Model for Radiation Pressure Damping

8.7 Experimental Walk-Through: Temperature Measurement Using the Detuned Probe

9.1.2 Model for Radiation Pressure Damping

on the power spectral density of Langevin force fluctuations,S_FL(Ω).

Sb(Ω) =|χef f(Ω)|²SF_L(Ω) (9.8a)

S_FL(Ω) = 4kBTbath

Ω Im

χ₀(Ω)⁻¹

= 4k_BT_bathΓ_m(Ω), (9.8b) where χ₀(Ω) is the natural susceptibility of the system (F_rad = 0). The latter expression is a consequence of the Fluctuation-Dissipation Theorem, as described in Section 2.2.

In the weak damping limit, the thermal fluctuations are described by a Lorentzian with a shifted frequency (due to the optical spring) and linewidth (due to the optical damping):

S_b(Ω) =|χef f(Ω)|²×4k_BT_bathΓ_mm≈ 4k_BT_bathΓ_m/m

((Ωm+ ∆Ωopt)²−Ω²)²+ (Γm+ Γopt)²Ω². (9.9) An important consequence of the modified mechanical response is the phenomenon of optical cooling/heating, which corresponds to a reduction/enhancement of the effective temperature ofb.

For weak damping:

T_{ef f} = m kB

Z ∞ 0

Ω²S_bb(Ω)dΩ

2π ≈T_bath Γ_m Γm+ Γopt

, (9.10)

or in terms of thermal occupation number:

¯

n≡ kBTef f

~(Ω_m+ ∆Ω_opt) ≈ kBTef f

~Ω_m = ¯n_bath Γm

Γ_m+ Γ_opt. (9.11)

The sign of Γ_optdepends on the sign of the correlations between the mirror position and the radiation pressure force. For positive optical damping Γ_opt > 0, the effective temperature of the oscillator is reduced. This is analagous to the suppression of noise in an electronic circuit using negative feedback.

Variables {g₀, a(t), E_in(t), E_out(t), E_ref(t), κ, κ₁, κ₂, ω_c, ω₀} correspond to the end-mirror optome-

chanical coupling, the amplitude of the intracavity standing wave, the amplitude of the input/output/reflected electric field, the total amplitude decay rate of the intracavity field, the amplitude decay rate of the

intracavity field through the input (mirror 1) and output (mirror 2) mirror, the resonance frequency of the cavity, and the frequency of the input field, respectively. a(t) andEin(t) are complex am- plitudes expressed in the frame rotating atω0 and normalized so that |a(t)|²=Uc(t) is the slowly varying envelope of the intracavity energy and|Ein(t)|² is the slowly varying envelope of the power coupled to the cavity.

The radiation pressure forceF_rad(t) can be computed from the gradient of the intracavity energy, with respect to the mirror position. For a low-loss cavity: F_rad=dU_c/db≈ −(dωc/db)·U_c/ω_c (Eq.

2.39). In the canonical two-mirror system, this reduces to the familar expression in terms of the circulating power (Eq. 2.37):

Frad(t) =−g0Uc(t)/ωc= 2Pcirc(t)/c. (9.13) (Sign convention discussed in Section 2.4.)

Solving Eq. 9.12 to first order in the small parameter g0b/κ gives the following expressions for the optical spring shift and damping rates in the “weak damping” approximation (see derivation in Section 2.4, Eq. 2.51):

Γopt= 2hncig₀²b²_zp κ

1

1 + (∆ + Ωm)²/κ² − 1

1 + (∆−Ωm)²/κ²

(9.14a)

∆Ωopt=hncig²₀b²_zp κ

(∆ + Ωm)/κ

1 + (∆ + Ωm)²/κ²+ (∆−Ωm)/κ 1 + (∆−Ωm)²/κ²

, (9.14b)

where ∆ =ω0−ωc is the detuning of the input field,hnci ≡ h|a|²i/~ωc is the intracavity photon number andxzp=p

~/2mΩmis the zero-point displacement.

The optical damping rate is proportional to the difference in the strength of the red and blue sidebands generated by modulation of the cavity resonance frequency. When the input field is red- detuned (∆ < 0), the damping rate is positive, Γopt > 0. Microscopically, this corresponds to a situation in which photons scattered from the mirror surface preferentially receive a blue shift before exiting the cavity.

9.1.2.2 Extension to the MIM System

The optomechanical interaction between the amplitudebijof a membrane vibrational mode and the amplitudea of a single mode of the intracavity field is described by the following pair of coupled

differential equations in the limit that the membrane reflection coefficient is “small” (Section 8.2):

mij¨bij(t) +mijΓijb˙ij(t) +mijΩ²_ijbij(t) =F_L^ij(t) +F_rad^ij (t) (9.15a)

˙

a(t) =−



κ+i(ω₀−ω_c−X

ij

g_ijb_ij(t))



a(t) +√

2κ₁E_in(t); g_ij ≡g_mη_ij (9.15b) E_out(t) =√

2κ₂a(t) (9.15c)

Eref(t) =√

2κ1a(t)−Ein(t). (9.15d)

These formulas are a formally equivalent to the coupled equations of motion for the canonical two- mirror optomechanical system, Eq. 9.12. The generalizations that have been made are as follows:

• Generalized displacement amplitude bij(t) is the amplitude of the displacement vector field φ~ij(x, y, zm) = ˆzsin(iπx/wm) sin(jπx/wm) describing drum vibrations of thewm×wmsquare membrane surface (Section 7.1.1).

• For vibrational modes of the membrane, effective mass m_ij is defined relative to the energy normalization condition m_ij=hU_iji/Ω²_ijhb²_iji=R

V ρ~φ_ij·φ~_ijdV =m_phys/4 (Section 7.1.1).

• For the membrane resonator, Ωij is the eigenfrequency of the (i, j) vibrational mode and Γij = Ωij/Qij is the energy damping rate of the of the (i, j) vibrational mode, characterized by mechanical quality Qij – Γij andQij may in general depend on the vibrational frequency (Section 7.3.1.2).

• Brownian motion of generalized amplitude bij(t) is described by a thermal force with spec- tral density S_Fij

L(Ω) = 4kBTbathmijΓij(Ω). The frequency dependence of the damping term accounts for the “structural” damping behavior in bulk elastic resonators (Section 7.3.1.2).

• In the MIM cavity,{a, E_in, E_out, E_ref, κ, κ₁, κ₂, ω_c, ω₀}have the same interpretation as in Eq.

9.12 with two important subtleties: (a) the intracavity mode is a more complicated mode- shape that has different amplitudes on the left and right of the membrane (the sum of the energy on both sides is |a|²); (b) the total amplitude decay rate, κ, and the amplitude decay rate through the input mirror κ1 and output mirror κ2 are all functions of the equilibrium membrane position,zm(See discussion in Section 8.2).

• In the MIM system, the optomechanical coupling for a single mode (i, j) is described by δωc = gijbij = ηijgm(zm)bij. gm(zm) gives the resonance frequency shift resulting from a small displacement of the membrane equilibrium position δz_m, and in general depends on the absolute membrane position relative to the intracavity standing wave (Section 3.3.1). Fac- tor η_ij accounts for the spatial overlap between the TEM_mn cavity mode and membrane

vibrational mode, and is given by the ratio of the antinode displacement and the displace- ment of the membrane surface S averaged over the intensity profile of the cavity mode:

ηij≡R

S|ψmn(x, y, z)|²φ(x, y, z)~ ·zdσ/ˆ R

S|ψmn(x, y, z)|²dσ (see Section 7.1.3).

In the MIM system, the generalized radiation force experienced by coordinatebij is given by:

F_rad^ij (t) =−g_ijU_c(t)/ω_c =η_ijg_m(z_m)U_c(t)/ω_c. (9.16)

By direct analogy to the two mirror system, the thermal noise spectrum for a single, underdamped vibration of the membrane in the presence of a weak optical spring is given by (assuming the membrane mode in question is well isolated in frequency from other modes):

Sb_ij(Ω)≈4kBTbΓij(Ω) mij

1

((Ω_ij+ ∆Ω^ij_opt)²−Ω²)²+ Ω²(Γ_ij(Ω) + Γ^ij_opt)² (9.17a) Γ^ij_opt= 2hnci(g_ijb^ij_zp)²

κ

1

1 + (∆ + Ωij)²/κ² − 1

1 + (∆−Ωij)²/κ²

(9.17b)

∆Ω^ij_opt≡=hnci(gijb^ij_zp)² κ

(∆ + Ωij)/κ

1 + (∆ + Ω_ij)²/κ²+ (∆−Ωij)/κ 1 + (∆−Ω_ij)²/κ²

(9.17c) T_{ef f}^ij ≈mij

kB

Z ∞ 0

Ω²Sb_ijb_ij(Ω)dΩ/2π≈ Γij

Γ_ij+ Γ^ij_optTbath, (9.17d) where hnci ≡ h|a|²i/~ωc is the generalized intracavity photon number and b^ij_zp = p

~/2mijΩij is the zero-point displacement. To deal with multimode thermal noise, we here retain the frequency dependence of the Γ_ij(Ω) associated with structural damping: Γ_ij(Ω) = Γ_ij(Ω_ij)Ω_ij/Ω = Ω²_ij/Q_ijΩ (Section 7.3.1.2). The integration R

Ωij is here understood to exclude off-resonant displacement of other vibrational modes for the multimode resonator.