NOISE SPECTRUM OF A QUANTUM POINT CONTACT COUPLED TO A NANO-MECHANICAL OSCILLATOR

Abstract

With the advance in nanotechnology, we are more interested in the "smaller worlds". One of the practical applications of this is to measure a very small displacement or the mass of a nano-mechanical object. To measure such properties, one needs a very sensitive detector. A quantum point contact (QPC) is one of the most sensitive detectors. In a QPC, electrons tunnel one by one through a tunnel junction (a "hole"). The tunnel junction in a QPC consists of a narrow constriction (nm-wide) between two conductors. To measure the properties of a nano-mechanical object (which acts as a harmonic oscillator), we couple it to a QPC. This coupling effects the electrons tunneling through the QPC junction. By measuring the transport properties of the tunneling electrons, we can infer the properties of the oscillator (i.e. the nano-mechanical object). However, this coupling introduces noise, which reduces the measurement precision. Thus, it is very important to understand this source of noise and to study how it effects the measurement process. We theoretically study the transport properties of electrons through a QPC junction, weakly coupled to a vibration mode of a nano-mechanical oscillator via both the position and the momentum of the oscillator. %We study both the position and momentum based coupling. The transport properties that we study consist of the average flow of current through the junction, given by the one-time correlation of the electron tunneling event, and the current noise given by the two-time correlation of the average current, i.e, the variance. The first comprehensive experimental study of the noise spectrum of a detector coupled to a QPC was performed by the group of Stettenheim et al. Their observed spectral features had two pronounced peaks which depict the noise produced due to the coupling of the QPC with the oscillator and in turn provide evidence of the induced feedback loop (back-action). Benatov and Blencowe theoretically studied these spectral features using the Born approximation and the Markovian approximation. In this case the Born approximation refers to second order perturbation of the interaction Hamiltonian. In this approximation, the electrons tunnel independently, i.e., one by one only, and co-tunneling is disregarded. The Markovian approximation does not take into account the past behavior of the system under time evolution. These two approximations also enable one to study the system analytically, and the noise is calculated using the MacDonald formula. Our main aim for this thesis is to find a suitable theoretical model that would replicate the experimental plots from the work of Stettenheim et al. Our work does not use the Markovian approximation. However, we do use the Born approximation. This is justified as long as the coupling between the oscillator and QPC is weak. We first obtain the non-Markovian unconditional master equation for the reduced density matrix of the system. Non-Markovian dynamics enables us to study, in principle, the full memory effects of the system. From the master equation, we then derive analytical results for the current and the current noise. Due to the non-Markovian nature of our system, the electron tunneling parameters are time-dependent. Therefore, we cannot study the system analytically. We thus numerically solve the current noise expression to obtain the noise spectrum. We then compare our noise spectrum with the experimental noise spectrum. We show that our spectral noise results agree better with the experimental evidence compared to the results obtained using the Markovian approximation. We thus conclude that one needs non-Markovian dynamics to understand the experimental noise spectrum of a QPC coupled to a nano-mechanical oscillator.