  Coupling a quantum particle to a fermionic bath suppresses the particle's
amplitude to tunnel, even at zero temperature. While this effect can generally
be neglected for gapped baths -- a key feature for superconducting qubits -- ,
it is possible for the bath to be gapped near the potential minima between
which the particle tunnels, but different minima to correspond to different
bath topologies. This enforces the bath to undergo gap closing along the
tunneling path. In this work, we investigate quantum tunneling in the presence
of such a topology-changing fermionic bath. We develop a field theory for this
problem, linking the instantons describing tunneling in a bath of $d$ space
dimensions to topological boundary modes of systems in $d+1$ dimensions, thus
stepping a level higher in a dimensional hierarchy. We study in detail a $d=1$
example, inspired by planar Josephson junctions where the particle coordinate
is the superconducting phase whose value sets the electronic topology. We find
that the topology change suppresses tunneling by a factor scaling exponentially
with the system size. This translates to a correspondingly enhanced suppression
of the energy splitting for the lowest-lying states, despite these being linear
combinations of states near potential minima where the bath is gapped. Our
results help to estimate the influence of charging energy on topological phases
arising due to the Josephson effect and, conversely, to assess the potential
utility of such topological systems as superconducting qubits. For
moderate-sized baths, the incomplete suppression of tunneling opens the
prospects of quantum-mechanical superpositions of many-body states of different
topology, including superpositions of states with and without Majorana
fermions.
