We demonstrate atomic force microscopy (AFM) imaging with a microcantilever force transducer where an integrated superconducting microwave resonant circuit detects cantilever deflection using the principles of cavity optomechanics. We discuss the detector responsivity and added noise, pointing to its crucial role in the context of force sensitivity. Through analysis of noise measurements we determine the effective temperature of the cantilever eigenmode and we determine the region of detector operation in which the sensor is thermal-noise-limited. Our analysis shows that the force-sensor design is a significant improvement over piezoelectric force sensors commonly used in low-temperature AFM. We discuss the potential for further improvement of the sensor design to achieve optimal detection at the standard quantum limit. We demonstrate AFM operation with surface-tracking feedback in both amplitude-modulation and frequency-modulation modes.

Electromechanics with micromachined vacuum gap capacitors have brought optomechanical techniques to the palm of our hand. Cooling these devices down to the millikelvin regime forms the basis for developing quantum memories and allows us to delve into basic science experiments. To date, only aluminium has been used as device material, and the design has not evolved significantly since it was introduced. Even more, the potential provided by the integration of sensing capabilities was not in the focus of any application driven developments so far. Here, we present several improvements comprising of niobium as base material, a double electrical cavity design and integrated piezoelectric actuators, creating a platform for sensing. We achieve a vacuum optomechanical coupling rate of g₀/2π = 160 Hz, which can be employed for back-action evading techniques and mechanical noise squeezing of the first mechanical mode. We show coupling between the two electrical cavities and the first four mechanical eigenmodes. Furthermore, using the piezoelectric actuator for phononic pumping, we investigate coupling between the mechanical modes among each other. Based on these achievements, it is strongly believed that more research activities will be stimulated in the field of quantum sensing for up to now unexplored and unprecedented application scenarios.

In this thesis we investigate force gradient sensing based on the principles of cavity optomechanics, specifically in the context of atomic force microscopy (AFM). Tip-surface forces perturb the motion of a cantilever and the cantilever's change of motion is detected with an integrated compact low-noise motion sensor based on cavity optomechanics.  

To this end we design and fabricate probes on a wafer scale. The probes consist of a cantilever in the shape of a triangular Si-N cantilever released from a substrate of Si. We fabricate low-frequency, 700 kHz, cantilevers as well as high-frequency, 5 MHz, cantilevers allowing for operation in the resolved sideband regime. At the base of the cantilever we fabricate a superconducting microwave resonant circuit (cavity) patterned from a thin film of Nb-Ti-N. The microwave circuit has a resonance frequency of 4-5 GHz. The coupling between the cantilever and the microwave circuit is based on strain modulation of the kinetic inductance of a meandering nanowire placed at the base of the cantilever. The mechanical motion can thus be detected through the modulation of the cavity resonance frequency. 

We characterise the mechanical mode as well as the microwave circuit and demonstrate the novel strain dependent coupling. We investigate the losses of the microwave circuit and find that in the temperature range 1.7-6 K the losses are not dominated by thermal-equilibrium quasi-particles. We also explore the possibility of using the nonlinear current dependence of the kinetic inductance as means to parametrically amplify the motional sidebands produced by the optomechanical interaction. Finally, with a prototype scanner assembly for low-temperature AFM we detect force gradients and image a surface implementing a two-tone pumping scheme for the microwave circuit while actuating the mechanical resonator.

I denna avhandling undersöker vi kraftgradientavkänning baserat på principerna av optomekanik, specifikt i kontexten av atomkraftmikroskopi (AFM). Tipp-ytkrafter förändrar en vipparms rörelse. Denna förändring detekteras med hjälp av en integrerad kompakt kraftsensor med lågt brus baserat på kavitetsoptomekanik.

För detta ändamål designar och tillverkar vi sensorer på kiselplattor. Sensorerna består av triangulära Si-N vipparmar frigjorda från ett substrat av Si. Vi tillverkar låg-frekventa, 700 kHz, vipparmar såväl som hög-frekventa, 5 MHz, vipparmar där de senare placerar sensorn i den 'goda kavitetsgränsen'. Vid basen av vipparmen tillverkar vi en supraledande mikrovågsresonanskrets formad från en tunn film av Nb-Ti-N. Mikrovågskretsen har en resonansfrekvens av 4-5 GHz. Kopplingen mellan vipparmen och mikrovågskretsen är baserat på töjningmodulering av den kinetiska induktansen i den slingrande nanotråden placerad vid basen av vipparmen. Den mekaniska rörelsen kan på så sätt detekteras genom moduleringen av kavitetens resonansfrekvens.

Vi karakteriserar den mekaniska moden såväl som mikrovågskretsen och visar på den nya töjningsberoende kopplingen. Vi undersöker förlusterna av mikrovågskretsen och finner att i temperaturintervallet 1.7-6 K så domineras förlusterna inte av kvasi-partiklar i termisk jämvikt. Vi undersöker också möjligheten till att använda det icke-linjära strömberoendet från den kinetiska induktansen som medel att parametriskt förstärka rörelsesidobanden producerade från den optomekaniska interaktionen. Slutligen, med en prototyp skannerenhet för AFM vid låga temperaturer detekterar vi kraftgradienter och avbildar en yta genom att implementera ett pump-arrangemang med två toner för mikrovågskretsen samtidigt som den mekaniska resonatorn drivs.

We describe a transducer for low-temperature atomic force microscopy based on electromechanical coupling due to a strain-dependent kinetic inductance of a superconducting nanowire. The force sensor is a bending triangular plate (cantilever) whose deflection is measured via a shift in the resonant frequency of a high-Q superconducting microwave resonator at 4.5 GHz. We present design simulations including mechanical finite-element modeling of surface strain and electromagnetic simulations of meandering nanowires with large kinetic inductance. We discuss a lumped-element model of the force sensor and describe the role of an additional shunt inductance for tuning the coupling to the transmission line used to measure the microwave resonance. A detailed description of our fabrication is presented, including information about the process parameters used for each layer. We also discuss the fabrication of sharp tips on the cantilever using focused electron beam-induced deposition of platinum. Finally, we present measurements that characterize the spread of mechanical resonant frequency, the temperature dependence of the microwave resonance, and the sensor's operation as an electromechanical transducer of force. 

We describe a transducer for low-temperature atomic force microscopy based on electromechanical coupling due to a strain-dependent kinetic inductance of a superconducting nanowire. The force sensor is a bending triangular plate (cantilever) whose deflection is measured via a shift in the resonant frequency of a high-Q superconducting microwave resonator at 4.5 GHz. We present design simulations including mechanical finite-element modeling of surface strain and electromagnetic simulations of meandering nanowires with large kinetic inductance. We discuss a lumped-element model of the force sensor and describe the role of an additional shunt inductance for tuning the coupling to the transmission line used to measure the microwave resonance. A detailed description of our fabrication is presented, including information about the process parameters used for each layer. We also discuss the fabrication of sharp tips on the cantilever using focused electron beam-induced deposition of platinum. Finally, we present measurements that characterize the spread of mechanical resonant frequency, the temperature dependence of the microwave resonance, and the sensor’s operation as an electromechanical transducer of force.

Electromechanical transduction gain of 21 dB is realized in a micro-cantilever resonant force sensor operated in the unresolved-sideband regime. Strain-dependent kinetic inductance weakly couples cantilever motion to a superconducting nonlinear resonant circuit. A single pump generates motional sidebands and parametrically amplifies them via four-wave mixing. We study the gain and added noise, and we analyze potential benefits of this integrated amplification process in the context force sensitivity.

We study force-gradient sensing with a coherently driven mechanical resonator and phase-sensitive detection of motion through the two-Tone backaction evading measurement of cavity optomechanics. The response of the optomechanical system, solved by numerical integration of the classical equations of motion, shows an extended region which is monotonic to changes in force gradient. We use Floquet theory to model the fluctuations, which rise only slightly above that of the usual backaction evading measurement in the presence of the mechanical drive. The monotonic response and minimal backaction are advantageous for applications such as atomic force microscopy.

We study the response of several microwave resonators made from superconducting NbTiN thin-film meandering nanowires with large kinetic inductance, having different circuit topology and coupling to the transmission line. Reflection measurements reveal the parameters of the circuit and analysis of their temperature dependence in the range 1.7-6 K extract the superconducting energy gap and critical temperature. The lumped-element LC resonator, valid in our frequency range of interest, allows us to predict the quasiparticle contribution to internal loss, independent of circuit topology and characteristic impedance. Our analysis shows that the internal quality factor is limited not by thermal-equilibrium quasiparticles, but an additional temperature-dependent source of internal microwave loss. 

We present a new test of non-randomness that tests both the lower and the upper critical limit of a chi 2-statistic. While checking the upper critical value has been employed by other tests, we argue that also the lower critical value should be examined for non-randomness. To this end, we prepare a binary sequence where all possible bit strings of a certain length occurs the same number of times and demonstrate that such sequences pass a well-known suite of tests for non-randomness. We show that such sequences can be compressed, and therefore are somewhat predictable and thus not fully random. The presented test can detect such non-randomness, and its novelty rests on analysing fixed-length bit string frequencies that lie closer to the a priori probabilities than could be expected by chance alone.

We use the principles of cavity optomechanics to design a resonant mechanical force sensor for atomic force microscopy. The sensor is based on a type of electromechanical coupling, dual to traditional capacitive coupling, whereby the motion of a cantilever induces surface strain that causes a change in the kinetic inductance of a superconducting nanowire. The cavity is realized by a compact microwave-plasma mode with an equivalent LC circuit involving the kinetic inductance of the nanowire. The device is fully coplanar and we show how to transform the cavity impedance for optimal coupling to the transmission line and the following amplifier. For the device presented here, we estimate the bare kinetic inductive mechanoelectric coupling (KIMEC) rate g0/2π in the range 3–10 Hz. We demonstrate phase-sensitive detection of cantilever motion using a multifrequency pumping and measurement scheme.