Later, Euler showed the existence of a repulsive force in the context of the longitudinal wave theory of light (credited to Huygens). In the inverse, sideband-unresolved regime (κ/4ωm>1), the speed of the off-resonance transition (red arrow in Figure 1.1) is sufficient to induce significant mode heating, making the quantum ground state of the mechanical mode out of reach.
Previous Work within the Painter Group
A change in the position of the end mirror modifies the cavity length, shifting the frequency of the optical cavity mode by ∆ωo=ωoˆx/Leff. In the cavity-bath mode interaction limit, the bath noise operators satisfy.
Semiclassical Derivations of Observed Spectra
Sideband-Resolved, Large Detuning Limit
For coherent mechanical oscillations where β0 is a simple complex number, we can write the single-sided power spectral density of the detected optical output power close to ωm (given by the autocorrelation of P(ω) =~ωo|αout|2) as. For ∆/κ >1 the measured sideband frequency (ω`+ωm) is blue of the pumping frequency, so from quantum theory we will consider normally ordered operators (ˆb†ˆb).
Modified Result for a Weak Detuned Probe
Classically, the ensemble averages of these products can be calculated for a thermal state from the Bose-Einstein distribution, given in the frequency domain by hβ∗0(ω)β0(ω))i= 1/(e~ω/kBTb −1) where kBer is the Boltzmann constant andTbis bath temperature. The transmission frequency response of the cavity at the probe frequency ω`+∆p for a fixed ∆/κ >1 is simple.
Quantum Mechanical Derivations of Dynamics
Sideband-Resolved, Mechanical Frequency Detuned Limit
We then see the importance of κ/4ωm<1 (κ/4ωm1 ideally) in the attempt to cool the phonon occupation of the mechanical state below unity. Similarly, we can say that at ∆ =ωm the optical bath is not sensitive to the zero-point fluctuations in the mechanical system.
Crystal Theory
- Plane Waves
- Translational Symmetry
- Bandgaps and Localization
- Phononic Bloch States
The range k can be further restricted to the irreducible Brillouin zone 0≤k < π/a with time-reversal symmetry (valid for all systems studied here). In principle, therefore, we can derive the phonon Bloch state (analogous to the photon Bloch state), denoted by the wavenumber, k, with the corresponding spectrum of eigenvalue solutions, ωn(k).
Index Guiding
Similar to above, we can assume a time harmonic solution and write the governing equation as. The modes below the light line have imaginary wave vector amplitudes (k2⊥ = (n2ω/c)2−kk2 < 0) in ε2, resulting in exponential decay away from the interface.
Optomechanical Coupling Rate
Moving Dielectric Boundary
Returning to the original problem and parameterizing the surface with (u, v), we can define the surface unit normal, n(u, v), at the interface ˆ ε1–ε2. Still in the local coordinate system, the electric field can be divided into components perpendicular to the boundary, E⊥, and parallel to the boundary, Ek.
![Figure 3.3: Moving dielectric boundary. a, From [77], schematic showing an infinite space of permittivity ε 2 in which we place a closed volume of permittivity ε 1 (solid line) with a perturbation specified by the surface function h(α, u, v) = Q(α, u, v) ·](https://thumb-ap.123doks.com/thumbv2/123dok/10405248.0/41.918.339.636.83.215/dielectric-boundary-schematic-infinite-permittivity-permittivity-perturbation-specified.webp)
Photoelastic Effect
The symmetry of ε and S also requires pijkl =pjikl =pjilk =pjilk, so it is convenient to use a contracted index notation (AppendixA.8) where. If not, we can construct a rotated form of the photoelastic tensor, shown in Appendix C.
Designing Nanobeams with Numerical Simulations
Unit Cell Design
Finally, the frequency tuning of d, the optical X-point, ande, the Γ-point mechanical state of interest, corresponding to change from the nominal to the defective unit cell, while keeping the kvector constant, is shown. Based on this intuition, we choose an optical mode drawn from the X symmetry point of the dielectric band and a mechanical “breathing” mode (with a shear field predominantly along their axis) drawn from the Γ point (from the highlighted bands in Figure 3.4c , f).
Cavity Design
For a computationally expensive fitness function with a large parameter space, a good choice of optimization algorithm is the Nelder-Mead method [91, 92]. Since the Nelder-Mead method is inherently an unbounded search algorithm, we must ensure that the .
Mechanical Quality Factor
Finally, we consider losses attributed to acoustic energy leaking into the substrate through the clamping points of the nanobeam. The 2D “cross” phononic crystal design [62] has previously demonstrated this with great success [63], and the wide tunability of the gap essentially allows the mechanical loss technique to be decoupled from the nanobeam design.
Nanobeam Designs
Summary of device designs and associated parameters for backaction cooling, presented in chronological order (left to right). FEM simulations corresponding to the normalized a, Ey field of the fundamental optical mode, b, displacement field,|Q|/max{|Q|}, of the fundamental breathing mode,c, surface integrand in (3.39), from which shows the individual contributions to gmb, end, volume integrand in (3.47), where the individual contributions togpe are shown.
Lab Procedure
Lithography Adjustments
The red line is the mapping between the realized dimensions and the dimensions of the lithographic features, and the black line shows how close the realized dimensions are to the correct dimensions. Inverting this function (Figure 4.3b–d) and entering the desired feature dimensions should then produce a lithography pattern that results in fabricated nanobeams of the correct size.
Silicon Surface Passivation
With careful calibration of the laser power arriving at the optical cavity, the optomechanical coupling rate, g, can be derived in two ways: via the broadening of the transparency window in electromagnetically induced transparency (EIT) reflection spectroscopy, and by broadening the mechanical mode peak in the power spectral density (PSD) of the optical signal amplitude measured on a high speed detector connected to a real-time spectrum analyzer (RSA). Calibration of the optical and electronic gains of the components to the optical cavity also allows the phonon occupation, ¯n, of the mechanical mode to be extracted from the RSA trace mechanical mode thermometry.
Experimental Setup
A small percentage (10%) of the laser intensity is sent to a wavemeter (λ-meter) for passive frequency stabilization of the laser (frequency/detonation stabilization is described in Appendix D.4). The alternate position of SW2 is used to calibrate the optical gain of the EDFA, and the nanosecond photodetector (D3) is used to measure the cavity DC transmission response.
EIT Measurements
If in the narrow regime we assume that ∆ = ωm (that is, we are detuned by a mechanical frequency, on the red side of the cavity), then in the sideband resolved regime this is the case. This quantitatively describes the EIT transparency window induced by the interaction of the optical field with the mechanics, and is spectrally an inverse Lorentzian width γ = γi + γOM with vanishing resonant reflection in the large cooperativity limit.
Mechanical Mode Thermometry
- Calibration of Input Power
- Calibration of EDFA Gain
- Calibration of Electronic Gain
- Optical Characterization
- Mechanical Characterization
- Optomechanical Coupling Rate Characterization
Then we lock the pump on the red side of the cavity (at ∆ = +ωm) and measure the entire width of the line, γ(red)=γi+γOM. Plot of measured (#) mechanical bath temperature (Tb) versus cryostat sample installation temperature (Tc).
Results
- Bath Temperature Characterization
- Device Characterization
- Backaction Cooling
- Error Analysis
Due to the low effective temperature of the laser drive, the mechanical state is not only dampened, but also cooled significantly. To evaluate the efficiency of the optical transduction of the mechanical motion, we also plot in Figure 5.5f the measured background noise PSD or level of imprecision.
Modifications to Intrinsic Mechanical Damping
Temperature Dependence
Using the calibration procedure discussed in section 5.3.1 for the pin, the error lies in determining the L0 and L1 losses. As such, we can estimate the cavity temperature by looking at the shift in the cavity frequency, starting from a known temperature.
Intracavity Photon Number Dependence
This analysis can be applied to the wavelength shift data for various input powers at low temperature (Figure 5.6b, red circles) to determine the heating due to the photon population inside the cavity. These measurements are performed at a low number of photons inside the cavity, which makes the effects of free carriers negligible (see Section 5.5.2).
Combined Loss Model
The deviation of the data from this prediction indicates an additional loss channel associated with the presence of free carriers. The addition of a free carrier-related loss channel is confirmed by pumping the silicon sample above the band gap with a 532 nm solid-state green laser, which directly stimulates the production of free carriers.
Noise Considerations
Phase Noise
An exaggerated plot of the impact of phase noise on measured power spectral density. The individual contributions to the phase noise-modified normalized spectral density are shown in Figure 5.10.
Amplifier Noise
This is plotted with (purple dotted curve) and without (black dotted curve) the optical insertion loss between the cavity exit and photodetection. The deviation of the measured SNR curve shape from the other cases is mainly attributed to the variance in the excess noise added by the EDFA amplification.
Measurement Imprecision
FEM simulations of the mechanical breathing mode of the nanobeam cavity yield a moving mass meff = 311 fg and a corresponding zero-point oscillation amplitude xzpf = 2.7 fm.
Future Directions
Further Device Improvements (5GHF Design)
Device Characterization
Backaction Cooling
FEM simulations corresponding to the normalized a, Ey field of the fundamental optical mode, b, displacement field, |Q|, of the fundamental breathing mode, c, surface integrand in (3.39), showing the individual contributions to gmb, and d, volume integrand in (3.47), showing the individual contributions togpe. The black trace corresponds to the measured noise floor where the driving laser is detuned far from the cavity resonance.
Free Space Couplers
Numerical Simulations
A direct finite element far-field simulation of an optical cavity mode is quite computationally expensive (requiring a simulation volume comparable to the far-field distance, typically several microns). Each cavity mode can thus be characterized by a best-fit set of beam parameters and a normalized far-field mode overlap.
Fabrication and Preliminary Results
The W1 waveguide acts as a generic gate for free-space couplers, allowing them to be easily combined with different photonic crystal cavity geometries, such as asb, face-to-face coupling to a 5G device, so that we have the desired single-sided coupling scheme, orc, side-coupled to a nanobeam that allows maintaining a phononic shield around the device. The broad peak indicated by the dashed red line is the lossy free-space coupler cavity mode (with a 3 dB bandwidth of ~8 nm or ~100 GHz and a maximum input-output coupling efficiency of 20%).
Motional Sideband Asymmetry
Observing Zero-Point Motion
For ∆ =ωm (of the probe) the resulting spectrum is given by (2.63) with a change in the cooling beam damped only phonon occupancy, ¯n, due to opposition of the probe beam (expressed as the cooperativity of the probe, CP) ,. Integrating (6.6) and (6.7) near ωm (with the appropriate proportionality constants and after subtracting the shot noise background) yields the respective spectral power near ωm, Pωm,+ (anti-Stokes sideband power; proportional to 1+ Cn¯ .p ) and Pωm,− (Stokes sideband power; proportional to 1−Cn+1¯ .p).
![Figure 6.7: Measuring zero-point motion. From [140]. a, The mechanical mode occupancy,](https://thumb-ap.123doks.com/thumbv2/123dok/10405248.0/98.918.171.804.95.460/figure-measuring-zero-point-motion-mechanical-mode-occupancy.webp)
Fourier Transform
Delta Functions
Spectral Density
Commutation Relations
Differential Equations
Trigonometric Identities
Lorentzian Function
Contracted Index Notation
Careful consideration of what to consider the "system", which contains the degrees of freedom of interest, and what to consider the environment (everything else) is essential to providing an accurate description of dynamics. One solution is to approach the environment as a "heat bath" and consider its action on the system as a random perturbing force, an idea originally used to describe the classical Brownian motion of a particle in a viscous to describe liquid [144].
Quantum Langevin Equation
We first assume that in the absence of ˆHint the time evolution of ˆ is simply ˆd(t) = ˆde−iωot, where ωo1 is the resonant frequency of the system. The former oscillate rapidly compared to the latter, have a vanishing cycle average on the 1/ω time scale and can be neglected: the rotating wave approximation.
Input–Output Operator Correlations
Single Mode Cavity Coupled to a Thermal Bath
If the crystal unit cell coordinate system is not aligned with the cavity coordinate system, we can apply a rotational transformation to the photoelastic tensor to align the two. From a practical point of view, the silicon wafers used in the fabrication process are (100) wafers, so we are only concerned with rotational transformations of the in-plane shape.
Equipment Listing
JDSU Switches
Electro-optic Amplitude Modulator
Laser Frequency Stabilization
Lock-in Method
We can fit the signal measured at lock-in using this model, which makes it possible to determine|∆|(the sign can be determined by optical characterization ofωo. and knowledge of the laser wavelength). In practice, the uncertainty in the tuning is dominated by fluctuations in the optical cavity and the resolution of the lock-in measurement.
Network Analyzer Method
The amplitude (√ .X2+Y2) of the reflected, modulated (at ωLI) probe signal as a function of probe detuning, ∆p, measured by a lock-in amplifier (LI in Figure 5.1). For higher driving powers at room temperature, there is additional noise in the signal related to the thermal relaxation time of the nanobeams (on the order of kilohertz to megahertz).
Continuous Flow Liquid Helium Cryostat
From this view you can see the copper thermal braiding that connects the sample holder to the cold finger via copper blocks. In the image, the sample is attached to the copper stage with copper tape, but we have since added a copper sample clip for better thermalization.
Dimpled Fiber Taper
Instrument Control
We note in this image that the area of the taper that touches the silicon is on the left side of the device (discolored area in the dimple) and that the taper is at a slight angle compared to the nanobeams. The way we have touched in the figure allows a part of the taper to be firmly planted on the side of the nanobeam (in the phonic shield region), while maintaining a sufficient distance from the optical cavity (the taper starts bend upwards as we pass the discolored part moves to the right) so as not to adversely disturb the optical field.
Functions
Geometry Indices
4% base: ('center', 'corner') (default 'center') which specifies where the anchor point is relative to the bounding box. 5% Returns the indices, the geometric elements of dimension dim inside a bounding box of size (dx, dy,dz), anchored at pos.
Perturbation Theory
3% Calculates the optomechanical coupling degree between the RF module/electromagnetic waves solution object and the structural mechanics module/solid, stress-strain solution object. 5% Returns the calculated optomechanical coupling rate g and an S object containing a breakdown of the individual contributions in first-order perturbation theory for moving boundaries and the photoelastic effect.
Miscellaneous
Zendri, "Feedback cooling of the normal modes of a massive electromechanical system at submillikelvin temperature," Phys. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys.
