Experimental distinction of Autler-Townes splitting from electromagnetically induced transparency using coupled mechanical oscillators system

Here we experimentally demonstrated the electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS) effects in mechanical coupled pendulums. The analogue of EIT and ATS has been studied in mechanical systems and the intrinsic physics between these two phenomena are also been discussed. Exploiting the Akaike Information Criterion, we discern the ATS effect from EIT effect in our experimental results.


Results
Experimental set-up and oscillation spectrums. Here the experiment setup is shown in Fig. 1. The system consists of two torsion pendulums made of plexiglass with the diameters of 350 mm, the thickness is 10 mm, and the mass is 0.975 kg. As shown in Fig. 1, the two oscillators are coupled by a spring and driven by a stepper motor (42BYG250Bk), here the stepper motor is controlled by the function generator. The data acquisition card (NI, usb6251) is used to collect the oscillation spectrum. The stepper motor is controlled with the function generator through a motor driver (SH2034D). A torsion spring connects the stepper motor to the pendulum 1. In our experiment, the driving frequency is chosen between 0.10-1. 20 Hz, and each of the pendulums connect with an angle encoder coaxially. The angle encoder is an 10 bit absolute rotary encoder, and its resolution is 1024 plus per round. We can get the swing angle and the amplitude from the output signal of angle encoder by data acquisition card and VI program. The front panel and block diagram of the VI program are shown also in Fig. 1.
The amplitude-frequency characteristics curves (AFC curves) of the pendulum 1 are measured as shown in Fig. 2 when the pendulum 2 is fixed. By cutting off the connection between pendulum 1 and torsion spring, our apparatus can work as a simple driven torsion pendulum. Here we set the driving frequency of 0.10 Hz, and when the pendulum 1 goes into a steady state, we record its amplitude by the VI program. Then by increasing  the driving frequency until 1.20 Hz, the steady amplitudes of the pendulum 1 are recorded, and we can get the no coupling spring line. Then, reconnecting the pendulum 1 to torsion spring and fixed the pendulum 2 at its equilibrium position, we can measure the AFC curves with deferent torsion spring, as shown in Fig. 2. By increasing the coupling strength of the coupling spring, we find the increment of the resonance frequency of the pendulum.

Amplitude-Frequency characteristics curves of coupled pendulums.
To study the coupled system further, we measured the AFC curves of coupled pendulums with the largest coupling strength of spring 4. As shown in Fig. 3 Fig. 2) to change the coupling strength. And the experiment results are shown in Fig. 4. There is a mode-splitting effect of the absorption spectrum if we turn on the coupling. According to the increment of the coupling strength, the absorption band becomes larger. Meanwhile, the mode splitting effect will be disappeared if we turn off the  coupling between the two pendulums. Here the pendulum 1 and 2 are connected with coupling spring, and pendulum 2 is fixed at its equilibrium position. It can be seen obviously that the resonance frequency of the system is increased by coupling spring. And the resonance frequency of a torsion pendulum can be described as Here, f, k and J are the natural frequency, stiffness and moment of inertia of a torsion pendulum, respectively. In our system, the coupling spring works as an added torsion spring when pendulum 2 is fixed. So the stiffness could be written as = + k k K i 1 , and the moment of inertia is = J J 1 .
Here, f 1 , k 1 and J 1 are the natural frequency, stiffness and moment of inertia of pendulum 1, respectively; and K i is the stiffness of coupling spring i. According to the parameters of pendulum 1, we can calculate its moment of inertia as = / = .  16). We find the EIT and ATS exhibits similar lineshapes in the oscillation spectrum. As shown in our experiment, under the driving of the pump field, the quasi-atomic system shows the EIT effect for the absorption spectrum. It is obvious that such mechanical analogue of EIT phenomenon could be explained as a three-level atomic system: here the three energy levels of the Λ -type atom could be described as the ground state, the excited state and a metastable state. The excitation from the ground state to the excited state and the metastable state to the excited state could be described as the mechanical oscillation of the two oscillators, respectively. Each mechanical system works as the excitation pathways of a three-level atomic system, and the coupling of the two pendulums provides the interaction of the two fields. Due to the destructive interference, the absorption spectral line-shape exhibits a single 'peak' at zero frequency detuning. The ATS effect can also exhibit in this system as it attributes to the strong coupling between the two system and splitting the degenerated eigenmodes into two modes. As both EIT and ATS effects exhibit similar transparency window in the transmission spectrum, discerning whether a transmission spectrum is the signature of EIT or ATS is an important issue in quantum optics. Here in the experiment, the amplitude-frequency characteristics (AFC) of the coupled pendulums is measured under different coupling strength, and the AFC curves are shown as the blue lines in Fig. 5. To describe the absorption properties of the system, we draw the fitting lines of the experimental results here using EIT and ATS models, respectively. Here we find both EIT and ATS effect in the coupled system. According to refs 10,12, we can get the absorption profiles equations of EIT and ATS as, Here, + C , − C , C 1 , and C 2 are the amplitudes of the Lorentzian curves; γ + , γ − , γ 1 , and γ 2 are their respective linewidths; , δ 1 , and δ 2 are the shifts from natural frequency.
Exploiting the Akaike's information criterion 10 to fit our data, we could quantitatively discerning the EIT and ATS models. AIC criterion quantifies the information loss when model Here σ i satisfies the relation σ ε ε = / + / 2 2 2 1 2 2 2 , and we use K i to denote the number of unknown parameters. i k , as presented in Fig. 6. Figure 6 describes the system in the ATS domain as the AIC value of the ATS model is larger than the EIT model. By decreasing the coupling strength between the two mechanical oscillators, we found the difference between the AIC values of the ATS model and the EIT model is becoming smaller. And the system could approach the transition point from the ATS domain to the EIT domain. However in our classical system, the coupled oscillation in the undriven system under the condition of weak coupling will be annihilated in the lower frequency domain due to the damping effect. Thus, the transparency window will disappear by continuously decreasing the coupling between the two oscillators, and the EIT domain could hardly be observed.

Discussion
In conclusion, we have experimentally studied the analogue of EIT effect and ATS effect in coupled mechanical systems and provided a classical analogue of EIT and ATS in this work. We use AIC to test the transmission spectrum of the mechanical system and discerning the ATS effect from EIT effect. By changing the coupling strength, the system could approach the transition point from the ATS domain to the EIT domain, but hardly into the EIT  Fig. 2(a)), 0.0143( Fig. 2(b)), 0.0255 (Fig. 2(c)) and 0.0319( Fig. 2(d)), respectively. domain. Furthermore, we theoretically analyzed the intrinsic physics of such results in the mechanical system. In addition, our work can be extended to study the parity-time-symmetric system 23,24 by inducing the gain of mechanical oscillation, and the gain assisted nonlinear mechanical effects.
, and the parameters are chosen as η = / K J, = / F M J 0 , then the equations of the system could be described as