Introduction

Cavity optomechanics1,2 describes the interaction between the electromagnetic radiation and nanomechanical or micromechanical motion, which has been developing rapidly. During the past decades, it leads to various important applications, such as gravitational-wave detection3, cooling of mechanical oscillators to the ground-state of motion4,5,6,7,8,9,10,11, optomechanically induced transparency (OMIT) and slow light12,13,14,15,16,17,18, precision measurements19,20, squeezing of light21, quantum information processing22,23,24, and so on.

Most of the recent developments in cavity optomechanics are based on the perturbative interaction between the driving light fields and the optomechanical system. For example, in the context of OMIT, the optomechanical system is coherently driven by both a control field and a probe field. If the strength of the probe field is far less than that of the control field, the perturbation method can be used and the OMIT can be properly described by the linearization of the Heisenberg-Langevin equations. Aside from OMIT, the linearization of optomechanical interaction has also been adopted in many other studies, such as optomechanical dark state25 and normal mode splitting26.

One key aspect is that if the strength of the probe field becomes comparable with that of the control field, the perturbative description breaks down, and some novel nonlinear and non-perturbative effects come to appear27,28. Therefore, extending the studies of cavity optomechanics from the linear and perturbation regime to the nonlinear and non-perturbative regime is of great interest. On the other hand, as a natural extension of the generic optomechanical system, the composite optomechanical system which consists of two directly coupled whispering-gallery-mode microcavities (called a photonic molecule29,30) with optomechanical oscillation in one microcavity has attracted much attention. In the composite system, the interplay between the optomechanical interaction and the tunable photon tunnelling forms the basis of some interesting phenomena, such as phonon lasing31,32, enhanced quantum nonlinearities33, coherent optical wavelength conversion34, optomechanical quantum gates22,23, coherent control of light transmission35,36,37,38,39,40,41, chaos42, and ground-state cooling of mechanical modes43,44.

In this work, we investigate the tunable high-order sideband spectra generation using a photonic molecule optomechanical system coupled to a waveguide in the non-perturbation regime. The composite system is coherently driven by a two-tone laser consisting of a continuous-wave (CW) control field and a pulsed driving field which propagates through the waveguide. We analyze the frequency spectral feature of the output field by performing fast Fourier transformation (FFT), and confirm that under the condition of intense and nanosecond pulse driving, the output spectrum exhibits the properties of high-order sideband frequency spectra. We find that the output spectrum can be efficiently tuned by using the power of the driving pulse and the coupling rate between the cavities. In addition, we show the carrier-envelop phase-dependent (CEP-dependent) effect of high-order sideband generation in the output spectra, and the results indicate that the CEP of laser pulse which contains huge numbers of cycles can also cause profound effects. Our study may provide a further insight of the properties of cavity optomechanics in the nonlinear and non-perturbative regime, and may have potential applications in optical frequency comb and communication based on the optomechanical platform45. Also the proposed whispering-gallery-mode photonic molecule-waveguide structure is compatible with large-scale integration for implementing complex photonic devices on a chip.

Results

Model

As shown in Fig. 1, we consider a system of two directly coupled whispering-gallery-mode microcavities. The first cavity with an effective mass m supports an optical mode and a mechanical mode with angular frequency Ωm. The second cavity only supports an optical mode which is coupled to the first cavity through an evanescent field. The cavity-cavity coupling rate J can be efficiently tuned by changing the distance between them. A tapered fiber is used to excite the cavity modes as the optical waveguide. The first cavity is side coupled to the fiber with the coupling rate κe. Sin and Sout represent the input and the output fields propagating in the waveguide, respectively. The Hamiltonian of this composite system can be divided into three parts, i.e.,

Figure 1: Schematic diagram of a double-cavity photonic molecule optomechanical system coupled to a waveguide.
figure 1

Sin (Sout) represents the input (output) field propagating in the waveguide. denotes the bosonic annihilation operator of the first (second) cavity field. The first cavity supports a mechanical mode with angular frequency Ωm. J is the cavity-cavity coupling rate, which can be efficiently adjusted by the distance between the cavities. κe denotes the coupling rate between the first cavity and the waveguide. κi1 and κi2 are the intrinsic cavity decay rates.

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where , and are the free Hamiltonian, interaction Hamiltonian and the driven Hamiltonian, respectively, which can be written as

where and represent the bosonic annihilation and creation operators of the first (second) cavity mode. ωc is the resonance frequency of the two cavities. and denote the mechanical position and momentum operators. G is the optomechanical coupling constant. Here we focus on the mean response of the composite system. Assuming , , , , , and by using the mean-field approximation , the Heisenberg-Langevin equations of the operators can be reduced to the mean value equations as:

where κi1 and κi2 are the cavity intrinsic decay rates. Γm represents the mechanical decay rate, which is introduced classically. The quantum and thermal noise terms are dropped as their mean values are 0. Eqs (3)–(6), are coupled ordinary differential equations of complex functions which describe the time evolution of the composite system.

In the following, we consider the case that the input field contains a CW control field and a pulsed driving field, i.e., . s1 and spε(t) are the amplitudes of the two fields, which are related to the optical powers propagating in the waveguide by and , respectively. ω1 and ωp are the frequency of the control field and the mean frequency of the driving pulse, respectively. ε(t) is the normalized pulse envelope which is assumed to be Gaussian here, i.e., , with t0 being the center time of the pulse and tp being the full width of half maximum (FWHM) of the intensity envelop. ϕ is the so-called CEP of the driving pulse. In this case, the maximum power of the driving pulse is . In a rotating frame at the frequency of ω1, the evolution equations of the composite system can be rewritten as:

where is the detuning between the frequency of the control field ω1 and the cavity resonance frequency ωc. Ω = ωp − ω1 is the detuning between the mean frequency of the driving pulse ωp and the frequency of the control field ω1. In such a rotating frame, can be seen as the effective input field, and the mean frequency of the effective driving pulse becomes Ω. Here, we set Ω equals to the low-frequency mechanical mode Ωm, which has been widely employed in OMIT.

High-order sideband frequency spectra generation

Figure 2 shows the frequency spectra of the output field with different powers of the driving pulse as: (a) , (b) , (c)  mW, and (d)  mW. Here the CW control field has wavelength 532 nm and power , with . The driving pulse has center time , FWHM tp = 12 ns, and CEP ϕ = 0. The cavity-cavity coupling rate is J = 2π × 0.5 MHz. As the Eqs (7)–(10), describe the evolution of the optical fields in a frame rotating at the frequency ω1, the output spectra exhibit a frequency shift of ω1. Moreover, there are positive frequencies and negative frequencies during FFT, and we only show the positive frequency components here. From Fig. 2, one can see that the output spectra contain two input field frequency components (the CW control field ω1 and the pulsed driving field ωp = ω1 + Ω), and a series of new components (higher-order sidebands). That is, when the driving pulse is incident upon the composite system which has been driven by the CW control field, the spectral components with frequencies ω = ω1 ± nΩ can be generated in the output field, where n = 0,1,2,… represents the order of the sidebands. When the driving pulse is relatively weak, e.g., in Fig. 2(a), there are only a few sidebands appear in the spectrum, and the intensity of individual sideband is decreased rapidly as the order of the sidebands is increased. As the power of the driving pulse is increased, e.g., in Fig. 2(b) and  mW in Fig. 2(c), the effect of high-order sideband generation could be observed. When the pulse power is sufficiently high, e.g.,  mW in Fig. 2(d), a robust high-order sideband generation can be achieved. In this case, the spectrum decreases rapidly for the first few order sidebands, followed by a plateau where the sideband intensity is slowly varying, and ends up with a sharp cutoff. Such typical spectral feature indicates the non-perturbative nature of high-order sideband generation27, which is very similar to the high-order harmonic generation in strongly driven atoms or molecules46,47.

Figure 2
figure 2

Frequency spectra of the output field with different values of as: (a) , (b) , (c)  mW, and (d)  mW. Here the CW control field has wavelength 532 nm and power P1 = 37.3 μW. The driving pulse has center time t0 = 0.2 μs, FWHM tp = 12 ns, and CEP ϕ = 0. The cavity-cavity coupling rate is J = 2π × 0.5 MHz. The other system parameters used for calculations are κe = 2π × 45.5 MHz, κi1 = κi2 = 2π × 0.5 MHz, m = 2.0 pg, Ωm = 1.4 GHz, Γm = 2π × 35 kHz, G = 485.0 GHz/nm, and Δ = −Ωm, respectively.

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In experiment, the cavity-cavity coupling rate J can be efficiently adjusted by changing the distance between the cavities. In order to show the influence of the parameter J on the high-order sideband spectra generation, Fig. 3 shows the frequency spectra of the output field with different values of J as: (a) J = 2π × 8 MHz, (b) J = 2π × 37 MHz, (c) J = 2π × 81 MHz, and (d) J = 2π × 278 MHz. Note that the power of the CW control field is P1 = 37.3 μW, the maximum power of the driving pulse is  mW, and the other system parameters used for calculations are the same as in Fig. 2. When J is relatively small, e.g., J = 2π × 8 MHz in Fig. 3(a), a lot of high-order sidebands can be obtained and the corresponding sideband intensities are large. With the increment of J, e.g., J = 2π × 37 MHz in Fig. 3(b) and J = 2π × 81 MHz in Fig. 3(c), the intricate competition between the two kinds of couplings, i.e., the cavity-cavity coupling and the optomechanical coupling, becomes obvious. Due to the linear cavity-cavity coupling indirectly influences the optomechanical coupling strength, the nonlinear effect of high-order sideband generation is decreased48,49. Last, for a large J, e.g., J = 2π × 278 MHz in Fig. 3(d), the photon tunnelling between the two cavities dominates the evolution process. As a result, only a few sidebands appear in the spectrum and the intensity of individual sideband is decreased rapidly as the sideband order is increased. This phenomenon provides us a potentially useful method to tune the output spectrum by using the photon tunnelling of the coupled cavities.

Figure 3
figure 3

Frequency spectra of the output field with different values of J as: (a) J = 2π × 8 MHz, (b) J = 2π × 37 MHz, (c) J = 2π × 81 MHz, and (d) J = 2π × 278 MHz. Here the power of the CW control field is P1 = 37.3 μW, the maximum power of the driving pulse is  mW, and the other parameters are the same as in Fig. 2.

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On the other hand, the CEP is another key parameter in describing the characteristics of the driving pulse. According to Eqs (7)–(10), when the CEP of the driving pulse ϕ goes to ϕ + 2π, the time evolution of the composite system remains unchanged. Thus there is a periodicity upon the CEP, where ϕ goes to ϕ + 2π leads to the same output spectra. However, within a range of 2π, the spectra may also be different for different values of ϕ. Figure 4 shows such a CEP-dependent effect of high-order sideband generation with different values of ϕ as: (a) ϕ = π/2, (b) ϕ = π, (c) ϕ = 3π/2, and (d) ϕ = 2π. Here we use P1 = 37.3 μW,  mW, and the other parameters are the same as in Fig. 2. Usually, the CEP only strongly affects the processes involving few-cycle light pulses50,51. For tp = 12 ns, the number of cycles in the driving pulse can be estimated to be , which means the driving pulse contains huge numbers of cycles and seems almost impossible to have the CEP-dependent effects. However, the dynamics of the composite system in our scheme is in a rotating frame, the mean frequency of the effective driving pulse is Ω, which equals to the low-frequency mechanical mode Ωm. One can estimate the number of cycles in the effective driving pulse to be by choosing tp = 12 ns and Ω = 1.4 GHz, which means the effective driving pulse only contains a few cycles. We can conclude that the CEP of the driving pulse becomes important and can produce obvious influences on the output spectra, even though the driving pulse contains huge numbers of cycles in reality. For longer driving pulses, the influence of the CEP on high-order sideband generation would become smaller49.

Figure 4
figure 4

Frequency spectra of the output field with different values of ϕ as: (a) ϕ = π/2, (b) ϕ = π, (c) ϕ = 3π/2, and (d) ϕ = 2π. Here we use P1 = 37.3 μW,  mW, and the other parameters are the same as in Fig. 2.

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Before ending, it is worth emphasizing that the linewidth of the high-order sidebands can be well described by the time-frequency uncertainty relation ΔωΔt ~ 2π27,28,48,49. In our scheme, the driving pulse lasts about 2tp, so Δt ≈ 2tp. Making use of the relationship Δω ~ 2π/Δt, we can roughly estimate the linewidth of the generated high-order sidebands to be Δω ~ Ωm/5. Therefore, the high-order sidebands are relatively narrow and clearly resolved in the spectra.

Summary

In summary, we have theoretically and numerically analyzed the nonlinear optical transmission characteristics in a waveguide-coupled photonic molecule optomechanical system. The composite system is coherently driven by a CW control field and an intense nanosecond driving pulse, and the numerical method is employed instead of the perturbation method in such a non-perturbative regime. By performing FFT, the frequency spectrum of the output field of the system can be obtained. We confirmed that the output spectrum exhibits the properties of high-order sideband frequency spectra, and the typical spectral feature reveals the non-perturbative nature of high-order sideband generation. The results clearly show that the output spectrum is tunable by changing the power of the driving pulse and also by using the photon tunnelling of the coupled cavities. In addition, we investigated the CEP-dependent effect of high-order sideband generation in the output spectra. The results may have potential applications in optical frequency comb and communication based on the optomechanical platform, and may open up a promising perspective for implementing complex photonic devices on a chip.

Method

In order to show the non-perturbative signals explicitly, we directly solved the evolution equations to study the dynamics of the system. The Runge-Kutta method is employed and the initial conditions are set as: a1(0) = 0, a2(0) = 0, x(0) = 0, and p(0) = 0, which can be achieved by cooling the mechanical mode to the ground-state of motion. The relevant system parameters used for calculations are κe = 2π × 45.5 MHz, κi1 = κi2 = 2π × 0.5 MHz, m = 2.0 pg, Ωm = 1.4 GHz, G = 485.0 GHz/nm, Γm = 2π × 35 kHz, and Δ = −Ωm, respectively. All these parameters are within the experimentally available parameter range52, and are used through out the paper.

Firstly, we consider sp = 0 and only a CW control field with wavelength 532 nm and power P1 = 37.3 μW is incident upon the composite system. In this case, the system would evolve to a steady state provided by the effective field s1. Figure 5 shows the time evolution of the composite system by solving Eqs (7)–(10), with the numerical method. One can see that after a transient process, the system reaches the steady state at about t = 0.1 μs.

Figure 5: Intensities of the optical fields and the mechanical oscillation in time domain by directly solving Eqs (7), (8), (9), (10) with sp = 0 and J = 2π 
figure 5

× 0.5 MHz. The time evolution of the (a) photon number in the first cavity , (b) photon number in the second cavity , and (c) mechanical position x in 0.2 μs are shown.

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Next, the input field contains a CW control field and a nanosecond driving pulse is taken into consideration. The center time of the driving pulse is adjusted to t0 = 0.2 μs to make sure that when the pulse is incident upon the composite system, the system has reached the steady state. The output field transmitting through the waveguide can be obtained by using the standard input-output formalism as . The output spectrum can be numerically obtained by performing FFT on , where is the spectrometer frequency. Here we do the FFT from t = 0.15 μs which is after the composite system reaches the steady state and before the driving pulse is incident upon the composite system, so that the complicated transient process has no influence on the spectral characters.

Additional Information

How to cite this article: Cao, C. et al. Tunable high-order sideband spectra generation using a photonic molecule optomechanical system. Sci. Rep. 6, 22920; doi: 10.1038/srep22920 (2016).