Optical resonators powered by low-noise lasers provide a quiet and well-understood means of sensing and controlling mechanical motion using the force of radiation pressure. Our main experimental result is the radiation pressure-based cooling of a high-order, 4.8 MHz drum drum mode from room temperature to 100 mK (500 phonons).
Background
Using this system, we have been able to cool a radio frequency (Ωm/2π= 4.8 MHz) vibrational state of the membrane from room temperature to an occupation number of ¯n≈500. In this thesis, I will elaborate on the development of our first-generation "membrane-in-the-middle" experiment in the Kimble group.
Outline
In Chapters 2-3 we discuss the fundamental dynamic properties of optomechanical systems and the peculiar steady-state properties of the "membrane-in-the-middle". In Chapter 10, we investigate a fundamental roadblock to ground-state cooling associated with Brownian motion of the mirror substrates.
Mechanical Oscillator: 1D Model
An important utility of power spectral density is that it can be used to predict the outcome of passing a noisy signal through a linear system. The step response function features a Fourier transform, g(Ω), the so-called “frequency response function” of the system.
Optical Cavity: 1D Model
This property defines a recursion relation between the total forward propagation field inside the cavity at time t and the forward propagation field at a later time+Trt [31, 32]:. Now consider the special case where the envelope function of the input field has a constant value, corresponding to a monochromatic input field with frequency ω0.
Optomechanical Interaction: 1D Model
Reflection/Transmission Coefficient for a Dielectric Plate
In our system, we will often be interested in the case where the dielectric sheet (eg the film inside our optical cavity) is embedded in air or vacuum: n1=n3≈1. We will be interested in the optical properties of the membrane in the near-infrared, especially between 800 and 1064 nm, where it is assumed that Im[nm] 1.
Reflection/Transmission Coefficient for a Dielectric Mirror Coating
The effect of the substrate can be absorbed in the effective index by defining n0ef f =nef f/√ n0nS. The penetration away from the center of the coating curve is even greater (at 935 nm, it appears to penetrate a few hundred nanometers deeper for the D1306 coating).
Fabry-Perot Cavity
The cavity transmission coefficient plays a central role in several aspects of the experiment. The finesse is also a measure of the resonant build-up of the circulating power in the cavity.
Dielectric Film inside a Fabry-Perot Cavity
MIM Cavity Resonance Condition, Optomechanical Coupling
To solve this transcendental equation, we can first assume that the membrane is close to the center of the cavity, zm= 0. The optomechanical coupling reaches a maximum when the membrane is located halfway between a node and an antinode in the intracavity field.
Linewidth and “Finesse” of the MIM Cavity
The single, dotted, light green line corresponds to the analytical expression for a lossless membrane (Eq. 3.24). We find that in the absence of absorption, the cavity finesse is indeed modulated according to the envelope function given in Eq.
Power Transmission/Reflection for the MIM Cavity
The reduction is most pronounced when the membrane is placed close to an antinode of the intracavitary field. The effect is most pronounced when the membrane is placed close to an antinode of the intracavitary field.
Conclusion
Fabrication
High stress silicon nitride is produced using a slightly modified recipe [41] resulting in a stoichiometric Si3N4 chemistry. In contrast, the stress involved in our commercial high-stress films is closer to T≈900 MPa.
Material Properties
SiN Membranes: Mechanical Description
Transverse Vibrational Modes
The right and left sides of the elastic wave equation (Eq. 4.1) compare the kinetic energy with the potential deformation energy due to bending stiffness and stress, respectively. To compare the roles of stress and elasticity, we can examine the dispersion relation of the mechanical frequency Ωij in the case of D = 0 and T = 0.
Dynamical Equation of Motion
Characterizing Mechanical Dissipation: Concept of Q
Steady-State Approach
- Driven Vibration
- Brownian Vibration
Thermal fluctuations of the membrane are characterized by their power spectral density (eq. 2.7), as described in section 2.2 (see also [29]). The power spectral density can be measured with a spectrum analyzer and fit to a Lorentzian, which requires the resolution bandwidth of the analyzer to be less than Γm.
Transient Approach: Ringdown Technique
As discussed below, we found this method to be difficult for high-quality mechanical resonances (Qm >106, Ωm >2π·1 MHz) due to the slow mechanical frequency shift.
Displacement Readout
Measurement Sensitivity
The optical field is collected in a photodiode, which produces a photocurrent(t) =RPout, where Ris is the responsivity of the detector. To determine the sensitivity of the displacement measurement, we can compare the power spectral density Si(Ω) of the photocurrent fluctuations produced by the etalon length fluctuations SL(Ω) with the photocurrent shot noise fluctuation spectral density Sishot(Ω ) [53, 54] .
Apparatus for Characterizing Membrane Mechanics
- Etalon
- Vacuum Chamber
- Optical Layout
- Spectrum/Network Analyzer
- Spectrum Analyzer Measurement
- Network Analyzer Measurement
Rough transverse alignment is obtained by monitoring scattering from the chip surface on a CCD camera mounted on the side of the chamber. Because in this case B < Γm/2π, the shape of the power spectrum is proportional to the underlying thermal noise power spectral density, given by Eq.
Measurements
Thermal Noise Spectrum: Static Mechanical Properties
In the top frame of Figure 4.4, we hone in on the fundamental drum mode of the membrane. In the figure, we show a fit to a Lorentzian whose linewidth is obtained using an off-ring measurement, as described in the next section.
Ringdown Measurement of Q m
- Q-factors of Higher-Harmonics — A Single Trial
A ringdown measurement of the 823 kHz fundamental mode corresponding to the thermal noise peak in Figure 4.4 is shown in Figure 4.6. The noise in the long term measurement is due to the Brownian motion of the diaphragm.
Compilation of Mechanical Q Measurements
Influence of Chip Mounting
For the measurements described in this chapter, we had to somehow attach the chip to the mirror base of the standard. The extent of the influence of Q depends on the size of the chip and membrane, which is more sensitive for thin membranes and chips.
Influence of Membrane Thickness and Substrate Thickness
We tried to glue the chip firmly over its entire surface using a liquid UV epoxy (Norland 81). Without exception, the best results we have observed are for a vertically oriented setup where the chip simply rests under its own gravity ("free standing") on the surface of a smooth mirror or on three points provided by resting on a curved mirror or a disc ring (see left side of Figure 4.3, each geometry gives similar results).
Influence of Membrane Window Size
We have found that at similar mechanical frequencies, the Qm for the wm = 0.25 mm membrane modes are approximately half that of the wm = 0.5 mm membrane. A subset of the results fordm= 50 nm membranes in which the chip dimensions are fixed atdchip= 0.2 mm thick and wchip= 5 mm wide is shown in Figure 4.10.
Summary of Q Measurements, Comparison to Clamping Models
Clamping Mechanisms
298 K (4.21) in the molecular flow regime, which is characterized by the mean free path ` between gas particles (with a diameter d0), which is greater than the largest dimension of the oscillator. Taking into account this discrepancy by adding an ad-hoc internal loss term, they can predict the differences in the quality factors of the various higher harmonics for the same square diaphragm with an accuracy of a factor of ~2.
Concluding Remarks
In this chapter, I present a top-down description of our first-generation "membrane in the middle." In this chapter, we explain the architecture and operation of the membrane-in-the-middle apparatus shown here.
Cavity Design
Mirrors: Coating and Substrate
At the center of the deposition curve, approximately λc ≈850 nm, the mirror reflectance is Rmirror≈0.999985. However, as usual, handling mirrors resulted in a slightly larger loss of ≈5 ppm.
Cavity Construction
- Cavity Parameters
The edge of the super-polished surface was then beveled at a 45-degree angle to reduce the diameter of the mirror surface to 1mm. For many of the measurements discussed in this thesis, an operating wavelength of 935.5 nm was used, in which case the cavity waist is 35.7 µm.
Nanopositioning System
- Picomotors
- Membrane Holder
To incorporate tip/tilt, we sourced the stiffest kinematic mount available from Newport (Suprema SN100) for the top of the translation stage. A HeNe fringe formed between the membrane chip and the back of one of the cavity mirrors (see Figure 5.7) has been used to verify the unidirectional step repeatability of the Picomotor over multiple fringes.
Hardware: Vibration Isolation, Vacuum System, and Optical Layout
Vibration Isolation
Grey, black, pink, red and green traces correspond respectively to acceleration on top of the laboratory floor, optical table before re-balancing, optical table after re-balancing, original VIS (after re-balancing) and new VIS. Grey, black, pink, red and green traces correspond respectively to acceleration on top of the laboratory floor, optical table before re-balancing, optical table after re-balancing, original VIS (after re-balancing) and new VIS.
Vacuum System
Optical Layout: Locking and Probing the Cavity at Variable Detuning
- Two-Probe Scheme
- Lock Error Signal and Displacement Measurement
- Stabilizing the MIM System
The main disadvantage is that the majority of the input field is reflected directly from the cavity. Stabilizing the laser-cavity tuning of the cavity to within 10% of the linewidth (FWHM) requires reducing the effective longitudinal noise of the cavity to δL = 10%×λ/2F ≈5 pm×(104/F), which is within an order of magnitude of the Brownian displacement at room temperature (rms) for the ground state of the 50 nm x 500 µm x 500 µm membrane used in the initial cooling experiment (Chapter 9).
Characterizing the End-Mirrors: Cavity Linewidth and Finesse
To measure the cavity linewidth, we monitor the transmitted cavity power while sweeping the cavity length across the resonance for the TEM00 cavity mode. The probe is slowly moved through resonance (~1 ms 1/γ) by applying a sinusoidal voltage to a shear piezo below one of the end mirrors (Figure 5.2).
Optomechanical Coupling of the Membrane
However, we have measured time and time again by stepping the membrane along the cavity axis and following the cavity resonance by manually tuning the ECDL wavelength. The laser frequency was monitored using a commercial optical spectrometer with 100 MHz resolution (Burleigh WA-1600).
Characterization of Membrane Optical Absorption
Ringdown Measurement in a Long Cavity with High Finesse
Ringdown measurements were made with the membrane repeatedly translated into and out of the cavity state. The fine scale (inset) corresponds to translation of the membrane between successive nodes in the intracavity field.
Linewidth Measurement in a Short Cavity with Moderate Finesse
Finesse is defined as the free spectral range of the bare cavity (c/2L) divided by the measured linewidth of the MIM cavity, γ= 2κ (FWHM):F =πc/Lγ (here γ is in angular units). The coarse scale corresponds to displacement by a significant fraction of the cavity length, L (here in steps of ~0.05 L).
MIM Cavity Transmission Vs. Membrane Position
Solid pink and gray correspond to the extrema for a model assuming {Im[nm],Re[nm], dm, L,F0}. For comparison with the model, we assume a transmission value of unity when the membrane is located at a node.
Concluding Remarks
Internal Modes of an Elastic Body: Displacement and Effective Mass
We will also assume that the eigenmodes comprise a complete, orthogonal set, so that each vibration of the elastic body can be written in the form ~u(x, y, z, t) = P. We will ultimately be interested in the displacement of a small piece of the membrane surface, defined by the size and location of the cavity mode that pierces the membrane.
Internal Modes of a Fabry-Perot Cavity: Hermite-Gaussian Modes
After this we will approximate the Gaussian beam as planar, so that the polarization vector is perpendicular to the axes of the cavity. For most of the experiments we've done in the lab, we use the lowest order cavity mode.
Optomechanical Coupling, “Effective Displacement”, and “Spatial Overlap” . 103
Mode φij is then described by an effective displacement δzijm ≡ ηijbij; δzmij corresponds to the displacement from the equilibrium position of the membrane required to shift the resonant frequency of the cavity by gm(zm)ηijby. This corresponds to a displacement of the equilibrium position of the membrane by an amount of δzm66=η66b66.
Multimode Vibration of the MIM Cavity and “Effective Displacement”
δzm,1,2 is called the (total) effective displacement of the membrane, mirror 1 and mirror 2, respectively.
Multimode Thermal Noise Spectrum
Membrane Thermal Noise: Examples
- Role of Spatial Overlap and Mechanical Quality
- Structural Vs. Velocity Damping
Gray and black lines correspond to measured and modeled internal vibrations of the cavity end mirror substrates, described in Section 7.3.2. Gray and black lines correspond to measured and modeled internal vibrations of the cavity end mirror substrates, described in Section 7.3.2.
Mirror Substrate Thermal Noise
- Effective Mass Coefficients
- Mirror Substrate Noise
- End-Mirror Coupling in a MIM Cavity
The optomechanical coupling (g−) with the antisymmetric (cavity length changing) displacement of the mirrors (δz−) is a complex function of the membrane position, but reduces to twice the canonical coupling (2g0) when the membrane is located in the middle of the cavity. In the laboratory, with the help of the cavity transfer function, we transform the frequency fluctuations of the cavity into
Input-Output Model of the MIM Cavity
Two-Mirror Model
By noise, we mean an arbitrary signal, for example a noisy photocurrent, described by a single-sided power spectral density, Si(Ω) (equation 2.9), with units of A2/Hz, normalized such that R photocurrent noise is characterized by: For a stationary cavity (δωc = 0) and monochromatic input field, Ein(t) =hEini, detuned from resonance with ∆≡ω0− hωci, the amplitude reflection and transmission coefficients of the cavity are given by:.
Extension of Two-Mirror Model to MIM System
Response Function of the Detuned Probe (DP) Transmission Measurement
Slow Modulation: Steady-State Treatment
The response function in this case is independent of Ω, since the cavity field inside the cavity is assumed to respond instantaneously to the change in the cavity resonance frequency.
Fast Modulation: Perturbative Treatment
When ωc is slowly modulated compared to the linewidth κ of the cavity, the resulting photocurrent modulation is approximated by differentiating the steady-state expression (Equation 8.8), yielding . Note that this equation holds for both the standard Fabry-Perot cavity and the MIM cavity in the "slow modulation" limit.
Response Function of the Pound-Drever-Hall (PDH) Measurement
Fast Modulation
The corresponding oscillating part of the reflected power is contributed by terms oscillating at frequencies ±Ω0±Ωm. 1 and neglecting terms fluctuating at 2Ω0, we get. 8.23d). The relationship between the error signal and the oscillations of the resonant frequency of the cavity is given by the response function G,ωc(Ω):. 8.25) In the limit where the membrane reflectance is small, our symmetric (r1=r2) MIM cavity is described by κ1≈κ2≈κ/2 and α≈1, which gives the transfer function:.
The Effect of Mode-Mismatch
In the limit that the reflectivity of the membrane is small, our symmetric cavity (r1=r2) MIM is described by κ1≈κ2≈κ/2 andα≈1, which gives the transfer function:. the one that is reflected directly, E~bin:. For the decoupled probe measurement, only the light that is coupled to the cavity passes to the transmission photodetector.
Shot Noise Sensitivity: What to Expect
8.37–8.38 to calculate the expected shock noise-limited sensitivity of the disassembled probe and PDH measurements to thermal drift. We express this sensitivity in terms of the “shot noise equivalent” cavity resonance frequency noise, Sωshotc (Ω), and the “shot noise equivalent” cavity length noise, SshotL (Ω) = Sωshotc (Ω )/g20.
Calibration of the Measurement Response Function by Phase Modulating the Input
Detuned Probe Measurement: Response to PM of the Input Field
In fact, the DP response function for instantaneous frequency modulation and resonant frequency modulation of the cavity are equivalent at a small modulation limit:
PDH Measurement: Response to PM of the Input Field
Experimental Walk-Through: Temperature Measurement Using the Detuned Probe
Spectrum Analyzer: Effective Noise Bandwidth
However, a laboratory spectrum analyzer has a finite filter bandwidth, and since we are dealing with narrow spectral characteristics (Γm/2π ~ 1 Hz for {Ωm/(2π), Qm), we must be careful in our interpretation of measured SV(Ω) The power transmitted through the cavity is monitored as a function of detonation, which varies by sweeping the length of the cavity.
Calibrating the Phase Modulation Depth
Fluctuations in transmitted power are monitored via the power spectral density of the transimpedance-amplified photocurrent produced by the cavity transfer photodetector, SV(Ω0). This value is monitored as a function of the separation between the probe and the secondary field to which the cavity is locked at resonance (this field is polarized along the opposite birefringent axis of the cavity).
Characterizing the Transfer Function
For this purpose, a phase modulation with a small modulation depth β <<2π at a frequency of Ω0≈Ω66≈2π·4.84 MHz is applied to the input field of the cavity ("probe" field). The method for adjusting the cavity declaration is described in section 5.3.3.1; it consists of locking the cavity to another beam with an adjustable frequency difference from the probe beam.
Uncertainties
Spatial overlap factor, η66: To determine the "spatial overlap factor", η66, between the (6,6) membrane vibration mode and the TEM00 mode of the cavity (Section 7.1.3), we measure the. Uncertainty in the determination ofη66 depends on the uncertainty in the derived position (x0, y0) and in the value used for the waist size wc of the cavity mode.
Calibrating the Electronics Downstream of the Photodetector
The natural frequencies of the drum modes are consistent with the value of 2.7 g/cm3 if the transverse dimensions and stress are assumed to match the values specified by Norcada (wm = 500 µm and T = 900 MPa, respectively). Using SEM imaging, the transverse dimensions of the membrane, {wm,x, wm,y}, have been shown to be accurate to <1% of the design value of {500 µm, 500 µm}.
Measurement Result
The span shown contains a Lorentzian feature proportional to the power spectrum of displacement of a single membrane vibrational mode SV(Ω) = |GT(Ω66)|2gm2η662 |Gi,ωc(Ω66)|6Sb66(Ω) and a narrow feature (delta) function convoluted with a Hanning window function) corresponding to phase modulation of the laser SV(Ω0). For the used ∼1 µW power and ∆/(2π) = 100 MHz detuning, we expect a negligible reduction in effective temperature due to radiation pressure feedback (we elaborate on this in Chapter 9).
Thermal Noise “Spectroscopy” to Determine the Spatial Overlap Coefficients 143
S is the surface area of the membrane and N is the number of independent ratios measured. Plot (a) is a typical power spectrum of a transimpedance amplified PDH error signal proportional to the cavity length noise spectrum.
Laser Frequency Noise
- Diode Laser
- Titanium-Sapphire Laser
All three curves are multiplied by a calibration factor obtained from a known phase modulation of the input beam (not shown). The broad peak at ~600 kHz is believed to correspond to relaxation-oscillations of the ti-sapp crystal.
Substrate Noise
Frad(b(t), t)≈ −koptb(t)−mΓoptb(t).˙ (9.2) From the point of view of linear response theory (section 2.1), the optical spring and damping forces appear as a modification of the mechanical sensitivity, χ(Ω) , defined as follows: In the presence of the position-dependent radiation pressure force,. the susceptibility of the mirror to the Langevin force becomes:.
Model for Radiation Pressure Damping
- Two-Mirror Resonator
- Extension to the MIM System
The sign of Γopt depends on the sign of the correlations between the mirror position and the radiation pressure. The radiation pressure force Frad(t) can be calculated from the gradient of the intracavity energy, with respect to the mirror position.
Experimental Parameters: What to Expect
- Intracavity Photon Number
- Effective Mass and Zero-Point Amplitude
- Optomechanical Coupling and Spatial Overlap
- Optical Damping and Spring Shift
- Effective Temperature and Thermal Occupation Number
However, with the membrane in the cavity, decay rates {κ, κ1, κ2} are all functions of the membrane position. The ratio 2κ1/κ can be determined from the resonant transmission/reflection of the MIM cavity (Eq. 8.5).
Measurements of Optomechanical Cooling
Optomechanical Cooling with a Diode Laser
Figure 9.2 shows a thermal noise measurement of the (6,6) mode with a "strong" input field declared red and provided by a 935 nm diode laser. For the measurements shown, hPouti is .23µW, so rather than shot noise, the progressively decreasing noise floor here reflects an apparent shift in the constant noise output of the photodetector transimpedance amplifier.
Multimode Cooling: Comparison to Model
Indeed, as shown in Figure 8.9, our 935 nm diode laser exhibits an effective cavity length noise of This background suggests a practical limit for cooling with the diode laser using current parameters at the level of ¯n66-1000.
In Figure 9.5 we change both the detuning and power to minimize and maximize the damping of the (6,6) mode. To illustrate this point, we have added in Figure 9.7 the measurement (black) and model (grey) of the end mirror substrate noise for the bare cavity, as discussed in Section 8.8.2.
Limits to Optomechanical Cooling in our System
Note κ= 2π×2.5 MHz corresponds to F = 40,000 for aL= 0.74 mm cavity, and smaller values of κ can be achieved either by increasing the finesse or using a longer cavity. Another outstanding goal is to achieve strong coupling between the ground state mechanical resonator and the cavity field.
Concluding Remarks
Our approach reduces cavity radiation pressure fluctuations associated with extraneous thermal motion, which can otherwise cause noisy membrane heating. In an optomechanical system with a cavity, this corresponds to structural oscillations other than the studied mode, which lead to extraneous oscillations of the resonant frequency of the cavity.
Extraneous Thermal Noise: Illustrative Example
This "substrate noise" forms a strange background to the "membrane-in-the-middle" system conceptualized in Figure 10.1 and detailed in [14]. Substrate thermal motion therefore constitutes an important roadblock for observing quantum behavior in our system [14].
Strategy to Suppress Extraneous Thermal Noise
Otherwise, all three are functions of the axial position of the diaphragm relative to its position in the cavity. As in Figure 10-3, the detuning and power of the science field are chosen to optically dampen the (3,3) membrane.
Experiment
- Substrate Noise Suppression with the Membrane Removed
- Combined Substrate and Membrane Thermal Noise
- Differential Sensing of Membrane and Substrate Motion
- Substrate Noise suppression With the Membrane Inside the Cavity
In Section 10.4.4 we combine these results to realize substrate noise suppression in the presence of the membrane. To “differentially sense” the noise shown in Figure 10.8, we use the probe field to monitor the resonant frequency of the TEM01 mode.
Extraneous Noise Suppression and Optical Damping: An Application
Discussion
Optical Cooling Limits
Summary and Conclusions
Appendix: Radiation Pressure Stiffening/Damping with Electro-Optic Feedback
