  Traditionally, strong-field physics explores phenomena in matter (atoms,
molecules, and solids) driven by an extremely strong laser field
nonperturbatively. However, even in the complete absence of an external
electromagnetic field, strong-field phenomena can arise when matter strongly
couples with the zero-point field of the quantum vacuum state, i.e.,
fluctuating electromagnetic waves whose expectation value is zero. Some of the
most striking examples of this occur in a cavity setting, in which an ensemble
of two-level atoms resonantly interacts with a single photonic mode of vacuum
fields, producing vacuum Rabi splitting. In particular, the nature of the
matter-vacuum-field coupled system fundamentally changes when the coupling rate
(equal to one half of the vacuum Rabi splitting) becomes comparable to, or
larger than, the resonance frequency. In this so-called ultrastrong coupling
regime, a non-negligible number of photons exist in the ground state of the
coupled system. Furthermore, the coupling rate can be cooperatively enhanced
(via so-called Dicke cooperativity) when the matter is comprised of a large
number of identical two-level particles, and a quantum phase transition is
predicted to occur as the coupling rate reaches a critical value. Low-energy
electronic or magnetic transitions in many-body condensed matter systems with
large dipole moments are ideally suited for searching for these predicted
phenomena. Here, we discuss two condensed matter systems that have shown
cooperative ultrastrong interactions in the terahertz frequency range: a
Landau-quantized two-dimensional electron gas interacting with
high-quality-factor cavity photons, and an Er$^{3+}$ spin ensemble interacting
with Fe$^{3+}$ magnons in ErFeO$_3$.

Mesoscale and Nanoscale Physics