Structural phase transitions in low-dimensional ion crystals

A chain of singly charged particles, confined by a harmonic potential, exhibits a sudden transition to a zigzag configuration when the radial potential reaches a critical value, depending on the particle number. This structural change is a phase transition of second order, whose order parameter is the crystal displacement from the chain axis. We study analytically the transition using Landau theory and find full agreement with numerical predictions by Schiffer [Phys. Rev. Lett. 70, 818 (1993)] and Piacente et al. [Phys. Rev. B 69, 045324 (2004)]. Our theory allows us to determine analytically the system’s behavior at the transition point.

Figure 1

Structural phase transition in a string of equidistant trapped ions from a linear to a planar zigzag configuration. For ions in a harmonic trap, close to the transition point the zigzag configuration is evident about the center of the trap, where the density of ion is larger (Ref. 22).

Figure 2

(Color online) Excitation frequencies of the oscillations modes of a three-ion string, in units of the axial trap frequency $\nu$, as a function of the trap anisotropy $\alpha$. The eigenmodes for $\alpha >{\alpha }^{*}$ are the longitudinal (top) and the transverse ones (bottom). For $\alpha <{\alpha }^{*}$ the eigenmodes are combination of longitudinal and transverse modes with opposite parity by reflection about $x=0$.

Figure 3

Equilibrium position of the external ions of a string of three particles as a function of the trap anisotropy $\alpha$. The solid and dashed lines display the longitudinal and transverse variables, $\overline{x}$ and $\overline{y}$, in units of the characteristic length $l$. The vertical dotted line indicates the transition value ${\alpha }^{*}$, where the equilibrium configuration makes an abrupt change from a linear chain to a zigzag structure.

Figure 4

(Color online) Excitation spectrum of the uniform chain. The eigenfrequencies $\omega$, in units of ${\omega }_0=\sqrt{Q^2∕m{a}^3}$, are plotted as a function of the quasimomentum $k$, in units of $\pi ∕a$. The axial spectrum (solid line) and the transverse spectrum (dashed line) are obtained from Eqs. (11, 12), respectively. Here, ${\nu }_t=1.1{\nu }_t^{\left(c\right)}$.

Figure 5

Transverse equilibrium displacement $b$, in units of the interparticle spacing $a$, as a function of the transverse frequency ${\nu }_t$ in units of ${\nu }_t^{\left(c\right)}$. On the right of the curve the ion crystal is a linear chain. In the region on the left of the curve it exhibits a zigzag structure.

Figure 6

(Color online) Branches of the excitation spectrum of a zigzag structure for the modes on the $x-y$ plane, as obtained from Eq. (23). The curves display the frequencies ${\omega }_{2,+}\left(k\right)$ (solid), ${\omega }_{2,-}\left(k\right)$ (dotted), ${\omega }_{1,+}\left(k\right)$ (dot-dashed), and ${\omega }_{1,-}\left(k\right)$ (dashed), in units of ${\omega }_0$, as a function of $k$, in units of $\pi ∕a$. The Brillouin zone is now half the Brillouin zone of the linear chain due to the doubling of the crystal periodicity. Here, ${\nu }_t=0.9{\nu }_t^{\left(c\right)}$.

Figure 7

(Color online) Branches of the excitation spectrum at $\left(a\right)$ ${\nu }_t={\nu }_t^c+0^{+}$ (just above the critical value) and $\left(b\right)$ ${\nu }_t={\nu }_t^c+0^{-}$ (just below the critical value). In $\left(b\right)$ the equilibrium structure is a zigzag, the periodicity is doubled with respect to the linear chain and the new Brillouin zone is halved. The four branches of the spectrum are obtained at this point by “folding” the two branches of the linear chain in $\left(a\right)$. The units and style codings in $\left(a\right)$ and $\left(b\right)$ are the same as in Figs. 4, 6, respectively. Here, ${\nu }_t^{\left(c\right)}\simeq 2.05{\omega }_0$.

Figure 8

Phase diagram close to the linear-zigzag transition in the thermodynamic limit, for ${}^{40}\mathrm{Ca}^{+}$ ions. The horizontal axis is the interparticle spacing $a$ in $\mu$m; the vertical axis the corresponding critical frequency ${\nu }_t^{\left(c\right)}$ in MHz. This graphic does not report further curves in the left region, giving the transition to more complex structures. A detailed study of the transitions to these structures can be found in Refs. 19, 20, 23.