Nanofriction in Cavity Quantum Electrodynamics

The dynamics of cold trapped ions in a high-finesse resonator results from the interplay between the long-range Coulomb repulsion and the cavity-induced interactions. The latter are due to multiple scatterings of laser photons inside the cavity and become relevant when the laser pump is sufficiently strong to overcome photon decay. We study the stationary states of ions coupled with a mode of a standing-wave cavity as a function of the cavity and laser parameters, when the typical length scales of the two self-organizing processes, Coulomb crystallization and photon-mediated interactions, are incommensurate. The dynamics are frustrated and in specific limiting cases can be cast in terms of the Frenkel-Kontorova model, which reproduces features of friction in one dimension. We numerically recover the sliding and pinned phases. For strong cavity nonlinearities, they are in general separated by bistable regions where superlubric and stick-slip dynamics coexist. The cavity, moreover, acts as a thermal reservoir and can cool the chain vibrations to temperatures controlled by the cavity parameters and by the ions’ phase. These features are imprinted in the radiation emitted by the cavity, which is readily measurable in state-of-the-art setups of cavity quantum electrodynamics.

Figure 1

(a) An array of cold ions in the optical lattice of a high-finesse cavity is an exotic realization of the Frenkel-Kontorova (FK) model. The cavity is pumped by a laser with amplitude $\eta$ and decays at rate $\kappa$, and the intracavity photon number is determined by the ion density; this gives rise to a globally deformable potential which depends nonlinearly on the ions’ positions. (b) The functional form of the cavity-induced potential for one particle. In the pinned phase, the shape is approximately sinusoidal for $|C|<1$, and for $|C|>1$ it becomes flat everywhere except in the vicinity of minima ($C>0$) or maxima ($C<0$). (c) Sketch of the phase diagram for the stationary state as a function of $\eta$ and of the cavity nonlinearity $C$ (cooperativity). Typically for $|C|>1$ bistable phases can be observed (shown by the hatched region), signifying that superlubric or stick-slip dynamics are found depending on the variation of $\eta$ with time.

Figure 2

Phase diagram as a function of $C$ and $\eta$ for (a) ${\mathrm{\Delta }}_c=0$, (b) ${\mathrm{\Delta }}_c=-2\kappa$. In (c) the phase diagram is plotted as a function of ${\mathrm{\Delta }}_c$ and $\eta$ for cooperativity $C=-2$. Parameters $\eta$ and ${\mathrm{\Delta }}_c$ are in units of $\kappa$. The solid red line indicates the symmetry-breaking transition from the sliding ($S$) to the pinned ($P$) phase, and the hatched white region is an area of bistability. The color code gives the value of $B_N$ for $C>0$, and of $1-B_N$ when $C<0$. Subplots (d) and (e) display $B_{N}N$ vs $N$ for $C=0.5$, ${\mathrm{\Delta }}_c=0$, where (d) $\eta =50\kappa$ and (e) $\eta =500\kappa$, respectively. (f) The individual contributions ${\cos }^2(k{x}_j)$ to $B_N$ for $N=81$ particles, deep in the pinned phase ($B_N=1.5\times 10^{-3}$) for $C=0.5$. The ions are ${{}^{174}\mathrm{Yb}}^{+}$ and the parameters are $|{\mathrm{\Delta }}_0|=2\pi \times 12\mathrm{GHz}$ and $\kappa =2\pi \times 0.2\mathrm{MHz}$; for $N=11$ the trap frequency is $\omega =2\pi \times 1.12\mathrm{MHz}$, while $g$ is varied in order to sweep over different values of $C$. The cavity wavelength is $\lambda =369\mathrm{nm}$ and the ratio $2d/\lambda =7.3507$. The center of the harmonic trap corresponds with a maximum of $V_{\text{cav}}$, ensuring a symmetry-breaking transition. The grey area in (a)–(c) indicates where the mean phonon number $\overline{n}<1$; outside of this region, our semiclassical analysis is reliable.

Figure 3

Spectrum at the cavity output $S(\nu )$ (in arbitrary units) for $C<0$ (a) in the sliding phase with $B_N=1–0.45$ and (b) in the pinned phase with $B_N=1–0.05$, for $C=-0.5$ (black line) and $C=-2$ (red line), and for 11 ${{}^{174}\mathrm{Yb}}^{+}$ ions, ${\mathrm{\Delta }}_c=0$, ${\mathrm{\Gamma }}_n=0.1\kappa$, and $T=100\mu \mathrm{K}$. The resonances correspond to vibrational eigenmodes coupling with the cavity field and change in the pinned phase as $C$ is increased. The elastic peak at ${\omega }_p$ (corresponding to $\nu =0$) is not reported.