Superconducting quantum fluctuations in one dimension

We review some recent developments in the field of quasi-one-dimensional superconductivity. We demonstrate that low temperature properties of superconducting nanowires are essentially determined by quantum fluctuations. Smooth (Gaussian) fluctuations of the superconducting phase (also associated with plasma modes propagating along the wire) may significantly affect the electron density of states in such nanowires and induce persistent current noise in superconducting nanorings. Further interesting phenomena such as, e.g., non-vanishing resistance and shot noise of the voltage in current-biased superconducting nanowires, are caused by non-Gaussian fluctuations of the order parameter -- quantum phase slips (QPS). Such phenomena may be interpreted in terms of tunneling of fluxons playing the role of effective quantum "particles" dual to Cooper pairs and obeying complicated full counting statistics which reduces to Poissonian one in the low frequency limit. We also demonstrate that QPS effects may be particularly pronounced in thinnest wires and rings where quantum phase slips remain unbound and determine a non-perturbative length scale $L_c$ beyond which the supercurrent gets suppressed by quantum fluctuations. Accordingly, for $T \to 0$ such nanowires should become insulating at scales exceeding $L_c$, whereas at shorter length scales they may still exhibit superconducting properties. We argue that certain non-trivial features associated with quantum fluctuations of the order parameter may be sensitive to specific circuit topology and may be observed in structures like, e.g., a system of capacitively coupled superconducting nanowires.