Single-ion nonlinear mechanical oscillator

Abstract

We study the steady-state motion of a single trapped ion oscillator driven to the nonlinear regime. Damping is achieved via Doppler laser cooling. The ion motion is found to be well described by the Duffing oscillator model with an additional nonlinear damping term. We demonstrate here the unique ability of tuning both the linear as well as the nonlinear damping coefficients by controlling the laser-cooling parameters. Our observations pave the way for the investigation of nonlinear dynamics on the quantum-to-classical interface as well as mechanical noise squeezing in laser-cooling dynamics.Nonlinear dynamics prevails in many dynamical systems in nature; it introduces a rich behavior that includes criticality, bifurcations, and chaos. Nonlinear dynamics on the microscopic scale is especially interesting because it can shed light on the quantum-to-classical transition [1] as well as provide a means to amplify or suppress thermal and quantum noise [2].All mechanical oscillators exhibit nonlinearity when driven far away from equilibrium. The simplest of such nonlinear oscillators is the Duffing oscillator, which includes a cubic term in the restoring force [3]. Duffing nonlinear dynamics has been studied with electrons in a Penning trap [4] and recently with nano-electromechanical beam resonators. The basins of attraction of a nanobeam oscillator were mapped [5]. Noise squeezing and stochastic resonances were observed close to the Duffing instability [6,7]. Noise squeezing was predicted to enable mass and force detection with precision below the standard thermal limit [8-10] and possibly below the standard quantum limit when operating close to the oscillator ground state [11].The mechanical motion of trapped ions is highly controllable and can be efficiently laser-cooled to the quantum ground state [12]. High-fidelity production of Fock, squeezed, and Schrödinger-cat states was demonstrated with a single trapped ion [13,14]. At the temperature range obtained with laser-cooling techniques, quadruple radiofrequency (rf) Paul traps are excellently approximated as harmonic. Nonlinearity in ion motion was observed when several ions were trapped due to their mutual Coulomb repulsion. Here, nonlinearity couples between the ion-crystal normal modes, even at the single quantum level [15,16]. Trap nonlinearities are important in the context of resonance ejection in high-resolution mass spectrometry [17] and were shown to introduce instabilities in the ion motion at certain trapping parameters [18]. Recently, amplification saturation of a single-ion "phonon laser," resulting from optical forces that are nonlinear in the ion velocity, was demonstrated [19].Here, we study the nonlinear mechanical response of a single laser-cooled 88 Sr + ion in a linear rf Paul trap. The nonlinearity originates from the higher than quadrupolar order terms in the trapping potential. We find that the ion's steady-state response is well described by the Duffing model with an additional nonlinear damping term [20]. Unlike other realizations of nonl...

Controlling the transport of an ion: classical and quantum mechanical solutions

“…Further miniaturization is expected to yield trapping potentials where the wavepacket samples regions of space in which the potential, or potential fluctuations, are strongly anharmonic. Also, for large motional excitations, recent experiments have shown nonlinear Duffing oscillator behavior [50], nonlinear coupling of modes in linear ion crystals [51,52] and amplitude dependent modifications of normal modes frequencies and amplitude due to nonlinearities [53]. In these cases, numerical optimization of the ionʼs quantum dynamics presents itself as a well-adapted and efficient approach capable of providing high-fidelity control solutions.…”

Quantum control of 88Sr+ in a miniature linear Paul trap

“…This term is measured by the study of the ion non-linear response to a strong, near resonance, drive. The cubic force term coefficient is found to be α/(2π) 2 = 1.74 ± 0.03 × 10 −7 kg Hz 2 /m 2 [22].…”

Kinetic phase transition with global coupling in the resonantly driven atomic trap

“…The sgn(z) indicates the sign function and z j represents the position of the jth atom of a stable state. The nonlinear oscillator in the periodically perturbed magneto-optical trap consists of a number of atoms, and there is a long-range attractive interaction between atoms, unlike in the single nonlinear mechanical oscillator [32]. The atom-atom interaction modifies the activation energy R n in equation (1), including the interaction effect, and thus the switching probability W nm is changed.…”