Competing many-body interactions in systems of trapped ions

We propose and theoretically analyze an experimental configuration in which lasers induce three-spin interactions between trapped ions. By properly choosing the intensities and frequencies of the lasers, three-spin couplings may be dominant or comparable to two-spin terms and magnetic fields. In this way, trapped ions can be used to study exotic quantum phases which do not have a counterpart in nature. We study the conditions for the validity of the effective three-spin Hamiltonian and predict qualitatively the quantum phase diagram of the system.

Figure 1

(Color online) Linear array of microtraps subjected to three laser beams along the axial and radial directions.

Figure 2

(Color online) Phonon frequencies ${\Omega }_{n\alpha }$ of (a) radial and (b) axial modes in the stiff limit ${\beta }_{\alpha }⪡1$, where the corrections due to the Coulomb energy $\frac{1}{2}{c}_{\alpha }{\beta }_{\alpha }{\omega }_{\alpha }{\mathcal{V}}_n$ have opposite signs for transverse $c_{x,y}=1$ or longitudinal $c_z=-2$ phonons. The coupling of spin-dependent dipole forces to vibrational modes is also shown (a) in the radial stiff limit, we can simultaneously blue detune every mode to the first sideband ${\omega }_{L_1^x}\gtrsim {\Omega }_{nx}$ or to the second sideband ${\omega }_{L_2^x}\gtrsim 2{\Omega }_{nx}$. (b) In the axial stiff limit, every mode shall be red detuned to the first sideband ${\omega }_{L_3^z}\lesssim {\Omega }_{nz}$.

Figure 3

(Color online) (a) Quantum phase diagram with P, AF, and F phases that coexist in the tricritical point (in red). Order parameters as a function of the couplings $J_2$ and $J_3$ for a chain with $N=15$ spins (b) antiferromagnetic order parameter (c) ferrimagnetic order parameter.