Signal amplification through bifurcation in nanoresonators

Abstract

The small size of microand nanomechanical resonators, combined with the slow decay of their vibrational modes, enable numerous applications of these devices—from the studies of macroscopic quantum phenomena [1, 2], to mass detection with single-atom sensitivity [3, 4], to imaging with single-atom and single-spin resolution [5]. An important feature of nanoresonators is the vibration nonlinearity, which becomes significant even at small vibration amplitudes. It allows investigating a broad variety of nonlinear phenomena, including basic phenomena such as bifurcations, where the number of stable states of the system, or the character of motion, change with varying parameters. A typical example is the onset of parametric resonance, where a resonator starts vibrating at half the frequency at which it is modulated. Nonlinear dynamics near bifurcation points is particularly interesting as it is controlled by one, or a few, slow variables [6] and displays universal systemindependent features, such as scaling behavior. In a paper published in Physical Review Letters, Rassul Karabalin and colleagues [7] from Caltech, US, and Tel Aviv University, Israel, propose and implement a new bifurcation-based amplification scheme using coupled nanoresonators and demonstrate that it can be employed for robust small-signal amplification. The results bear not only on the fast expanding field of bifurcationbased amplification [8], but also on the physics of classical and quantum fluctuation phenomena in nonlinear vibrational systems, and on the dynamics and fluctuations near bifurcation points. Conventional bifurcation amplifiers utilize bifurcations where, as the input changes, the state occupied by the system disappears and the system switches to a different state. For example, in Josephson bifurcation amplifiers used in quantum measurements, the system switches between vibrational states with different amplitudes and phases [8]. This is similar to magnetization switching in single-domain magnets once an external magnetic field exceeds a critical value. The bifurcation amplifier by Karabalin et al. works in a different way. Here, as the control parameter is varied, the system goes to the state with either large or small output, depending on the input signal; that is why the authors call the device a bifurcation-topology amplifier. Their system is sketched in the top panel of Fig. 1. It consists of two nearly identical nanoresonators, 1 and 2. The eigenfrequencies ω1 and ω2 of their fundamental flexural (bending) modes are close in value by design. The modes are underdamped, allowing a strong response in resonant conditions. The stiffness of the resonators, and thus the values of ω1 and ω2, is modulated at a frequency ωp close to 2ω1,2. For a single resonator, such modulation can lead to parametric resonance with the onset of two vibrational states that have equal amplitudes and frequency ωp/2 but differ in phase by π. In the experiment, the vibrations are detected by optical interferometry using a laser beam reflected from the resonators. The key feature of the amplifier is that the distance between the resonators is small compared to the laser beam diameter. Therefore, the optical response is large if the resonators are vibrating in phase, as in this case the reflected signals coherently interfere; if the vibrations are in counterphase, the reflection is suppressed. The idea put forward by Karabalin et al. is that the relative phase of the resonators can be controlled by a weak input signal, which will then trigger a large response with amplitude determined by the amplitude of the parametrically excited vibrations. The resonators are parametrically excited by ramping up the modulation frequency ωp across the bifurcational values ωB1,2. Let us assume that resonator 1 is slightly “softer” than resonator 2, i.e., ω1 < ω2 and ωB1 < ωB2. Then, if we start from low ωp, resonator 1 will be excited first, once ωp reaches ωB1. If the resonators are uncoupled, resonator 2 will be excited later, at ωp = ωB2. Its phase can be close or opposite to the phase of resonator 1, with equal probability. Because the