R. Gerritsma

We study the occurrence of long-lived metastable states in the 4f$^{13}$5d6s electron configuration of Yb$^+$. By optical pumping of a single trapped ion on the $^2F^\text{o}_{7/2}\rightarrow (7/2,0)_{7/2}$ transition at 377.5 nm, we prepare a wide range of metastable electronic states. We use a co-trapped control ion to sympathetically cool the spectroscopy ion, allowing us to accurately time its subsequent decay. We record a strong decay signal corresponding to a lifetime of 0.92(8) s, a weaker decay signal with lifetime 9.8(+2.9, -2.0) s, and find evidence for a much longer lifetime, $>$ 30 s. We identify the metastable states with these lifetimes qualitatively, and corroborate our results with atomic structure calculations that support the observed lifetimes and decay paths. These long-lived states provide new opportunities in qubit and qudit state detection and optical clocks.

Tightly-focused laser beams, or optical tweezers, are essential for analogue and digital quantum simulation with neutral atoms and trapped ions. Despite this, most of the current intuition and theoretical treatment utilizes the paraxial approximation, which breaks down at the focus of optical tweezers. We develop an analytic model, which we use in tandem with numerical simulations, to quantify how non-paraxial effects will manifest in the next-generation of scalable quantum hardware, where tightly focused beams are used for individual qubit control. In particular, we calculate the light potentials of Gaussian and Laguerre-Gaussian beams driving the quadrupole $^2$S$_{1/2}\rightarrow$ $^2$D$_{5/2}$ transition in $^{40}$Ca$^+$. Longitudinal field components in the beam center cause spatially-dependent Rabi frequencies and AC Stark shifts, leading to unexpected qubit-motion coupling. We characterize single- and two-qubit gate infidelities due to this effect with an analytic model and numerical simulation. We identify regimes where non-paraxial effects should be taken into account for high-precision quantum control. Finally, we highlight that non-paraxial effects are potentially more severe in the case of neutral atom and molecule addressing.

We consider the quantum simulation of quadratic spin-phonon coupling in a crystal of trapped ions. The coupling is implemented using tightly focused optical tweezers on each ion that change the local trapping potential in a state-dependent way. By encoding spins in the internal states of the ions and adding a tunneling term via Mølmer-Sørensen-type interactions, we calculate the emergence of mobile bipolarons driven by the zero-point energy of the ion crystal phonons. We show that thermal occupation may pin the bipolarons for ion crystals at finite temperature. Our scheme can be used to study and illustrate the emergence of mobile bipolarons as a function of temperature.

This paper presents a routine to align an optical tweezer on a single trapped ion and use the ion as a probe to characterize the tweezer. We find a smallest tweezer waist of $2.3(2)\,μ$m, which is in agreement with the theoretical minimal attainable waist of $2.5(2)\,μ$m in our setup. We characterize the spatial dependence of the tweezer Rabi frequency which is suppressed by a factor of 19(3) in the immediate surrounding of the ion. We investigate the effects of optical forces and coherent population trapping on the ion. Finally, we show that the challenges posed by these forces can be overcome, and that the number of tweezers can be easily scaled up to reach several ions by using a spatial light modulator.

We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers. Strong polarization gradients near the tweezer focus lead to qubit-state dependent forces on the ion. We show that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal. The qubit-state dependent forces generated by this effect live on the plane perpendicular to the direction of propagation of the laser beams opening new ways of coupling to motional modes of an ion crystal. The proposed gate does not require ground state cooling of the ions and does not rely on the Lamb-Dicke approximation, although the waist of the tightly focused beam needs to be comparable with its wavelength in order to achieve the needed field curvature. Furthermore, the gate can be performed on both ground state and magnetic field insensitive clock state qubits without the need for counter-propagating laser fields. This simplifies the setup and eliminates errors due to phase instabilities between the gate laser beams. Finally, we show that imperfections in the gate execution, in particular pointing errors $<30$ nm in the tweezers reduce the gate fidelity from $\mathcal F\gtrsim 0.99998$ to $\gtrsim 0.999$.

We theoretically investigate the trap-assisted formation of complexes in atom-ion collisions and their impact on the stability of the trapped ion. The time-dependent potential of the Paul trap facilitates the formation of temporary complexes by reducing the energy of the atom, which gets temporarily stuck in the atom-ion potential. As a result, those complexes significantly impact termolecular reactions leading to molecular ion formation via three-body recombination. We find that complex formation is more pronounced in systems with heavy atoms, but the mass has no influence on the lifetime of the transient state. Instead, the complex formation rate strongly depends on the amplitude of the ion's micromotion. We also show that complex formation persists even in the case of a time-independent harmonic trap. In this case, we find higher formation rates and longer lifetimes than the Paul trap, indicating that the atom-ion complex plays an essential role in atom-ion mixtures in optical traps.

We measure chemical reactions between a single trapped $^{174}$Yb$^+$ ion immersed in an ultracold bath of $^6$Li atoms containing trace amounts of Li$_2$ dimers. This produces LiYb$^+$ molecular ions that we detect via mass spectrometry. We explain the reaction rates by modelling the dimer density as a function of the magnetic field and obtain excellent agreement when we assume the reaction to follow the Langevin rate. Our results present a novel approach towards the creation of cold molecular ions and point to the exploration of ultracold chemistry in ion molecule collisions. What is more, with a detection sensitivity below molecule densities of $10^{14}\,\mathrm{m}^{-3}$, we provide a new method to detect low-density molecular gases.

For exploration of quantum effects with hybrid atom-ion systems, reaching ultracold temperatures is the major limiting factor. In this work, we present results on numerical simulations of trapped ion buffer gas cooling using an ultracold atomic gas in a large number of experimentally realistic scenarios. We explore the suppression of micromotion-induced heating effects through optimization of trap parameters for various radio-frequency (rf) traps and rf driving schemes including linear and octupole traps, digital Paul traps, rotating traps and hybrid optical/rf traps. We find that very similar ion energies can be reached in all of them even when considering experimental imperfections that cause so-called excess micromotion. Moreover we look into a quantum description of the system and show that quantum mechanics cannot save the ion from micromotion-induced heating in an atom-ion collision. The results suggest that buffer gas cooling can be used to reach close to the ion's groundstate of motion and is even competitive when compared to some sub-Doppler cooling techniques such as Sisyphus cooling. Thus, buffer gas cooling is a viable alternative for ions that are not amenable to laser cooling, a result that may be of interest for studies into quantum chemistry and precision spectroscopy.

We propose a new scalable architecture for trapped ion quantum computing that combines optical tweezers delivering qubit state-dependent local potentials with oscillating electric fields. Since the electric field allows for long-range qubit-qubit interactions mediated by the center-of-mass motion of the ion crystal alone, it is inherently scalable to large ion crystals. Furthermore, our proposed scheme does not rely on either ground state cooling or the Lamb-Dicke approximation. We study the effects of imperfect cooling of the ion crystal, as well as the role of unwanted qubit-motion entanglement, and discuss the prospects of implementing the state-dependent tweezers in the laboratory.

We theoretically investigate the possibility of creating phonon-mediated spin-spin interactions between neutral atoms trapped in optical tweezers. By laser coupling the atoms to Rydberg states, collective modes of motion appear. We show that these can be used to mediate effective spin-spin interactions or quantum logic gates between the atoms in analogy to schemes employed in trapped ions. In particular, we employ Rydberg dressing in a novel scheme to induce the needed interaction, and we show that it is possible to replicate the working of the Mølmer-Sørensen entanglement scheme. The Mølmer-Sørensen gate is widely used in emerging quantum computers using trapped ion qubits and currently features some of the highest fidelities of any quantum gate under consideration. We find arbitrarily high fidelity for the coherent time evolution of the two-atom state even at non-zero temperature.

We describe and characterize an experimental apparatus that has been used to study interactions between ultracold lithium atoms and ytterbium ions. The preparation of ultracold clouds of Li atoms is described as well as their subsequent transport and overlap with Yb$^+$ ions trapped in a Paul trap. We show how the kinetic energy of the ion after interacting with the atoms can be obtained by laser spectroscopy. From analyzing the dynamics of the ion in the absence of atoms, we conclude that background heating, due to electric field noise, limits attainable buffer gas cooling temperatures. We suspect that this effect can be mitigated by noise reduction and by increasing the density of the Li gas, in order to improve its cooling power. Imperfections in the Paul trap lead to so-called excess micromotion, which poses another limitation to the buffer gas cooling. We describe in detail how we measure and subsequently minimize excess micromotion in our setup. We measure the effect of excess micromotion on attainable ion temperatures after buffer gas cooling and compare this to molecular dynamics simulations which describe the observed data very well.

The family of $n$-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary $n$-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to $n$ other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely strongly coupling $n+1$ qubits and a driving frequency that scales with $n$, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and on superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-\textsc{not}-gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.

Great advances in precision quantum measurement have been achieved with trapped ions and atomic gases at the lowest possible temperatures. These successes have inspired ideas to merge the two systems. In this way one can study the unique properties of ionic impurities inside a quantum fluid or explore buffer gas cooling of the trapped ion quantum computer. Remarkably, in spite of its importance, experiments with atom-ion mixtures remained firmly confined to the classical collision regime. We report a collision energy of 1.15(0.23) times the $s$-wave energy (or 9.9(2.0)~$μ$K) for a trapped ytterbium ion in an ultracold lithium gas. We observed a deviation from classical Langevin theory by studying the spin-exchange dynamics, indicating quantum behavior in the atom-ion collisions. Our results open up numerous opportunities, such as the exploration of atom-ion Feshbach resonances, in analogy to neutral systems.

We report on the observation of interactions between ultracold Rydberg atoms and ions in a Paul trap. The rate of observed inelastic collisions, which manifest themselves as charge transfer between the Rydberg atoms and ions, exceeds that of Langevin collisions for ground state atoms by about three orders of magnitude. This indicates a huge increase in interaction strength. We study the effect of the vacant Paul trap's electric fields on the Rydberg excitation spectra. To quantitatively describe the exhibited shape of the ion loss spectra, we need to include the ion-induced Stark shift on the Rydberg atoms. Furthermore, we demonstrate Rydberg excitation on a dipole-forbidden transition with the aid of the electric field of a single trapped ion. Our results confirm that interactions between ultracold atoms and trapped ions can be controlled by laser coupling to Rydberg states. Adding dynamic Rydberg dressing may allow for the creation of spin-spin interactions between atoms and ions, and the elimination of collisional heating due to ionic micromotion in atom-ion mixtures.

We perform numerical simulations of trapped $^{171}$Yb$^+$ ions that are buffer gas cooled by a cold cloud of $^6$Li atoms. This species combination has been suggested to be the most promising for reaching the quantum regime of interacting atoms and ions in a Paul trap. Treating the atoms and ions classically, we compute that the collision energy indeed reaches below the quantum limit for a perfect linear Paul trap. We analyze the effect of imperfections in the ion trap that cause excess micromotion. We find that the suppression of excess micromotion required to reach the quantum limit should be within experimental reach. Indeed, although the requirements are strong, they are not excessive and lie within reported values in the literature. We analyze the detection and suppression of excess micromotion in our experimental setup. Using the obtained experimental parameters in our simulation, we calculate collision energies that are a factor 2-11 larger than the quantum limit, indicating that improvements in micromotion detection and compensation are needed there. We also analyze the buffer-gas cooling of linear and two-dimensional ion crystals. We find that the energy stored in the eigenmodes of ion motion may reach 10-100 $μ$K after buffer-gas cooling under realistic experimental circumstances. Interestingly, not all eigenmodes are buffer-gas cooled to the same energy. Our results show that with modest improvements of our experiment, studying atom-ion mixtures in the quantum regime is in reach, allowing for buffer-gas cooling of the trapped ion quantum platform and to study the occurrence of atom-ion Feshbach resonances.

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 presence of excess micromotion.

We investigate an atomic ensemble of interacting bosons trapped in a symmetric double well potential in contact with a single tightly trapped ion which has been recently proposed [R. Gerritsma et al., Phys. Rev. Lett. 109, 080402 (2012)] as a source of entanglement between a Bose-Einstein condensate and an ion. Compared to the previous study, the present work aims at performing a detailed and accurate many-body analysis of such combined atomic quantum system by means of the ab-initio multi-configuration time-dependent Hartree method for bosons, which allows to take into account all correlations in the system. The analysis elucidates the importance of quantum correlations in the bosonic ensemble and reveals that entanglement generation between an ion and a condensate is indeed possible, as previously predicted. Moreover, we provide an intuitive picture of the impact of the correlations on the out-of-equilibrium dynamics by employing a natural orbital analysis which we show to be indeed experimentally verifiable.

We demonstrate excitation of a single trapped cold $^{40}$Ca$^+$ ion to Rydberg levels by laser radiation in the vacuum-ultraviolet at 122 nm wavelength. Observed resonances are identified as 3d$^2$D$_{3/2}$ to 51 F, 52 F and 3d$^2$D$_{5/2}$ to 64F. We model the lineshape and our results imply a large state-dependent coupling to the trapping potential. Rydberg ions are of great interest for future applications in quantum computing and simulation, in which large dipolar interactions are combined with the superb experimental control offered by Paul traps.

We propose and theoretically investigate a hybrid system composed of a crystal of trapped ions coupled to a cloud of ultracold fermions. The ions form a periodic lattice and induce a band structure in the atoms. This system combines the advantages of scalability and tunability of ultracold atomic systems with the high fidelity operations and detection offered by trapped ion systems. It also features close analogies to natural solid-state systems, as the atomic degrees of freedom couple to phonons of the ion lattice, thereby emulating a solid-state system. Starting from the microscopic many-body Hamiltonian, we derive the low energy Hamiltonian including the atomic band structure and give an expression for the atom-phonon coupling. We discuss possible experimental implementations such as a Peierls-like transition into a period-doubled dimerized state.

The ability to detect the interaction of light and matter at the single-particle level is becoming increasingly important for many areas of science and technology. The absorption or emission of a photon on a narrow transition of a trapped ion can be detected with near unit probability, thereby enabling the realization of ultra-precise ion clocks and quantum information processing applications. Extending this sensitivity to broad transitions is challenging due to the difficulty of detecting the rapid photon scattering events in this case. Here, we demonstrate a technique to detect the scattering of a single photon on a broad optical transition with high sensitivity. Our approach is to use an entangled state to amplify the tiny momentum kick an ion receives upon scattering a photon. The method should find applications in spectroscopy of atomic and molecular ions and quantum information processing.

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 by optical dipole forces. As the proposed method relies on techniques well established in ion trap quantum information processing it is within reach of current technology.

We theoretically investigate the properties of a double-well bosonic Josephson junction coupled to a single trapped ion. We find that the coupling between the wells can be controlled by the internal state of the ion, which can be used for studying mesoscopic entanglement between the two systems and to measure their interaction with high precision. As a particular example we consider a single $^{87}$Rb atom and a small Bose-Einstein condensate controlled by a single $^{171}$Yb$^+$ ion. We calculate inter-well coupling rates reaching hundreds of Hz, while the state dependence amounts to tens of Hz for plausible values of the currently unknown s-wave scattering length between the atom and the ion. The analysis shows that it is possible to induce either the self-trapping or the tunneling regime, depending on the internal state of the ion. This enables the generation of large scale ion-atomic wavepacket entanglement within current technology.

A digital quantum simulator is an envisioned quantum device that can be pro- grammed to efficiently simulate any other local system. We demonstrate and investigate the digital approach to quantum simulation in a system of trapped ions. Using sequences of up to 100 gates and 6 qubits, the full time dynamics of a range of spin systems are digitally simulated. Interactions beyond those naturally present in our simulator are accurately reproduced and quantitative bounds are provided for the overall simulation quality. Our results demon- strate the key principles of digital quantum simulation and provide evidence that the level of control required for a full-scale device is within reach.

We propose the quantum simulation of a fermion and an antifermion field modes interacting via a bosonic field mode, and present a possible implementation with two trapped ions. This quantum platform allows for the scalable add-up of bosonic and fermionic modes, and represents an avenue towards quantum simulations of quantum field theories in perturbative and nonperturbative regimes.

We consider the quantum simulation of relativistic quantum mechanics, as described by the Dirac equation and classical potentials, in trapped-ion systems. We concentrate on three problems of growing complexity. First, we study the bidimensional relativistic scattering of single Dirac particles by a linear potential. Furthermore, we explore the case of a Dirac particle in a magnetic field and its topological properties. Finally, we analyze the problem of two Dirac particles that are coupled by a controllable and confining potential. The latter interaction may be useful to study important phenomena as the confinement and asymptotic freedom of quarks.

A quantum simulator is a device engineered to reproduce the properties of an ideal quantum model. It allows the study of quantum systems that cannot be efficiently simulated on classical computers. While a universal quantum computer is also a quantum simulator, only particular systems have been simulated up to now. Still, there is a wealth of successful cases, such as spin models, quantum chemistry, relativistic quantum physics and quantum phase transitions. Here, we show how to design a quantum simulator for the Majorana equation, a non-Hamiltonian relativistic wave equation that might describe neutrinos and other exotic particles beyond the standard model. The simulation demands the implementation of charge conjugation, an unphysical operation that opens a new front in quantum simulations, including the discrete symmetries associated with complex conjugation and time reversal. Finally, we show how to implement this general method in trapped ions.

We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external potentials we are able to simulate text book relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator. We describe extensions to solve problems that are beyond current classical computing capabilities.

We propose the quantum simulation of the Dirac equation with potentials, allowing the study of relativistic scaterring and the Klein tunneling. This quantum relativistic effect permits a positive-energy Dirac particle to propagate through a repulsive potential via the population transfer to negative-energy components. We show how to engineer scalar, pseudoscalar, and other potentials in the 1+1 Dirac equation by manipulating two trapped ions. The Dirac spinor is represented by the internal states of one ion, while its position and momentum are described by those of a collective motional mode. The second ion is used to build the desired potentials with high spatial resolution.

A basic assumption behind the inequalities used for testing noncontextual hidden variable models is that the observables measured on the same individual system are perfectly compatible. However, compatibility is not perfect in actual experiments using sequential measurements. We discuss the resulting "compatibility loophole" and present several methods to rule out certain hidden variable models which obey a kind of extended noncontextuality. Finally, we present a detailed analysis of experimental imperfections in a recent trapped ion experiment and apply our analysis to that case.

We experimentally demonstrate a quantum walk on a line in phase space using one and two trapped ion. A walk with up to 23 steps is realized by subjecting an ion to state-dependent displacement operations interleaved with quantum coin tossing operations. To analyze the ion's motional state after each step we apply a technique that directly maps the probability density distribution onto the ion's internal state. The measured probability distributions and the position's second moment clearly show the non-classical character of the quantum walk. To further highlight the difference between the classical (random) and the quantum walk, we demonstrate the reversibility of the latter. Finally, we extend the quantum walk by using two ions, giving the walker the additional possibility to stay instead of taking a step.

The Dirac equation is a cornerstone in the history of physics, merging successfully quantum mechanics with special relativity, providing a natural description of the electron spin and predicting the existence of anti-matter. Furthermore, it is able to reproduce accurately the spectrum of the hydrogen atom and its realm, relativistic quantum mechanics, is considered as the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects such as Klein's paradox and Zitterbewegung, an unexpected quivering motion of a free relativistic quantum particle first examined by Schrödinger. These and other predictions would be difficult to observe in real particles, while constituting key fundamental examples to understand relativistic quantum effects. Recent years have seen an increased interest in simulations of relativistic quantum effects in different physical setups, where parameter tunability allows accessibility to different physical regimes. Here, we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion, which is set to behave as a free relativistic quantum particle. We measure as a function of time the particle position and study Zitterbewegung for different initial superpositions of positive and negative energy spinor states, as well as the cross-over from relativistic to nonrelativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate elegant textbook examples of relativistic quantum physics.

The question of whether quantum phenomena can be explained by classical models with hidden variables is the subject of a long lasting debate. In 1964, Bell showed that certain types of classical models cannot explain the quantum mechanical predictions for specific states of distant particles. Along this line, some types of hidden variable models have been experimentally ruled out. An intuitive feature for classical models is non-contextuality: the property that any measurement has a value which is independent of other compatible measurements being carried out at the same time. However, the results of Kochen, Specker, and Bell show that non-contextuality is in conflict with quantum mechanics. The conflict resides in the structure of the theory and is independent of the properties of special states. It has been debated whether the Kochen-Specker theorem could be experimentally tested at all. Only recently, first tests of quantum contextuality have been proposed and undertaken with photons and neutrons. Yet these tests required the generation of special quantum states and left various loopholes open. Here, using trapped ions, we experimentally demonstrate a state-independent conflict with non-contextuality. The experiment is not subject to the detection loophole and we show that, despite imperfections and possible measurement disturbances, our results cannot be explained in non-contextual terms.

We perform precision measurements of the branching ratios of the 4P3/2 level decay of a single 40Ca+ ion suspended in a linear Paul trap. High precision is achieved by a novel technique based on monitoring the population transfer when repeatedly pumping the ion between different internal states. The branching fractions into the 4S1/2, 3D5/2 and 3D3/2 levels are found to be 0.9347(3), 0.0587(2) and 0.00661(4), respectively. For the branching ratio A(P3/2-S1/2)/\sum_J A(P3/2-D_J)=14.31(5), we find a forty-fold improvement in accuracy as compared to the best previous measurement.