Phase-coherent sensing of the center-of-mass motion of trapped-ion crystals.

Trapped ions are sensitive detectors of weak forces and electric fields that excite ion motion. Here measurements of the center-of-mass motion of a trapped-ion crystal that are phase coherent with an applied weak external force are reported. These experiments are conducted far from the trap motional frequency on a two-dimensional trapped-ion crystal of approximately 100 ions, and determine the fundamental measurement imprecision of our protocol free from noise associated with the center-of-mass mode. The driven sinusoidal displacement of the crystal is detected by coupling the ion crystal motion to the internal spin degree of freedom of the ions using an oscillating spin-dependent optical dipole force. The resulting induced spin precession is proportional to the displacement amplitude of the crystal, and is measured with near-projection-noise-limited resolution. A 49 pm displacement is detected with a signal-to-noise ratio of 1 in a single experimental determination, which is an order-of-magnitude improvement over prior phase-incoherent experiments. This displacement amplitude is 40 times smaller than the zero-point fluctuations. With our repetition rate, an 8.4   pm / Hz displacement sensitivity is achieved, which implies 12   ( yN/ion ) / Hz and 77   ( µ V/m ) / Hz sensitivities to forces and electric fields, respectively. This displacement sensitivity, when applied on-resonance with the center-of-mass mode, indicates the possibility of weak force and electric field detection below 10-3 yN/ion and 1 nV/m, respectively.

Author Correction: Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Quantum amplification of mechanical oscillator motion.

Detection of the weakest forces in nature is aided by increasingly sensitive measurements of the motion of mechanical oscillators. However, the attainable knowledge of an oscillator's motion is limited by quantum fluctuations that exist even if the oscillator is in its lowest possible energy state. We demonstrate a technique for amplifying coherent displacements of a mechanical oscillator with initial magnitudes well below these zero-point fluctuations. When applying two orthogonal squeezing interactions, one before and one after a small displacement, the displacement is amplified, ideally with no added quantum noise. We implemented this protocol with a trapped-ion mechanical oscillator and determined an increase by a factor of up to 7.3 (±0.3) in sensitivity to small displacements.

Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model.

Scrambling is the process by which information stored in local degrees of freedom spreads over the many-body degrees of freedom of a quantum system, becoming inaccessible to local probes and apparently lost. Scrambling and entanglement can reconcile seemingly unrelated behaviors including thermalization of isolated quantum systems and information loss in black holes. Here, we demonstrate that fidelity out-of-time-order correlators (FOTOCs) can elucidate connections between scrambling, entanglement, ergodicity and quantum chaos (butterfly effect). We compute FOTOCs for the paradigmatic Dicke model, and show they can measure subsystem Rényi entropies and inform about quantum thermalization. Moreover, we illustrate why FOTOCs give access to a simple relation between quantum and classical Lyapunov exponents in a chaotic system without finite-size effects. Our results open a path to experimental use FOTOCs to explore scrambling, bounds on quantum information processing and investigation of black hole analogs in controllable quantum systems.

Verification of a Many-Ion Simulator of the Dicke Model Through Slow Quenches across a Phase Transition.

We use a self-assembled two-dimensional Coulomb crystal of ∼70 ions in the presence of an external transverse field to engineer a simulator of the Dicke Hamiltonian, an iconic model in quantum optics which features a quantum phase transition between a superradiant (ferromagnetic) and a normal (paramagnetic) phase. We experimentally implement slow quenches across the quantum critical point and benchmark the dynamics and the performance of the simulator through extensive theory-experiment comparisons which show excellent agreement. The implementation of the Dicke model in fully controllable trapped ion arrays can open a path for the generation of highly entangled states useful for enhanced metrology and the observation of scrambling and quantum chaos in a many-body system.

Amplitude Sensing below the Zero-Point Fluctuations with a Two-Dimensional Trapped-Ion Mechanical Oscillator.

We present a technique to measure the amplitude of a center-of-mass (c.m.) motion of a two-dimensional ion crystal of ∼100 ions. By sensing motion at frequencies far from the c.m. resonance frequency, we experimentally determine the technique's measurement imprecision. We resolve amplitudes as small as 50 pm, 40 times smaller than the c.m. mode zero-point fluctuations. The technique employs a spin-dependent, optical-dipole force to couple the mechanical oscillation to the electron spins of the trapped ions, enabling a measurement of one quadrature of the c.m. motion through a readout of the spin state. We demonstrate sensitivity limits set by spin projection noise and spin decoherence due to off-resonant light scattering. When performed on resonance with the c.m. mode frequency, the technique demonstrated here can enable the detection of extremely weak forces (<1 yN) and electric fields (<1 nV/m), providing an opportunity to probe quantum sensing limits and search for physics beyond the standard model.

Phase-coherent detection of an optical dipole force by Doppler velocimetry.

We report phase-coherent Doppler detection of optical dipole forces using large ion crystals in a Penning trap. The technique is based on laser Doppler velocimetry using a cycling transition in 9Be+ near 313 nm and the center-of-mass (COM) ion motional mode. The optical dipole force is tuned to excite the COM mode, and measurements of photon arrival times synchronized with the excitation potential show oscillations with a period commensurate with the COM motional frequency. Experimental results compare well with a quantitative model for a driven harmonic oscillator. This technique permits characterization of motional modes in ion crystals; the measurement of both frequency and phase information relative to the driving force is a key enabling capability--comparable to lockin detection - providing access to a parameter that is typically not available in time-averaged measurements. This additional information facilitates discrimination of nearly degenerate motional modes.

Decoherence due to elastic Rayleigh scattering.

We present theoretical and experimental studies of the decoherence of hyperfine ground-state superpositions due to elastic Rayleigh scattering of light off resonant with higher lying excited states. We demonstrate that under appropriate conditions, elastic Rayleigh scattering can be the dominant source of decoherence, contrary to previous discussions in the literature. We show that the elastic-scattering decoherence rate of a two-level system is given by the square of the difference between the elastic-scattering amplitudes for the two levels, and that for certain detunings of the light, the amplitudes can interfere constructively even when the elastic-scattering rates from the two levels are equal. We confirm this prediction through calculations and measurements of the total decoherence rate for a superposition of the valence electron spin levels in the ground state of 9Be+ in a 4.5 T magnetic field.

Entangled mechanical oscillators.

Hallmarks of quantum mechanics include superposition and entanglement. In the context of large complex systems, these features should lead to situations as envisaged in the 'Schrödinger's cat' thought experiment (where the cat exists in a superposition of alive and dead states entangled with a radioactive nucleus). Such situations are not observed in nature. This may be simply due to our inability to sufficiently isolate the system of interest from the surrounding environment-a technical limitation. Another possibility is some as-yet-undiscovered mechanism that prevents the formation of macroscopic entangled states. Such a limitation might depend on the number of elementary constituents in the system or on the types of degrees of freedom that are entangled. Tests of the latter possibility have been made with photons, atoms and condensed matter devices. One system ubiquitous to nature where entanglement has not been previously demonstrated consists of distinct mechanical oscillators. Here we demonstrate deterministic entanglement of separated mechanical oscillators, consisting of the vibrational states of two pairs of atomic ions held in different locations. We also demonstrate entanglement of the internal states of an atomic ion with a distant mechanical oscillator. These results show quantum entanglement in a degree of freedom that pervades the classical world. Such experiments may lead to the generation of entangled states of larger-scale mechanical oscillators, and offer possibilities for testing non-locality with mesoscopic systems. In addition, the control developed here is an important ingredient for scaling-up quantum information processing with trapped atomic ions.

Microfabricated surface-electrode ion trap for scalable quantum information processing.

Individual laser-cooled 24Mg+ ions are confined in a linear Paul trap with a novel geometry where gold electrodes are located in a single plane and the ions are trapped 40 microm above this plane. The relatively simple trap design and fabrication procedure are important for large-scale quantum information processing (QIP) using ions. Measured ion motional frequencies are compared to simulations. Measurements of ion recooling after cooling is temporarily suspended yield a heating rate of approximately 5 motional quanta per millisecond for a trap frequency of 2.83 MHz, sufficiently low to be useful for QIP.

Rapid heating of a strongly coupled plasma near the solid-liquid phase transition.

Between 10(4) and 10(6) 9Be+ ions were trapped in a Penning trap and laser cooled to approximately 1 mK, where they formed a crystalline plasma. We measured the ion temperature as a function of time after turning off the laser cooling and observed a rapid temperature increase as the plasma underwent the solid-liquid phase transition at T approximately 10 mK (Gamma approximately 170). We present evidence that this rapid heating is due to a sudden release of energy from weakly cooled degrees of freedom involving the cyclotron motion of trapped impurity ions. This equilibration of cyclotron motion with motion parallel to the magnetic field is more than 10 orders of magnitude faster than that predicted by currently available theory, which is valid only in the absence of correlations (Gamma<<1).

Laser-generated waves and wakes in rotating ion crystals.

Locally excited plasma waves are generated in a Coulomb crystal by "pushing" with radiation pressure on a rotating cloud of laser-cooled 9Be+ ions. The waves form a stationary wake that is directly imaged through the dependence of the ion fluorescence on Doppler shifts, and theoretical calculations in a slab geometry are shown to accurately reproduce these images. The technique demonstrates a new method of exciting and studying waves in cold ion clouds.

Primary Atomic Frequency Standards at NIST.

The development of atomic frequency standards at NIST is discussed and three of the key frequency-standard technologies of the current era are described. For each of these technologies, the most recent NIST implementation of the particular type of standard is described in greater detail. The best relative standard uncertainty achieved to date for a NIST frequency standard is 1.5×10(-15). The uncertainties of the most recent NIST standards are displayed relative to the uncertainties of atomic frequency standards of several other countries.

Progress at NIST toward absolute frequency standards using stored ions.

Experiments directed toward the realization of frequency standards of high accuracy using stored ions are briefly summarized. In one experiment, an RF oscillator is locked to a nuclear spin-flip hyperfine transition (frequency approximately 3.03x10(8) Hz) in (9 )Be(+) ions that are stored in a Penning trap and sympathetically laser-cooled. Stability is better than 3x10(-12)tau(-(1/2)) and uncertainty in Doppler shifts is estimated to be less than 5x10(-15). In a second experiment, a stable laser is used to probe an electric quadrupole transition (frequency approximately 1.07x10(15) Hz) in a single laser-cooled (199)Hg(+) ion stored in a Paul trap. The measured Q value of this transition is approximately 10(13). Future possible experiments are discussed.