Bose-Einstein Condensation and Strong-Correlation Behavior of Phonons in Ion Traps

We show that the dynamics of phonons in a set of trapped ions interacting with lasers is described by a Bose-Hubbard model whose parameters can be externally adjusted. We investigate the possibility of observing several quantum many-body phenomena, including Bose-Einstein condensation as well as a superfluid-Mott insulator quantum phase transition.

Figure 1

Left: Phonon trapping potential (2) for the radial modes of a Coulomb chain with $N=50$ ions, and ${\beta }_x={10}^{-2}$, as a function of the ion position along the chain. Left, inset: Spectrum of the radial collective modes that diagonalize $H_{x0}$. We plot the energy of the modes relative to the minimum phonon energy, ${\Delta }_{x,q}=({\Omega }_{x,q}-{\Omega }_{x,0})/{\omega }_x$, where ${\Omega }_{x,q}$ is the energy of the collective modes, and $q$ is the mode number. Right: Mean phonon-number $⟨n_i⟩=⟨a_i^{†}{a}_i⟩$ along Coulomb chains with different number of ions, $N$, in the state with $N_{\mathrm{p}\mathrm{h}}=N$ phonons in the radial lowest mode. The width of the wave function in units of $N$, is $\propto 1/\sqrt{N}$, in accord with the scaling for the phonon trapping frequency, ${\omega }_c\propto 1/N$.

Figure 2

Mean phonon number at each ion in the ground state of a Coulomb chain with $N=6$, $N_{\mathrm{p}\mathrm{h}}=6$. ${\beta }_x=0.01$, so that the nearest-neighbor hopping terms are $t\approx {510}^{-3}{\omega }_x$. Black circles: phonon density without standing wave ($U=0$), which shows the confinement due to the phonon trapping potential. Empty circles: Mott phase when a standing wave is applied with $F{\eta }^2=0.1{\omega }_x$, ${\eta }^2=0.1$, and $U\approx 0.02{\omega }_x>t$. Squares: $U=0.01{\omega }_x$, Triangles: $U=5\times {10}^{-3}{\omega }_x$.