This thesis consists of two parts. In the first part, the properties of excess electrons in water clusters are studied via a hybrid quantum and classical mechanics method. The existence of the solvated electron in water was experimentally demonstrated long ago, and it is among the most interesting charged species. However, a satisfactory characterization of the water clusters has always been a challenge. In our simulation, we treat a region of the cluster nearest to the centroid of the excess electron distribution quantum mechanically, while the rest of the water molecules are treated classically. The binding energies of a localized excess electron are calculated in clusters with sizes ranging from 16 to 300. The density distributions of the excess electrons verify the existence of both surface localization mode and interior localization model. We studied the energetically favored localization modes depending on the sizes of the clusters and the transition point. In the second part, the energy spectra, spin configurations, and entanglement characteristics of a system of four electrons in lateral double quantum dots are investigated using exact diagonalization (EXD), as a function of interdot separation, applied magnetic field, and strength of interelectron repulsion. As a function of the magnetic field, the energy spectra exhibit a low-energy band consisting of a group of six states, with the number six being a consequence of the conservation of the total spin of the four electrons and the ensuing spin degeneracies. These six states appear to cross at a single value of the magnetic field, with the crossing point becoming sharper for larger interdot distances. As the strength of the Coulomb repulsion increases, the six states tend to become degenerate and a well defined energy gap separates them from the higher-in-energy excited states. The appearance of the low-energy band is a consequence of the formation of a Wigner supermolecule. Using the spin-resolved pair-correlation functions, one can map the EXD many-body wave functions onto the spin functions associated with the four localized electrons. The ability to determine associated spin functions enables investigations concerning entanglement properties of the system of four electrons.

Part I of this thesis is concerned with cavity optomechanical systems subject to periodic driving. We develop a Floquet approach to solve time-periodic quantum Langevin equations in the steady state, show that two-time correlation functions of system operators can be expanded in a Fourier series, and derive a generalized Wiener-Khinchin theorem that relates the Fourier transform of the autocorrelator to the noise spectrum. Weapply our framework to optomechanical systems driven with two tones. In a setting used to prepare mechanical resonators in quantum squeezed states, we nd and study the general solution in the rotating-wave approximation. In the following chapter, we show that our technique reveals an exact analytical solution of the explicitly time-periodic quantum Langevin equation describing the dual-tone backaction-evading measurement of a single mechanical oscillator quadrature due to Braginsky, Vorontsov, and Thorne [Science 209, 547 (1980)] beyond the commonly used rotating-wave approximation and show that our solution can be generalized to a wide class of systems, including to dissipatively or parametrically squeezed oscillators, as well as recent two-mode backaction-evading measurements. In Part II, we study nonreciprocal optomechanical systems with several optical and mechanical modes. We show that an optomechanical plaquette with two cavity modes coupled to two mechanical modes is a versatile system in which isolators, quantum-limited phase-preserving, and phase-sensitive directional ampliers for microwave signals can be realized. We discuss the noise added by such devices, and derive isolation bandwidth, gain bandwidth, and gain-bandwidth product, paving the way toward exible, integrated nonreciprocal microwave ampliers. Finally, we show that similar techniques can be exploited for current rectication in double quantum dots, thereby introducing fermionic reservoir engineering. We verify our prediction with a weak-coupling quantum master equation and the exact solution. Directionality is attained through the interference of coherent and dissipative coupling. The relative phase is tuned with an external magnetic eld, such that directionality can be reversed, as well as turned on and off dynamically.

We report on conductance measurements in carbon nanotube based double quantum dots connected to two normal electrodes and a central superconducting finger. By operating our devices as Cooper pair beam splitters, we provide evidence for Crossed Andreev Reflection (CAR). We inject Cooper pairs in the superconducting electrode and measure the differential conductance at both left and right arm. The contacts split the device into two coupled quantum dots. Each of the quantum dots can be tuned by a lateral sidegate. If the two sidegates are tuned such that both quantum dots are at a transmission resonance, a considerable part of the injected Cooper pairs splits into different normal contacts. On the contrary, if only one of the two dots is at resonance, nearly all pairs tunnel to the same normal contact. By comparing different triple points in the double dot stability diagram, we demonstrate the contribution of split Cooper pairs to the total current. In this manner, we are able to extract a splitting efficiency of up to 50% in the resonant case. Carbon Nanotubes ensure ballistic transport and long spin-flip scattering lengths. Due to these properties they are promising candidates to investigate EPR-type correlations in solid state systems.