  Temporal cavity solitons (CSs) are stable, localized particle-like objects in
the form of optical pulses that circulate indefinitely in coherently driven
nonlinear resonators. In the spectral domain, they form highly coherent
frequency combs. Owing to their remarkable stability, they are attracting
attention for applications in sensing, metrology, or optical signal synthesis.
In this work, we report on the dynamics of CSs interacting with a trapping
potential. We demonstrate that this interaction provides a powerful means to
control their properties such as position, speed, and central frequency. Our
theoretical analysis predicts fundamental limitations on the spectral shift of
CSs relative to the driving frequency. Specifically, it reveals that within a
broad range of detunings, frequency-shifted CSs encounter destabilization
through a Hopf bifurcation. Moreover, we find that with periodic potentials,
the Kelly sidebands emitted by trapped solitons undergo Bloch oscillations. In
our experiments, we use an intracavity phase modulator to create the equivalent
of an external real potential. We observe stable blue- and red- shifted
solitons up to a limit close to our theoretical predictions. We then show
theoretically and experimentally that this unprecedented level of control over
the CS spectrum can be leveraged to cancel the Raman-induced self-frequency
shift and even to stabilize CSs beyond the limitation imposed by stimulated
Raman scattering. Our results provide valuable insights for applications
requiring robust and potentially rapid tunable control over the cavity soliton
properties.
