Radiation Pressure Induced Oscillations of an Optically Levitating Mirror

Optical Fabry-Perot cavity with a movable mirror is a paradigmatic optomechanical systems. While usually the mirror is supported by a mechanical spring, it has been shown that it is possible to keep one of the mirrors in a stable equilibrium purely by optical levitation without any mechanical support. In this work we expand previous studies of nonlinear dynamics of such a system by demonstrating a possibility for mechanical parametric instability and emergence of the ``phonon laser'' phenomenon.


I. INTRODUCTION
The idea that cavity optomechanical phenomena can be observed in a cavity formed by optically levitating mirror has been proposed and realized by a number of authors [1][2][3].The authors of Ref. [4] derived main equations describing the motion of the center of mass of a levitated mirror and studied stability of its dynamics.They found that even if the system possess a stable equlibrium in the quasi-static approximation, the oscillations around this equilibrium become unstable if one takes into account corrections to the quasi-stationary approximation.These corrections result in optically induced amplification of mechanical motion and run-away instability.In this paper we show that the mechanical dissipation of the mirror, which is present due to the air resistance and can be controlled, stabilizes the nonlinear dynamics of the mirror resulting in a multimode phonon lasing-like behavior [5][6][7][8][9][10][11][12][13].
In this note we present the results of our analysis of the periodic oscillations of the levitating mirror in this regime.
Equations of motion for optical and mechanical degrees of freedom can be written down as where in Eq. 1 x, M, γ m are coordinate, mass, and mechanical damping coefficients of the levitated mirror, g is the acceleration of gravity, ω c is the optical resonance frequency dependent on the mirror coordinate x (N is the order of the optical resonance), and a is the amplitude of the optical mode.
Additionally, parameter ω L in Eq. 2 is the frequency of the driving laser, F is the cavity finesse, c-speed of light, and the term π 2 c/Fx represents the cavity decay rate, while a in describes the amplitude of the driving field normalized such that where P in is the input power.Unlike standard optomechanical models [14,15], Eq. 1 lacks a mechanical spring force, so that the oscillations of the mirror occur solely due to socalled "optical spring" effect [16].It shall be noted that while mechanical spring is linear and instantaneous, the "optical spring" is non-linear with respect to mirror's displacement, and is also characterized by a time delay determined by the lifetime of the optical cavity mode.
If the input power P in exceeds the critical value P cr = M gcπ 2 /(2F), the levitated mirror can be in the stable equilibrium in the position with coordinate x eq defined as where and parameter characterizes the linewidth of the cavity resonance expressed in terms of the wavelength of the laser λ L rather than inn terms of the frequency.The corresponding equilibrium field amplitude is given by It is convenient to rewrite the equations of motion in terms of real and imaginary parts of the relative deviation of the field amplitude from its equilibrium value a eq : w = Re [(a − a eq ) /a eq ] and y = Im [(a − a eq ) /a eq ], and dimensionless mechanical displacement expressed in terms of the cavity line width ξ, u = (x − x eq ) /ξ : Here τ is dimensionless time defined as τ = tΩ max ,where is the maximum mechanical frequency of linear oscillations of the mirror, 2η = γ m /Ω max is the dimensionless mechanical damping parameter, and r is dimensionless detuning from the equlibrium position of the mirror x eq defined as Parameter r also serves as a measure of input power and is the main parameter controlling the behavior of the system.Eq. 9 -11 are similar to equations derived in Ref. [4] with one significant difference: Eq. 9 -11 contain the term proportional to mirror's velocity du/dτ , which is responsible for mechanical damping.In the absence of this term the authors of Ref. [4] correctly predicted that optomechanical interaction results in amplification and the loss of stability of mechanical oscillations.However, as we will show below, the mechanical damping stabilizes nonlinear dynamics of the mirror resulting in a behavior similar to phonon lasing [8, 10-12, 17, 18].
It is justified, therefore, to analyze the system of Eq. 9 -11 using a perturbation expansion in small parameter ϵ.The same approach was used in Ref. [4] and earlier in Ref. [15].
The zero-order solution results in the well known quasi-stationary approximation, where the optical amplitude is assumed to follow the mechanical displacement.In terms of the variables used in this work it is written as In the absence of the damping, the mechanical motion in this approximation can be characterized by potential shown in the Fig. 1 for several values of the detuning r.In the linear approximation we have harmonic oscillations with frequency where the maximum frequency corresponds to P in = 2P cr .These oscillations arise solely due to the "optical spring" effect as no mechanical springs are present in the system.The first order correction in ϵ introduces an optical amplification, and the nonlinear dynamics of the mirror in this approximation is described by equation where effective dissipation/gain parameter η ef f is given by (16) If η = 0, this parameter is always negative and the optomechanical interaction results in an unsaturated mechanical gain rendering the system unstable [4].If, however, η ̸ = 0, the strength of the initial linear amplification occuring when the linear gain parameter becomes negative is limited by nonlinear terms, and one can expect an initial growth of the mechanical amplitude to saturate and the system to settle in stable oscillations.The range of parameters allowing for amplification to happen is determined by inequality [19] The function on the right-hand side of this inequality has a maximum value 3 √ 3/16 at r 2 = 1/3, so that the amplification regime is possible only if η/2ϵ does not exceed this value.
For each value of η/2ϵ<3 √ 3/16, the amplification region is limited by r min < r < r max ,where r min and r max are solution to equation Not surprisingly, for r close to one of the boundaries, when amplification is weak, the time it takes for system to reach the steady state oscillations is longer than for r closer to the center of the amplification region.This point is illustrated in Fig. 2, which shows the saturation of the initial amplification as the nonlinear terms in Eq. 15 stabilize the effective gain for different values of r.One can clearly see the tendency for the increased time to steady state oscillations for values of r closer to the boundaries of the amplification region, which for the chosen values of parameters η/ϵ = is between r min = 0.194 and r max = 1.298.
The observed nonlinear stabilization of oscillations is reminiscent to the population inversion saturation in regular lasers, but there is also a significant difference.In a simplest single mode laser the population inversion saturates to a constant value such that the effective gain in the steady lasing regime remains zero and the steady state oscillations are harmonic.In the case considered in this work the situation is more complicated -the effective gain oscillates around zero as shown in Fig. 3.As a result of these oscillations the steady state regime of the mirror's oscillations is not monochromatic, and the degree of deviation from purely harmonic behavior depends on the input power via detuning parameter r.
This trend is illustrated in Fig. 4 presenting the time dependence of the mirror's displacement for two different values of r.Oscillations with larger amplitude correspond to a deeper potential well (see Fig. 1), which is better approximated by a quadratic behavior, and, therefore, are more harmonic then the oscillations with smaller amplitude: the former are characterized by at least three clearly discernible harmonics, while the latter's spectrum consists of one main frequency with a weak contribution from another harmonic.Similar  trend is also seen in oscillations of the effective gain parameter shown in Fig. 3.The increase in the depth of the potential well with increasing r also explains the corresponding increase in the amplitude of the oscillations as seen in Figure 5.
To further illustrate the nature of the nonlinear oscillations of the mirror we have constructed the phase trajectories of the oscillations by plotting velocity ḋu/dτ versus displacement u, shown in Figure 6.The circular and elliptic phase trajectories in the left figure are indicative of weakly anharmonic oscillations, while more complicated shapes in the right panel corresponds to the stronger anharmonicity.Finally, we present the phase portraits of

III. CONCLUSION
In this work we analyzed the motion of a center of mass of a levitated mirror in the presence of mechanical damping.We showed that the damping stabilizes nonlinear dynamics of mirror resulting in a behavior reminiscent to phonon lasing.Unlike the simplest lasing dynamics, however, the time dependence of the mirror's displacement in the steady state is in our case is anharmonic with the effective gain parameter oscillating around rather then being pinned to zero.

Figure 1 :
Figure 1: Optomechanical potential for different values of the dimensionless detuning (input power parameter) r.Obviously larger r results in the deeper potential well.

Figure 2 :
Figure 2: Mechanical displacement u as a function of time for different value of r.The values of mechanical damping η and the cavity dynamical parameter ε are given in the figures.

Figure 3 :
Figure 3: Time-dependence of the effective nonlinear gain parameter for two different values of detuning (power) parameter r.One can see that the effective gain does not saturate to zero but keeps oscillating resulting in the energy flowing from light to mechanical degrees of freedon when the gain is negative and in the opposite direction when it is positive.

Figure 4 :
Figure 4: Mirror oscillations in the steady state regime for different values of the detuning and the corresponding Fourier spectra.

Figure 6 :
Figure 6: Phase trajectories of the oscillating mirror in the steady state regime for strong (left) and weak (right) nonlinear regimes.

Figure 7 :
Figure 7: (Left) The phase portrait of mechanical oscillations.The stable oscillations seen in this figure as limiting cycle, approached by the phase trajectories (Right) The separation of the phase space into regions of stable limiting cycles, and unstable regions.