Abstract Super-oscillation is a counterintuitive phenomenon describing localized fast variations of functions and fields that happen at frequencies higher than the highest Fourier component of their spectra. The physical implications of this effect have been studied in information theory and optics of classical fields, and have been used in super-resolution imaging. As a general phenomenon of wave dynamics, super-oscillations have also been predicted to exist in quantum wavefunctions. Here we report the experimental demonstration of super-oscillatory behavior of a single-quantum object, a photon. The super-oscillatory behavior is demonstrated by tight localization of the photon wavefunction after focusing with an appropriately designed slit mask to create an interference pattern with a sub-diffraction hotspot (~0.45 λ). Such quantum super-oscillation can be used for low-intensity far-field super-resolution imaging techniques even down to single-photon counting regime, which would be of interest to quantum physics and non-invasive and label-free biological studies.
We study the mixed state in an extreme type-II lattice ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductor in the experimentally most relevant regime of intermediate magnetic fields
Hc1⪡H⪡Hc2. We analyze the low energy spectrum of the problem, dominated by nodal Dirac-like quasiparticles with momenta near ${\mathbf{k}}_{F}=(\ifmmode\pm\else\textpm\fi{}{k}_{D},\ifmmode\pm\else\textpm\fi{}{k}_{D})$, and find that the spectrum exhibits characteristic oscillatory behavior with respect to the product of
kD and magnetic length
l. The Simon-Lee scaling, predicted in this regime, is satisfied only on average, with the magnitude of the oscillatory part of the spectrum displaying the same ${l}^{\ensuremath{-}1}$ dependence as its monotonous ``envelope'' part. In general, the spectrum obeys a scaling law ${E}_{n\mathbf{k}}=(\ensuremath{\hbar}{v}_{F}∕l){\mathcal{E}}_{n}(\mathbf{k}l,t∕\ensuremath{\Delta},{k}_{D}l)$, where
E is a dimensionless universal $2\ensuremath{\pi}$-periodic function of
kDl. The oscillatory behavior of the spectrum is due to the internodal interference enhanced by the singular nature of the low-energy eigenfunctions near vortices. Our results constitute an example of a finite size scaling of the Dirac-type quantum criticality. We also study a separate problem of a single vortex piercing an isolated superconducting grain of size $L\ifmmode\times\else\texttimes\fi{}L$. Here we find that the periodicity of the quasiparticle energy oscillations with respect to
kDL is doubled relative to the case where the field is zero and the vortex is absent, both such oscillatory behaviors being present at the leading order in ${L}^{\ensuremath{-}1}$. Finally, we review the overall features of the tunneling conductance experiments in $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ (YBCO) and ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ (BSCCO), and suggest an interpretation of the peaks at
5−−20meV observed in the tunneling local density of states in these materials. We find that in the case of a pure
d-wave superconducting order parameter with featureless vortex cores, the zero bias conductance peak (ZBCP) appears only on the sites that are the immediate nearest neighbors of vortex locations, while all the other sites in the close vicinity of vortices exhibit no such ZBCP and instead display pronounced peaks at subgap energies, typically at about a half or less of the coherence peak energy. Furthermore, we find that the on-site ZBCP can be strongly suppressed by enhanced local pairing near a vortex.
The capability of controlling light at scales that are much smaller than the operating wave-length enables new optical functionalities, and opens up a wide range of applications. Such a capability is out of the realm of conventional optical approaches. This dissertation aims to explore the light-matter interactions at nanometer scale, and to investigate the novel scien-tific and industrial applications. In particular, we will explain how to detect nanoparticles using an ultra-sensitive nano-sensor; we will also describe a photonic diode which gener-ates a unidirectional flow of single photons; Moreover, in an one-dimensional waveguide QED system where the fermionic degree of freedom is present, we will show that strong photon-photon interactions can be generated through scattering means, leading to photonic bunching and anti-bunching with various applications. Finally, we will introduce a mecha-nism to achieve super-resolution to discern fine features that are orders of magnitude smaller than the illuminating wavelength. These research projects incorporate recent advances in quantum nanophotonics, nanotechnologies, imaging reconstruction techniques, and rigorous numerical simulations.