Nanomechanics coupled to SET
One of the challenges of nanomechanics is detecting the motion where the sensitivity is limited by fundamental physics. It is well known since the 1980s in the context of gravitational wave detection, that quantum mechanics limits the position resolution when continuously monitoring a mechanical oscillator; this is known as the standard quantum limit. Furthermore, given the high frequencies that are possible with nanomechanical resonators, cooling such a device to a temperature where it will occupy its ground state is possible. In the end, the ultimate sensitivity should be limited by both the quantum fluctuations of the mechanical oscillator and the quantum fluctuations of the linear amplifying system.
Approaching the Heisenberg Uncertainty Principle
Single electron transistors (SET) were expected to be near perfect amplifiers for radio frequency nanomechanical systems, capable of performing near the Uncertainty Principle limit. Given the geometry, we expected to be able to couple tightly enough to realize this sensitivity. This optimistic expectation lead us and others to study devices with SET's coupled to nanomechanical resonators.
Our first success with these devices was our achievement of two records, which we reported in Science in 2004. We realized a position sensitivity a factor of 5.8 from the Heisenberg Uncertainty Principle Limit, with a detection noise temperature of 16mK at 20MHz. With this sensitivity we were able to detect the thermal motion of the mechanical resonator at milli-kelvin temperatures and perform noise thermometry on the mechanical modes. We found that the mechanical mode followed the temperature of the dilution refrigerator down to ~60mK, where we were limited by heating due to the SET. This mode temperature corresponded to an occupation factor of N=60. This was, and still remains the closest approach to both the Heisenberg Uncertainty Principle for any position detection.
For comparison, similar work performed at Andrew Cleland's laboratory at UCSB showed a position sensitivity a factor of 100 from the Heisenberg-limit and a detection noise temperature of 30K, which was insufficient to observe the nanomechanical mode temperature. (This sensitivity is within reach of standard room temperature pre-amplifiers.) The major difference between the sensitivity of the two experiments is a result of the much higher mechanical quality factor in our device (Q=50,000 compared to 1,700), and the much lower SET charge noise we achieved (10 micro-e/rtHz compared to 325 micro-e/rtHz.)
Quantum Measurement Back-action and Cooling
The Uncertainty Principle requires that measuring position must be accompanied by a disturbance in momentum. By engineering a device with tighter coupling between the SET and the mechanical mode, we set-out to detect what is called quantum measurement back-action. As it turned out, this experiment became more complex than we first anticipated due to the complex tunneling processes at play in a superconducting SET. With the theoretical help of Miles Blencowe (Dartmouth), Andrew Armour(Nottingham), and Aash Clerk (McGill) we began to understand exactly how to bias the SET. Most interestingly, the theory told us that not only should the SET provide stochastic back-action to the mechanical mode, it should also be capable of cooling a hot mode.
We observed these effects and reported our results in Nature in 2006. We observed both the back-action forces from the SET which are responsible for driving the nanomechanical mode, and the cooling processes which are possible with carefully chosen points of bias. Furthermore, we showed a mechanical mode with a thermal occupation of N=25, a position resolution of 0.6 fm/rtHz, and a force sensitivity of 0.6 aN/rtHz. This is the closest measured approach to the quantum ground state for a mechanical device, the highest position resolution achieved on a radio-frequency mechanical resonator, and force sensitivity equaling the best achieved with audio frequency resonators to detect a single electron spin (Rugar and Mamin.)
The SEM micrograph above shows the layout of a typical device. The nanomechanical resonator is ~8 micron X 200nm X 200nm and is composed of SiN coated with Au or Al films. Fabricated nearby, we place a single electron transistor (SET) which interacts with the mechanical mode through capacitance. The tunnel junctions of the SET are located at the ends of the Al line marked with 'J.' The device shown has been fabricated on a SiN membrane; our more recent devices are surface micromachined using RIE. This was essential to avoid heating of the membrane to ~60mK by the 100fW dissipated by the SET. Our most recent work showed that by fabricating the SET on the crystalline substrate, nanomechanical mode temperatures of 25mK are possible, with no evidence that this is the lower limit.
