  Superconducting order in a sufficiently narrow and infinitely long wire is
destroyed at zero temperature by quantum fluctuations, which induce $2\pi$
slips of the phase of the order parameter. However, in a finite-length wire
coherent quantum phase-slips would manifest themselves simply as shifts of
energy levels in the excitations spectrum of an electrical circuit
incorporating this wire. The higher the phase-slips probability amplitude, the
larger are the shifts. Phase-slips occurring at different locations along the
wire interfere with each other. Due to the Aharonov-Casher effect, the
resulting full amplitude of a phase-slip depends on the offset charges
surrounding the wire. Slow temporal fluctuations of the offset charges make the
phase-slips amplitudes random functions of time, and therefore turn energy
levels shifts into linewidths. We experimentally observed this effect on a long
Josephson junction array acting as a "slippery" wire. The slip-induced
linewidths, despite being only of order 100 kHz, were resolved from the
flux-dependent dephasing of the fluxonium qubit.
