Cold, macroscopic mechanical systems are expected to behave contrary to our usual classical understanding of reality; the most striking and counterintuitive predictions involve the existence of states in which the mechanical system is located in two places simultaneously. Various schemes have been proposed to generate and detect such states, and all require starting from mechanical states that are close to the lowest energy eigenstate, the mechanical ground state. Here we report the cooling of the motion of a radio-frequency nanomechanical resonator by parametric coupling to a driven, microwave-frequency superconducting resonator. Starting from a thermal occupation of 480 quanta, we have observed occupation factors as low as 3.8 +/- 1.3 and expect the mechanical resonator to be found with probability 0.21 in the quantum ground state of motion. Further cooling is limited by random excitation of the microwave resonator and heating of the dissipative mechanical bath. This level of cooling is expected to make possible a series of fundamental quantum mechanical observations including direct measurement of the Heisenberg uncertainty principle and quantum entanglement with qubits.

When performing continuous measurements of position with sensitivity approaching quantum mechanical limits, one must confront the fundamental effects of detector back-action. Back-action forces are responsible for the ultimate limit on continuous position detection, can also be harnessed to cool the observed structure [1,2,3,4], and are expected to generate quantum entanglement [5]. Back-action can also be evaded[6,7,11], allowing measurements with sensitivities that exceed the standard quantum limit, and potentially allowing for the generation of quantum squeezed states. We realize a device based on the parametric coupling between an ultra-low dissipation nanomechanical resonator (Q~10 6 ) and a microwave resonator.[20] Here we demonstrate back-action evading (BAE) detection of a single quadrature of motion with sensitivity 4 times the quantum zero-point motion,, back-action cooling of the mechanical resonator to 12  NR n quanta, and a parametric mechanical pre-amplification effect which is harnessed to achieve position resolution a factor 1.3 times ZP x  .When attempting to obtain complete knowledge of the dynamics of a simple harmonic oscillator, x(t), back-action effects combined with the quantum zero-point motion of the oscillator limit the ultimate resolution to the standard quantum limit (SQL) [8,9]. The origin of this limit is the primitive fact that position and momentum are non-commuting, and are then linked through the equations of motion, , but are not linked dynamically and are constants of the motion:Furthermore, it was realized how to couple 1 X to a detector in a way which does not * Correspondence should be sent to schwab@caltech.edu perturb the dynamics of 1, where int H is the interaction Hamiltonian.[12,13] Thus, while a measurement of 1 X necessarily disturbs 2 X , this disturbance has no effect on the subsequent dynamics or measurements of X1. One can thus increase the coupling strength arbitrarily without fear of back-action, meaning that the sensitivity of such an ideal single quadrature probe is not fundamentally limited The expected quantum Hamiltonian of our parametrically coupled system is given by:

A capacitive-comb transduced micromechanical resonator using aggressive lithography to occupy only 0.0154-mm 2 of die area has been combined via bond-wiring with a custom ASIC sustaining amplifier and a supply voltage of only 1.65V to realize a 32.768-kHz real-time clock oscillator more than 100× smaller by area than miniaturized quartz crystal implementations and at least 4× smaller than other MEMSbased approaches, including those using piezoelectric material.The key to achieving such large reductions in size is the enormous rate at which scaling improves the performance of capacitive-comb transduced folded-beam micromechanical resonators, for which scaling of lateral dimensions by a factor S provides an S 2 × reduction in both motional resistance and footprint for a given resonance frequency. This is a very strong dependency that raises eyebrows, since the size of the frequency-setting tank element may soon become the most important attribute governing cost in a potential MEMS-based or otherwise batch-fabricated 32.768-kHz timing oscillator market. In addition, unlike quartz counterparts, the size reduction demonstrated here actually reduces power consumption, allowing this oscillator to operate with only 2.1μW of DC power.

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A 61-MHz Pierce oscillator constructed in 0.35-µm CMOS technology and referenced to a polysilicon surface-micromachined capacitive-gap-transduced wineglass disk resonator has achieved phase noise marks of −119 dBc/Hz at 1-kHz offset and −139 dBc/Hz at far-fromcarrier offsets. When divided down to 13 MHz, this corresponds to −132 dBc/Hz at 1-kHz offset from the carrier and −152 dBc/Hz far-from-carrier, sufficient for mobile phone reference oscillator applications, using a single MEMS resonator, i.e., without the need to array multiple resonators. Key to achieving these marks is a Pierce-based circuit design that harnesses a MEMS-enabled input-to-output shunt capacitance more than 100× smaller than exhibited by macroscopic quartz crystals to enable enough negative resistance to instigate and sustain oscillation while consuming only 78 µW of power-a reduction of ∼4.5× over previous work. Increasing the bias voltage of the resonator by 1.25 V further reduces power consumption to 43 µW at the cost of only a few decibels in far-from-carrier phase noise. This oscillator achieves a 1-kHz-offset figure of merit (FOM) of −231 dB, which is now the best among published chipscale oscillators to date. A complete linear circuit analysis quantifies the influence of resonator input-to-output shunt capacitance on power consumption and predicts further reductions in power consumption via reduction of electrode-to-resonator transducer gaps and bond pad sizes. The demonstrated phase noise and power consumption posted by this tiny MEMS-based oscillator are attractive as potential enablers for low-power "set-and-forget" autonomous sensor networks and embedded radios.