Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity

Abstract

Mechanical resonators are promising systems for storing and manipulating information. To transfer information between mechanical modes, either direct coupling or an interface between these modes is needed. In previous works, strong coupling between different modes in a single mechanical resonator and direct interaction between neighboring mechanical resonators have been demonstrated. However, coupling between distant mechanical resonators, which is a crucial request for long-distance classical and quantum information processing using mechanical devices, remains an experimental challenge. Here, we report the experimental observation of strong indirect coupling between separated mechanical resonators in a graphene-based electromechanical system. The coupling is mediated by a far-off-resonant phonon cavity through virtual excitations via a Raman-like process. By controlling the resonant frequency of the phonon cavity, the indirect coupling can be tuned in a wide range. Our results may lead to the development of gate-controlled all-mechanical devices and open up the possibility of long-distance quantum mechanical experiments.

Introduction

The rapid development of nanofabrication technology enables the storage and manipulation of phonon states in micro- and nano-mechanical resonators^1,2,3,4,5. Mechanical resonators with quality factors⁶ exceeding 5 million and frequencies^7,8 in the sub-gigahertz range have been reported. These advances have paved the route to controllable mechanical devices with ultralong memory time⁹. To transfer information between different mechanical modes, tunable interactions between these modes are required¹⁰. While different modes in a single mechanical resonator can be coupled by parametric pump^{3,4,11,12,13,14,15,16} and neighboring mechanical resonators can be coupled via phonon processes through the substrate² or direct contact interaction¹⁷, it is challenging to directly couple distant mechanical resonators.

Here, we observe strong effective coupling between mechanical resonators separated at a distance via a phonon cavity that is significantly detuned from these two resonator modes. The coupling is generated via a Raman-like process through virtual excitations in the phonon cavity and is tunable by varying the frequency of the phonon cavity. Typically, a Raman process can be realized in an atom with three energy levels in the Λ form^18,19. The two lower energy levels are each coupled to the third energy level via an optical field with detunings. When these two detunings are tuned to be equal to each other, an effective coupling is formed between the lower two levels. To our knowledge, tunable indirect coupling in electro-mechanical systems has not been demonstrated before. The physical mechanism of this coupling is analogous to the coupling between distant qubits in circuit quantum electrodynamics^20,21, where the interaction between qubits is induced by virtual photon exchange via a superconducting microwave resonator.

Results

Sample characterization

The sample structure is shown in Fig. 1a, where a graphene ribbon^22,23 with a width of ~1 μm and ~5 layers is suspended over three trenches (2 μm in width, 150 nm in depth) between four metal (Ti/Au) electrodes. This configuration defines three distinct electromechanical resonators: R₁, R₂ and R₃. The metallic contacts S and D₃ are each 2 μm wide and D₁ and D₂ are each 1.5 μm wide, which leads to a 7-μm separation between the centers of R₁ and R₃ (see Supplementary Methods and Supplementary Fig. 1). All measurements are performed in a dilution refrigerator at a base temperature of approximately 10 mK and at pressures below 10⁻⁷ torr. The suspended resonators are biased by a dc gate voltage ($V_{{\rm g}i}^{{\mathrm{DC}}}$ for the ith resonator) and actuated by an ac voltage ($V_{{\rm g}i}^{{\mathrm{AC}}}$ for the ith resonator with driving frequency f_gi = ω_d/2π) through electrodes (g_i for the ith resonator) underneath the respective resonators. To characterize the spectroscopic properties of the resonators, a driving tone is applied to one or more of the bottom gates with frequency ω_d, and another microwave tone with frequency ω_d + δω is applied to the contact S. A mixing current (I_mix = I_x + jI_y) can then be obtained at D₃ (D₁ and D₂ are floated during all measurements) by detecting the δω signal with a lock-in amplifier fixed at zero phase during all measurements (see Supplementary Methods and Supplementary Fig. 2).

Fig. 1

Sample structure and device characterization. a Scanning electron microscope photograph of a typical sample. An ~1-μm-wide graphene ribbon was suspended over four contacts, labeled as S, D₁, D₂, and D₃, respectively. These contacts divide the ribbon into three sections, each with a gate of ~150 nm beneath the ribbon. A driving microwave with frequency ω_d + δω is applied to contact S and is detected at contact D₃ after mixing with another driving tone with frequency ω_d applied to one or more of the control gates. Scale bar is 1 μm. b The differentiation of the mixed current dI_x/dω_d as a function of driving frequency ω_d and gate voltage $V_{{\mathrm{g}}3}^{{\mathrm{DC}}}$ with $V_{{\mathrm{g}}1}^{{\mathrm{DC}}} = V_{{\mathrm{g}}2}^{{\mathrm{DC}}} = 0$ V. Here, the frequencies of all resonators can be tuned from several tens of MHz to ~100 MHz by adjusting the dc gate voltages. c The mixing current as a function of the driving frequency ω_d at voltage $V_{{\mathrm{g}}3}^{{\mathrm{DC}}} = 10\,{\mathrm{V}}$. Using a fitting process (see Supplementary Fig. 8), we extract the linewidth of the mechanical mode. The data were obtained at a driving power of −5 dBm. d, e Spectra of coupled modes R₁ and R₂ (d, where $V_{{\mathrm{g}}1}^{{\mathrm{DC}}} = 10.5$ V and $V_{{\mathrm{g}}3}^{{\mathrm{DC}}} = 0$ V) and R₂ and R₃ (e, where $V_{{\mathrm{g}}1}^{{\mathrm{DC}}} = 0$ V and $V_{{\mathrm{g}}3}^{{\mathrm{DC}}} = 10.5$ V). Strong couplings between these modes are manifested as avoided level crossings in the plots. Coupling strengths Ω₁₂/2π ~ 240 kHz and Ω₂₃/2π ~ 200 kHz are extracted from the plots. f The spectrum of R₂ coupled to both R₁ and R₃. In this case, the gate voltages $V_{{\mathrm{g}}1}^{{\mathrm{DC}}} = 10.45$ V and $V_{{\mathrm{g}}3}^{{\mathrm{DC}}} = 8.35$ V are fixed

Figure 1b shows the measured mixing current as a function of the dc gate voltage and the ac driving frequency on R₃, where the oblique lines represent the resonant frequencies of the resonator modes. We denote the resonant frequency of the ith resonator as f_mi = ω_mi/2π. This plot shows that ${\rm d}f_{{\mathrm{m}}3}{\mathrm{/}}{\rm d}V_{{\mathrm{g}}3}^{{\mathrm{DC}}} \sim 7.7\,{\mathrm{MHz/V}}$ when $|V_{{\mathrm{g}}3}^{{\mathrm{DC}}}| > 5\,{\mathrm{V}}$. The frequencies of the resonators can hence be tuned in a wide range (see Supplementary Note 1 and Supplementary Fig. 3 for results of R₁ and R₂), which allows us to adjust the mechanical modes to be on or off resonance with each other. The quality factors (Q) of the resonant modes are determined by fitting the measured spectral widths (see Supplementary Fig. 8) at low driving powers (typically −50 dBm). Figure 1c shows the spectral dependence of R₃, which gives a linewidth of γ₃/2π ~ 28 kHz at a resonant frequency of f_m3 ~ 98.05 MHz. The resulting quality factor is Q ~ 3500. The quality factors of the other two resonators are similar, at ~3000.

Strong coupling between neighboring resonators

Neighboring resonators in this system couple strongly with each other, similar to previous studies on gallium arsenide² and carbon nanotube¹⁷. Figure 1d, e shows the spectra of the coupled modes (R₁, R₂) and (R₂, R₃), respectively, by plotting the mixed current I_x as a function of gate voltages and driving frequencies. In Fig. 1e, the voltage $V_{{\mathrm{g}}3}^{{\mathrm{DC}}}$ is fixed at 10.5 V, with a corresponding resonant frequency f_m3 = 101.15 MHz, and $V_{{\mathrm{g}}2}^{{\mathrm{DC}}}$ is scanned over a range with f_m2 being near-resonant to f_m3. A distinct avoided level crossing appears when f_m2 approaches f_m3, which is a central feature of two resonators with direct coupling. From the measured data, we extract the coupling rate between these two modes as Ω₂₃/2π ~ 200 kHz, which is the energy splitting when f_m2 = f_m3. In Supplementary Note 2 and Supplementary Fig. 4, we fit the measured spectrum with a single two-mode model using this coupling rate. Similarly in Fig. 1d, by fixing $V_{{\mathrm{g}}1}^{{\mathrm{DC}}}$ at 10.5 V and scanning the voltage $V_{{\mathrm{g}}2}^{{\mathrm{DC}}}$, we obtain the coupling rate between R₁ and R₂ as Ω₁₂/2π ~ 240 kHz. There are several possible origins for the coupling between two adjacent resonators in this system. One coupling medium is the substrate and the other medium is the graphene ribbon itself. Mechanical energy can be transferred in a solid-state material by phonon propagation, as demonstrated in several experiments^2,17,24. Second, because adjacent resonators share lattice bonds, the phonon energy can transfer in the graphene ribbon. The dependence of the coupling strength on the width of the drain contacts is still unknown (see Supplementary Fig. 5 for another sample).

The measured coupling strength satisfies the strong coupling condition with ${\rm{\Omega }}_{23} \gg \gamma _2,\gamma _3$. Defining the cooperativity for this phonon–phonon coupling system as $C = {\rm{\Omega}}_{23}^2/\gamma _2\gamma _3$, we find that C = 44. A similar strong coupling condition can be found between modes R₁ and R₂. By adjusting the gate voltages of these three resonators, R₂ can be successively coupled to both R₁ and R₃ (see Fig. 1f).

For comparison, we study the coupling strength between modes R₁ and R₃. The frequency f_m2 of resonator R₂ is set to be detuned from f_m1 and f_m3 by 700 kHz in Fig. 2a. In the dashed circle, we observe a near-perfect level crossing when f_m1 approaches f_m3, which indicates a negligible coupling between these two modes, with ${\rm{\Omega }}_{13} \ll \gamma _1,\gamma _3$ (also see Supplementary Fig. 6).

Fig. 2

Hybridization between all three modes. a Measured spectrum of the three-mode system when the frequency of R₂ is far off-resonance from that of mode R₁ by a detuning Δ₁₂/2π ~ 70 kHz (here, $V_{{\mathrm{g}}1}^{{\mathrm{DC}}} = 10.5$ V and $V_{{\mathrm{g}}2}^{{\mathrm{DC}}} = 7.64$ V). The dc voltage $V_{{\mathrm{g}}3}^{{\mathrm{DC}}}$ is scanned over a wide range, crossing both f_m1 and f_m2. An avoided level crossing is observed when f_m3 approaches f_m2. A level crossing is observed when f_m3 approaches f_m1. b Measured spectrum of the three-mode system when the detuning is Δ₁₂/2π ~ 180 kHz (here, $V_{{\mathrm{g}}1}^{{\mathrm{DC}}} = 10.5$ V and $V_{{\mathrm{g}}2}^{{\mathrm{DC}}} = 7.56$ V, and here the ranges of the axes are set to be the same as the black dashed box shown in a). Here, a strongly avoided level crossing appears when f_m3 approaches f_m1. The strengths of the direct couplings extracted from the measured spectrum are Ω₁₂/2π = 240 kHz and Ω₂₃/2π = 170 kHz. c, d Spectra calculated using the theoretical model for the three modes (Eq. (2)) and coupling constants Ω₁₂ and Ω₂₃. Δ₁₂/2π = 700 kHz in c and Δ₁₂/2π = 180 kHz in d

Raman-like coupling between well-separated resonators

The three resonator modes in our system are in the classical regime. The Hamiltonian of these three classical resonators can be written as:

$${\cal H}_{c} = \mathop {\sum }\limits_i^3 \frac{1}{2}(p_i^2 + \omega _{{\rm m}i}^2x_i^2) + {\mathrm{\Lambda }}_{12}x_1x_2 + {\mathrm{\Lambda }}_{23}x_2x_3,$$

(1)

where ${\mathrm{\Lambda }}_{ij} = {\rm{\Omega }}_{ij}\sqrt {\omega _{{\rm m}i}\omega _{{\rm m}j}}$ is a coupling parameter between i- and jth resonators, p_pi is the effective momentum and x_i is the effective coordinate of the oscillation for the ith resonator, respectively. Let $x_i = \sqrt {\frac{1}{{2\omega _{{\rm m}i}}}} (\alpha _i^ \ast + \alpha _i)$ and $p_i = {\mathrm{i}}\sqrt {\frac{{\omega _{{\rm m}i}}}{2}} (\alpha _i^ \ast - \alpha _i)$, with α_i and $\alpha _i^ \ast$ being complex numbers. The Hamiltonian in Eq. (1) can be written as

$${\cal H}_t = {\sum} {\omega _{{\rm m}i}\alpha _i^ \ast \alpha _i + \frac{{{\rm{\Omega }}_{12}}}{2}\left( {\alpha _1^ \ast \alpha _2 + \alpha _1\alpha _2^ \ast } \right) + \frac{{{\rm{\Omega }}_{23}}}{2}\left( {\alpha _2^ \ast \alpha _3 + \alpha _2\alpha _3^ \ast } \right)} .$$

(2)

Here, we have applied the rotating-wave approximation and neglected the $\alpha _i\alpha _j$ and $\alpha _i^ \ast \alpha _j^ \ast$ terms. This approximation is valid when $\omega _{{\rm m}i} \gg {\rm{\Omega }}_{12},{\rm{\Omega }}_{23}$. This Hamiltonian describes the direct couplings between neighboring resonators (R₁, R₂) and (R₂, R₃). Through these couplings, the mechanical modes hybridize into three normal modes, and an effective coupling between modes R₁ and R₃ can be achieved. If the resonators work in the quantum regime, $\alpha _i$ and $\alpha _i^ \ast$ can be quantized into the annihilation and creation operators of a quantum harmonic oscillator, respectively.

We study the hybridization of this three-mode system by fixing the gate voltages (mode frequencies) of modes R₁ and R₂, and sweeping the gate voltage of R₃ over a wide range. The spectrum of this system depends strongly on the detuning between modes R₁ and R₂, which is defined as Δ₁₂ = 2π(f_m2−f_m1). In Fig. 2a, Δ₁₂/2π ~ 70 kHz. Similar to Fig. 1f, modes R₃ and R₁ show a level crossing. Moreover, we observe a large avoided level crossing between modes R₂ and R₃ when the frequency f_m3 approaches f_m2, indicating strong coupling between these two modes. Hence, even with strong couplings between all neighboring resonators, the effective coupling between the distant modes R₁ and R₃ is still negligible when the frequency of mode R₂ is significantly far off resonance from the other two modes. On the contrary, when the detuning Δ₁₂/2π is lowered to ~180 kHz, a distinct avoided level crossing between modes R₁ and R₃ is observed, as shown inside the dashed circle in Fig. 2b.

With coupling strengths Ω₁₂/2π = 240 kHz and Ω₂₃/2π = 170 kHz extracted from the measured data, we plot the theoretical spectra of the normal modes in this three-mode system given by Eq. (2), for Δ₁₂/2π = 700 and 180 kHz in Fig. 2c, d, respectively. Our result shows good agreement between theoretical and experimental results.

With direct couplings between neighboring resonators, an effective coupling between the two distant resonators R₁ and R₃ can be obtained via their couplings to mode R₂. The effective coupling can be viewed as a Raman process, as illustrated in Fig. 3a. Here mode R₂ functions as a phonon cavity that connects the mechanical resonators R₁ and R₃ via virtual phonon excitations. The physical mechanism of this effective coupling is similar to that of the coupling between distant superconducting qubits via a superconducting microwave cavity²⁰. The detuning between the phonon cavity and the other two modes Δ₁₂ can be used as a control parameter to adjust this effective coupling.

Fig. 3

Indirect coupling between separated resonators via a phonon cavity. a Raman-like coupling between modes R₁ and R₃ via virtual excitation of the phonon cavity R₂. The coupling strength can be controlled by changing the detuning Δ₁₂. b Effective coupling as a function of Δ₁₂. The error bars are obtained from the s. e. m. of the measured data and are extracted from the statistical deviation of the estimated values at different detunings from Supplementary Fig. 7. The red line is given by Ω₁₃ = Ω₁₂Ω₂₃/2Δ₁₂, with Ω₁₂/2π = 240 kHz and Ω₂₃/2π = 170 kHz

To derive the effective coupling, we consider the case of Δ₁₂ = Δ₃₂ = Δ, where Δ₃₂/2π = f_m2−f_m3 and $\left| {\rm{\Delta }} \right| \gg {\rm{\Omega }}_{12},\,{\rm{\Omega }}_{23}$. The avoided level crossing between modes R₁ and R₃ can be extracted at this point. Using a perturbation theory approach, we obtain the effective Hamiltonian between modes R₁ and R₃ as (see Methods for details)

$${\cal H}_{{\mathrm{eff}}} = (\Delta + \frac{{{\rm{\Omega }}_{12}^2}}{{4\Delta }})\alpha _1^ \ast \alpha _1 + (\Delta + \frac{{{\rm{\Omega }}_{23}^2}}{{4\Delta }})\alpha _3^ \ast \alpha _3 + \frac{{{\rm{\Omega }}_{13}}}{2}(\alpha _1^ \ast \alpha _3 + \alpha _3^ \ast \alpha _1).$$

(3)

Here, an effective coupling is generated between R₁ and R₃ with magnitude Ω₁₃ = Ω₁₂Ω₂₃/2Δ, and the resonant frequencies of each mode are shifted by a small term. The effective coupling Ω₁₃ in the Hamiltonian depends strongly on the detuning Δ. Thus, the effective coupling between R₁ and R₃ can be controlled over a wide range by varying the frequency (gate voltage) of resonator R₂.

The effective coupling strength Ω₁₃ between R₁ and R₃ as a function of Δ₁₂ is shown in Fig. 3b. Each data point is obtained by changing the gate voltage of R₂ and repeating the measurements in Fig. 2a, b (see Supplementary Fig. 7). Over a large range of detuning, the effective coupling is larger than the linewidths of the resonators γ_1,2,3/2π, with Ω₁₃ > 30 kHz. The red line shows the results using perturbation theory. The experimental data indicate good agreement with the theoretical results.

Discussion

In summary, we have demonstrated indirect coupling between separated mechanical resonators in a three-mode electromechanical system constructed from a graphene ribbon. Our study suggests that coupling between well-separated mechanical modes can be created and manipulated via a phonon cavity. These observations hold promise for a wide range of applications in phonon state storage, transmission, and transformation. In the current experiment, the sample works in an environment subjected to noise and microwave heating with typical temperatures as high as 100 mK and phonon numbers reaching ~24. By cooling the mechanical resonators to lower temperatures^25,26,27,28, quantum states could be manipulated via this indirect coupling^29,30. Furthermore, in the quantum limit, by coupling the mechanical modes to solid-state qubits, such as quantum-dots and superconducting qubits^17,31,32, this system can be utilized as a quantum data bus to transfer information between qubits^33,34. Future work may lead to the development of graphene-based mechanical resonator arrays as phononic waveguides²⁴ and quantum memories³⁵ with high tunabilities.

Methods

Theory of three-mode coupling

We describe this three-mode system with the Hamiltonian (ħ = 1)

$${\cal H}_{\mathrm{t}} = {\sum} {\omega _{mi}\alpha _i^ \ast \alpha _i + \frac{{{\rm{\Omega }}_{12}}}{2}(\alpha _1^ \ast \alpha _2 + \alpha _1\alpha _2^ \ast ) + \frac{{{\rm{\Omega }}_{23}}}{2}(\alpha _2^ \ast \alpha _3 + \alpha _2\alpha _3^ \ast )} ,$$

(4)

where ${\rm{\Omega }}_{ij}$ is the coupling between mechanical resonators i and j. The couplings between the resonators induce hybridization of the three modes. The hybridized normal modes under this Hamiltonian can be obtained by solving the eigenvalues of the matrix

$$M = \left( {\begin{array}{*{20}{c}} {{\mathrm{\Delta }}_{12}} & {\frac{{{\rm{\Omega }}_{12}}}{2}} & 0 \\ {\frac{{{\rm{\Omega }}_{12}}}{2}} & 0 & {\frac{{{\rm{\Omega }}_{23}}}{2}} \\ 0 & {\frac{{{\rm{\Omega }}_{23}}}{2}} & {{\mathrm{\Delta }}_{23}} \end{array}} \right),$$

(5)

where Δ_ij/2π = f_mi−f_mj is the frequency difference between R_i and R_j. The eigenvalues of this matrix correspond to the frequencies of the normal modes, i.e., the peaks in the spectroscopic measurement.

We consider the special case of Δ₁₂ = Δ₂₃ = Δ, with $\left| {\mathrm{\Delta }} \right| \gg {\rm{\Omega }}_{12},\,{\rm{\Omega }}_{23}$, in the three-mode system. Here, the eigenvalues of the normal modes can be derived analytically. One eigenvalue is ω_Δ = Δ, which corresponds to the eigenmode

$$\alpha _{\mathrm{\Delta }} = - \frac{{{\rm{\Omega }}_{23}}}{{\sqrt {{\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2} }}\alpha _1 + \frac{{{\rm{\Omega }}_{12}}}{{\sqrt {{\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2} }}\alpha _3.$$

(6)

This mode is a superposition of the end modes α₁ and α₃, and does not include the middle mode. The two other eigenvalues are

$${\mathrm{\omega }}_{{\mathrm{\Delta }} \pm } = \frac{1}{2}({\mathrm{\Delta }} \pm \omega _{{\mathrm{\Delta }}0})$$

(7)

with $\omega _{\Delta 0} = \sqrt {\Delta ^2 + {\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2}$. The corresponding normal modes are

$$a_{{\mathrm{\Delta }} \pm } = \frac{{({\rm{\Omega }}_{12}a_1 \pm \left( {\omega _{\Delta 0} \mp \Delta } \right)a_2 + {\rm{\Omega }}_{23}a_3)}}{{\sqrt {2\omega _{{\mathrm{\Delta }}0}(\omega _{{\mathrm{\Delta }}0} \mp \Delta )} }}.$$

(8)

With $\left| {\mathrm{\Delta }} \right| \gg {\rm{\Omega }}_{12},\,{\rm{\Omega }}_{23}$, for Δ > 0, ${\mathrm{\omega }}_{{\mathrm{\Delta }} + } \approx \Delta + ({\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2)/4\Delta$. The mode α_Δ+ is nearly degenerate with α_Δ, and

$$\alpha _{{\mathrm{\Delta }} + } \approx \frac{{{\rm{\Omega }}_{12}\alpha _1 + {\rm{\Omega }}_{23}\alpha _3}}{{\sqrt {{\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2} }}.$$

(9)

The mode α_Δ− has frequency ${\mathrm{\omega }}_{{\mathrm{\Delta }} - } \approx - ({\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2)/4\Delta$, with α_Δ− ≈ α₂. The normal modes now become separated into two nearly degenerate modes {α_Δ,α_Δ+}, which are superpositions of modes α₁ and α₃, and a third mode α_Δ− that is significantly off resonance from the other two modes. The nearly degenerate modes can be viewed as a hybridization of α₁ and α₃ with an effective splitting $({\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2){\mathrm{/}}2{\mathrm{\Delta }}$. A similar result can be derived for Δ < 0, where ${\mathrm{\omega }}_{{\mathrm{\Delta }} - } \approx {\mathrm{\Delta }} + ({\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2){\mathrm{/}}4{\mathrm{\Delta }}$, with α_Δ− given by the expression in Eq. (8), and ${\mathrm{\omega }}_{{\mathrm{\Delta }} + } \approx - ({\rm{\Omega }}_{12}^2 + {\rm{\Omega }}_{23}^2){\mathrm{/}}4{\mathrm{\Delta }}$ with α_Δ+ ≈ α₂.

The effective coupling rate can be derived with a perturbative approach on the matrix M. When |Δ| $\gg$ Ω₁₂, Ω₂₃, the dynamics of α₁ and α₃ is governed by matrix

$${M_{\mathrm{eff}}} = \left( {\begin{array}{*{20}{c}} {\Delta + \frac{{{\rm{\Omega }}_{12}^2}}{{4\Delta }}} & {\frac{{{\rm{\Omega }}_{12}{\rm{\Omega }}_{23}}}{{4{\mathrm{\Delta }}}}} \\ {\frac{{{\rm{\Omega }}_{12}{\rm{\Omega }}_{23}}}{{4{\mathrm{\Delta }}}}} & {\Delta + \frac{{{\rm{\Omega }}_{23}^2}}{{4\Delta }}} \end{array}} \right).$$

(10)

This matrix tells us that because of their interaction with the middle mode α₂, the frequency of mode α₁ (α₃) is shifted by $\frac{{{\rm{\Omega }}_{12}^2}}{{4\Delta }}$ ($\frac{{{\rm{\Omega }}_{23}^2}}{{4{\mathrm{\Delta }}}}$), which is much smaller than |Δ|. Meanwhile, an effective coupling is generated between these two modes with magnitude ${\rm{\Omega }}_{13} = \frac{{{\rm{\Omega }}_{12}{\rm{\Omega }}_{23}}}{{2{\mathrm{\Delta }}}}$. The effective Hamiltonian for α₁ and α₃ can be written as

$${\cal H}_{{\mathrm{eff}}} = (\Delta + \frac{{{\rm{\Omega }}_{12}^2}}{{4\Delta }})\alpha _1^ \ast \alpha _1 + (\Delta + \frac{{{\rm{\Omega }}_{23}^2}}{{4\Delta }})\alpha _3^ \ast \alpha _3 + \frac{{{\rm{\Omega }}_{13}}}{2}(\alpha _1^ \ast \alpha _3 + \alpha _3^ \ast \alpha _1).$$

(11)

The effective coupling can be controlled over a wide range by varying the frequency of the second mode α₂.

Data availability

The remaining data contained within the paper and Supplementary files are available from the author upon request.