Phonon localization in an ion chain under irradiation with a Bessel laser beam

Trapped ions offer an excellent platform to investigate fascinating phenomena in quantum physics. Here we propose an experimental scheme to achieve Anderson localization of phonons using an ion chain under irradiation of a laser Bessel beam. The quasiperiodicity comes from the characteristic of the Bessel beam, which is only relevant to the positions of the individual ions with respect to the standing wave of the Bessel beam, rather than the internal degrees of freedom of the ions. We demonstrate, by the inverse participation ratio, the delocalization-localization transition, which is sensitive to the strength and the phase of the Bessel beam laser. Our model involving long-range phonon hopping could present richer phenomena than the Aubry-André model including only nearest-neighbor coupling.

Figure 1

Schematic of the phonon localization in a linear ion chain confined in an electromagnetic trap and experienced by a Bessel beam standing wave.

Figure 2

Phononic probability distribution (left-hand side panels) and IPR (right-hand side panels) with variation of ${\mathrm{\Omega }}_{\text{A}}$, where we consider 101 ions and $i$ represents the $i\mathrm{th}$ ion. Other parameters are set as $d=11.12\mu \mathrm{m}$, ${\omega }_x/2\pi =2$ MHz, $m_{C{a}^{+}}=6.6575\times {10}^{-26}$ kg, $\delta =\pi /7$, and $1/\beta =14.559/2\pi \mu \mathrm{m}$. The panels correspond to the eigenstates (a), (b) ${\mathrm{\Psi }}^{\left(1\right)}$, (c), (d) ${\mathrm{\Psi }}^{\left(53\right)}$, and (e), (f) ${\mathrm{\Psi }}^{\left(85\right)}$.

Figure 3

IPRs for different eigenstates with variation of ${\mathrm{\Omega }}_{\text{A}}$, where we consider 1000 ions in confinement and the parameters are set as ${\omega }_x/\sqrt{C}=10$ and $\chi =\frac{1}{2}(\sqrt{5}+1)$. (a) The ion-trap model with hopping between any two ions. (b) The reduced model with the nearest- and next-to-nearest-neighbor hopping. (c) The reduced model only with the nearest-neighbor hopping.

Figure 4

Phononic localization for certain eigenmodes, where we consider 1000 ions in confinement and the parameters are set as ${\omega }_x/\sqrt{C}=10$ and $\chi =\frac{1}{2}(\sqrt{5}+1)$. (a) The case of ${\mathrm{\Psi }}^{\left(1\right)},{\mathrm{\Omega }}_{\text{A}}/\sqrt{C}=1$. (b) The case of ${\mathrm{\Psi }}^{\left(950\right)},{\mathrm{\Omega }}_{\text{A}}/\sqrt{C}=1$. (c) The case of ${\mathrm{\Psi }}^{\left(1\right)},{\mathrm{\Omega }}_{\text{A}}/\sqrt{C}=3$. (d) The case of ${\mathrm{\Psi }}^{\left(950\right)},{\mathrm{\Omega }}_{\text{A}}/\sqrt{C}=3$. (e) The case of ${\mathrm{\Psi }}^{\left(1\right)},{\mathrm{\Omega }}_{\text{A}}/\sqrt{C}=5$. (f) The case of ${\mathrm{\Psi }}^{\left(950\right)},{\mathrm{\Omega }}_{\text{A}}/\sqrt{C}=5$.

Figure 5

Energy eigenvalues of $H_e$ in Eq. (6) and the IPR of the corresponding wave functions for 1000 trapped ions as a linear chain, where we set $\chi =\frac{1+\sqrt{5}}{2}$ and ${\omega }_x/\sqrt{C}=8$. The shading of the energy curves presents the amplitude of the IPR. The black line is for the mobility edge, separating the localized region (below the line) from the delocalized region (above the line). The critical regions of the phase transition are plotted in green.