strength was small. Then the likelihood of the EIT model decreased and that of the ATS model increased as the coupling strength was increased up to 50 MHz, where we observed . Further increase in the coupling strength beyond this point revealed a transition to an ATSdominated regime . This results agree well with the predictions of the model: in the EITdominated regime we had κ<κ_{T}, in the transition regime we had κ~κ_{T} and finally in the ATSdominated regime we had κ>>κ_{T}. In Fig. 5d–f, we also present (w_{EIT}, w_{ATS}) as a function of the coupling strength. As expected, these weights exhibit a binary behaviour with an abrupt change from the EITdominated regime to the ATSdominated regime.
Since we collected 10 sets of data at each specific condition, we could assign s.d.’s to the AIC weights as seen in Fig. 5. The technical noise in the experimental data points plays an accumulated role in the model fitting, which blurs the distinction between the models. This is clearly seen in the comparison of the AIC weights obtained from the experimental data with the theoretical weights. When the coupling strength was very small, in particular for the EIT regime, the noise had a larger blurring effect. This is attributed to the fact that in the very weak coupling regime, the EIT transparency window is so small that it is buried in the noise; thus, the contribution of the transparency band to the whole fitting decreases. The factors that affect the fitting and hence the modelassignment according to AIC weights are thermal noise, the probe laser frequency and amplitude fluctuations, the limited number of data points acquired for each spectrum, the resolution of the measurement equipments and the efficiency of the detector. We estimated that the s.d. of the total noise in our experiments is 1–2.5% of the measured signal.
Finally in Fig. 6 we show examples of typical transmission spectra obtained in our experiments in the EITdominated (Figs 5a and 6a), the ATSdominated (Figs 5b and 6b) and the EITtoATS transition regimes (Figs 5c and 6c), together with the bestfitting curves using the expressions T_{EIT} and T_{ATS} derived from the theoretical model. It is clear that for the spectrum for which the AIC assigned the EIT model, the function T_{EIT} provided a better fit than T_{ATS}. In particular, the T_{ATS} fitting performed poorly around the narrow transparency window (Fig. 6a inset). In the spectrum for which the ATS model was assigned according to AIC weights, T_{ATS} performed extremely well, whereas the T_{EIT} fitting was very poor (Fig. 6b). The experimental conditions for the data shown in Fig. 5c revealed a transition from EIT to ATS. We chose a spectrum obtained in the vicinity of the transition point and performed curvefitting using T_{EIT} and T_{ATS}. It is clearly seen in Fig. 6c that T_{EIT} and T_{ATS} functions perform equally well and one cannot conclusively assign a model to it: we cannot conclusively show EIT (or ATS) nor rule EIT (or ATS) out. These curvefitting tests (Fig. 6) agree well with the predictions of the AIC weights (Fig. 5).
As we showed in the previous sections and in the Supplementary Note 1 and Supplementary Fig. 1, EIT/ATS model (intermediatedriving regime) has two terms, one of which is exactly the same as the ATS model (strongdriving regime) and the other is an interference term. We found theoretically and experimentally that and exhibit the same values because the contribution of the interference term is set to zero or to much lower values than the contribution from the ATS part during the curvefitting because C_{1} of equation (12) is a freeparameter (Supplementary Note 2 and Supplementary Figs 4–6). Moreover, curvefitting using the intermediatedriving regime model to the experimentally obtained transmission spectra revealed that for the transmission spectra for EIT case, the EIT/ATS model do not provide a good fit, although it has more free parameters. The discrepancy is significant around the zerodetuning where the transparency window exists. For the spectra obtained for the ATS and EITtoATS transition, we see that the EIT/ATS model provides a curvefitting at least as good as the ATS model (Supplementary Note 2 and Supplementary Fig. 7).
Discussion
Although initially proposed, observed and used in atomic and molecular systems, Fano interference, EIT and ATS are among many quantum phenomena that have classical and, more importantly, alloptical analogues. Their demonstrations in onchip physical systems using optical microresonators, metamaterials or plasmonics offer great promises for a wide range of applications including controlling the flow of light onchip, highperformance sensors and studying the effects of many parameters that are difficult to test in atomic and molecular systems that need sophisticated and hardtoaccess experimental environment and techniques. In particular, the capability of creating EIT and controlling the features of the transparency window in onchip coupled optical microcavities is important for onchip alloptical slowing and stopping of light, tunable optical filters, switching and nonlinear optics.
Our approach, demonstrated in this work, provides a highly accessible and controllable platform that allows to test the effects of all relevant parameters (coupling strength, decay rates and frequency detunings) on the same coupledresonator system for the observation and control of Fano resonances, EIT and ATS, as well as to probe the transitions among them. The capability to finetune the parameters at a high level, as demonstrated here, enabled us to show the avoided crossing in the ATS process as the frequency of one of the resonances were steered.
In order to make good use of the observed transparency windows in coupled microresonators for the practical applications mentioned above, it is crucial that we know whether Fano interferences have played a role or not (that is, is the transparency the result of EIT or ATS?). Here we applied an objective test to characterize the parameters involved in Fano, EIT or ATS processes in coupled optical resonators and clearly discerned between EIT and ATS. The test used here is the AIC proposed by Anisimov et al.^{17} This test clearly and with high confidence revealed whether the EIT or the ATS was involved in the experimentally obtained transmission spectra under different operating conditions. In addition to its capability to discriminate between EIT and ATS, the test revealed the sensitivity of the parameters involved. Our study demonstrates the suitability of the AIC method to characterize EIT and ATS in coupled microresonator systems and to study the effects of the system parameters on the observed spectra in the transition regime.
Additional information
How to cite this article: Peng, B. et al. What is and what is not electromagnetically induced transparency in whisperinggallery microcavities. Nat. Commun. 5:5082 doi: 10.1038/ncomms6082 (2014).
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Acknowledgements
This work is supported by the NSF under grant number 0954941 and the US Army Research Office under grant number W911NF1210026. F.N. is supported by the RIKEN iTHES Project, MURI Center for Dynamic MagnetoOptics and a GrantinAid for Scientific Research (S).
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B.P. and S.K.O. contributed equally to this work. S.K.O. and F.N. conceived the idea, S.K.O. and L.Y. designed the experiments, B.P. performed the experiments and processed the data with help from S.K.O. and W.C., L.Y., S.K.O, and F.N. wrote the paper with contributions from all authors.
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Peng, B., Özdemir, Ş., Chen, W. et al. What is and what is not electromagnetically induced transparency in whisperinggallery microcavities. Nat Commun 5, 5082 (2014). https://doi.org/10.1038/ncomms6082
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DOI: https://doi.org/10.1038/ncomms6082
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