Experimental Parameters: What to Expect

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

9.1.3 Experimental Parameters: What to Expect

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.

values very with the exact position of λ on the mirror coating curve). Variables {Ω_m, Q_m} are the typical frequency and mechanical quality of a higher-order mode of the membrane, which has material properties{dm, wm,T, ρ}, corresponding to thickness, square width, tension, and density.

9.1.3.1 Intracavity Photon Number

We first compute the intracavity photon number, which depends on the steady-state input and output power,{hPini,hPouti={h|Ein|²i,h|Eout|²i}(we here assume that the input field is monochromatic), and the magnitude of cavity decay rates{κ, κ₁, κ₂}. With the membrane removed, our Fabry-Perot cavity is nearly symmetric and lossless: κ₁≈κ₂≈κ/2 (Section 6.2). However, with the membrane in the cavity, decay rates{κ, κ1, κ2} are all functions of the membrane position. The steady-state intracavity photon number in this more generic case is given by solving Eq. 9.15 with ˙a = 0 and bij = 0 (Eq. 8.5):

hn_ci=h|a|²i

~ω_c =hPouti

~ω_cκ κ

2κ₂ = hPini

~ω_cκ 2κ1

κ 1

1 + ∆²/κ² (9.19a)

hPouti=hPout(∆ = 0)i

1 + ∆²/κ² . (9.19b)

The ratio 2κ1/κcan be determined from the resonant transmission/reflection of the MIM cavity (Eq. 8.5). Assuming negligible internal loss, so thatκ1+κ2=κ:

hPref(∆ = 0)i

hP_ini = hPini − hPout(∆ = 0)i hP_ini =

2κ1

κ −1 ²

(9.20a)

→ 2κ₁

κ = 2−2κ₂ κ =

s

1−hP_out(∆ = 0)i

hPini + 1. (9.20b)

The magnitude ofhPout(∆ = 0)i/hPinidepends on the position of the membrane in the cavity. It can be computed numerically by the method discussed in Section 3.4, and has been measured for our

“science” cavity as discussed in Section 6.5. For our low-loss, 50-nm-thick membrane with reflectivity r_m= 0.42 at 935 nm, the magnitude ofhPout(∆ = 0)i/hPinivaries between 0.8 and 1.0 and has a magnitude of ≈0.9 when the membrane is positioned halfway between a node and an antinode of the intracavity field. At this position, for which g_m= 2r_mg₀, we find 2κ₁/κ≈(g_m/g₀)²≈0.7 and 2κ₂/κ≈2−(g_m/g₀)²≈1.3.

For relevant cavity parameters, we have:

hnci= 0.750×10⁶×

hP_ini 10µW

λ 935 nm

10 MHz κ/2π

2κ₁ κ

1 1 + ∆²/κ²

(9.21a)

= 0.750×10⁶×

hPout(∆ = 0)i 10µW

λ 935 nm

10 MHz κ/2π

κ 2κ₂

1 1 + ∆²/κ²

. (9.21b)

It’s useful to use the quantity hPout(∆ = 0)i because in the lab we can measure this number

without worrying about input mode-matching. Also, for a perfectly mode-matched, symmetric cavity,hP_out(∆ = 0)i=hP_ini.

9.1.3.2 Effective Mass and Zero-Point Amplitude

The vibrational modes of the membrane were discussed in detail in Sections 4.2 and 7.1.1. The effective mass of generalized displacement amplitude b_ij coinciding with the displacement of an antinode on the membrane surface is given bym_ij =m_phys/4. Thus we have, for the effective mass and the zero-point amplitude:

m_ij = 8.44 ng× ρ 2.7 g/cm²

! dm

50 nm

wm

500µm ²

(9.22a)

b^ij_zp = s

~

2m_ijΩ_ij = 4.45×10⁻¹⁶ m×

8.44 ng m_ij

^1/25 MHz Ω_ij/2π

^1/2

. (9.22b)

9.1.3.3 Optomechanical Coupling and Spatial Overlap

For a nearly lossless 50-nm-thick film at 935 nm, the reflection coefficient isr_m= 0.42. Using Eq.

3.22 derived forgm(zm) and defining the vibrational amplitudeb as the amplitude of an antinode, we have for a single membrane mode (dropping the indicesiandj):

g^ij_m≡ δω_c bij

=η_ijg_m(z_m) =η_ij× −2|r_m|sin(2kz)

p1− |rm|²cos²(2kz)×g₀≤η_ij×2r_m×ω_c

L (9.23a)

= 2π×0.363MHz pm ×ηij

1

rm

0.42

743µm L

935 nm λ

. (9.23b)

9.1.3.4 Optical Damping and Spring Shift

The full expressions for “weak” optical damping and spring rates given in 9.17 can be expressed:

Γ^ij_opt= 2π×3.92 kHz×

hPout(∆ = 0)i 10µW

10 MHz κ/2π

²743µm L

² rm

0.42 2ηij

1 2

(9.24a)

×

935 nm λ

5 MHz Ωij/2π

8.44 ng mij

κ 2κ2

1 1 + ∆²/κ²

κ²

(Ωij+ ∆)²+κ² − κ² (Ωij−∆)²+κ²

(9.24b) and

∆Ω^ij_opt= 2π×1.96 kHz×

hPout(∆ = 0)i 10µW

10 MHz κ/2π

²743µm L

² rm

0.42 2ηij

1 2

(9.25a)

×

935 nm λ

5 MHz Ω_ij/2π

8.44 ng m_ij

κ 2κ₂

1 1 + ∆²/κ²

κ(Ωij+ ∆)

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

.

(9.25b)

Two important limits exist for the unitless terms at the far right, corresponding to the difference of the strength of the blue sideband and the red sideband generated on the intracavity field by the vibrating membrane. In the sideband unresolved or “bad” cavity limit (Ωij << κ), they reduce to

κ²

(Ωij+ ∆)²+κ² − κ² (Ωij−∆)²+κ²

Ω_ij<<κ

−−−−−→ 4Ω_ij κ

∆/κ

(1 + (∆/κ)²)² =±Ω_ij

κ for ∆ =±κ (9.26a) κ(Ωij+ ∆)

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

Ω_ij<<κ

−−−−−→ 2∆/κ

1 + (∆/κ)² ≤ ±1 for ∆ =±κ. (9.26b) In the sideband resolved or “good cavity” limit, Ωij >> κ, with ∆ =±Ωij (in which case only one sideband is resonant), the prefactors reduce to

κ²

(Ωij+ ∆)²+κ² − κ² (Ωij−∆)²+κ²

Ωij>>κ

−−−−−→ ±1 for ∆ =±Ωij (9.27a) κ(Ω_ij+ ∆)

(Ωij+ ∆)²+κ² − κ(Ω_ij−∆) (Ωij−∆)²+κ²

Ωij>>κ

−−−−−→ ± κ 2Ωij

for ∆ =±Ωij. (9.27b) In practice, we currently operate the cavity in between the “good” and “bad” cavity limit, using the (6,6) mode with Ω66/2π≈5 MHz and the cavity at a wavelength for whichκ/2π≈10 MHz. In this case the unitless factors (Eqs. 9.27a and 9.27b) are roughly 0.5 for both the spring and damping rates.

9.1.3.5 Effective Temperature and Thermal Occupation Number

For parameters used in the previous section, we expect substantial radiation pressure damping rate for ∼ 10 microwatts of power coupled into the cavity. For damping rates much smaller than the mechanical frequency or the cavity linewidth, we can naively applyT_{ef f}^ij =Tbath×Γij/(Γij+ Γ^ij_opt) (Eq. 9.17d) to predict the effective temperature and occupation number ¯nij =kBT_{ef f}^ij /~Ωij of the damped mode. In the limit Γ^ij_opt>>Γij:

T_{ef f}^ij = 0.38 K·

10µW hP_out(∆ = 0)i

κ/2π 10MHz

² L 743µm

²0.42 r_m

² 1 η_ij

² λ 935 nm

(9.28a)

· 10⁶

Qij

Ωij/2π 5 MHz

2 mij

8.44 ng 2κ2

κ

1 1 + ∆²/κ²

−1

κ²

(Ωij+ ∆)²+κ² − κ² (Ωij−∆)²+κ²

−1

(9.28b)

¯

n_ij = 1590·

10µW hPout(∆ = 0)i

κ/2π 10MHz

² L 743µm

²0.42 rm

² 1 ηij

² λ 935 nm

10⁶ Qij

(9.28c)

·

Ω_ij/2π 5 MHz

T_bath 300 K

m_ij 8.44 ng

2κ₂ κ

1 1 + ∆²/κ²

−1 κ²

(Ωij+ ∆)²+κ² − κ² (Ωij−∆)²+κ²

−1

.

(9.28d)

For cavity and membrane parameters used in the measurements below, {κ/2π, L,∆, λ,Ω₆₆/2π, Q₆₆, m₆₆, r_m, η} ≈ {10 MHz, 742 µm, 5 MHz, 935 nm, 5 MHz, 1.5×10⁶, 8.44 ng, 0.42, 0.64}, the full expression for weak optical damping (Eq. 9.24) gives Γ⁶⁶_opt/2π≈ 4.0(κ/2κ2) kHz for hPouti= 100µW. The resulting effective temperature is and T_{ef f}⁶⁶ ≈ 0.25(2κ2/κ) K, corresponding to a thermal occupation number ¯n66 ≈ 1.0×10³(2κ2/κ), which represents significant optomechanical cooling for moderate circulating powers of∼1 W. We now test these predictions.