Substrate Noise

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

8.8.2 Substrate Noise

To measure the substrate noise, we remove the membrane from the cavity and perform the detuned probe measurement using the low noise ti-sapph laser (in this caseSL(Ω)|sub >> S_L⁰(Ω)|T iS). For this measurement, the operating wavelength was λ≈805 nm, the power on the transmission pho- todetector was hPouti ≈ 100µW, and the cavity linewidth was measured to be κ ≈ 9 MHz. A spectrum analyzer bandwidth ofEN BW ≈30 kHz was used. The inferred substrate noise is shown in Figure 8.9. Three major contributions to the measurement noise spectrum are shown: detector noise, shot noise, and the desired substrate thermal noise. Detector noise is recorded with the laser field blocked and shot noise is estimated by directly coupling 100 µW into the photodetector (by-

(a)

100 (b)

10

^-19

10

^-18

10

^-17

10

^-16

10

^-15

5 4

3 2

1

0.1 1 10

Diode laser FM noise Ti-Sapph laser FM noise Substrate thermal noise (model) 6

10

^-188 2 4 6

10

^-178 2 4 6

10

^-168

5 4

3 2

1

4 6 8

1

2 4 6 8

10

2 Substrate thermal noise 4 Shot noise

Photodetector noise

Figure 8.9: Measurement of substrate thermal noise and comparison to laser frequency noise. Sub- strate noise was measured by coupling ≈ 100µW of radiation from the low noise Ti-Sapph laser through the science cavity (TEM₀₀ mode) at detuning ∆ =−κ≈2π·9 MHz and monitoring the transmitted intensity fluctuations. In the upper plot we compare the relative contributions of detec- tor noise (black) and shot noise (green), which have been subtracted from the raw signal to obtain the red curve shown. In the lower plot we compare the measured substrate noise to the measured frequency noise of our diode (red) and ti-sapph (blue) laser. We also compare the substrate noise to a numerical finite element model, shown in pink.

passing the cavity). The shot noise curve shown in green has been corrected by subtracting the measured detector noise (black). For the detuned probe measurement we coupledhP_outi= 100µW through the science cavity at a detuning of ∆≈ −κ. The resulting spectrum shown in red is cor- rected by subtracting both the black and green curves. All shown curves are calibrated using the phase modulation method summarized in Eq. 8.73.

The substrate noise measurement reveals a dense spectrum of vibrational peaks with the lowest order peak occuring at 785 kHz. The vibrational peaks between 785 kHz and 5 MHz are spaced by

∼10–100 kHz and appear to have a common quality factor of roughlyQm= 650. The off-resonant noise is at a level of 1–2×10⁻¹⁸ m/√

Hz. The noise peaks reach a level of 1–4×10⁻¹⁷ m/√

Hz. The peaks shown here are slightly smoothed by the large bandwidth of 30 kHz used. In Figure 8.9 we compare the substrate noise to the diode laser and ti-sapph laser noise as well as a numerical finite element model developed by our post-doc Kang-Kuen Ni. This model — which is in remarkable agreement — is discussed in detail in Section 7.3.2. Also note that the inferred shot noise sensitivity of 1–1.5 m/√

Hz is a factor of two larger than the predicted sensitivity of 0.7 m/√

Hz based on incorporating the above parameters into Eq. 8.42 (using a photodetector responsivity ofR ≈0.5 at 805 nm).

Chapter 9

Optomechanical Cooling

In Chapter 2 we derived the optical spring and damping rates experienced by a compliant end-mirror in a Fabry-Perot cavity. In this chapter we apply that model to our “membrane-in-the-middle”

apparatus (Figure 9.1), consisting of a short, high-finesse (F ≈10⁴) Fabry-Perot cavity coupled to a stiff, high mechanical quality (Ω_m/2π >10⁶, Q_m >10⁶) Si₃N₄ membrane. We anticipate that our system can be used to realize significant optomechanical cooling rates, possibily even to access the quantum regime from room temperature [14]. In the lab, however, we are presently limited to

¯

n > 100, owing to various technical challenges described below. Significantly, we note that as of the time of this writing several groups have managed to prepare the quantum mechanical ground state of a NEMS-scale mechanical oscillator using a combination of cryogenic and optomechanical cooling [15, 6, 10], while the Yale group’s efforts with SiN membranes in a cryogenic version of [23]

is currently limited by laser phase noise [61].

To help summarize the reasoning leading up to this experiment, this chapter begins with a review of optical spring and damping forces, with an emphasis on extending the canonical two-mirror system described in Section 2.4 to the MIM system. We carefully step through the application of a simple “weak damping” model to our system, showing that significant optomechanical cooling can be observed for only microwatts of input optical power. We then present measurements we have performed to validate the weak damping model. Operationally, the experiment consists of measuring the membrane’s displacement using a strong red-detuned probe (Section 8.3). In our main example we demonstrate that with ∼ 1 W of power circulating in the cavity, the mean- squared Brownian vibration amplitude of a higher-order membrane mode (the (6,6) mode, with frequency Ω66/2π = 4.83 MHz and mechancial quality Q66 = 1.5×10⁶) can be suppressed by a factor ≈ 3×10³. This corresponds to a reduction in temperature from Troom ≈ 300 K to an effective value of T_{ef f}⁶⁶ ≈ 100 mK, or equivalently, a reduction in mean thermal occupation from

¯

n^room₆₆ =k_BT_room/~Ω₆₆≈1.3×10⁶ to ¯n₆₆ ≈500 phonons. The amount of “refrigeration” we have achieved is limited by the amount of power we can couple into the cavity for technical reasons and — we anticipate — the background intracavity intensity fluctuations associated with thermal motion

-250 0 250

(ar b)

(a)

(b)

(c)

(ar b)

(�m)

Figure 9.1: (a) Experimental overview: membrane window (thicknessdm= 50 nm, square dimension wm = 500 µm) in a short cavity (lenth L = 742 µm, finesse (F) ≈10⁴). (b) Illustration of (3,3) membrane mode amplitude with b33φ33(x, y). (c) Intensity profile of TEM00 optical mode |ψ00|² (red solid line) compared to amplitude of membrane modes (2,2) (top) and (6,6) (bottom) with a cavity spot size ofwc= 35.6µm.

of the end-mirror substrates, which we address in the next chapter.

9.1 Radiation Pressure Back-Action: Compressed Review and Extension to MIM System

9.1.1 Compressed Review

In Chapter 2 we described the effect of radiation pressure on a compliant end-mirror in a Fabry-Perot cavity. The end-mirror in this — the “canonical” — optomechanical system (Figure 2.1) is envisioned as a rigid plate with effective (= physical) mass m attached to a massless pendulum spring. The mass-spring system is modeled as a driven, damped harmonic oscillator with displacement amplitude b(coinciding with the position of the mirror surface), resonance frequency Ωm, and energy damping rate, Γm. The driving force has two contributions: a Langevin forceFL, which describes Brownian motion of b, and a radiation pressure Frad = 2Pcirc/c, wherePcirc is the optical power circulating between the cavity end-mirrors. Crucially, optomechanical coupling between the position of the mirror and the resonance frequency of the cavity implies that the circulating power is a function of

mirror positionb(t):

m¨b(t) + Γmmb(t) +˙ mΩ²_mb(t) =FL(t) +Frad(b(t), t). (9.1) For sufficiently weak optomechanical coupling, the position-dependent radiation pressure force can be described as the sum of a force proportional to the position and the velocity of the mirror; these correspond to the the “optical spring” and “optical damping” forces, respectively:

Frad(b(t), t)≈ −koptb(t)−mΓoptb(t).˙ (9.2) From the standpoint of linear response theory (Section 2.1), the optical spring and damping forces manifest themselves as a modification to the mechanical susceptibility,χ(Ω), defined as follows:

χ(Ω)≡ b(Ω)

F(Ω); {b(Ω), F(Ω)}= Z ∞

−∞

{b(t), F(t)}e^−iΩtdt. (9.3)

In the presence of the position-dependent radiation pressure force,

F_rad(Ω) =−k_opt(Ω)b(Ω)−imΓ_opt(Ω)b(Ω), (9.4)

the susceptibility of the mirror to the Langevin force becomes:

χef f(Ω) = b(Ω)

FL(Ω) = 1/m

Ω²_m−Ω²+kopt(Ω)/m−i(Γm(Ω) + Γopt(Ω))Ω. (9.5a) The frequency dependence of{Γ_m(Ω),Γ_opt(Ω), k_opt(Ω)} is a consequence of the fact that a lossy mechanical resonator does not have a well-defined frequency. When both the intrinsic and optical spring are underdamped, then this frequency dependence can be approximately neglected. We will refer to this as the “weak damping” approximation

{Γ_m(Ω),Γ_opt(Ω), k_opt(Ω)} ≈ {Γ_m(Ω_m),Γ_opt(Ω_m), k_opt(Ω_m)} ≡ {Γ_m,Γ_opt, k_opt}, (9.6)

and we will approximate the optical spring as a small frequency shift:

χef f(Ω)≈ 1/m

(Ωm+ ∆Ωopt)²−Ω²−i(Γm+ Γopt)Ω (9.7a)

∆Ωopt≡kopt(Ωm)/(2m). (9.7b)

Thermal flucutuations ofb are described by the mechanical transfer function|χef f(Ω)|² acting

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.