Electromechanical cooling

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Input-output formalism

As the eld leaks into the readout port a measurable current is generated. The output of the circuit can be predicted by the input-output relation [35], giving the output eld out in terms of and in. This relation depends on the readout scheme [20]. Within the context of this work, two main congurations will be presented: the re ection and the hanger congurations. In both cases the input and output elds pass through the same circuit port.
The re ection conguration simply consists of a transmission line which capacitively couples to the circuit{see g. 1.8a. This line carries the input eld and the output circuit signal; its behavior is analogous to that of an optical circuit measured in re ection, and will have the same sort of response. In this conguration the input-output relation reads out = 􀀀in;c + p c: (1.36). In the hanger conguration, the circuit is inductively coupled to a transmission line which carries the information, as shown in gure g. 1.8b. This is almost equivalent to reading the re ection of a circuit, except that the output signal radiates symmetrically in both directions of the transmission line, meaning half of the infor mation put out by the circuit is lost. This conguration is nevertheless practical for measuring several cavities in a series, and is used in this work. For this conguration the input-output relation reads out = 􀀀in;c + rc 2.

Electromechanical cooling

As one might expect from the uctuation-dissipation theorem, the modication in the damping rate has consequences for the motion spectrum of the mechanical resonator. Interestingly, although we have previously found that the dissipation rate played no role in the RMS uctuations of the resonator (i.e. did not aect its temperature), the situation is now dierent. To show this, we solve for ^x in eqs. (1.50) to (1.52), and nd x[ ] = 􀀀1 e ( ) ^ Fth[ ] + ^ Fcav[ ] .

Silicon-nitride membrane electromechanics

Owing to the generic nature of the dispersive interaction, no fundamental constraint is imposed either on the shape, size, or frequency of the mechanical resonator. As a result, a plethora of resonators can be found in the literature of cavity electromechanics. In this section we brie y review the various possible options, and show the advantages of the system we ultimately opted for: a device based on silicon nitride (SiN) membranes.
First and foremost, to have the capacity to prepare quantum states of mechanical motion, the resonator must be in its ground state. If it must be cooled by the optoor electromechanical interaction, a necessary (but insucient) condition is that the mechanical frequency f m=2 must be larger than the thermal decoherence rate [44]. This can be expressed in terms of the so-called Qf-product, which must satisfy Qf > kBTenv h : (2.1).
Note that higher temperatures also result in low coherence times. Experiments are therefore often performed in cryogenic environments to ease the requirements on the device parameters. For cooling at Tenv = 20 mK as in this work, Qf & 109 Hz is theoretically required. This is in fact readily achievable but the Qf- product still serves as a good gure of merit for the mechanical resonator, and we will base our choice on this factor. Having long coherence times is conditional for highly perfomant in quantum memories [10], for coupling incompatible quantum systems such as for optical-to-microwave photon conversion [6{9], or for studying non-classical states of motion [16, 45, 46]. Maximizing this product starts with the right choice of material.

Choosing the mechanical resonator

Crystalline materials are an intuitive choice for obtaining high Q: they can be almost perfectly cleaved or etched along their at faces, resulting in low surface roughness and low loss [49]. Many ground-breaking experiments have been pursued using for instance quartz [40, 50] (Qf 1010 Hz in [40]) or silicon [12, 22, 51,52] (Qf 1015 Hz in [22] at 25 mK){an example of a Si resonator is shown in g. 2.1a. Amongst crystalline materials those exhibiting piezoelectric properties are interesting [9, 53] (Qf 1013 Hz in [9]) because they are intrinsically coupled to electromagnetic elds, which allows for integrated designs. For this reason, metals are also an interesting materials to make mechanical resonators out of [13, 17, 47, 54] (Qf 1012 Hz in [54]). Figure 2.1b shows a microwave cavity where the capacitance is made of a mechanically compliant aluminium plate; in g. 2.1c a nonlinear cavity is shown where the frequency is shifted by mechanically varying the coupling to a ux line. Other kinds of materials such as carbon nanotubes [55, 56], graphene [57], or super uid helium [58] have also demonstrated promising features.
Among the numerous possible platforms the current record holder for the best Qf-product is, perhaps surprisingly, made from an amorphous dielectric: SiN. The typical geometries for SiN resonators are thin beams (g. 2.1d) or membranes (g. 2.1e), which can yield moderate Qf-products (for their fundamental modes, Qf 1013 Hz [44] at room temperature). Yet in recent years engineered SiN resonators have demonstrated quality factors approaching the billions [24], and even at room temperature Qf > 1014 Hz [31]1; an example of such an engineered resonator is shown in g. 2.1f. So far, these engineered resonators have mainly been employed in the optical domain [59, 60], and no experiments have demonstrated quantum behaviors in an electromechanical system. Nevertheless, promising results regarding the implementation of SiN resonators in microwave circuits have been obtained [6, 61].
Furthermore, it it well understood that mechanical quality factors of systems limited by two-level-system losses, as is the case for SiN, increase at cryogenic temperatures [9, 62{65]. This suggests that the new generation of engineered SiN resonators  could signicantly exceed the performance of the best crystalline siliconresonators once cooled to cryogenic temperatures. SiN resonators can only be found in the form of beams or membranes, due to the fact that the mechanical properties of SiN only excel when it is clamped and highly stressed [48, 67, 68] (see chapter 3 for a more detailed discussion). As a general rule, beams are often used for compact, integrated designs; devices with membranes on the other hand are larger but modular: their components can be fabricated and characterized separately before being coupled, allowing each element to be characterized individually before being implemented. In this work the latter approach was preferred. We employ membranes, thin square sheets of SiN, with a typical side length of a millimeter and a thickness of approximately 100-200 nm. An example of the membranes fabricated for this work is shown in g. 2.2.

Electromechanical devices with SiN membranes

To couple a SiN membrane to an electromechanical cavity, essentially two approaches have been taken in the past, diering in the way in which the microwave cavity is created. One option is to fabricate it from a superconducting metal box (g. 2.3a). Inside of it the electromagnetic eld can resonate at specic frequencies. The mode is coupled to an antenna which can address the membrane motion. Alternatively, the cavity can made of a planar superconducting circuit fabricated on-chip, as illustrated in g. 2.3b. In both cases the membrane is fabricated on its own separate chip, and can be functionalized by the deposition of metal on its surface. The functionalized area is aligned to the metal plate of a separate chip, which is either the antenna of the 3D cavity or the capacive plate of the 2D circuit. thereby forming a capacitor whose capacitance varies with the position of the membrane (for a 3D cavity, that chip contains the antenna; in the 2D case, it contains the resonant circuit). This assembly is called a \ ip-chip » and is illustrated in g. 2.3c.
Since the mechanical resonator remains the same in both cases, we can compare them based on the mechanical loss 􀀀m, the microwave loss , and the cavity frequency shift per mechanical displacement2 G. The values of these parameters for devices found in the literature, for both 2D and 3D systems, are reported in table 2.1. We nd that overall, the 2D and the 3D approaches yield comparable devices, with one exception: in Ref. [66], an exceptionally low mechanical dissipation is found. That that is not unique to 3D cavities however, and we show below that a low mechanical dissipation rate can be obtained in 2D circuits as well. This being said the 2D approach presents the advantage of being more streamlined in design, as it does not require an intermediary antenna to couple microwaves and mechanics. Furthermore, it was expected that using 2D circuits would ultimately yield lower , as the eld is more strongly conned than 3D cavities, reducing the participation ratio of the amorphous SiN in microwave losses.

Fabricating the electromechanical device

The design of our microwave circuit is based on the optimized designs of Geerlings et al. [70]. It is a lumped-element circuit, meaning the inductor and the capacitor are discrete entities (see g. 2.4), and the sizes of the individual elements are well below the microwave photon wavelength. The cavity is addressed through a coplanar  waveguide (CPW) feedline: a metallic strip surrounded by two planes connected to the ground conducts the signal,. Note that placing the circuit in this manner with respect to the feedline places it in a hanger conguration.
We fabricate such circuits by evaporating a uniform niobium (Nb) layer on a bare Si substrate, and patterning the metal by UV lithography{see Ref. [43] for further details. To ensure that to be low the superconducting metal Nb is used, with a transition temperature of 9:3 K. It is used for instance in Ref. [6] and shown to allow for high coherence times [71]. The substrate used here is high resistivity Si ( > 10 k .cm).
The electromechanical device is then fabricated with a ip-chip design: the microwave circuit chip and the membrane chip are aligned and stacked, giving the overall structure shown in g. 2.5a. The alignment of the chips is done using a mask aligner, such that the metallic pad on the membrane faced (g. 2.5b) the capacitive plates of the circuit. They are then xed together with two drops of epoxy on diagonally opposite corners of the chips. Aluminium pillars approximately 300-nm thick serve the role of spacers (g. 2.5c).
As mentioned in the previous section the two main gures of merit for the system are the electromechanical coupling G and the Qf-product. To estimate these parameters m, 􀀀m, , and g0 are measured, and the results are given below. The measurements are taken in a wet Helium-3 cryostat, at a temperature of approximately 400 mK, unless stated otherwise.

Table of contents :

Introduction
1 Introduction to opto- and electromechanics 
1.1 Thermal motion of a harmonic oscillator
1.1.1 Equation of motion of a mode
1.1.2 The autocorrelation function
1.1.3 Noise spectral density
1.1.4 The uctuation-dissipation theorem
1.1.5 Noise spectrum of the harmonic oscillator
1.2 Optical measurement of motion
1.2.1 Position-dependent phase shift
1.2.2 The Mach-Zehnder interferometer
1.3 Cavity electromechanics
1.3.1 The intracavity light eld of a bare circuit
1.3.2 Input-output formalism
1.3.3 Dispersive coupling
1.3.4 Dynamical backaction
1.3.5 Electromechanical cooling
1.4 Summary
2 First electromechanical experiments 
2.1 Silicon-nitride membrane electromechanics
2.1.1 Choosing the mechanical resonator
2.1.2 Electromechanical devices with SiN membranes
2.1.3 Fabricating the electromechanical device
2.2 Characterization experiments
2.2.1 Characterizing the cavity resonance
2.2.2 Optomechanically Induced Transparency (determining m) .
2.2.3 Measuring g0
2.2.4 Ringdown measurement of 􀀀m
2.2.5 Summary
2.3 Cooling experiments
2.4 Discussion
2.4.1 The microwave cavity
2.4.2 The mechanical resonator
2.5 Summary
3 Membrane design and simulations 
3.1 Key parameters of SiN nanomembranes
3.1.1 Quality factor of a harmonic oscillator
3.1.2 Mode prole of a vibrating plate
3.1.3 Bending losses
3.1.4 Radiation losses
3.1.5 Scaling of the parameters with the membrane geometry
3.2 Phononic crystal membrane design
3.2.1 Engineering bandgaps in periodic structures
3.2.2 Localized defect states in PnC membranes
3.2.3 The curvature prole of defect and edge modes
3.2.4 Computing Qb of D1 and VE1
3.2.5 A \nal » consistency check
3.3 Mode coupling in PnC membranes
3.3.1 Coupling of lossy harmonic oscillators
3.3.2 Numerical analysis of edge mode coupling
3.3.3 Edge mode engineering
3.3.4 Double-defect membranes
3.4 Concluding remarks
4 Membrane fabrication and characterization 
4.1 Fabrication procedure
4.1.1 Wafer details
4.1.2 Releasing free-standing PnC membranes
4.1.3 Additional fabrication details
4.2 Experimental Setup
4.2.1 The vacuum chamber
4.2.2 A shot-noise limited optical interferometer
4.2.3 Driving the mechanical motion
4.3 Experimental methods
4.3.1 Measuring the spatially-dependent thermal spectrum
4.3.2 Measuring the mode prole
4.3.3 Ringdown measurement of the quality factor
4.4 Results for defect and edge modes
4.4.1 Measured thermal spectra
4.4.2 Measured mode proles
4.4.3 Measured Qb
4.5 Results for dimer membranes
4.5.1 Measured thermal spectra and mode proles
4.5.2 Determining the dimer coupling rate
4.6 Concluding remarks
5 Conclusion and outlook 
5.1 Single-defect PnC membranes
5.2 Preparing non-Gaussian states of motion
5.2.1 Direct coupling
5.2.2 Extrinsic nonlinearity
5.2.3 Counting single microwave photons
5.3 Concluding remarks
Appendix A Calibrating the resonator population 
Appendix B Scaling of the eective population 
Appendix C Coupled damped harmonic oscillators 
C.1 Eigenfrequencies
C.2 Eigenvectors
Appendix D Fabrication recipes 
D.1 Plain SiN membrane fabrication
D.2 Patterned SiN membrane fabrication
Appendix E Symbol list

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