  Laser cooled ions trapped in a linear Paul trap are long-standing ideal
candidates for realizing quantum simulation, especially of many-body systems.
The properties that contribute to this also provide the opportunity to
demonstrate unexpected quantum phenomena in few-body systems. A pair of ions
interacting in such traps exchange vibrational quanta through the Coulomb
interaction. This linear interaction can be anharmonically modulated by an
elementary coupling to the internal two-level structure of one of the ions.
Driven by thermal energy in the passively coupled oscillators, which are
themselves coupled to the internal ground states of the ions, the nonlinear
interaction autonomously and unconditionally generates entanglement between the
mechanical modes of the ions. We examine this counter-intuitive thermally
induced entanglement for several experimentally feasible model systems, and
propose parameter regimes where state of the art trapped ion systems can
produce such phenomena. In addition, we demonstrate a multiqubit enhancement of
such thermally induced entanglement.
