ARTICLE
Received 23 Apr 2014 | Accepted 27 Aug 2014 | Published 24 Oct 2014
There has been an increasing interest in all-optical analogues of electromagnetically induced transparency and AutlerTownes splitting. Despite the differences in their underlying physics, both electromagnetically induced transparency and AutlerTownes splitting are quantied 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 AutlerTownes splitting it is the result of strong eld-driven 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 AutlerTownes 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 whispering-gallery-mode resonators. Moreover, we report the application of the Akaike Information Criterion discerning between all-optical analogues of electromagnetically induced transparency and AutlerTownes splitting and clarifying the transition between them.
1 Department of Electrical and Systems Engineering, Washington University, St. Louis, Missouri 63130, USA. 2 Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan. 3 Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA. Correspondence and requests for materials should be addressed to S.K.. (email: mailto:[email protected]
Web End [email protected] ) or to L.Y. (email: mailto:[email protected]
Web End [email protected] ).
NATURE COMMUNICATIONS | 5:5082 | DOI: 10.1038/ncomms6082 | http://www.nature.com/naturecommunications
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DOI: 10.1038/ncomms6082
What is and what is not electromagnetically induced transparency in whispering-gallery microcavities
Bo Peng1, Sahin Kayazdemir1, Weijian Chen1, Franco Nori2,3 & Lan Yang1
ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6082
Coherent interactions of light with multilevel atoms can dramatically modify their optical response via quantum interferences between various excitation pathways, or via
strong-coupling-eld-induced energy level splitting. The process known as electromagnetically induced transparency14 (EIT) is the result of Fano interferences5,6 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 propagation7, light storage8,9, dissipation-free light transmission and nonlinear optics with weak light. AutlerTownes splitting10 (ATS) involves eld-induced 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 prole. It has been used for measuring transition dipole moments11 and quantum control of spinorbit interactions12.
Coherent processes leading to EIT and ATS have been studied in: atomic gases7,13, atomic and molecular systems14, solid-state systems15, superconductors16,17, plasmonics18, metamaterials19, optomechanics20,21, electronics22, photonic crystals23 and whispering-gallery-mode microresonators (WGMRs)2433. Systems in which EIT and ATS have been studied are listed in Fig. 1. Using the identications 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 on-chip 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 low-temperature environment, the need for stable lasers matching the atomic transitions or propagation-scaling limitations because of control-eld absorption).
WGMRs have been a fruitful platform to study various aspects of classical and all-optical analogues of EIT and ATS. Fano resonances and EIT have been observed in a silica microsphere28, a polydimethyl-siloxane-coated silica microtoroid with two WGMs29, two directly coupled silica microspheres25,30, two indirectly coupled silicon microrings26, and indirectly coupled microdisk and microtoroid31. In these EIT implementations, two frequency-degenerate WGMs of high- and low-quality 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 microspheres25, directly coupled silica microtoroids34, hybrid systems formed by directly coupling polydimethyl-siloxane-coated silica microtoroids with silica microtoroids or microspheres35 and directly coupled polyethylene and quartz disks in the THz domain36. In these systems, ATS originates from the lifting of the frequency degeneracy of the eigenmodes, hence their splitting into two resonances because of strong inter-resonator coupling. The spectral region between the split modes corresponds to a transparency window. A scatterer-induced coupling between the frequency-degenerate clockwise and counter-clockwise travelling modes of a WGMR can also lead to mode-splitting3739. This has been used to detect and measure nanoscale objects with single particle resolution37 and to directly measure the Purcell factor40.
The effects of ATS in the absorption prole resemble that of EIT, in that both of the processes display a transparency window, that is, a reduction in the absorption prole. This similarity has led to much confusion16,41,42 and many discussions17,4346 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 rst decide whether to focus on the sharpness of the transmission or the sharpness of susceptibility/absorption. Moreover, system-specic 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 orders-of-magnitude less than their atomic counterparts; hence, the dip in the susceptibility of superconducting systems is lesser sharp than that of alkaline atoms47. Therefore, objective and system-independent methods and tests are needed to make claims of EIT. Since identifying whether an experiment involves EIT
Probe
Pump
Probe
A1
Atom
A2
[afii9837]
1
[afii9828]1
3
[afii9828]13
Control
[afii9829]
2
1
[afii9837]
[afii9853]probe
[afii9837]
[afii9846]13 [afii9846]12
2
2
1
Probe
[afii9828]12
[afii9828]2
0
1 2
m1 m2
X1 X2
k1 k2
K
Probe Control
Probe A
2Gaxzpf
m
Fs(t)
[afii9828]A
X(t)
SW
C
R2 R1
Cavity 2 Probe
H
b
E
Vs
~ q2 q1
a
[afii9828]a
L C2 C1 L1
2
[afii9828]b
d
Cavity 1
Figure 1 | Different platforms used for realizing electromagnetically induced transparency. (a) Atomic ensembles with L-type three-level atoms: ground state |1S; dark state |2S; excited state |3S; density matrix elements s12 and s13; pump eld Rabi frequency O; dephasing rates g12 and g13; detuning d (ref. 17). (b) Coupled system of microresonators: intracavity eld amplitudes A1 and A2; cavity coupling strength k; decay rates g1 and g2 of the optical modes A1 and A2; resonance detuning D; detuning of the probe eld d (ref. 24). (c) Mechanical systems: spring constants K, k1, k2;
masses m1 and m2; displacements X1 and X2 of mass 1 and mass 2, respectively; mechanical dampings G1 and G2; driving force Fs(t) (ref. 22).
(d) Optomechanical systems: intracavity eld amplitude A ; mechanical displacement X; optomechanical coupling strength 2Ga
%xzpf; decay gA of A ; mechanical damping Gm (ref. 20). (e) Electronic circuits: inductors L1 and L2; capacitors C, C1 and C2; resistors R1 and R2; voltage source Vs; charges q1 and q2 in the resonance circuits; switch SW (ref. 22).
(f) Plasmonics: the radiative plasmonic state |aS; the dark plasmonic state |bS; plasmonic coupling strength k; damping rates ga and gb (ref. 18).
Correspondences and analogies among these system-specic parameters are shown in Table 1.
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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6082 ARTICLE
Table 1 | Correspondences among parameters of various systems (Fig. 1) in which EIT and ATS have been experimentally observed.
Atomic Optomechanics Resonators Electronics Mechanical Plasmonics Density matrix element (Radiative state) s13 A A1 q1 X1 |aS
Density matrix element (Dark state) s12 X A2 q2 X2 |bS Coupling strength (Rabi Frequency) O 2Ga%xzpf k
1=L2C
p
K=m1 p
k
Dephasing rate (Radiative state) g13 gA- g1 R1/L1 G1 ga Dephasing rate (Dark state) g12 Gm g2 R2/L2 G2 gb
Tunable laser
Computer
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 ATS4347. 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 one-dimensional (1D) superconducting transmission line coupled to a ux qubit, concluding that the reported data16 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.
ResultsExperimental set-up. Our system consists of two directly coupled silica microtoroidal WGM resonators mR1 and mR2, with mR1 coupled to a bre taper (Fig. 2). We fabricated the silica micro-toroids 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 nely tuned to control the coupling strength k between them. The coupling strength k decreases exponentially with increasing distance. The probe light in the 1,550-nm band from a narrow linewidth tunable laser was coupled into a WGM of mR1 via the bre taper. The same bre 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. Fibre-based 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 thermo-optic effect, to control the frequency detuning of the chosen WGMs in the two micro-toroids. 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 Q-contrasts. In addition to the ability of choosing different WGM pairs, our set-up 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 (QmR1 and QmR2) of (1.91 105, 7.26 107),
(1.63 106, 1.54 106) and (1.78 106, 4.67 106). 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 mR1 side with a bre taper, the Q of the mR1 is smaller than the above intrinsic Q-values because of the additional coupling losses (that is, mR1 has more loss than mR2).
Analogy between coupled resonators and three-level atoms. Here we will elucidate the analogy between atomic and photonic coherence effects leading to EIT and ATS. Using coupled-mode theory, we nd the equations of motion for the complex intra-cavity eld amplitudes A1 and A2 in the steady state as
d1
ig1=2
A1 kA2 i
gc
1
Transmission
0.8
0.6
Function generator
Detuning (MHz)
Polarization controller
Oscilloscope
Fibre taper
200 100 0 100 200
Photodetector
Coupled microtoroids
Figure 2 | Experimental conguration for the coupled WGM microcavity system. Set-up with a microscope image of the coupled microtoroid system. The system consists of two directly coupled WGM resonators (mR1 and mR2), and a bre taper waveguide coupled to mR1. The probe laser is in the 1,550-nm band. Inset: typical transmission spectra showing the resonant mode in mR1.
p Ap 1
d2 ig2=2A2 kA1 0 2
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where g1 g01 gc and g2 denote the total losses in mR1 and
mR2, respectively, g01 is the intrinsic loss of mR1 and gc is the coupling loss between the bre taper and mR1, d1 o o1 and
d2
o o2 denote the detuning between the frequency o of the
probe light eld Ap and the resonance frequencies o1 and o2 of the WGMs, and k is the coupling strength between the WGMs. In the EIT and ATS experiments, we set o1 o2 o0
via the thermo-optic 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 o1 and o2 we have D o2 o1 0, and in the rotated frame (o0-0)
we have d1 d2 o. In addition, note that in the system
depicted in Figs 1b and 2 the input and output ports are at the side of mR1; hence, the output eld is given as
Aout Ap
gc
1
C1
o2
o2i
o iw r
o2
o2 i
G21
C2
o2
G22
8
where all the parameters whose values cannot be determined precisely are incorporated into the coefcients Ck and Gk that can be used as free parameters to perform curve-tting to experimentally obtained transmission spectra. Clearly, the opposite signs of the Lorentzians lead to a destructive interference that results in a transmission prole exhibiting a transparency window similar to that of EIT.
In the strong-driving regime, where k44kT is satised, b 2k
is real (that is, bi 0 and br 2k) implying o izk, that
is the resonances are located at frequencies k with a spectral distance of 2k between them. The resonance linewidths are quantied by Im(o) z. Approximating w as w 1/2
we nd the imaginary part of w as
wi
1
2
p A1 where the intracavity eld A1 can be written as A1 i
gc
p Apw with
w
o ia2
3
where we used ak iok gk/2, with k 1, 2. This solution w has a
form similar to the response of an EIT medium (three-level atom) to a probe eld. Then, we can write the normalized transmission T |Aout/Ap|2 as
T 1 g2c w
j j2 2gcwi 4 where wi is the imaginary part of w. Since 144g2c w
j j2 and
g2c w
j j2oogcwi we can rewrite the transmission as
T 1 2gcwi 5 Thus, it is sufcient to analyse the behaviour of wi 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 o
( ia1 ia2b)/2, with b2 4k2 (a1 a2)2. This reveals a
transition at the threshold coupling strength 2|kT| a1 a2
(g1 g2)/2, where we have used the fact that in our system g14g2,
as stated in the previous section. We dene the regimes where kokT and k4kT as the weak- and strong-driving regimes, respectively, and k kT as the transition point.
Using the eigenfrequencies, we can rewrite the expression in equation (3) as
w
w
o o
k2
o ia1o ia2
9
which implies that wi is the sum of two same-sign Lorentzians centred at k. The transmission in this regime is then given by
TATS 1 gc
z o k
2 z2
z o k
2 z2
2 z2
z o k
1
2 z2
z o k
C
2 G2
C
o d
2 G2
10
where C, G and d are the free parameters that can be used in curve-tting to experimentally obtained transmission spectra. Clearly, the transmission in this strong-driving regime presents a symmetric doublet spectra and the observed transparency is because of the contribution of two Lorentzians.
In the intermediate-driving regime51 quantied by k4kT, b br is real (that is, bi 0). This leads to complex
eigenfrequencies o ( ig1 ig22br)/4 and complex
w
1/2i(g1 g2)/4br. Thus, the supermodes have
different resonance frequencies located at br/2 but have the same linewidths quantied by their imaginary parts Im(o) oi ( g1 g2)/4. Consequently, we have
wi
o o
o d
r
w i o
2 o2i
o o
iw r
r
w i o
iw r
o o
r
o o
r
2 o2 i
11
w
o o
6
where w 8(o ia2)/b 1/2ix/b satisfying w w
1 and x (g1 g2)/4.
In the weak-driving regime quantied by kokT, b is imaginary, that is b ibi and Re(b) br 0. This leads to real w
(that is, Im(w) wi 0) with Re(w) wr 1/2x/|b|,
and imaginary eigenfrequencies with Re(o) or 0 and
Im(o) oi z|b|/2, where z (g1 g2)/4. Thus, the
supermodes have the same resonance frequencies and are located at the centre of the frequency axis but have different linewidths quantied by their imaginary parts. The imaginary parts of w are then given by
wi
o iw r
o2
and
TEIT=ATS 1
o eC1o e2 G2
o eC1 o e2 G2
12
where e br/2. The expression in the second bracket of
equation (12) is the sum of two Lorentzians, similar to the expression obtained for the strong-driving regime in equation (10), implying the contribution of ATS. The expression in the rst bracket corresponds to the interference term and can be controlled by choosing the loss of the coupled modes. For example, choosing two modes satisfying g1 g2 will
lead to C1 0, and hence the expression T
C2
o e2 G2
C2
o e2 G2
o2i
o iw r
o2
o2 i
7
which consists of two Lorentzians centred at o 0 with different
signs (that is, the rst term in the equation is negative, whereas
the second term is positive). The transmission in this regime becomes
TEIT 1 2gcwi 1 2gc
o iw r
o2
EIT/ATS will become the same as TATS. 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.
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The theoretical shapes and more detailed discussions of these operating regimes are given in Supplementary Note 1 and Supplementary Figs 13.
Fano resonances and EIT in experiments with coupled WGMRs. EIT is a result of strong Fano interferences and takes place when a high-Q WGM of one microresonator is directly coupled to a low-Q WGM of a second microresonator with zero-detuning in their resonance frequencies. In order to demonstrate this, we chose a low-Q mode in mR1 (QmR1 1.91 105) and a
high-Q mode in mR2 (QmR2 7.26 107). 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 high-Q mode in mR2 such that it approached to the frequency of the low-Q mode in mR1. As the frequency-detuning between the modes gradually decreased, the modes became spectrally overlapped. Consequently, we rst observed an asymmetric Fano lineshape with the peak located closer to the lower-detuning side (Fig. 3a, upper panels), and then a transparency window appeared at zero-detuning o2 o1 0 (Fig. 3a, middle panel). The linewidth of
the transparency window was 5 MHz. The asymmetry of Fano resonances decreased as we approached to zero-detuning. As the frequency of the high-Q mode was further increased, detuning started to increase again leading to the emergence of Fano line-shapes whose peaks were also located closer to zero-detuning (Fig. 3a, lower panels). Finally, when the frequency was increased such that there was no overlap between the modes, Fano line-shapes were lost and we observed two independent Lorentzian lineshapes corresponding to the two modes in mR1 and mR2.
Next, we studied the effect of coupling on the transparency window by rst setting the frequency detuning of the low- and high-Q 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 eld 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 strong-coupling between two WGMs of similar Q. In order to demonstrate this, we chose the mode with QmR1 1.63
106 in mR1 and the mode QmR2 1.54 106 in mR2. We rst
tuned the resonance frequencies of the modes to have o1 o2 (that
is, zero-detuning) when the microtoroids were sufciently 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 well-tted by two Lorentzian resonances.
Next, we chose two detuned resonance lines and set the coupling to strong-coupling 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 mR2, the split modes started to approach each other (that is, decreasing mode-splitting) and the difference between their transmission dips decreased. At zero-detuning 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 mR2 was further increased beyond the zero-detuning 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 micro-resonators (that is, similar to a three-level quantum system). However, EIT is because of Fano interference and hence requires coupling between a high-Q mode (that is, discrete system) and a low-Q mode (that is, continuum) in the weak-driving regime, while ATS is because of strong-coupling induced splitting of resonance modes and requires the interaction between modes with similar Q in the strong-driving regime.
a
b
c
Coupling strength (MHz)
Fano interference EIT
20 40 60 80
1
0.8
3
3
Normalized transmission
Transparency-window
linewidth (MHz)
40
2.5
2.5
Transmission
2
2
1.5
1.5
20
1
1
0.6
0.5
0.5
2,000 0 2,000
2,000 0 2,000
20 40 60 80
0
Detuning (MHz)
Detuning (MHz)
Coupling strength (MHz)
Figure 3 | Fano interference and EIT in coupled WGM microcavities. (a) Spectral tuning of the system from asymmetric Fano resonance to EIT. When the resonance modes of the microresonators have non-zero frequency-detuning, the transmission exhibits asymmetric Fano resonances. At zero detuning, a transparency window appears. (b) Effect of coupling strength on the EIT spectra (that is, zero detuning between resonance modes of the resonators). The coupling strength (increasing from the bottom to the top curve) depends on the distance between the resonators. (c) Effect of the coupling strength on the linewidth (red circles) and the peak transmission (blue squares) of the transparency window. The curves are the best t to the experimental data.
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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6082
a
3.5
3
2.5
Transmission
2
1.5
1
j1 e Ij=2, where
i Ii/n, can be applied. In our study involving only two models
(N 2), we can rewrite wi and
wi as
0.5
400 200 0 200 400
Detuning (MHz)
b
wEIT
e IEIT=2
e IEIT=2 e I
ATS=2 ;
5
13
wEIT
e IEIT=2n
e IEIT=2n e I
ATS=2n
with wEIT wATS 1 and
wEIT
wATS 1.
In our experiments, we acquired 10,000 data points (n 10,000) to form a transmission spectrum at each setting of
coupling strength and frequency-detuning, and we repeated the measurements to obtain 10 transmission spectra at each setting, to take into account the uctuations and uncertainty in the measurements. We used TEIT and TATS given in equation (8) and
equation (10), respectively, to t the transmission spectra obtained in the experiments. Then we used the AIC tests proposed by Anisimov et al.17 by calculating (wEIT, wATS) and
wEIT;
wATS 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 (wEIT, wATS) and
wEIT;
wATS obtained at three different experimental settings,
corresponding to three different regimes of operation, as the coupling strength was increased: EIT-dominated (Fig. 5a), ATS-dominated (Fig. 5b) and EIT-to-ATS 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 rst case (Fig. 5a), the WGMs in the resonators were chosen such that their decay rates, quantied by QmR1 (that is, g1)
and QmR2 (that is, g2), signicantly differed from each other (that is, QmR2/QmR1B400). We calculated kT |g1 g2|/4
312.8 MHz, which was larger than the coupling strengths used in the experiments (ko100 MHz). Starting from
wEIT
wATS
0:5 (that is, both the EIT and the ATS models are equally likely) for very weak coupling strength (0rkr15 MHzookT), 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 weak-driving regime (kokT), data obtained in the experiments favour the EIT model, as revealed by
wEIT4
wATS.
In the second case (Fig. 5b), the decay rates of the coupled WGMs were very similar to each other (that is, QmR2/QmR1
B0.95), and we estimated the critical coupling strength as kT
16.2 MHz, which was smaller than the coupling strengths considered k460 MHz. Therefore, as demonstrated in the model, this falls in the strong-driving regime (k44kT), where ATS is expected. Indeed, in this experimental setting, starting from
4
Transmission
3
2
1
0
1,000 500 0 500 1,000
Detuning (MHz)
Figure 4 | ATS and avoided crossing in coupled WGM microcavities. (a) Effect of the coupling strength on the ATS spectra. During the process, the resonance modes had zero frequency detuning, and as the coupling strength was increased (from the bottom to the top curve) the splitting increased. (b) Effect of the frequency detuning of the coupled resonant modes on the ATS spectra. When the two resonant modes had non-zero detuning, the transmission exhibited asymmetric ATS (unequal transmission dips). As the frequency of one of the resonant modes was increased (frequency detuning approaches zero), the split modes started to approach each other and the transmission exhibited more symmetric ATS spectrum. With further increase in the frequency, the split modes exhibited avoided crossing. The smallest splitting was B300 MHz, at which the frequency detuning between the resonant modes was zero. Further increase in the resonance frequency then led back to an asymmetric ATS.
Thus, depending on the relative Qs of the interacting modes and their coupling strength, the system of coupled micro-resonators 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 coupled-resonator 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 KullbackLeibler (KL) distance between the model and the truth. The KL distance quanties the amount of information lost when approximating the truth. Thus, a good model is the one that minimizes the information loss and hence the KL distance. Then AIC quanties the amount of information lost when the model li with ki unknown parameters out of N models is used to t the data x x1, x2, ..., xn obtained in the measurements, and is given
as Ii 2logLi 2ki, where Li is the maximum likelihood for the
candidate model li and 2ki accounts for the penalty for the number of parameters used in the tting. Then, the relative likelihood of the model li is given by the Akaike weight wi e Ii=2= PN
j1 e Ij=2. In the case of least-squares and the
presence of technical noise in the experiments, a tness test using per-point (mean) AIC weight
wi e Ii=2= PN
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Figure 5 | AIC to discriminate EIT from ATS in optical microcavities. (a) The AIC per-point weight for the pair of modes chosen in the rst and second microresonators with QB(1.91 105, 7.26 107). As the coupling strength was increased, the EIT model dominated. (b) The AIC per-point weight
for pair of modes with QB(1.63 106, 1.54 106). As the coupling strength was increased above 60 MHz, the ATS model dominated. (c) The AIC
per-point weight for the pair of modes with QB(1.78 106, 4.67 106). A transition from EIT-dominated to ATS-dominated regime is clearly seen.
(df) The AIC weights for the mode pairs studied in (ac). Data for EIT and ATS are given in blue and red marks/curves, respectively, and the green and orange marks/curves represent the theoretical AIC per-point weights (that is, these do not take into account the experimental noise). Each data point is the average of 10 measurements, and the error bars represent the s.d.s.
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 tting decreases. The factors that affect the tting and hence the model-assignment according to AIC weights are thermal noise, the probe laser frequency and amplitude uctuations, the limited number of data points acquired for each spectrum, the resolution of the measurement equipments and the efciency of the detector. We estimated that the s.d. of the total noise in our experiments is 12.5% of the measured signal.
Finally in Fig. 6 we show examples of typical transmission spectra obtained in our experiments in the EIT-dominated (Figs 5a and 6a), the ATS-dominated (Figs 5b and 6b) and the EIT-to-ATS transition regimes (Figs 5c and 6c), together with the best-tting curves using the expressions TEIT and TATS derived
from the theoretical model. It is clear that for the spectrum for which the AIC assigned the EIT model, the function TEIT
provided a better t than TATS. In particular, the TATS tting performed poorly around the narrow transparency window (Fig. 6a inset). In the spectrum for which the ATS model was assigned according to AIC weights, TATS performed extremely well, whereas the TEIT tting 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 curve-tting using TEIT and TATS. It is clearly seen in Fig. 6c that TEIT and TATS 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 curve-tting tests (Fig. 6) agree well with the predictions of the AIC weights (Fig. 5).
wEIT
wATS 0:5, 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
wATS4
wEIT.
The third case (Fig. 5c) revealed an interesting phenomenon: transition from an EIT-dominated regime to an ATS-dominated regime through an inconclusive regime, where both EIT and ATS are equally likely. The decay rates of the chosen WGMs were similar (that is, QmR2/QmR1 B2.6); 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 kT 29.5 MHz. In this
case, the EIT model rst dominated
wEIT4
wATS 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
wEIT
wATS 0:5. Further increase in the coupling strength
beyond this point revealed a transition to an ATS-dominated regime
wATS4
wEIT . This results agree well with the predictions of
the model: in the EIT-dominated regime we had kokT, in the transition regime we had kBkT and nally in the ATS-dominated regime we had k44kT. In Fig. 5df, we also present (wEIT, wATS) as a function of the coupling strength. As expected, these weights exhibit a binary behaviour with an abrupt change from the EIT-dominated regime to the ATS-dominated regime.
Since we collected 10 sets of data at each specic 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 tting, 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,
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Figure 6 | Experimentally observed transmission spectra with EIT and ATS model ttings. The transmission spectra shown here are chosen to represent the three regimes (EIT-dominated, ATS-dominated and EIT-to-ATS transition regimes) observed in Fig. 5 according to AIC weights and per-point weights. Data obtained in the experiments are given in blue curves (black in the insets), and the best ts using the TEIT model and the TATS model are given by the red and green curves, respectively. (a) The mode pair with QB(1.91 105, 7.26 107) shows a clear transparency window, and the EIT model
provides the best t. This agrees with the result in Fig. 5a,d. (b) The mode pair with QB(1.63 106, 1.54 106) exhibits a broader transparency
window with the best t provided by the ATS model. This agrees with the result in Fig. 5b,e. (c) At the transition region, both the EIT and the ATS models t equally well to the transmission spectrum obtained for the mode pair with QB(1.78 106, 4.67 106). This agrees well with the results of Fig. 5c,f.
Insets are enlarged versions of the spectra around the central frequency (zero detuning).
As we showed in the previous sections and in the Supplementary Note 1 and Supplementary Fig. 1, EIT/ATS model (intermediate-driving regime) has two terms, one of which is exactly the same as the ATS model (strong-driving regime) and the other is an interference term. We found theoretically and experimentally that
wATS and
wEIT=ATS 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 curve-tting because C1 of equation (12) is a free-parameter (Supplementary Note 2 and Supplementary Figs 46). Moreover, curve-tting using the intermediate-driving 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 t, although it has more free parameters. The discrepancy is signicant around the zero-detuning where the transparency window exists. For the spectra obtained for the ATS and EIT-to-ATS transition, we see that the EIT/ATS model provides a curve-tting at least as good as the ATS model (Supplementary Note 2 and Supplementary Fig. 7).
DiscussionAlthough 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, all-optical analogues. Their demonstrations in on-chip physical systems using optical microresonators, meta-materials or plasmonics offer great promises for a wide range of applications including controlling the ow of light on-chip, high-performance sensors and studying the effects of many parameters that are difcult to test in atomic and molecular systems that need sophisticated and hard-to-access experimental environment and techniques. In particular, the capability of creating EIT and controlling the features of the transparency window in on-chip coupled optical microcavities is important for on-chip all-optical slowing and stopping of light, tunable optical lters, 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 coupled-resonator system for the observation and control of Fano resonances, EIT and ATS, as well as to probe the transitions among them. The capability to ne-tune 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 condence 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.
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Acknowledgements
This work is supported by the NSF under grant number 0954941 and the US Army Research Ofce under grant number W911NF-12-1-0026. F.N. is supported by the RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics and a Grant-in-Aid for Scientic Research (S).
Author contributions
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.
Additional information
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How to cite this article: Peng, B. et al. What is and what is not electromagnetically induced transparency in whispering-gallery microcavities. Nat. Commun. 5:5082 doi: 10.1038/ncomms6082 (2014).
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Abstract
There has been an increasing interest in all-optical 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 field-driven 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 whispering-gallery-mode resonators. Moreover, we report the application of the Akaike Information Criterion discerning between all-optical analogues of electromagnetically induced transparency and Autler-Townes splitting and clarifying the transition between them.
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