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arXiv:1209.2133 · Physical Review A

Ion chains in high-finesse cavities

Cecilia Cormick, Giovanna Morigi

Published Jan 24, 2013• 13 citations
arXivHTMLPDF

Abstract

We analyze the dynamics of a chain of singly charged ions confined in a linear Paul trap and which couple with the mode of a high-finesse optical resonator. In these settings the ions interact via the Coulomb repulsion and are subject to the mechanical forces due to scattering of cavity photons. We show that the interplay of these interactions can give rise to bistable equilibrium configurations, into which the chain can be cooled by cavity-enhanced photon scattering. We characterize the resulting equilibrium structures by determining the stationary state in the semiclassical limit for both cavity field and crystal motion. The mean occupation of the vibrational modes at steady state is evaluated, showing that the vibrational modes coupled to the cavity can be simultaneously cooled to low occupation numbers. It is also found that at steady state the vibrations are entangled with the cavity field fluctuations. The entanglement is quantified by means of the logarithmic negativity. The spectrum of the light at the cavity output is evaluated and the features signaling entanglement are identified.

DOI
10.1103/physreva.87.013829

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We analyze the dynamics of a chain of singly charged ions confined in a linear Paul trap and which couple with the mode of a high-finesse optical resonator. In these settings the ions interact via the Coulomb repulsion and are subject to the mechanical forces due to scattering of cavity photons. We show that the interplay of these interactions can give rise to bistable equilibrium configurations, into which the chain can be cooled by cavity-enhanced photon scattering. We characterize the resulting equilibrium structures by determining the stationary state in the semiclassical limit for both cavity field and crystal motion. The mean occupation of the vibrational modes at steady state is evaluated, showing that the vibrational modes coupled to the cavity can be simultaneously cooled to low occupation numbers. It is also found that at steady state the vibrations are entangled with the cavity field fluctuations. The entanglement is quantified by means of the logarithmic negativity. The spectrum of the light at the cavity output is evaluated and the features signaling entanglement are identified.
We analyze the stability and dynamics of an ion chain confined inside a high-finesse optical resonator. When the dipolar transition of the ions strongly couples to one cavity mode, the mechanical effects of light modify the chain properties close to a structural transition. We focus on the linear chain close to the zigzag instability and show that linear and zigzag arrays are bistable for certain strengths of the laser pumping the cavity. For these regimes the chain is cooled into one of the configurations by cavity-enhanced photon scattering. The excitations of these structures mix photonic and vibrational fluctuations, which can be entangled at steady state. These features are signaled by Fano-like resonances in the spectrum of light at the cavity output.
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.
We show that the vibrations of an ion Coulomb crystal can be cooled to the zero-point motion through the optomechanical coupling with a high-finesse cavity.Cooling results from the interplay between coherent scattering of cavity photons by the ions, which dynamically modifies the vibrational spectrum, and cavity losses, that dissipate motional energy.The cooling mechanism we propose requires that the length scales of the crystal and the cavity are mismatched so that the system is intrinsically frustrated, leading to the formation of defects (kinks).When the pump is strong enough, the anti-Stokes sidebands of all vibrational modes can be simultaneously driven.These dynamics can be used to prepare ultracold chains of dozens of ions within tens of milliseconds in state-of-the-art experimental setups.In addition, we identify parameter regimes of the optomechanical interactions where individual localized modes can be selectively manipulated, and monitored through the light at the cavity output.These dynamics exemplify robust quantum reservoir engineering of strongly-correlated mesoscopic systems and could find applications in optical cooling of solids.
Dissipative quantum systems can under appropriate conditions exhibit bi- or multipartite entanglement at the steady state. The presence and properties of these quantum correlations depend on the relevant model parameters. Here, we characterize the steady-state entanglement in connection with the spatial structure of a small chain of three ions dispersively coupled with a pumped optical cavity. Within a semiclassical approximation, we describe the relation between entanglement, spatial organization, and vibrational modes of the ion chain. Upon increasing the pumping strength, our system undergoes a transition from a sliding to a pinned configuration, in which ions are expelled from the maxima of the optical potential. The features of the steady-state entanglement strongly depend on the kind of pinned configuration reached. We identify scenarios leading to entangled steady states, analyze the effect of defect formation upon entanglement between different system partitions, and observe the presence of multipartite quantum correlations.
We consider an optomechanical system composed of a trapped ion dispersively coupled to a single mode of a pumped optical cavity. We focus on a parameter range for which the semiclassical description predicts two clearly distinct equilibrium configurations in the limits of small and large photon pumping, while a bistable regime is found for intermediate pumping. This semiclassical description, however, is not valid in close proximity to the system transitions or when the mean photon number is low. Here, we provide a numerical analysis of the fully quantum state in the few-photon regime, exploring the features of the asymptotic state across the transition and analyzing possible markers of semiclassical bistability. We find an increase in the entropy of the system and of the entanglement in the transition region but no clear signatures of metastability in the spectrum of the evolution.
The semiclassical dynamics of atoms, when the atoms are confined inside a standing-wave high-finesse resonator, is theoretically studied. The atoms are cooled by scattering processes in which the photons of a transverse laser are coherently scattered into the cavity mode. We derive a Fokker-Planck equation for the atomic center-of-mass variables which allows us to determine the equations of motion in the semiclassical limit for any value of the intensity of the laser field. We extract its prediction for the dynamics when the resonator is essentially in the vacuum state and the atoms are cooled by scattering photons into the cavity mode, which then decays. Its predictions for the stationary atomic distribution are compared with the ones of the Fokker-Planck equation by Domokos et al. [J. Phys. B 34, 187 (2001)], which has been derived under different assumptions. We find full agreement in the considered parameter regime.