We report a precise search for correlation effects in linear chains of 2 and 3 trapped Ca+ ions. Unexplained correlations in photon emission times within a linear chain of trapped ions have been reported, which, if genuine, cast doubt on the potential of an ion trap to realize quantum information processing. We observe quantum jumps from the metastable 3d 2D_{5/2} level for several hours, searching for correlations between the decay times of the different ions. We find no evidence for correlations: the number of quantum jumps with separations of less than 10 ms is consistent with statistics to within errors of 0.05%; the lifetime of the metastable level derived from the data is consistent with that derived from independent single-ion data at the level of the experimental errors 1%; and no rank correlations between the decay times were found with sensitivity to rank correlation coefficients at the level of |R| = 0.024.
We demonstrate a double-trap system well suited to study cold collisions between trapped ions and trapped atoms. Using Yb$^+$ ions confined in a Paul trap and Yb atoms in a magneto-optical trap, we investigate charge-exchange collisions of several isotopes for collision energies down to 400 neV (5 mK). The measured rate coefficient of $6 \times 10^{-10}$ cm$^{3}$s$^{-1}$, constant over four orders of magnitude in collision energy, is in good agreement with that derived from a semiclassical Langevin model for an atomic polarizability of 143 a.u.
We demonstrate an ultraviolet diode laser system for cooling of trapped ytterbium ions. The laser power and linewidth are comparable to previous systems based on resonant frequency doubling, but the system is simpler, more robust, and less expensive. We use the laser system to cool small numbers of ytterbium ions confined in a linear Paul trap. From the observed spectra, we deduce final temperatures < 270 mK.
The direct photodissociation of trapped $^{85}$Rb$_2^+$ (rubidium) molecular ions by the cooling light for the $^{85}$Rb magneto-optical trap (MOT) is studied, both experimentally and theoretically. Vibrationally excited Rb$_{2}^{+}$ ions are created by photoionization of Rb$_{2}$ molecules formed photoassociatively in the Rb MOT and are trapped in a modified spherical Paul trap. The decay rate of the trapped Rb$_{2}^{+}$ ion signal in the presence of the MOT cooling light is measured and agreement with our calculated rates for molecular ion photodissociation is observed. The photodissociation…
Various techniques are used to detect the presence of charged particles stored in electromagnetic traps, their energy, their mass, or their internal states. Detection methods can rely on the variation of the number of trapped particles (destructive methods) or the use of the ion's interaction with electromagnetic radiation as a non-destructive tool to probe the trapped particles. This review gives an introduction into various methods, discussing the basic mode of operation completed by the description of recent realizations.
We propose and demonstrate a new method for Doppler cooling trapped-ion crystals where the distribution of micromotion amplitudes may be large and uneven. The technique uses pulses of Doppler cooling light synchronized with the trap RF that selectively target ions when their velocity is near a node, leading to more uniform cooling across a crystal by a single tone of cooling light. We lay out a theoretical framework that describes where this technique is practical, and provide a simple experimental demonstration.
We demonstrate a method for precision spectroscopy on trapped ions in the limit of unresolved motional sidebands. By sympathetic cooling of a chain of crystallized ions we suppress adverse temperature variations induced by the spectroscopy laser that usually lead to a distorted line profle and obtain a Voigt profile with negligible distortions. We applied the method to measure the absolute frequency of the astrophysically relevant D2 transition in single 24-Mg+ ions and find 1072082934.33(16)MHz, a nearly 400fold improvement over previous results. Further, we find the excited state lifetime to…
Trapped diatomic molecular ions could prove to be a sensitive probe for a permanent electron electric dipole moment (eEDM). We propose to use a ground or metastable $^3Δ_1$ level, due to its high polarizability and large EDM enhancement factor. Ions allow for simple trapping and long interrogation times, but require a time-varying electric bias field in order to probe the eEDM. We discuss experimental design as well as challenges in performing a precision spectroscopic measurement in rapidly time-varying electric fields.
We measured the ground-state hyperfine splitting of trapped ${}^{113}$Cd${}^{+}$ ions to be 15199862855.02799(27) Hz with a fractional uncertainty of $1.8\times10^{-14}$. The ions were trapped and laser-cooled in a linear quadrupole Paul trap. The fractional frequency stability was measured to be ${4.2} \times 10^{-13}/\sqrtτ $, obtained from Ramsey fringes of high signal-to-noise ratio and taken over a measurement time of nearly 5 hours, which is close to the short-term stability limit estimated from the Dick effect. Our result is consistent with previous reported values, but the measurement …
Doppler cooling with lasers is essential to ions' trapping and also a preliminary step towards achievement of ultracold ions. Due to lack of effective tools, experimentally monitoring the ions' temperature and the laser-ion coupling is difficult in Doppler cooling. Here we analytically explore the Doppler cooling process of trapped ions, exemplified by $^{40}$Ca$^{+}$, by solving the friction coefficient in the Doppler cooling with respect to a thermal bath, particularly, to a bath with large heating rate. We show four regions for cooling and heating induced by the three-level electromagnetica…
Sympathetic cooling of trapped ions through collisions with neutral buffer gases is critical to a variety of modern scientific fields, including fundamental chemistry, mass spectrometry, nuclear and particle physics, and atomic and molecular physics. Despite its widespread use over four decades, there remain open questions regarding its fundamental limitations. To probe these limits, here we examine the steady-state evolution of up to ten barium ions immersed in a gas of three-million laser-cooled calcium atoms. We observe and explain the emergence of nonequilibrium behavior as evidenced by bi…
We consider the practical feasibility of using the direct, electric dipole-dipole interaction between co-trapped molecular ions for robust quantum logic without the need for static polarizing fields. The use of oscillating dipole moments, as opposed to static electric dipoles, dynamically decouples the dipoles from laboratory fields, including the electric fields of the trap itself. Further, this implementation does not require quantum control of motion, potentially removing a major roadblock to ion trap quantum computing scalability. Since the polarizing field is electromagnetic radiation, ev…
Using a laser polarization gradient, we realize 3D Sisyphus cooling of $^{171}$Yb$^+$ ions confined in and near the Lamb-Dicke regime in a linear Paul trap. The cooling rate and final mean motional energy of a single ion are characterized as a function of laser intensity and compared to semiclassical and quantum simulations. Sisyphus cooling is also applied to a linear string of four ions to obtain a mean energy of 1-3 quanta for all vibrational modes, an approximately order-of-magnitude reduction below Doppler cooled energies. This is used to enable subsequent, efficient sideband laser coolin…
A mixed system of cooled and trapped, ions and atoms, paves the way for ion assisted cold chemistry and novel many body studies. Due to the different individual trapping mechanisms, trapped atoms are significantly colder than trapped ions, therefore in the combined system, the strong binary ion$-$atom interaction results in heat flow from ions to atoms. Conversely, trapped ions can also get collisionally heated by the cold atoms, making the resulting equilibrium between ions and atoms intriguing. Here we experimentally demonstrate, Rubidium ions (Rb$^+$) cool in contact with magneto-optically …
We examine the properties of cold ions confined by a Paul trap in a linear crystal configuration, a system of considerable current interest due to its application to practical quantum computation. Using a combination of theoretical and numerical calculation, a semi- empirical formula for the positions of the ions is derived.
In this chapter, we present an overview of experiments with trapped Rydberg ions and outline the advantages and challenges of developing applications of this new platform for quantum computing, sensing and simulation. Trapped Rydberg ions feature several important properties, unique in their combination: they are tightly bound in a harmonic potential of a Paul trap, in which their internal and external degrees of freedom can be controlled in a precise fashion. High fidelity state preparation of both internal and motional states of the ions has been demonstrated, and the internal states have be…
Radiative emission during cold collisions between trapped laser-cooled Rb atoms and alkaline-earth ions (Ca$^+$, Sr$^+$, Ba$^+$) and Yb$^+$, and between Li and Yb$^+$, are studied theoretically, using accurate effective-core-potential based quantum chemistry calculations of potential energy curves and transition dipole moments of the related molecular ions. Radiative association of molecular ions is predicted to occur for all systems with a cross section two to ten times larger than the radiative charge transfer one. Partial and total rate constants are also calculated and compared to availabl…
Distribution of the entangled state of trapped atomic ions to long distance using quantum repeater protocol is considered. Indeed, the long distance is divided into short parts, and then using entanglement generation and entanglement swapping techniques in optomechanical cavities, the entanglement is distributed. To do the task, we perform interaction between trapped atomic ions in optomechanical cavities, operate proper measurements on trapped ions and also make Bell state measurement as a well-known way to swap the entanglement. Accordingly, the entanglement is distributed between target ion…
Using trapped ions in an entangled state we propose detecting a magnetic dipole of a single atom at distance of a few $μ$m. This requires a measurement of the magnetic field gradient at a level of about 10$^{-13}$ Tesla/$μ$m. We discuss applications e.g. in determining a wide variation of ionic magnetic moments, for investigating the magnetic substructure of ions with a level structure not accessible for optical cooling and detection,and for studying exotic or rare ions, and molecular ions. The scheme may also be used for measureing spin imbalances of neutral atoms or atomic ensembles trapped …
We present a protocol for using trapped ions to measure rotations via matter-wave Sagnac interferometry. The trap allows the interferometer to enclose a large area in a compact apparatus through repeated round-trips in a Sagnac geometry. We show how a uniform magnetic field can be used to close the interferometer over a large dynamic range in rotation speed and measurement bandwidth without losing contrast. Since this technique does not require the ions to be confined in the Lamb-Dicke regime, thermal states with many phonons should be sufficient for operation.
We explore the possibility of implementing a Heisenberg-limited Mach-Zehnder interferometry via an array of trapped ions, which obey a quantum Ising model within a transverse field. Based upon adiabatic processes of increasing the Ising interaction and then decreasing the transverse field, we demonstrate a perfect transition from paramagnetism to ferromagnetic states, which can be used as the beam splitter for the multi-ion Mach-Zehnder interferometry. The achieved NOON state of the ions enables the Heisenberg-limited interferometry. Using currently available techniques for ultracold ions, we …
We investigate the energy dynamics of non-crystallized (melted) ions, confined in a Paul trap. The non-periodic Coulomb interaction experienced by melted ions forms a medium for non-conservative energy transfer from the radio-frequency (rf) field to the ions, a process known as rf heating. We study rf heating by analyzing numerical simulations of non-crystallized ion motion in Paul trap potentials, in which the energy of the ions' secular motion changes at discrete intervals, corresponding to ion-ion collisions. The analysis of these collisions is used as a basis to derive a simplified model o…
Using discrete and continuous variable subsystems, hybrid approaches to quantum information could enable more quantum computational power for the same physical resources. Here, we propose a hybrid scheme that can be used to generate the necessary Gaussian and non-Gaussian operations for universal continuous variable quantum computing in trapped ions. This scheme utilizes two linear spin-motion interactions to generate a broad set of non-linear effective spin-motion interactions including one and two mode squeezing, beam splitter, and trisqueezing operations in trapped ion systems. We discuss p…
We present a theoretical analysis of the implementation of an entangling quantum gate between two trapped Ca$^+$ ions which is based on the dipolar interaction among ionic Rydberg states. In trapped ions the Rydberg excitation dynamics is usually strongly affected by mechanical forces due to the strong couplings between electronic and vibrational degrees of freedom in inhomogeneous electric fields. We demonstrate that this harmful effect can be overcome by using dressed states that emerge from the microwave coupling of nearby Rydberg states. At the same time these dressed states exhibit long r…
Direct frequency comb spectroscopy of trapped ions is demonstated for the first time. It is shown that the 4s^2S_(1/2)-4p^2P_(3/2) transition in calcium ions can be excited directly with a frequency comb laser that is upconverted to 393 nm. Detection of the transition is performed using a shelving scheme to suppress background signal from non-resonant comb modes. The measured transition frequency of f=761 905 012.7(0.5) MHz presents an improvement in accuracy of more than two orders of magnitude.
We demonstrate a trapped-ion entangling-gate scheme proposed by Bermudez et al. [Phys. Rev. A 85, 040302 (2012)]. Simultaneous excitation of a strong carrier and a single-sideband transition enables deterministic creation of entangled states. The method works for magnetic field-insensitive states, is robust against thermal excitations, includes dynamical decoupling from qubit dephasing errors, and provides simplifications in experimental implementation compared to some other entangling gates with trapped ions. We achieve a Bell state fidelity of 0.974(4) and identify the main sources of error.
This book chapter gives an introduction to, and an overview of, methods for cooling trapped ions. The main addressees are researchers entering the field. It is not intended as a comprehensive survey and historical account of the extensive literature on this topic. We present the physical ideas behind several cooling schemes, outline their mathematical description, and point to relevant literature useful for a more in-depth study of this topic.
We discuss a protocol for the analogue quantum simulation of superradiance and subradiance using a linear chain of N trapped qubit ions with a single sympathetic cooling ion. We develop a simple analytic model that shows the dynamics of the qubit subspace converge to those of a cloud undergoing Dicke superradiance and subradiance. We provide numerical simulations of the full ion chain and show that they converge to the dynamics predicted by our analytic model with no fitting parameters. We also map out the parameter regime needed to reach this convergence.
We propose and demonstrate phonon-number-resolving detection of the multiple local phonon modes in a trapped-ion chain. To mitigate the effect of phonon hopping during the detection process, the probability amplitude of each local phonon mode is mapped to the auxiliary long-lived motional ground states. Sequential state-dependent fluorescence detection is then performed. In the experiment, we have successfully observed the time evolution of two local phonon modes in two ions, including the phonon-number correlation between the two modes.
We theoretically study trapped ions that are immersed in an ultracold gas of Rydberg-dressed atoms. By off-resonant coupling on a dipole-forbidden transition, the adiabatic atom-ion potential can be made repulsive. We study the energy exchange between the atoms and a single trapped ion and find that Langevin collisions are inhibited in the ultracold regime for these repulsive interactions. Therefore, the proposed system avoids recently observed ion heating in hybrid atom-ion systems caused by coupling to the ion's radio frequency trapping field and retains ultracold temperatures even in the pr…
We provide a detailed theoretical and conceptual study of a planned experiment to excite Rydberg states of ions trapped in a Paul trap. The ultimate goal is to exploit the strong state dependent interactions between Rydberg ions to implement quantum information processing protocols and to simulate the dynamics of strongly interacting spin systems. We highlight the promises of this approach when combining the high degree of control and readout of quantum states in trapped ion crystals with the novel and fast gate schemes based on interacting giant Rydberg atomic dipole moments. We discuss antic…
We report on ground state laser cooling of single 111Cd+ ions confined in radio-frequency (Paul) traps. Heating rates of trapped ion motion are measured for two different trapping geometries and electrode materials, where no effort was made to shield the electrodes from the atomic Cd source. The low measured heating rates suggest that trapped 111Cd+ ions may be well-suited for experiments involving quantum control of atomic motion, including applications in quantum information science.
We report on a surface-electrode trap with SmCo magnets arranged in a quadrupole configuration underneath the trap electrode. Because the distance between the magnets and the trapped ions can be as little as several hundred micrometers, a large magnetic field is produced without any heat management. The magnetic-field gradient was measured using the Zeeman splitting of a single trapped $^{40}$Ca$^+$ ion at several positions, and a field gradient of 36 T/m was obtained. Such a field gradient is useful for the generation of a state-dependent force, which is important for quantum simulation and/o…
Spontaneous symmetry breaking can lead to the formation of time crystals, as well as spatial crystals. Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We …
Generating quantum entanglement in large systems on time scales much shorter than the coherence time is key to powerful quantum simulation and computation. Trapped ions are among the most accurately controlled and best isolated quantum systems with low-error entanglement gates operated via the vibrational motion of a few-ion crystal within tens of microseconds. To exceed the level of complexity tractable by classical computers the main challenge is to realise fast entanglement operations in large ion crystals. The strong dipole-dipole interactions in polar molecule and Rydberg atom systems all…
We implement a two-qubit entangling Mølmer-Sørensen interaction by transporting two co-trapped $^{40}\mathrm{Ca}^{+}$ ions through a stationary, bichromatic optical beam within a surface-electrode Paul trap. We describe a procedure for achieving a constant Doppler shift during the transport which uses fine temporal adjustment of the moving confinement potential. The fixed interaction duration of the ions transported through the laser beam as well as the dynamically changing ac Stark shift require alterations to the calibration procedures used for a stationary gate. We use the interaction to pr…
We present a realistic molecular-dynamics treatment of laser-cooled ions in radiofrequency ion traps which avoids previously made simplifications such as modeling laser cooling as a friction force and combining individual heating mechanisms into a single effective heating force. Based on this implementation, we show that infrequent energetic collisions of single ions with background gas molecules lead to pronounced heating of the entire ion ensemble and a time-varying secular ensemble temperature which manifests itself in a superstatistical time-averaged velocity distribution of the ions. The …
Resolved sideband cooling is a standard technique for cooling trapped ions below the Doppler limit to near their motional ground state. Yet, the most common methods for sideband cooling implicitly rely on low Doppler-cooled temperatures and tightly confined ions, and they cannot be optimized for different experimental conditions. Here we introduce a framework which calculates the fastest possible pulsed sideband cooling sequence for a given number of pulses and set of experimental parameters, and we verify its improvement compared to traditional methods using a trapped $^{171}$Yb$^+$ ion. Afte…
We experimentally demonstrate cooling of trapped ions by collisions with co-trapped, higher mass neutral atoms. It is shown that the lighter $^{39}$K$^{+}$ ions, created by ionizing $^{39}$K atoms in a magneto-optical trap (MOT), when trapped in an ion trap and subsequently allowed to cool by collisions with ultracold, heavier $^{85}$Rb atoms in a MOT, exhibit a longer trap lifetime than without the localized $^{85}$Rb MOT atoms. A similar cooling of trapped $^{85}$Rb$^{+}$ ions by ultracold $^{133}$Cs atoms in a MOT is also demonstrated in a different experimental configuration to validate th…
The two most widely used ion cooling methods are laser cooling and sympathetic cooling by elastic collisions (ECs). Here we demonstrate another method of cooling ions that is based on resonant charge exchange (RCE) between the trapped ion and the ultracold parent atom. Specifically, trapped Cs$^{+}$ ions are cooled by collisions with co-trapped, ultracold Cs atoms and, separately, by collisions with co-trapped, ultracold Rb atoms. We observe that the cooling of Cs$^{+}$ ions by Cs atoms is more efficient than cooling of Cs$^{+}$ ions by Rb atoms. This signals the presence of a cooling mechanis…
We study a hybrid quantum system composed of an ion and an electric dipole. We show how a trapped ion can be used to measure the small electric field generated by a classical dipole. We discuss the application of this scheme to measure the electric dipole moment of cold polar molecules, whose internal state can be controlled with ultrafast laser pulses, by trapping them in the vicinity of a trapped ion.
