Precise control of coupling strength in photonic molecules over a wide range using nanoelectromechanical systems

Photonic molecules have a range of promising applications including quantum information processing, where precise control of coupling strength is critical. Here, by laterally shifting the center-to-center offset of coupled photonic crystal nanobeam cavities, we demonstrate a method to precisely and dynamically control the coupling strength of photonic molecules through integrated nanoelectromechanical systems with a precision of a few GHz over a range of several THz without modifying the nature of their constituent resonators. Furthermore, the coupling strength can be tuned continuously from negative (strong coupling regime) to zero (weak coupling regime) and further to positive (strong coupling regime) and vice versa. Our work opens a door to the optimization of the coupling strength of photonic molecules in situ for the study of cavity quantum electrodynamics and the development of efficient quantum information devices.


Supplementary Note 1: Coupled-mode theory for photonic molecules
The mode coupling in a photonic crystal (PhC) nanobeam photonic molecule can be described by coupled-mode theory 1,2 . We first consider a PhC nanobeam photonic molecule, in which ω1 (ω2) and τ1 (τ2) are the resonance frequencies and photon lifetimes of the uncoupled high Q factor PhC nanobeam cavites, respectively. The mode amplitude in one cavity is denoted by A1, and the mode amplitude in the other cavity is denoted by A2. Then we have the following coupled-mode equations: where, к12 and к21 are the coupling strengths between the two PhC nanobeam cavities. Consider a coupled supermode at frequency , i.e. A1=a1· exp(jt) and A2=a2· exp(jt), based on Eqs. (S-1) and (S-2), we have the matrix equation To have non-trivial solutions of a1 and a2, the following determinant must be zero.    In the experiment, due to the high Q factor of the PhC nanobeam cavity, 1 − 1 2 is at least one order smaller than ω-ω2. Therefore, tan −1 is close to zero. According to Eq. (S-9), when x > 0, Δφ ≈ 0, which indicates the sign of the electric field in both of the cavity is the same, and the field in both of the cavity is opposite, and the corresponding mode is odd-like mode.
When the coupling strength (к) is zero, Eq. (S-5) becomes, In order words, the resonance frequencies and photon lifetimes of the PhC nanobeam photonic molecule supermodes are equal to those of the uncoupled PhC nanobeam cavities, i.e. (ω1, ω2) and (τ1, τ2) respectively, when the coupling strength is zero. The mode splitting width is thus equal to the initial frequency detuning (Δ = |ω1-ω2|).

Supplementary Note 2: Lateral displacement calibration
The electrostatic comb drive and folded beam suspensions are standard designs. The electrostatic force generated by the comb drive is given by where n is the number of movable fingers, ε is the permittivity, t and g are the finger thickness and finger gap spacing, and V is the applied voltage. The spring constant of a single set of the folded beam suspension is given by where b, and l are the flexural beam width and length respectively, and E is the Young's Modulus of the beam material. Consequently, the displacement d can be obtained as where m is number of folded beam suspensions and  denotes a constant. Therefore, the lateral displacement of the movable beam is linearly proportional to the square of applied voltage.
At initial zero center-to-center offset between two cavities, the mode splitting width of a pair of photonic molecule supermodes is at maximum. When the movable beam is laterally shifted, the mode splitting width of the photonic molecule is changed. When the lateral displacement is equal to one lattice period (a = 310 nm) of the PhC nanobeam cavity, the absolute value of coupling strength reaches a maximum again, so does the mode splitting width. In our experiment, we gradually increase the applied voltage and record those (V1 and V2) at two consecutive mode splitting maxima, where the corresponding lateral displacement difference is equal to 310 nm.
According to Eq. (S-15), the value of Ф can be obtained with  = 310 nm / (V2 2 -V1 2 ), and the 6 lateral displacements, or equivalently the cavities' center-to-center offsets, under various applied voltages are calculated and listed in the Table S1 below.
The coupling strength at every FDTD simulated center-to-center offset between two cavities is further calculated based on Eq. (S-5), and the relationship between the FDTD simulated resonant mode wavelength and coupling strength is plotted below. Figure S1: the relationship between the resonance mode wavelength and coupling strength. (a) The non-zero detuning (Δ ≠ 0) case corresponding to Fig. 3a; (b) the zero detuning case (Δ = 0) corresponding to Fig. 3b.  Fig. 5b and 5c). The cavity center-to-center lateral offset is zero when the voltage squared is 12.25 V 2
As shown in Fig. 1a, there are two waveguides feeding into cavity-1 and cavity-2, respectively.
These two waveguides are close and thus not completely decoupled (similarly for the two output waveguides). When the PhC photonic molecule is excited from the upper waveguide that is on cavity-1 side, a portion of light can also couple into the waveguide that is on cavity-2 side. The excited cavity-2 resonance can also couple back into the output waveguide in a similar way. In other words, both cavity-1 and cavity-2 are excited and detected in our experiment even when the coupling strength of the two cavities is 0.   Fig. S3a, the strong coupling between the two PhC nanobeam cavities leads to a mode splitting, and the splitting width decreases as the gap increases. When the gap is large enough (> 500 nm), the mode splitting disappears, corresponding to the weak coupling regime. During the coupling gap tuning process, the even mode always has a longer wavelength while the odd mode always has a shorter wavelength, which indicates that the sign of the coupling strength and mode parity do not inverse.
(see Fig. S3b). Furthermore, in order to tune the coupling strength to zero, the coupling gap has to increase by more than 400 nm, which requires a much larger displacement compared with our lateral tuning method (~150 nm).