Quantized Flux States of Superconducting Cylinders

We have measured the magnetic flux trapped in hollow superconducting cylinders ($14-56-\mu \mathrm{i}.\mathrm{d}.\mathrm{},0.1-5-\mu$ walls, 0.5-2.5 cm long) as a function of the applied magnetic field in which the cylinder was cooled through its transition temperature. For most applied fields, the entire cylinder is in the same quantum state; for some fields there is a mixed state with bands along the cylinder in states differing by one flux quantum. In cylinders with walls thick with respect to the penetration depth, the trapped flux is an integral multiple of $\frac{\mathrm{hc}}{2e}\pm \frac{1}{2}\%$. We have measured the temperature dependence of the trapped flux in a cylinder containing a single flux quantum and find a continuous reversible change by just the amount required to maintain the constancy of the fluxoid.