1. Introduction

Cavity optomechanics studies the coupling between optical cavity modes and mechanical degrees of freedom through the optical radiation pressure [1,2]. The optomechanical coupling imprints the mechanical motion into the light field fluctuations, enabling the detection of mechanical movement with exquisite sensitivity. At the same time, the light field within the optical cavity can also be utilized to manipulate the mechanical resonator. Over the past years, with the advent of optical microcavities and nano-fabrication techniques, cavity optomechanical systems have undergone rapid development, enabling advanced applications in various fields, such as quantum information processing [3,4], quantum ground state sideband and feedback cooling [5–8], ultra-sensitive sensing [9–12], non-reciprocity interactions [13,14], squeezing light generation [15,16], entanglement [17] and coherent state transfer [18]. In most of the above examples, the optomechanical coupling can be regarded as linear, where the mechanical motion is weak and the perturbation method can be used to approximate the system parameters.

When the input laser is blue-detuned and larger than a threshold power, the mechanical resonator can be driven into a self-oscillation state, a phenomenon known as the optomechanical oscillation (OMO) and analogous to the laser [19–21]. In this condition, the mechanical motion-induced cavity frequency shift is comparable to or larger than the optical linewidth, the linear approximation thus breaks down and nonlinear optomechanics is relevant [22]. This OMO process exhibits rich nonlinear dynamics characteristics, including limit cycles and multistability [23–25], periodic doubling and chaotic dynamic [26–28], and the self-modulated oscillations corresponding to stable limit tori [29,30], synchronization [31,32].

Apart from the coupling between optical and mechanical modes, thermal effects also play an important role in cavity optomechanical systems. The thermal effects originate from thermal-optic and thermal-expansion mechanisms due to the absorption of photons in the optical cavity. Previously, thermal effects in optical cavities have been thoroughly studied [33,34], which can be used to stabilize the laser frequency [35,36]. In the cavity optomechanical system, the input optical power is relatively high when OMO happens, thus resulting in significant thermal effects and creating a cavity opto-mechanical-thermal (OMT) system.

Many previous works have been devoted to the studies of the OMT system and complex system dynamics were found, including optomechanical thermal oscillation [37], photothermal force assisted heating [38], self-induced optical modulation [39] and optomechanical frequency comb generation [37,40]. It has also been found in these studies that the theoretical model incorporating the thermal effects can precisely predict the nonlinear dynamics. Recently, a new dynamical phenomenon called optomechanically induced bistability was discovered when sweeping the laser across the optical cavity resonance [41]. However, this paper did not show a full dynamic behaviors of the cavity OMT system when sweeping the laser frequency due to the relatively fast sweeping speed, which is very important when we want to design OMT devices accordingly. Therefore, systematic research of the nonlinear dynamics of an OMT system is needed to find out how the system parameters change for different detuning conditions.

This paper investigates the nonlinear dynamics of an OMT system and is organised as follows: Section 2 introduces the theoretical model based on dimensionless parameters to analytically and numerically investigate the dynamical process. Section 3 shows the dynamic processes when sweeping the laser in the optical cavity, namely the staircase and bistability effects. Section 4 systematically analyses the reason behind the two dynamic phenomena. Finally, Section 5 discusses the characteristics and potential applications of the discovered dynamics and gives the conclusion.

2. Theoretical background of optomechanical-thermal system

We consider a theoretical model of an OMT system as shown in Fig. 1, in which a microtoroid optical cavity is pumped by an optical tapered fiber. In the system, the mechanical deformation of the cavity changes the optical path, and at the same time, the optical radiation force generated by the enhanced optical field leads to mechanical deformation, thus realizing the cavity optomechanical coupling. In this model, we also need to consider the thermal effect. We assume the averaged temperature difference between the cavity and the surrounding environment is $\Delta T$. The dynamics of this OMT cavity system including the optical field amplitude $a(t)$, the mechanical displacement $x(t)$, and the temperature difference $\Delta T$ is thus governed by the following equations [37,41],

 

Fig. 1. Schematic diagram of an OMT system in the WGM form. The mechanical breathing mode with amplitude $x$ is coupled to an input optical field with amplitude $x$. The circulating optical field in the cavity has an amplitude $a$, and the averaged temperature difference between the cavity and the surrounding environment is $\Delta T$.

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Equation (1) describes the dynamics of the optical field [42], in which the optical field amplitude inside the cavity is $a(t)$ and $|a(t)|^2$ can be normalized as the number of photons circulating within the optical cavity. Here, $\Delta \omega _L=\omega _{L}-\omega _{\textrm{cav}}$ is the frequency detuning between the pump laser frequency $\omega _{L}$ and the cold optical cavity resonance frequency $\omega _{\textrm{cav}}$. $G$ represents the optomechanical coupling coefficient and is defined as $G=-\partial \omega _{\textrm{cav}}/\partial x$, thus $Gx(t)$ represents the optical resonance frequency shift induced by the mechanical motion. $\beta _T=-\partial \omega _{\textrm{cav}}/\partial \Delta T$ is the coefficient of optical resonance change induced by the temperature variation, which has included the influence of refractive index change through thermo-optic effect and cavity length change through the thermal expansion effect [33], thus $\beta _T \Delta T$ is the thermal-induced optical cavity resonance frequency shift. $\kappa$ is the photon cavity decay rate and $\kappa _{\textrm{ex}}$ represents the cavity decay rate caused by the external coupling. $s$ is the effective drive amplitude of the pump laser and can be written as $s=\sqrt {P_{\textrm{in}}/(\hbar \omega _{L})}$ ($P_{\textrm{in}}$ is the input laser power, $\hbar$ is the reduced Planck Constant). It should be noted that we have introduced a rotating frame with the laser frequency $\omega _{L}$, i.e., $a^{\textrm{real}}(t)=a^{\textrm{here}}(t)e^{-i\omega _{L}t}$. In Eq. (2), $m_{\textrm{eff}}$, $\Omega _{m}$, $\Gamma _m$ represent effective mass, resonance frequency and energy damping rate of the mechanical resonator, respectively. The term on the right side of Eq. (2) is the radiation pressure force acting on the mechanical resonator. Equation (3) describes the cavity temperature change during the dynamical process. $c_T$ (unit: K/s) characterizes the thermal-optic effect due to the optical absorption and $\gamma _T$ (unit: Hz) is the thermal relaxation rate depicting the thermal variation speed in the cavity.

The above three coupling equations can theoretically describe the dynamical process of OMT coupling process and have proved many experimental results, including the optical frequency comb generation [37] and optomechanically induced thermal bistability [41]. For generality, it is desirable to non-dimensionalise the above three coupled differential equations. Here, we rescale the original variables $t$, $a$, $x$, $\Delta T$ as $\tilde {t}=\Omega _mt$, $\tilde {a}=a\Omega _m/(2s\sqrt {k_{\textrm{ex}}})$, $\tilde {x}=Gx/\Omega _m$, $\widetilde {\Delta T}=\beta _T\Delta T/\Omega _m$ and introduce a new set of dimensionless parameters $\widetilde {\Delta \omega _L}=\Delta \omega _L/\Omega _m$, $\tilde {\kappa }=\kappa /\Omega _m$, $\widetilde {\Gamma _m}=\Gamma _m/\Omega _m$, $\widetilde {\kappa _{\textrm{ex}}}=\kappa _{\textrm{ex}}/\Omega _m$, $\widetilde {\gamma _T}=\gamma _T/\Omega _m$. The original coupled equations can thus be reduced to Eq. (4), Eq. (5) and Eq. (6) shown below:

By this dimensionless method, the quantitative dynamics of this system will only depend on 8 dimensionless parameters instead of the 12 parameters in the original equations, which greatly eases the complexity of the following analysis. Note that we will use dimensionless parameters in the subsequent parts by default.

3. Dynamic thermal behaviors of the OMT system

We here adopt the system parameters that have been used in the paper [41] in which the thermal effects have been included in cavity optomechanical system, with $\Omega _m/2\pi =19.8$ MHz, $\Gamma _m/2\pi =28.3$ KHz, $\kappa /2\pi =17.73$ MHz, $\kappa _{\textrm{ex}}/2\pi =12.5$ MHz, $m_{\textrm{eff}}=1.7\times 10^{-13}$ Kg, $G/2\pi =1.52\times 10^{17}$ Hz/m, $\lambda _{\textrm{cav}}=1540$ nm, $\gamma _T=90\times 10^3$ Hz, $\beta _T/2\pi =1.71\times 10^{9}$ Hz/K, $c_T=1.2\times 10^{-4}$ K/s and $P_{\textrm{in}}=0.48$ mW. This set of dimensional parameters can generate the following dimensionless parameters that have been defined in Section 2, namely $\tilde {\kappa }=0.8955$, $\widetilde {\kappa _{\textrm{ex}}}=0.6313$, $\widetilde {\Gamma _m}=0.00143$, $\widetilde {\gamma _T}=7.2343\times 10^{-4}$, $\widetilde {X}=1.1754\times 10^{-9}$, $\widetilde {Y}=2.9891\times 10^7$, $\widetilde {Z}=3.3322\times 10^{-10}$.

We first studied the dynamic behaviors of the OMT system by sweeping the laser frequency across the optical cavity resonance in forward (red-detuned regime to blue-detuned-regime) and backward direction (blue-detuned regime to red-detuned-regime). This can be achieved by numerically solving the dimensionless differential equations Eq. (4) to Eq. (6) with the explicit Runge Kutta method based on Dormand-Prince pair [43], note that the sweeping laser frequency can be set up by setting a time-dependent laser detuning value $\widetilde {\Delta \omega _L}(\tilde {t})$ when solving the equations. Figure 2 shows the OMT dynamics characteristics including cavity energy $|\tilde {a}|^2$, mechanical displacement $\tilde {x}$ and temperature $\widetilde {\Delta T}$ as we scan the wavelength across the optical cavity resonance in forward and backward direction.

 

Fig. 2. (a) The OMT system dynamics when sweeping the laser forward. (b) The system dynamics when sweeping the laser backward. The upper, middle and bottom chart shows the cavity energy change, mechanical displacement change and temperature change respectively. The insets in the upper and middle parts show the enlarged view of the corresponding charts which depict sinusoidal-like mechanical oscillation and pulse-shape cavity energy change. In the bottom chart, the points where the staircase effect happens are labelled. The insets in bottom show that temperature also experience oscillation during the sweeping process, with very tiny oscillation amplitude.

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In Fig. 2(a), we change the dimensionless laser detuning $\widetilde {\Delta \omega _L}$ forward from -15 to 10. In Fig. 2(b), we swept the laser frequency in a backward direction, with dimensionless laser detuning $\widetilde {\Delta \omega _L}$ varied from 10 to -15. The total sweeping time is $2.4\times 10^6$ mechanical oscillation periods, which corresponds to a dimensional sweeping speed of around 32 pm/s in this OMT system. This sweeping speed is much slower than that in [41] and thus allows enough time for the temperature change. A more slower sweeping speed was tried but no new dynamical characteristics was found, this sweeping speed is therefore slow enough to reveal all the nonlinear dynamical characteristics of the cavity OMT system. From the simulated OMT dynamics, we can clearly locate the point where the optomechanical oscillation starts to happen. From the insets of Fig. 2, it can be found that when the optomechanical oscillation happens, the cavity energy, mechanical displacement and temperature all experience oscillation processes, corresponding to limit cycles. For forward sweeping, the system suddenly enters the oscillation state when $\widetilde {\Delta \omega _L}\approx$-2.4. As the OMT cavity is backward swept, the cavity temperature gradually increases until reaching the point $\widetilde {\Delta \omega _L}\approx$-11, after which the cavity temperature suddenly drops to zero. This discrepancy between the two critical points is caused by the thermal nonlinear effect inside the cavity and has been theoretically and experimentally analysed previously [33].

More importantly, from Fig. 2, the dynamics of the OMT system show staircase behaviors for all three system parameters. In the forward sweeping process shown in Fig. 2(a), the temperature firstly experiences a sudden increase and then an overall gradual drop ($\widetilde {\Delta \omega _L}$ from around -2.4 to 1.4), during which we can see three sharp rises of cavity temperature, and this is what we call the up-staircase effect. It can also be found that the cavity energy and mechanical displacement also experience a dramatic change at these three laser detuning points. For the backward sweeping process shown in Fig. 2(b), the down-staircase effect exists. The temperature first experiences an overall gradual increase ($\widetilde {\Delta \omega _L}$ from around 5 to -11) and then a sudden drop. During the overall gradual increase phase, we can see the temperature change experiences two sharp declines, and the cavity energy and mechanical displacement also decrease at the corresponding laser detuning points. These multiple sudden decreases or increases during a slow transition process are the staircase effect that will be studied in this paper.

We then compare the forward and backward sweeping dynamics in OMT systems in the same figure. The results are shown in Fig. 3. The temperature change $\Delta T$ would cause a shift of the cavity resonance frequency with value $\beta _T\Delta T$, which will further change the effective laser detuning. Therefore, in addition to simulating the temperature change during the sweeping as shown in Fig. 3(a), we also calculate the effective detuning $\widetilde {\Delta \omega _{\textrm{eff}}}=\widetilde {\Delta \omega _{L}}+\widetilde {\Delta T}$ in Fig. 3(b) to observe the effective laser detuning change of the optical cavity. Also, we study the dynamics in an optical-thermal (OT) cavity ($G=0$) without considering the mechanical degrees of freedom to study the influence of the mechanical oscillation to a pure OT system. It can be found that the temperature and effective detuning change in the OMT cavity system generally align with that in OT cavity system. Due to the coupling to the mechanical resonator and the resulting mechanical oscillation, the OMT cavity shows a "warm" or "cold" effect compared to the OT cavity depending on the laser detuning value, which has been systematically analysed in [41]. A bistability region labelled with gray shadow ($\widetilde {\Delta \omega _L}$ ranges from around -11 to -2.4) is clearly shown, and this has been attributed to thermal-nonlinearity and has been systematically studied in [33]. We here mainly focus on the gray box area ($\widetilde {\Delta \omega _L}$ ranges from around -2.4 to 1.4) which has been enlarged in the inset view.

 

Fig. 3. (a) Temperature change and (b) Effective detuning change for an OMT cavity and on OT cavity in forward and backward sweeping process, with arrows indicating the sweeping direction. The inset is the enlarged view of the areas where staircase and bistability effects happen. Three bistability regions have been labelled with A and B.

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Except for the staircase effects, a bistability effect was discovered in the gray box area that has never been explained thoroughly before to our best knowledge. We can find two bistability regions where we have labelled A and B in Fig. 3. The temperature $\widetilde {\Delta T}$ and effective laser detuning $\widetilde {\Delta \omega _{\textrm{eff}}}$ in an OMT system in forward sweep differs from that in backward sweep condition for the same laser detuning value $\widetilde {\Delta \omega _L}$ in these regions. In the following section, we will study the staircase effect and bistability effect systematically and it will be found that both of them are caused by thermal instability between averaged cavity energy and laser detuning.

4. Staircase effect and bistability effect analysis

4.1 Staircase effect analysis

To explain the aforementioned staircase effects in an OMT system when sweeping the laser across the cavity resonance both forward and backward, we first study the dynamics of an optomechanical (OM) cavity without considering the effect of thermal effect ($\beta _T=\gamma _T=c_T=0$). By assuming a sinusoidal-like mechanical displacement with a dimensionless oscillation amplitude $\tilde {A}$ and using the optomechanical oscillation balance condition that mechanical energy gain due to radiation pressure force equals the energy loss caused by mechanical friction, we can theoretically obtain the relationship between dimensionless mechanical oscillation amplitude $\tilde {A}$ and laser detuning $\widetilde {\Delta \omega _{L}}$. This method has been well-studied in previous research [23,24] and the result is shown in Fig. 4, with the contour lines indicating the possible relationships between the mechanical oscillation amplitude $\tilde {A}$ and laser detuning $\widetilde {\Delta \omega _{L}}$ when optomechanical oscillation happens.

 

Fig. 4. Theoretical relationship between the mechanical oscillation amplitude and laser detuning in an OM cavity system. The color represents the ratio of mechanical gain to friction loss, with the contour line indicating the condition where the ratio is 1. The upper red part of contour line represents the stable trajectory, while the lower black trajectory is unstable.

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It should be noted that only the upper part with red color is stable, and the lower black part is unstable [24]. Therefore, when sweeping the laser forward, the mechanical oscillation initiates when the laser is blue-detuned ($\widetilde {\Delta \omega _{L}}>0$). The dimensionless oscillation amplitude $\tilde {A}$ then gradually increases with a wave-like pattern following the upper red route before reaching the critical point ($\widetilde {\Delta \omega _L} \approx 3.1$), and then the amplitude drops suddenly to zero. When sweeping the laser backward, the system dynamics evolve along a different path. The mechanical resonator starts to oscillate when $\widetilde {\Delta \omega _L} \approx 1.6$, and the dimensionless mechanical oscillation amplitude $\tilde {A}$ abruptly increases to around 3, and then descends following the upper red route. It can be found that the mechanical oscillation amplitude change for the OM cavity system is continuous and smooth along the red route both forward and backward, which is different to the staircase effects in OMT cavity system shown in Fig. 2.

In Fig. 5(a), we show the simulated cavity energy change when sweeping the laser forward and backward in an OM cavity by solving the differential equations Eq. (4) to Eq. (6) numerically ($\beta _T=\gamma _T=c_T=0$). The cavity energy starts to be modulated by the mechanical oscillation when the laser is swept forward to be blue-detuned, and then experiences a wave-like change before suddenly dropping when $\widetilde {\Delta \omega _L} \approx 3.1$. When backward sweeping the laser, the cavity energy begins to be modulated heavily when ${\widetilde {\Delta \omega _L}}$ approaches a value of around 1.6, and this is caused by the abrupt increase of mechanical oscillation amplitude. These numerically simulated results are in line with the theoretical results shown in Fig. 4.

 

Fig. 5. (a) Cavity energy change and (b) Averaged cavity energy change for an OM cavity in forward and backward sweeping process. The insets in (a) show the modulated cavity energy caused by the mechanical oscillation. The gray boxs in (b) indicate the thermally unstable regions.

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From now on, we need to take into account the thermal effect in the OM cavity. For the system parameters in this work, the thermal relaxation time ($\approx 11 \mu$s) is much longer than the mechanical oscillation period ($\approx$ 50 ns), therefore temperature changes cannot quickly respond to mechanical oscillation and the resulting cavity energy oscillation. Because the temperature change is directly related to the cavity energy, we, therefore integrate the cavity energy change in Fig. 5(a) during every mechanical oscillation period to study the thermal effect. Figure 5(b) depicts the averaged cavity energy $\left \langle |\tilde {a}|^2 \right \rangle$ change with laser detuning $\widetilde {\Delta \omega _{L}}$ in the OM cavity for forward sweep and backward sweep. If we only consider the OM coupling, the mappings of averaged cavity energy to every value of laser detuning along the trajectories are stable. However, when we include the thermal effect in the OM cavity system, the stability exhibits significant differences.

In the three regions I, II, and III that have been labelled with gray box in Fig. 5 (b), the averaged cavity energy increases with laser detuning. When the thermal effect is presented, the increased averaged cavity energy leads to an increase in the cavity temperature due to the optical absorption effects according to Eq. (6). The rising cavity temperature then causes an increase in the effective laser detuning, thus forming a positive feedback between averaged cavity energy $\left \langle |\tilde {a}|^2 \right \rangle$ and laser detuning $\widetilde {\Delta \omega _{L}}$. Therefore, these three regions are thermally unstable, during which the averaged cavity energy and laser detuning would quickly jump or drop to adjacent stable regions, where the averaged cavity energy $\left \langle |\tilde {a}|^2 \right \rangle$ decreases with laser detuning $\widetilde {\Delta \omega _{L}}$. The dimensionless laser detuning ranges for the three unstable regions I, II, and III are 0.36 to 0.89, 1.44 to 1.96 and 2.56 to 2.98 respectively. It should be noted that regions I, II are basically two sections of the averaged cavity energy curve in OM cavity system indicating the unstable thermal state, which lead to the generation of bistability region A and B enclosed by the forward sweep curves and backward sweep curves.

We can then look back to Fig. 3. When forward sweeping the laser, we found three up-staircases where the effective detuning $\widetilde {\Delta \omega _{\textrm{eff}}}$ rises sharply from around 0.36, 1.44 and 2.56 due to the thermal instability, which ideally match the values of laser detuning at the left edges of three unstable regions I, II, and III shown in Fig. 5(b). It should be noted that the results in Fig. 5 (b) is about OM cavities where temperature is not relevant, so that we compare the effective detuning value $\widetilde {\Delta \omega _{\textrm{eff}}}$ in OMT cavity systems with the laser detuning value $\widetilde {\Delta \omega _{L}}$ in OM cavity systems. For the down-staircase effect in the backward sweeping process, the temperature gradually increases and the effective detuning gradually decreases until reaching a critical point when the effective detuning $\widetilde {\Delta \omega _{\textrm{eff}}}$ is around 1.6 and after that point the mechanical resonator starts to oscillate according to the results in Fig. 4. The cavity then quickly builds up and the effective detuning abruptly increases to around 2.5 due to the optical-absorption-induced increase in effective detuning. After that, the averaged cavity energy will follow the blue route in Fig. 5(b) backward and pass across two unstable regions I and II. During this process, we can find that the temperature and effective detuning drop suddenly twice when the effective detuning is reduced to around 1.96 and 0.89. These two critical values perfectly coincide with the right edge of the two unstable regions I and II as shown in Fig. 5(b), and this indicates that the down-staircase effect is also caused by the thermal instability between averaged cavity energy and laser detuning in OMT system.

In summary, the up-staircase effect in forward-sweeping dynamics and the down-staircase effect in backward sweeping dynamics are both caused by the thermally unstable region. The unstable regions are formed due to the positive feedback between the averaged cavity energy and laser detuning.

4.2 Bistability effect analysis

From Fig. 3, we can find that there exist two bistability regions labelled with A and B. In the regions, the temperature and effective detuning experience sharp rise and drop, but the laser detuning point where the rise and drop happens has different values, thus leading to the bistability effect. Therefore, we performed a numerical simulation to observe the time-domain change process during the rise and drop phase. The results for the rise and drop in region A are shown in Fig. 6(a), with the upper figure showing the drop process and the lower figure showing the rise process.

 

Fig. 6. (a) The time-domain variation process of averaged cavity energy and effective detuning for the sharp dropping (upper) and sharp rising (lower) in the bistability region A. (b) The changing process in the bistability regions depicted in averaged cavity energy change curves for OM system (upper) and effective detuning change curves for OMT systems (lower). The sharp rising (black dotted arrow) and sharp dropping (red dotted arrow) is shown, with clear correspondence between the two views with dotted lines.

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The upper and lower figure in Fig. 6(a) show that the effective detuning value experiences quick changes before finally settling to a steady state for both the sharp dropping (upper) and rising (lower) processes. It has been analysed that the thermally unstable effect causes the staircase effect, and the laser detuning range for the unstable region I is 0.36 to 0.89. However, in the upper part, the effective detuning value decreases gradually from around 0.89 to a steady value of around 0.2, and in the lower part, it increases from around 0.36 to around 1.1. It is therefore clear that effective detuning value in the two processes both evolves beyond the edge of unstable region I into the adjacent stable regions.

To clearly show the changing processes in the bistability region, in the upper part of Fig. 6(b), we depict the rise and drop processes for the two bistability regions in the averaged cavity energy $\left \langle |\tilde {a}|^2 \right \rangle$ change curves for OM system, with the red dotted arrows indicating the drop process and black representing the rise process. In the lower part, we plot the transposed view of effective detuning $\widetilde {\Delta \omega _{\textrm{eff}}}$ change curves for OMT systems, which shows the direct correspondence between the two different figures with dotted lines. The rise and drop processes evolve along the trajectories in the unstable region and exceed the edge deep into the stable region. This overshoot can be understood by the averaged cavity energy change (the blue line in Fig. 6(a)): averaged cavity energy change experiences its local maximum and minimum at the edge of the unstable region so that the temperature and effective detuning change experience its steepest change at the edge and therefore will continue to change. After that, the averaged cavity energy gradually increases or decreases with the effective detuning varies into the stable region, in which the temperature and effective detuning can settle into a steady state. Therefore, the areas enclosed by the red dotted arrow, black dotted arrow and two stable trajectories represent the two bistability regions labelled with A and B in Fig. 3.

In summary, the bistability effect is caused by the overshoot fact that the OMT system would quickly evolve across the thermally unstable region and go deep into the stable region.

5. Discussion and conclusion

Here, we systematically studied the nonlinear dynamics of cavity optomechanical-thermal systems. We conducted an extensive analysis by solving a set of dimensionless differential equations describing the optomechanical-thermal cavity system. By forward and backward sweeping the laser frequency across the optical cavity resonance and driving the mechanical resonator into oscillation, we discovered two interesting dynamic effects: the staircase effect and bistability effect. The staircase effect can abruptly change the cavity characteristics including mechanical oscillation amplitude, cavity energy and cavity temperature. In addition, the bistability effect makes the abrupt staircase change happens in different laser detuning value for forward sweep and backward sweep. Extensive discussion and analysis were provided to explain the two effects. It is found that both of the effects are caused by the thermally unstable relationship between averaged cavity energy and laser detuning in cavity optomechanical-thermal systems, which is intrinsically stable for a cavity optomechanical system. Compared to the results in [41] where the same OMT system parameters were used, a slower sweeping speed in this work reveals more details about the nonlinear dynamics when sweeping the laser frequency. In addition, we provide a more quantitative analysis on the staircase and bistability effect using comprehensive numerical simulation methods.

This work provides a view of the nonlinear dynamics for cavity optomechanical-thermal systems and serves as guidance for leveraging the OMT cavity for specific applications. For example, the staircase effect can be used for switching the OMT systems between several discrete states, thus realizing a stepped manipulation on the optical frequency comb or optical pulse generation. In addition, a very sensitive control on OMT systems can be achieved when setting the system near the staircase up or down points, in which a tiny variation of the pump laser detuning can lead to a significant change in the characteristics of the generated optical frequency comb or optical pulse.

It should be noted that we only studied an OMT system with one set of system parameters, which is in the resolved-sideband regime. A full analysis of the system parameters on the dynamic characteristics and a bifurcation analysis in OMT systems are currently underway, which will provide a full view of the nonlinear dynamics of OMT systems and will obviously need a large number of computational resources. The prelimitary results show that a smaller value of $\widetilde {\kappa }$ and a larger value of $\widetilde {Y}$ would cause a more complicated distribution of thermally-unstable regions, thus leading to more staircases and bistability regions. On the contrary, with the increase of $\widetilde {\kappa }$ and decrease of $\widetilde {Y}$, these two effects would disappear. A larger thermal lifetime means a slower temperature change, and therefore needs a slower laser sweeping speed to reveal all the thermal-related nonlinear dynamics.