Motion of ions in the trap can be represented as a set of normal modes, each have its own frequency.
While these modes are usually considered independent, anharmonicity of the Coloumb
interaction between the ions introduces coupling between them.
This coupling is a nonlinear effect analogous to the well known model of parametric oscillator in optics, however
in case of the trapped ions interaction between the modes manifest itself even on the level of single quanta^{[1]}.

For example, a single quanta of motion (phonon) in the axial mode could be converted to two phonons in the radial mode and vise versa
if the frequency of radial mode is exactly twice the frequency of the radial mode^{[2]}. If the frequencies of the two modes do not satisfy the resonance condition, the interaction between them manifest itself as cross-Kerr effect: excitation of one mode can shift the frequency of the
other mode^{[3]}.

Our group explores applications of these nonlinear effects to quantum simulations, quantum computing, and quantum thermodynamics.

**Fig. 1.** Demonstration of cross Kerr effect between modes of motion.
One quanta of energy added to a motional mode results in the frequency shift of other mode.

- Shiqian Ding, Gleb Maslennikov, Roland Hablutzel, Dzmitry Matsukevich,
"Quantum Simulations with a Trilinear Hamiltonian"
*Phys. Rev. Lett.***121**130502 (2018). - Shiqian Ding, Gleb Maslennikov, Roland Hablutzel, Huanqian Loh, Dzmitry Matsukevich,
"Quantum parametric oscillator with trapped ions",
*Phys. Rev. Lett.***119**, 150404 (2017). - Shiqian Ding, Gleb Maslennikov, Roland Hablutzel, Dzmitry Matsukevich
"Cross-Kerr nonlinearity for phonon counting",
*Phys. Rev. Lett.***119**, 193602 (2017).

Dzmitry Matsukevich's Group