In the present work we solve the motion equations of a particle in a Paul trap embeded in the gravitational field of a spherically symmetric mass. One of the ideas behind this work concerns the analysis of the effects that the gravity--induced quantum noise, stemming from the bodies in the neighborhood of the Paul trap, could have upon the enhancement of the quantum behavior of this system. This will be done considering a series expansion for the gravitational field of the source, and including in the Hamiltonian of the Paul trap only the first two terms. Higher--order contributions will be introduced as part of the environment of the system, and in consequence will not appear in the Hamiltonian. In other words, we put forward an argument that allows us to differentiate those gravitational degrees of freedom that will appear as an uncontrollable influence on the Paul trap. Along the ideas of the so called restricted path integral formalism, we take into account the continuous monitoring of the position of our particle, and in consequence the corresponding propagators and probabilities, associated with the different measurements outputs, are obtained.
Afterwards, the differential equation related to a quantum nondemolition variable is posed and solved, i.e., a family of quantum nondemolition parameters is obtained. Finally, a qualitative analysis of the effects on the system, of the gravity--induced environment, will be done.
In this work a family of quantum nondemolition variables for the case of a particle caught in a Paul trap is obtained. Afterwards, in the context of the so called restricted path integral formalism, a continuous measuring process for this family of parameters is considered, and then the corresponding propagators are calculated. In other words, the time evolution of a particle in a Paul trap, when the corresponding quantum nondemolition parameter is being continuously monitored, is deduced. The probabilities associated with the possible measurement outputs are also obtained, and in this way new…
We show that quantum localization occurs in the center-of-mass motion of an ion stored in a Paul trap and interacting with a standing laser field. The present experimental state of the art makes the observation of this phenomenon feasible.
Using the symmetries of the three-dimensional Paul trap, we derive the solutions of the time-dependent Schrödinger equation for this system, in both Cartesian and cylindrical coordinates. Our symmetry calculations provide insights that are not always obvious from the conventional viewpoint.
Paul traps are ion traps that are widely used in spectroscopic experiments to confine and stabilize a charged particle within a small region using oscillating electric fields. The dynamics of the particle inside a Paul trap is described by Mathieu equations. It has been proposed that such traps can be used to detect the effects produced by spontaneous collapse of the associated wavefunction, as described by the model of CSL (Continuous Spontaneous Localization). This model is a non-linear, stochastic and non-relativistic modification to the Schrödinger equation which predicts an additional ran…
Using the Paul Trap as a model, we point out that the same wave functions can be variously coherent or squeezed states, depending upon the system they are applied to.
Recently predicted and observed Pauli crystals are structures formed by trapped ultracold non-interacting particles obeying Fermi statistics. The relative positions of the particles are determined by the trapping potential and the Pauli exclusion principle. Measuring all particles at once, their positions tend to be close to vertices of non-trivial polyhedrons. Similar shapes are found for systems of particles bound to the surface of a sphere.
Trapping ions in Paul traps requires high radio-frequency voltages, which are generated using resonators. When operating traps in a cryogenic environment, an in-vacuum resonator showing low loss is crucial to limit the thermal load to the cryostat. In this study, we present a guide for the design and production of compact, shielded cryogenic resonators. We produced and characterized three different types of resonators and furthermore demonstrate efficient impedance matching of these resonators at cryogenic temperatures.
We demonstrate trapping of electrons in a millimeter-sized quadrupole Paul trap driven at 1.6~GHz in a room-temperature ultra-high vacuum setup. Cold electrons are introduced into the trap by ionization of atomic calcium via Rydberg states and stay confined by microwave and static electric fields for several tens of milliseconds. A fraction of these electrons remain trapped longer and show no measurable loss for measurement times up to a second. Electronic excitation of the motion reveals secular frequencies which can be tuned over a range of several tens to hundreds of MHz. Operating a simila…
We calculate the propagator of a particle caught in a Paul trap and subject to the continuous quantum measurement of its position. The probabilities of the measurement outputs, the possible trajectories of the particle, are also found. This enables us to propose a series of experiments that would allow to confront the predictions of one of the models that describe the interaction between a measured quantum system and measuring device, namely the so called Restricted Path-Integral Formalism, with the experiment.
Probing high-frequency gravitational waves with entangled vibrational qubits in linear Paul traps
KEK Theory Center（高能加速器研究机构理论中心）；The Graduate University for Advanced Studies, SOKENDAI（高级研究大学院大学（SOKENDAI））；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
We modify the time-dependent electric potential of the Paul trap from a sinusoidal waveform to a square waveform. The exact quantum motion and the Berry's phase of an electron in the modified Paul trap are found in an analytically closed form. We consider a scheme to detect the Berry's phase by a Bohm-Aharonov-type interference experiment and point out a critical property which renders it practicable.
We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general three-dimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We derive analytically the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a method of infinite determinants to find the modes which diagonalize the linearized time-dependent dynamical problem. This allows obtaining explicitly the ('Floquet-Lyapunov') transformation to coordinates of decoupled linear oscillators. We …
The effect of geometrical perturbations on ion oscillation frequencies in a 3-D quadrupole trap is analyzed by theoretical simulations and compared with the experimental results. Theoretically, the distorted potential within this non-ideal trap caused by the insertion of filament has been simulated. An analytical expression for the potential is then fitted and numerical solution of the equations of trapped ion motion is obtained. Motional frequencies are then calculated from the Fourier transformation of the simulated ion trajectories at any given trapping potential and compared with our exper…
Single ions held in linear Paul traps are promising candidates for a future quantum computer. Here, we discuss a two-layer microstructured segmented linear ion trap. The radial and axial potentials are obtained from numeric field simulations and the geometry of the trap is optimized. As the trap electrodes are segmented in the axial direction, the trap allows the transport of ions between different spatial regions. Starting with realistic numerically obtained axial potentials, we optimize the transport of an ion such that the motional degrees of freedom are not excited, even though the transpo…
Two-ion crystals constitute a platform for investigations of quantum nature that can be extended to any ion species or charged particle provided one of the ions in the crystal can be directly laser-cooled and manipulated with laser radiation. This paper presents the formation of two-ion crystals for quantum metrology in a 7-tesla open-ring Penning trap. $^{40}$Ca$^+$ ions are produced either internally by photoionization or externally in a (Paul-trap) source, transported through the strong magnetic field gradient of the superconducting solenoid, and captured in-flight with a mean kinetic energ…
We present a study on the trapping of hard ferromagnetic particles using alternating magnetic fields, with a focus on planar trap geometries. First, we realize and characterize a magnetic Paul trap design for millimeter-size magnets based on a rotating magnetic potential. Employing a physically rotating platform with two pairs of permanent magnets with opposite poles, we show stable trapping of hard ferromagnets a centimeter above the trap and demonstrate that the particle shape plays a critical role in the trapping. Finally, we propose a chip trap design that will open a path to studies of gy…
We present a model as well as experimental results for a surface electrode radio-frequency Paul trap that has a circular electrode geometry well-suited for trapping of single ions and two-dimensional planar ion crystals. The trap design is compatible with microfabrication and offers a simple method by which the height of the trapped ions above the surface may be changed \emph{in situ}. We demonstrate trapping of single and few Sr+ ions over an ion height range of 200-1000 microns for several hours under Doppler laser cooling, and use these to characterize the trap, finding good agreement with …
We study the quantum stability of the dynamics of ions in a Paul trap. We revisit the results of Wang et al. [Phys. Rev. A 52, 1419 (1995)], which showed that quantum trajectories did not have the same region of stability as their classical counterpart, contrary to what is obtained from a Floquet analysis of the motion in the periodic trapping field. Using numerical simulations of the full wave-packet dynamics, we confirm that the classical trapping criterion are fully applicable to quantum motion, when considering both the expectation value of the position of the wave packet and its width.
We have developed an trapped ion system for producing two-dimensional (2D) ion crystals for applications in scalable quantum computing, quantum simulations, and 2D crystal phase transition and defect studies. The trap is a modification of a Paul trap with its ring electrode flattened and split into eight identical sectors, and its two endcap electrodes shaped as truncated hollow cones for laser and imaging optics access. All ten trap electrodes can be independently DC-biased to create various aspect ratio trap geometries. We trap and Doppler cool 2D crystals of up to 30 Ba+ ions and demonstrat…
We report techniques for the fabrication of multi-zone linear RF Paul traps that exploit the machinability and electrical conductivity of degenerate silicon. The approach was tested by trapping and laser cooling 24Mg+ ions in two trap geometries: a single-zone two-layer trap and a multi-zone surface-electrode trap. From the measured ion motional heating rate we determine an electric field spectral density at the ion's position of approximately 1E-10 (V/m)^2/Hz at a frequency of 1.125 MHz when the ion lies 40 micron above the trap surface. One application of these devices is controlled manipula…
We report on the construction and characterization of an apparatus for quantum information experiments using $^{88}$Sr$^+$ ions. A miniature linear radio-frequency (rf) Paul trap was designed and built. Trap frequencies above 1 MHz in all directions are obtained with 50 V on the trap end-caps and less than 1 W of rf power. We encode a quantum bit (qubit) in the two spin states of the $S_{1/2}$ electronic ground-state of the ion. We constructed all the necessary laser sources for laser cooling and full coherent manipulation of the ions' external and internal states. Oscillating magnetic fields …
A single Ca+ ion in a Paul trap has been cooled to the ground state of vibration with up to 99.9% probability. Starting from this Fock state |n=0> we have demonstrated coherent quantum state manipulation on an optical transition. Up to 30 Rabi oscillations within 1.4 ms have been observed. We find a similar number of Rabi oscillations after preparation of the ion in the |n=1> Fock state. The coherence of optical state manipulation is only limited by laser and ambient magnetic field fluctuations. Motional heating has been measured to be as low as one vibrational quantum in 190 ms.
Long range polarization forces between ions and neutral atoms result in large elastic scattering cross sections, e.g., 10^6 a.u. for Na+ on Na or Ca+ on Na at cold and ultracold temperatures. This suggests that a hybrid ion-neutral trap should offer a general means for significant sympathetic cooling of atomic or molecular ions. We present SIMION 7.0 simulation results concerning the advantages and limitations of sympathetic cooling within a hybrid trap apparatus, consisting of a linear rf Paul trap concentric with a Na magneto-optical trap (MOT). This paper explores the impact of various heat…
We report on the loading of large ion Coulomb crystals into a linear Paul trap incorporating a high-Finesse optical cavity (F~3200). We show that, even though the 3-mm diameter dielectric cavity mirrors are placed between the trap electrodes and separated by only 12 mm, it is possible to produce in situ ion Coulomb crystals containing more than 100.000 calcium ions of various isotopes and with lengths of up to several millimeters along the cavity axis. We show that the number of ions inside the cavity mode is in principle high enough to achieve strong collective coupling between the ion Coulom…
We describe the construction and operation of a segmented linear Paul trap, fabricated in printed-circuit-board technology with an electrode segment width of 500 microns. We prove the applicability of this technology to reliable ion trapping and report the observation of Doppler cooled ion crystals of Ca-40 with this kind of traps. Measured trap frequencies agree with numerical simulations at the level of a few percent from which we infer a high fabrication accuracy of the segmented trap. To demonstrate its usefulness and versatility for trapped ion experiments we study the fast transport of a…
We propose a simple idea for realizing a quantum gate with two identical fermions in a double well trap via external optical pulses without addressing the atoms individually. The key components of the scheme are Feshbach resonance and Pauli blocking, which decouple unwanted states from the dynamics. As a physical example we study atoms in the presence of a magnetic Feshbach resonance in a nanoplasmonic trap and discuss the constraints on the operation times for realistic parameters, reaching a fidelity above $99.9\%$ within $42μ$s.
We present and derive analytic expressions for a fundamental limit to the sympathetic cooling of ions in radio-frequency traps using cold atoms. The limit arises from the work done by the trap electric field during a long-range ion-atom collision and applies even to cooling by a zero-temperature atomic gas in a perfectly compensated trap. We conclude that in current experimental implementations this collisional heating prevents access to the regimes of single-partial-wave atom-ion interaction or quantized ion motion. We determine conditions on the atom-ion mass ratio and on the trap parameters…
All three motional modes of a charged dielectric nanoparticle in a Paul trap are cooled by direct feedback to temperatures of a few mK. We test two methods, one based on electrical forces and the other on optical forces; for both methods, we find similar cooling efficiencies. Cooling is characterized for both feedback forces as a function of feedback parameters, background pressure, and the particle's position.
We demonstrate a general technique to achieve a precise radial displacement of the nodal line of the radiofrequency (rf) field in a linear Paul trap. The technique relies on selective adjustment of the load capacitance of the trap electrodes, achieved through the addition of capacitors to the basic resonant rf-circuit used to drive the trap. Displacements of up to 100 micrometer with micrometer precision are measured using a combination of fluorescence images of ion Coulomb crystals and coherent coupling of such crystals to a mode of an optical cavity. The displacements are made without measur…
We investigate the dynamics of mixed-species ion crystals during transport between spatially distinct locations in a linear Paul trap in the diabatic regime. In a general mixed-species crystal, all degrees of freedom along the direction of transport are excited by an accelerating well, so unlike the case of same-species ions, where only the center-of-mass-mode is excited, several degrees of freedom have to be simultaneously controlled by the transport protocol. We design protocols that lead to low final excitations in the diabatic regime using invariant-based inverse-engineering for two differ…
We present measurements of the Electronic Spin Resonance (ESR) of Nitrogen Vacancy (NV) centers in diamonds that are levitating in a ring Paul trap under vacuum. We observe ESR spectra of NV centers embedded in micron-sized diamonds at vacuum pressures down to $2\times 10^{-1}$ mbar, limited by the green laser heating of the diamond. We also observe the NV photoluminescence at vacuum pressures down to $10^{-2}$ mbar and discuss the steps required to control the NV spin under ultra-high vacuum.
We investigate the problem of a single ion in a radio-frequency trap and immersed in an ultracold Bose gas either in a condensed or a non-condensed phase. We develop master equation formalism describing the sympathetic cooling and we determine the cooling rates of ions. We show that cold atomic reservoir modifies the stability diagram of the ion in the Paul trap creating the regions where the ion is either cooled or heated due to the energy quanta exchanged with the time-dependent potential.
We present a comprehensive phase-space treatment of the motion of charged particles in electrodynamic traps. Focusing on five-wire surface-electrode Paul traps, we study the details of integrable and chaotic motion of a single ion. We introduce appropriate phase-space measures and give a universal characterization of the trap effectiveness as a function of the parameters. We rigorously derive the commonly used (time-independent) pseudopotential approximation, quantify its regime of validity and analyze the mechanism of its breakdown within the time-dependent potential. The phase space approach…
Continuous spontaneous localisation (CSL) is a model that captures the effects of a class of extensions to quantum theory which are expected to result from quantum gravity, and is such that wavefunction collapse is a physical process. The rate of such a process could be very much lower than the upper bounds set by searches to date, and yet still modify greatly the interpretation of quantum mechanics and solve the quantum measurement problem. Consequently experiments are sought to explore this. We describe an experiment that has the potential to extend sensitivity to CSL by many orders of magni…
We investigate the evolution of quasi-static stray electric fields in a linear Paul trap over a period of several months. Depending on how these electric fields are initially induced we observe very different time scales for the field drifts. Photo-induced electric fields decay on time scales of days. We interpret this as photo-electrically generated charges on insulating materials which decay via discharge currents. In contrast, stray fields due to the exposure of the ion trap to a beam of Ba atoms mainly exhibit slow dynamics on the order of months. We explain this observation as a consequen…
Transient dynamics of parametric driving for single-electron image current detection in a Paul trap
University of California, Berkeley（加州大学伯克利分校）；Challenge Institute for Quantum Computation, University of California, Berkeley（加州大学伯克利分校量子计算挑战研究所）；School of Physics, Huazhong University of Science and Technology（华中科技大学物理学院）
It was recently proposed to use small groups of trapped ions as qubit carriers in miniaturized electrode arrays that comprise a large number of individual trapping zones, between which ions could be moved. This approach might be scalable for quantum information processing with a large numbers of qubits. Processing of quantum information is achieved by transporting ions to and from separate memory and qubit manipulation zones in between quantum logic operations. The transport of ion groups in this scheme plays a major role and requires precise experimental control and fast transport. In this pa…
We experimentally demonstrate fast separation of a two-ion crystal in a microstructured segmented Paul trap. By the use of spectroscopic calibration routines for the electrostatic trap potentials, we achieve the required precise control of the ion trajectories near the \textit{critical point}, where the harmonic confinement by the external potential vanishes. The separation procedure can be controlled by three parameters: A static potential tilt, a voltage offset at the critical point, and the total duration of the process. We show how to optimize the control parameters by measurements of ion …
We develop the theory to describe the equilibrium ion positions and phonon modes for a trapped ion quantum simulator in an oblate Paul trap that creates two-dimensional Coulomb crystals in a triangular lattice. By coupling the internal states of the ions to laser beams propagating along the symmetry axis, we study the effective Ising spin-spin interactions that are mediated via the axial phonons and are less sensitive to ion micromotion. We find that the axial mode frequencies permit the programming of Ising interactions with inverse power law spin-spin couplings that can be tuned from uniform…
The realization of high fidelity quantum gates in a multi-qubit system, with a typical target set at 99.9%, is a critical requirement for the implementation of fault-tolerant quantum computation. To reach this level of fidelity, one needs to carefully analyze the noises and imperfections in the experimental system and optimize the gate operations to mitigate their effects. Here, we consider one of the leading experimental systems for the fault-tolerant quantum computation, ions in an anharmonic linear Paul trap, and optimize entangling quantum gates using segmented laser pulses with the assist…
We investigate single ions of $^{40}Ca^+$ in Paul traps for quantum information processing. Superpositions of the S$_{1/2}$ electronic ground state and the metastable D$_{5/2}$ state are used to implement a qubit. Laser light on the S$_{1/2} \leftrightarrow$ D$_{5/2}$ transition is used for the manipulation of the ion's quantum state. We apply sideband cooling to the ion and reach the ground state of vibration with up to 99.9% probability. Starting from this Fock state $|n=0>$, we demonstrate coherent quantum state manipulation. A large number of Rabi oscillations and a ms-coherence time is ob…
Electrical Noise Produced by Micron-Sized Particles above a Surface Paul Trap
University of California, Berkeley（加州大学伯克利分校）；Challenge Institute for Quantum Computation, University of California, Berkeley（加州大学伯克利分校量子计算挑战研究所）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present the design and construction of a new experimental apparatus for the trapping of single Ba$^+$ ions in the center of curvature of an optical-quality hemispherical mirror. We describe the layout, fabrication and integration of the full setup, consisting of a high-optical access monolithic `3D-printed' Paul trap, the hemispherical mirror, a diffraction-limited in-vacuum lens (NA = 0.7) for collection of atomic fluorescence and a state-of-the art ultra-high vacuum vessel. This new apparatus enables the study of quantum electrodynamics effects such as strong inhibition and enhancement of…
We experimentally study two-dimensional (2D) Coulomb crystals in the "radial-2D" phase of a linear Paul trap. This phase is identified by a 2D ion lattice aligned entirely with the radial plane and is created by imposing a large ratio of axial to radial trapping potentials. Using arrays of up to 19 $^{171}$Yb$^+$ ions, we demonstrate that the structural phase boundaries and vibrational mode frequencies of such crystals are well-described by the pseudopotential approximation, despite the time-dependent ion positions driven by intrinsic micromotion. We further observe that micromotion-induced he…
We theoretically investigate the process of splitting two-ion crystals in segmented Paul traps, i.e. the structural transition from two ions confined in a common well to ions confined in separate wells. The precise control of this process by application of suitable voltage ramps to the trap segments is non-trivial, as the harmonic confinement transiently vanishes during the process. This makes the ions strongly susceptible to background electric field noise, and to static offset fields in the direction of the trap axis. We analyze the reasons why large energy transfers can occur, which are imp…
