El TIPO DE INDIVIDUO QUE LA LEY PRETENDE FORJAR

It is well known, especially in the gravitational wave community [228], that single- cavity configurations with a suspended mirror introduce tilt instabilities. The triple- cavity configuration of the optical tripod, however, combines the optical springs from three independent fields to create a fully stable system. Yet this is not enough: the self-locking that originates from the restoring e↵ect of the radiation pressure force gradient is tainted by the occurrence of anti-damping and parametric amplification of the oscillations due to the delayed response of the cavities [80].

As the tripod’s upper mirror does not have any mechanical supports acting as dissipative sinks, a remedy to the problem needs to be found in the interaction with the optical field. In particular, one can resort to the same optical spring e↵ect which introduces the dynamical instability in the first place, since detuning the field to the other side of the resonance induces a damping, anti-restoring force instead. The three blue-detuned beams creating the trap can therefore be combined with another set of red-detuned beams to make the system robust against parametric amplification and favour lasting stability. The newly introduced fields need not be comparable in intensity to the trapping ones. Because the dispersive and the dissipative attributes of the optical

spring scale di↵erently as a function of detuning (cf. Fig. 3.4), it is possible to use low- power damping beams with a di↵erent detuning than the trapping ones to introduce a dissipation capable of stabilizing the system without introducing substantial changes to the sti↵ness.

To implement the dual-beam configuration, rather than doubling the number of cavities to six it is certainly more convenient to let the original three cavities be dou- bly resonant [89]. When considering this option it should be remembered that, unless orthogonal polarizations are used, the injection of two di↵erent fields into the same cav- ity results into interference that will cause part of the intra-cavity power to beat. The beating consequently extends to the force experienced by the mirror, and the levita- tion dynamics may be a↵ected beyond control. The mechanical response of the system, however, is more or less receptive to the interference depending on the time scale of the beating. The susceptibility of the upper mirror is particularly prominent only at fre- quencies close to the motional eigenfrequency !_os, which for the optical tripod is fully determined by the optical spring. The bandwidth of the susceptibility is determined by the magnitude of the damping rate,_{| m|}. Even though we are trying to ultimately minimize the (anti-)damping, we may for now assume it to be finite but smaller than the frequency of oscillation, i.e. _{| m|}.!_os. The mirror’s motion is not driven by the beating when the beat frequency is many multiples of_{| m|}higher than !_os, when only a time-averaged e↵ect is perceived. Since the beat frequency is determined by the rela- tive detuning between the two fields, the complications emerging from the interference can thus be neglected if the two input beams are detuned sufficiently apart from each other. Whether this condition is naturally satisfied or not, it is always possible to take advantage of the periodicity of the cavity’s resonance and detune the two beams to independent free spectral ranges. As the beat frequency is up-shifted by one or more free spectral ranges, orders of magnitude higher than the peak in susceptibility, the mirror’s dynamics become clear from any undesired e↵ects. Numerical support to these claims is o↵ered in Appendix C. Under these circumstances the optical springs can be added together as if the two intra-cavity fields acted independently and without reciprocal interference.

The damping component of the optical spring is manifest only in the full dynamical expression of Eq. 3.70, in which case the optical spring is a function of the spectral frequency !. The generalized eigenvalue of the sti↵ness tensor at the origin is, in the

vertical direction, Kzz(!) = X ⌫ 8G0Pin,⌫ c⌧ _{⌫ ⌫} (2 ⌫+ 2⌫)2  1 ! 2 ⌫+ 2⌫ (! 2i_⌫) 1 cos2(✓_⌫). (8.14)

The frequency of the oscillations and the corresponding damping depend, respectively, on the real and the imaginary part as

!_m,z(!) = r Re(K_zz(!)) m , (8.15) m,z(!) = Im(K_zz(!)) m! . (8.16)

In ordinary optomechanical systems, the frequency dependence of the optically induced parameters is typically convolved with the mechanical susceptibility of the oscillator. For high mechanical quality factors the peak in susceptibility at the intrinsic mechanical frequency can be approximated as a delta function, implying that only the component at the natural frequency of the oscillator is relevant in the optical spring.

0 ∆𝜈 𝑃𝜈 ∆𝜈 = −𝜔m 𝑃in,𝜈 = 12.7% ∆𝜈 = ∆𝜈(bal) 𝑃in,𝜈 = 100% 0 -1 -2 -3 1 2 3 10 5 0 -5 -10 (𝜔m/2𝜋)2 (109 Hz2) 𝛾m /2 𝜋 (kHz )

Figure 8.5: The dual-beam configuration combines two optical springs to change the damping of the system. The two optical springs can be represented in the complex plane, where the horizontal axis is the square of the oscillation frequency and the vertical axis is the optical damping. In this representation they can be combined as any two vectors as long as beams’ relative detuning is large compared to the dynamics of the mirror. The curves trace the course of the optical spring of Eq. 8.15–8.16 as a function of detuning. The arrows point to the values of the optical spring of the blue-shifted trapping beam (in blue) and of the red-shifted damping beam (in red), which combine to a strictly real and positive optical spring (in black). The trapping beam in each cavity has an input power of 1 W and is detuned to satisfy the balancing condition as in Eq. 8.8. The damping beam has a negative detuning equivalent to the oscillation frequency, and the input power is adjusted to 12.7 % of the power of the trapping beam in order to make the overall damping component vanish.

Since the optomechanical system under analysis behaves like a free mass, however, this line of reasoning cannot be directly applied. The frequency of the oscillations is directly determined by the optical spring, which itself depends on the spectral frequency considered, and without a well-defined resonance there might not be a well-defined solution. The argument results into a recurrence relation for the frequency of the oscillations,

!_m,z[n+ 1] =

r

Re(_Kzz(!_m,z[n]))

m , (8.17)

with seed value !m,z[0] = 0. Numerical estimates suggest that in the regime of small

and medium finesses, where we operate, the recursion settles very rapidly after the first few iterations. For cavities with very high finesse the recursion fails to converge to a single solution, suggesting that the dynamical stability may additionally depend on the spatial extent of the tight-confinement.

With the same choice of parameters used to calculate the static stability (cf. Fig. 8.3), we have that the single-beam optical spring of the trapping beam converges, at balance, to !_m,z = 2⇡⇥50.1 kHz and _m,z = 2⇡⇥9.9 kHz. The second optical spring from the damping beam is tuned to have equal but positive damping _m,z when detuned by !_m,z. The two optical springs cooperate as shown in Fig. 8.5 to cancel any optical dissipation e↵ects. The modest anti-restoring component of the damping beam also combines to the sti↵ness of the original trapping beam, but because the power and the detuning have been tailored specifically to minimize this drawback the net sti↵ness is still largely positive. The radiation pressure force is thus purely restor- ing and conservative. The oscillation frequency modelled for the combined dual-beam configuration is !_m,z = 2⇡_⇥43.3 kHz. Similar corrections occur at the same time on the horizontal directions, where the e↵ects of the damping beam rescale as those of the trapping beam because they have equal projections onto the x and y axes. The frequency of the oscillations in the horizontal plane adjusts from 0.89 kHz to 0.77 kHz.