Dirac equation and quantum relativistic effects in a single trapped ion

Trembling Motion of Exciton Polaritons Close to the Rashba-Dresselhaus Regime

We report the experimental observation of trembling quantum motion, or Zitterbewegung, of exciton polaritons in a perovskite microcavity at room temperature. By introducing liquid-crystal molecules into the microcavity, we achieve spinor states with synthetic Rashba-Dresselhaus spin-orbit coupling and tunable energy splitting. Under a resonant excitation, the polariton fluid exhibits clear trembling motion perpendicular to its flowing direction, accompanied by a unique spin pattern resembling interlocked fingers. Furthermore, leveraging the sizable tunability of energy gaps by external electrical voltages, we observe the continuous transition of polariton Zitterbewegung from relativistic (small gaps) to nonrelativistic (large gaps) regimes. Our findings pave the way for using exciton polaritons in the emulation of relativistic quantum physics.

Quantum Metric Unveils Defect Freezing in Non-Hermitian Systems

Non-Hermiticity in quantum Hamiltonians leads to nonunitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this work, we study the dynamics of an exactly solvable non-Hermitian system, hosting both PT-symmetric and PT-broken modes subject to a linear quench. Employing a fully consistent framework, in which the Hilbert space is endowed with a nontrivial dynamical metric, we analyze the dynamics of the generated defects. In contrast to Hermitian systems, our study reveals that PT-broken time evolution leads to defect freezing and hence the violation of adiabaticity. This physics necessitates the so-called metric framework, as it is missed by the oft used approach of normalizing quantities by the time-dependent norm of the state. Our results are relevant for a wide class of experimental systems.

Quantum Simulation of the Two-Dimensional Weyl Equation in a Magnetic Field

Quantum simulation of 1D relativistic quantum mechanics has been achieved in well-controlled systems like trapped ions, but properties like spin dynamics and response to external magnetic fields that appear only in higher dimensions remain unexplored. Here we simulate the dynamics of a 2D Weyl particle. We show the linear dispersion relation of the free particle and the discrete Landau levels in a magnetic field, and we explicitly measure the spatial and spin dynamics from which the conservation of helicity and properties of antiparticles can be verified. Our work extends the application of an ion trap quantum simulator in particle physics with the additional spatial and spin degrees of freedom.

Correlations at PT-Symmetric Quantum Critical Point

We consider a PT-symmetric Fermi gas with an exceptional point, representing the critical point between PT-symmetric and symmetry broken phases. The low energy spectrum remains linear in momentum and is identical to that of a Hermitian Fermi gas. The fermionic Green's function decays in a power law fashion for large distances, as expected from gapless excitations, although the exponent is reduced from -1 due to the quantum Zeno effect. In spite of the gapless nature of the excitations, the ground state entanglement entropy saturates to a finite value, independent of the subsystem size due to the non-Hermitian correlation length intrinsic to the system. Attractive or repulsive interaction drives the system into the PT-symmetry broken regime or opens up a gap and protects PT symmetry, respectively. Our results challenge the concept of universality in non-Hermitian systems, where quantum criticality can be masked due to non-Hermiticity.

Kubo Formula for Non-Hermitian Systems and Tachyon Optical Conductivity

Linear response theory plays a prominent role in various fields of physics and provides us with extensive information about the thermodynamics and dynamics of quantum and classical systems. Here we develop a general theory for the linear response in non-Hermitian systems with nonunitary dynamics and derive a modified Kubo formula for the generalized susceptibility for an arbitrary (Hermitian and non-Hermitian) system and perturbation. We use this to evaluate the dynamical response of a non-Hermitian, one-dimensional Dirac model with imaginary and real masses, perturbed by a time-dependent electric field. The model has a rich phase diagram, and in particular, features a tachyon phase, where excitations travel faster than an effective speed of light. Surprisingly, we find that the dc conductivity of tachyons is finite, and the optical sum rule is exactly satisfied for all masses. Our results highlight the peculiar properties of the Kubo formula for non-Hermitian systems and are applicable for a large variety of settings.

Zitterbewegung of Moiré Excitons in Twisted MoS_{2}/WSe_{2} Heterobilayers

The moiré pattern observed in stacked noncommensurate crystal lattices, such as heterobilayers of transition metal dichalcogenides, produces a periodic modulation of their band gap. Excitons subjected to this potential landscape exhibit a band structure that gives rise to a quasiparticle dubbed the moiré exciton. In the case of MoS_{2}/WSe_{2} heterobilayers, the moiré trapping potential has honeycomb symmetry and, consequently, the moiré exciton band structure is the same as that of a Dirac-Weyl fermion, whose mass can be further tuned down to zero with a perpendicularly applied field. Here we show that, analogously to other Dirac-like particles, the moiré exciton exhibits a trembling motion, also known as Zitterbewegung, whose long timescales are compatible with current experimental techniques for exciton dynamics. This promotes the study of the dynamics of moiré excitons in van der Waals heterostructures as an advantageous solid-state platform to probe Zitterbewegung, broadly tunable by gating and interlayer twist angle.

Spin dynamics under the influence of elliptically rotating fields: Extracting the field topology from time-averaged quantities

Systems that can be effectively described as a localized spin-s particle subject to time-dependent fields have attracted a great deal of interest due to, among other things, their relevance for quantum technologies. Establishing analytical relationships between the topological features of the applied fields and certain time-averaged quantities of the spin can provide important information for the theoretical understanding of these systems. Here, we address this question in the case of a localized spin-s particle subject to a static magnetic field coplanar to a coexisting elliptically rotating magnetic field. The total field periodically traces out an ellipse which encloses the origin of the coordinate system or not, depending on the values taken on by the static and the rotating components. As a result, two regimes with different topological properties characterized by the winding number of the total field emerge: the winding number is 1 if the origin lies inside the ellipse, and 0 if it lies outside. We show that the time average of the energy associated with the rotating component of the magnetic field is always proportional to the time average of the out-of-plane component of the expectation value of the spin. Moreover, the product of the signs of these two time-averaged quantities is uniquely determined by the topology of the total field and, consequently, provides a measurable indicator of this topology. We also propose an implementation of these theoretical results in a trapped-ion quantum system. Remarkably, our findings are valid in the totality of the parameter space and regardless of the initial state of the spin. In particular, when the system is prepared in a Floquet state, we demonstrate that the quasienergies, as a function of the driving amplitude at constant eccentricity, have stationary points at the topological transition boundary. The ability of the topological indicator proposed here to accurately locate the abrupt topological transition can have practical applications for the determination of unknown parameters appearing in the Hamiltonian. In addition, our predictions about the quasienergies can assist in the interpretation of conductance measurements in transport experiments with spin carriers in mesoscopic rings.

Effect of zitterbewegung on the propagation of wave packets in ABC-stacked multilayer graphene: an analytical and computational approach

The time evolution of a low-energy two-dimensional Gaussian wave packet in ABC-stacked n-layer graphene (ABC-NLG) is investigated. Expectation values of the position (x, y) of center-of-mass and the total probability densities of the wave packet are calculated analytically using the Green's function method. These results are confirmed using an alternative numerical method based on the split-operator technique within the Dirac approach for ABC-NLG, which additionally allows to include external fields and potentials. The main features of the zitterbewegung (trembling motion) of wave packets in graphene are demonstrated and are found to depend not only on the wave packet width and initial pseudospin polarization, but also on the number of layers. Moreover, the analytical and numerical methods proposed here allow to investigate wave packet dynamics in graphene systems with an arbitrary number of layers and arbitrary potential landscapes.

Time-Rescaling of Dirac Dynamics: Shortcuts to Adiabaticity in Ion Traps and Weyl Semimetals

Only very recently, rescaling time has been recognized as a way to achieve adiabatic dynamics in fast processes. The advantage of time-rescaling over other shortcuts to adiabaticity is that it does not depend on the eigenspectrum and eigenstates of the Hamiltonian. However, time-rescaling requires that the original dynamics are adiabatic, and in the rescaled time frame, the Hamiltonian exhibits non-trivial time-dependence. In this work, we show how time-rescaling can be applied to Dirac dynamics, and we show that all time-dependence can be absorbed into the effective potentials through a judiciously chosen unitary transformation. This is demonstrated for two experimentally relevant scenarios, namely for ion traps and adiabatic creation of Weyl points.

Optical radiation induced chaotic dynamics of electrons in a uniform magnetic field

We investigate the effects of linearly polarized optical radiation on the cyclotron motion of an electron wave packet, considering the full quantum dynamics of the system. Analysis of the Landau-level (LL) spectrum reveals that only intra band cyclotron oscillation frequencies contribute to the effective oscillation frequency of the motion, whereas scattering between electron and hole Landau levels are forbidden. We find that the wave packet dynamics is significantly affected by varying the polarization direction of the electromagnetic radiation. The optical radiation is also affected by its interaction with electrons. Interestingly, we find that chaotic effects are induced by radiation in the dynamics of electron wave packet in an applied uniform magnetic field. Chaotic signatures in the dynamics are diagnosed by computing the relevant out-of-time-order correlation function and analyzed by using Poincaré maps. We attribute the appearance of such chaotic transport of electron wave packet to the nonlinear interaction between the optical radiation and internal cooperative oscillating mode produced by the interplay of relativistic (zitterbewegung) and cyclotron oscillations.

Protocol for Implementing Quantum Nonparametric Learning with Trapped Ions

Nonparametric learning is able to make reliable predictions by extracting information from similarities between a new set of input data and all samples. Here we point out a quantum paradigm of nonparametric learning that offers an exponential speedup over the sample size. By encoding data into quantum feature space, the similarity between the data is defined as an inner product of quantum states. A quantum training state is introduced to superpose all data of samples, encoding relevant information for learning in its bipartite entanglement spectrum. We demonstrate that a trained state for prediction can be obtained by entanglement spectrum transformation, using the quantum matrix toolbox. We further work out a feasible protocol to implement the quantum nonparametric learning with trapped ions, and demonstrate the power of quantum superposition for machine learning.

Relativistic Measurement Backaction in the Quantum Dirac Oscillator

An elegant method to circumvent quantum measurement backaction is the use of quantum mechanics free subsystems (QMFS), with one approach involving the use of two oscillators with effective masses of opposite signs. Since negative energies, and hence masses, are a characteristic of relativistic systems a natural question is to what extent QMFS can be realized in this context. Using the example of a one-dimensional Dirac oscillator we investigate conditions under which this can be achieved, and identify Zitterbewegung or virtual pair creation as the physical mechanism that fundamentally limits the feasibility of the scheme. We propose a tabletop implementation of a Dirac oscillator system based on a spin-orbit coupled ultracold atomic sample that allows for a direct observation of the corresponding analog of virtual pair creation on quantum measurement backaction.

Simulating para-Fermi oscillators

Quantum mechanics allows for a consistent formulation of particles that are neither bosons nor fermions. These para-particles are rather indiscernible in nature. Recently, we showed that strong coupling between a qubit and two field modes is required to simulate even order para-Bose oscillators. Here, we show that finite-dimensional representations of even order para-Fermi oscillators are feasible of quantum simulation under weak coupling. This opens the door to their potential implementation in different contemporaneous quantum electrodynamics platforms. We emphasize the intrinsic value of para-particles for the quantum state engineering of bichromatic field modes. In particular, we demonstrate that binomial two field mode states result from the evolution of para-Fermi vacuum states in the quantum simulation of these oscillators.

Dirac Equation in (1+1)-Dimensional Curved Spacetime and the Multiphoton Quantum Rabi Model

We introduce an exact mapping between the Dirac equation in (1+1)-dimensional curved spacetime (DCS) and a multiphoton quantum Rabi model (QRM). A background of a (1+1)-dimensional black hole requires a QRM with one- and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime. We illustrate our proposal with a numerical analysis of the free fall of a Dirac particle into a (1+1)-dimensional black hole, and find that the Zitterbewegung effect, measurable via the oscillatory trajectory of the Dirac particle, persists in the presence of gravity. From the duality between the squeezing term in the multiphoton QRM and the metric coupling in the DCS, we show that gravity generates squeezing of the Dirac particle wave function.

Experimental quantum simulation of fermion-antifermion scattering via boson exchange in a trapped ion

Quantum field theories describe a variety of fundamental phenomena in physics. However, their study often involves cumbersome numerical simulations. Quantum simulators, on the other hand, may outperform classical computational capacities due to their potential scalability. Here we report an experimental realization of a quantum simulation of fermion-antifermion scattering mediated by bosonic modes, using a multilevel trapped ion, which is a simplified model of fermion scattering in both perturbative and non-perturbative quantum electrodynamics. The simulated model exhibits prototypical features in quantum field theory including particle pair creation and annihilation, as well as self-energy interactions. These are experimentally observed by manipulating four internal levels of a ¹⁷¹Yb⁺ trapped ion, where we encode the fermionic modes, and two motional degrees of freedom that simulate the bosonic modes. Our experiment establishes an avenue towards the efficient implementation of field modes, which may prove useful in studies of quantum field theories including non-perturbative regimes.

Decoherence Control of Nitrogen-Vacancy Centers

Quantum mechanical systems lose coherence through interacting with external environments-a process known as decoherence. Although decoherence is detrimental for most of the tasks in quantum information processing, a substantial degree of decoherence is crucial for boosting the efficiency of quantum processes, for example, in quantum biology and other open systems. The key to the success in simulating those open quantum systems is therefore the ability of controlling decoherence, instead of eliminating it. Motivated by simulating quantum open systems with Nitrogen-Vacancy centers, which has become an increasingly important platform for quantum information processing tasks, we developed a new set of steering pulse sequences for controlling various coherence times of Nitrogen-Vacancy centers; our method is based on a hybrid approach that exploits ingredients in both digital and analog quantum simulations to dynamically couple or decouple the system with the physical environment. Our numerical simulations, based on experimentally-feasible parameters, indicate that decoherence of Nitrogen-Vacancy centers can be controlled externally to a very large extend.

Mapping curved spacetimes into Dirac spinors

We show how to transform a Dirac equation in a curved static spacetime into a Dirac equation in flat spacetime. In particular, we show that any solution of the free massless Dirac equation in a 1 + 1 dimensional flat spacetime can be transformed via a local phase transformation into a solution of the corresponding Dirac equation in a curved static background, where the spacetime metric is encoded into the phase. In this way, the existing quantum simulators of the Dirac equation can naturally incorporate curved static spacetimes. As a first example we use our technique to obtain solutions of the Dirac equation in a particular family of interesting spacetimes in 1 + 1 dimensions.

Quantum simulations and many-body physics with light

In this review we discuss the works in the area of quantum simulation and many-body physics with light, from the early proposals on equilibrium models to the more recent works in driven dissipative platforms. We start by describing the founding works on Jaynes-Cummings-Hubbard model and the corresponding photon-blockade induced Mott transitions and continue by discussing the proposals to simulate effective spin models and fractional quantum Hall states in coupled resonator arrays (CRAs). We also analyse the recent efforts to study out-of-equilibrium many-body effects using driven CRAs, including the predictions for photon fermionisation and crystallisation in driven rings of CRAs as well as other dynamical and transient phenomena. We try to summarise some of the relatively recent results predicting exotic phases such as super-solidity and Majorana like modes and then shift our attention to developments involving 1D nonlinear slow light setups. There the simulation of strongly correlated phases characterising Tonks-Girardeau gases, Luttinger liquids, and interacting relativistic fermionic models is described. We review the major theory results and also briefly outline recent developments in ongoing experimental efforts involving different platforms in circuit QED, photonic crystals and nanophotonic fibres interfaced with cold atoms.

Real-space imaging of a topologically protected edge state with ultracold atoms in an amplitude-chirped optical lattice

To describe a mobile defect in polyacetylene chains, Su, Schrieffer and Heeger formulated a model assuming two degenerate energy configurations that are characterized by two different topological phases. An immediate consequence was the emergence of a soliton-type edge state located at the boundary between two regions of different configurations. Besides giving first insights in the electrical properties of polyacetylene materials, interest in this effect also stems from its close connection to states with fractional charge from relativistic field theory. Here, using a one-dimensional optical lattice for cold rubidium atoms with a spatially chirped amplitude, we experimentally realize an interface between two spatial regions of different topological order in an atomic physics system. We directly observe atoms confined in the edge state at the intersection by optical real-space imaging and characterize the state as well as the size of the associated energy gap. Our findings hold prospects for the spectroscopy of surface states in topological matter and for the quantum simulation of interacting Dirac systems.

Topological states of matter cannot be distinguished on the basis of local measurements in the bulk of the material. Here the authors report on the observation of an edge state between two topological distinct phases of an ultracold atomic one-dimensional system using optical microscopy.

Digital-Analog Quantum Simulation of Spin Models in Trapped Ions

We propose a method to simulate spin models in trapped ions using a digital-analog approach, consisting in a suitable gate decomposition in terms of analog blocks and digital steps. In this way, we show that the quantum dynamics of an enhanced variety of spin models could be implemented with substantially less number of gates than a fully digital approach. Typically, analog blocks are built of multipartite dynamics providing the complexity of the simulated model, while the digital steps are local operations bringing versatility to it. Finally, we describe a possible experimental implementation in trapped-ion technologies.

Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices

Can high-energy physics be simulated by low-energy, non-relativistic, many-body systems such as ultracold atoms? Such ultracold atomic systems lack the type of symmetries and dynamical properties of high energy physics models: in particular, they manifest neither local gauge invariance nor Lorentz invariance, which are crucial properties of the quantum field theories which are the building blocks of the standard model of elementary particles. However, it turns out, surprisingly, that there are ways to configure an atomic system to manifest both local gauge invariance and Lorentz invariance. In particular, local gauge invariance can arise either as an effective low-energy symmetry, or as an exact symmetry, following from the conservation laws in atomic interactions. Hence, one could hope that such quantum simulators may lead to a new type of (table-top) experiments which will be used to study various QCD (quantum chromodynamics) phenomena, such as the confinement of dynamical quarks, phase transitions and other effects, which are inaccessible using the currently known computational methods. In this report, we review the Hamiltonian formulation of lattice gauge theories, and then describe our recent progress in constructing the quantum simulation of Abelian and non-Abelian lattice gauge theories in 1  +  1 and 2  +  1 dimensions using ultracold atoms in optical lattices.

Quantum Rabi Model with Trapped Ions

We propose the quantum simulation of the quantum Rabi model in all parameter regimes by means of detuned bichromatic sideband excitations of a single trapped ion. We show that current setups can reproduce, in particular, the ultrastrong and deep strong coupling regimes of such a paradigmatic light-matter interaction. Furthermore, associated with these extreme dipolar regimes, we study the controlled generation and detection of their entangled ground states by means of adiabatic methods. Ion traps have arguably performed the first quantum simulation of the Jaynes-Cummings model, a restricted regime of the quantum Rabi model where the rotating-wave approximation holds. We show that one can go beyond and experimentally investigate the quantum simulation of coupling regimes of the quantum Rabi model that are difficult to achieve with natural dipolar interactions.

Quantum Simulator for Transport Phenomena in Fluid Flows

Transport phenomena still stand as one of the most challenging problems in computational physics. By exploiting the analogies between Dirac and lattice Boltzmann equations, we develop a quantum simulator based on pseudospin-boson quantum systems, which is suitable for encoding fluid dynamics transport phenomena within a lattice kinetic formalism. It is shown that both the streaming and collision processes of lattice Boltzmann dynamics can be implemented with controlled quantum operations, using a heralded quantum protocol to encode non-unitary scattering processes. The proposed simulator is amenable to realization in controlled quantum platforms, such as ion-trap quantum computers or circuit quantum electrodynamics processors.

Time reversal and charge conjugation in an embedding quantum simulator

A quantum simulator is an important device that may soon outperform current classical computations. A basic arithmetic operation, the complex conjugate, however, is considered to be impossible to be implemented in such a quantum system due to the linear character of quantum mechanics. Here, we present the experimental quantum simulation of such an unphysical operation beyond the regime of unitary and dissipative evolutions through the embedding of a quantum dynamics in the electronic multilevels of a ¹⁷¹Yb⁺ ion. We perform time reversal and charge conjugation, which are paradigmatic examples of antiunitary symmetry operators, in the evolution of a Majorana equation without the tomographic knowledge of the evolving state. Thus, these operations can be applied regardless of the system size. Our approach offers the possibility to add unphysical operations to the toolbox of quantum simulation, and provides a route to efficiently compute otherwise intractable quantities, such as entanglement monotones.

Quantum simulation has the potential to enable experimentally studying problems which are not directly tractable in a laboratory or computationally. Here, the authors simulate the Majorana equation with a ytterbium ion, which requires them to simulate antiunitary and therefore unphysical operations.

Digital quantum Rabi and Dicke models in superconducting circuits

We propose the analog-digital quantum simulation of the quantum Rabi and Dicke models using circuit quantum electrodynamics (QED). We find that all physical regimes, in particular those which are impossible to realize in typical cavity QED setups, can be simulated via unitary decomposition into digital steps. Furthermore, we show the emergence of the Dirac equation dynamics from the quantum Rabi model when the mode frequency vanishes. Finally, we analyze the feasibility of this proposal under realistic superconducting circuit scenarios.

Probing the topological properties of the Jackiw-Rebbi model with light

The Jackiw-Rebbi model describes a one-dimensional Dirac field coupled to a soliton field and can be equivalently thought of as a model describing a Dirac field with a spatially dependent mass term. Neglecting the dynamics of the soliton field, a kink in the background soliton profile yields a topologically protected zero-energy mode for the field, which in turn leads to charge fractionalisation. We show here that the model, in the first quantised form, can be realised in a driven slow-light setup, where photons mimic the Dirac field and the soliton field can be implemented–and tuned–by adjusting optical parameters such as the atom-photon detuning. Furthermore, we discuss how the existence of the zero-mode and its topological stability can be probed naturally by studying the transmission spectrum. We conclude by analysing the robustness of our approach against possible experimental errors in engineering the Jackiw-Rebbi Hamiltonian in this optical setup.

Veselago lensing with ultracold atoms in an optical lattice

Veselago pointed out that electromagnetic wave theory allows for materials with a negative index of refraction, in which most known optical phenomena would be reversed. A slab of such a material can focus light by negative refraction, an imaging technique strikingly different from conventional positive refractive index optics, where curved surfaces bend the rays to form an image of an object. Here we demonstrate Veselago lensing for matter waves, using ultracold atoms in an optical lattice. A relativistic, that is, photon-like, dispersion relation for rubidium atoms is realized with a bichromatic optical lattice potential. We rely on a Raman π-pulse technique to transfer atoms between two different branches of the dispersion relation, resulting in a focusing that is completely analogous to the effect described by Veselago for light waves. Future prospects of the demonstrated effects include novel sub-de Broglie wavelength imaging applications.

From transistor to trapped-ion computers for quantum chemistry

Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.

First experimental realization of the Dirac oscillator

We present the first experimental microwave realization of the one-dimensional Dirac oscillator, a paradigm in exactly solvable relativistic systems. The experiment relies on a relation of the Dirac oscillator to a corresponding tight-binding system. This tight-binding system is implemented as a microwave system by a chain of coupled dielectric disks, where the coupling is evanescent and can be adjusted appropriately. The resonances of the finite microwave system yield the spectrum of the one-dimensional Dirac oscillator with and without a mass term. The flexibility of the experimental setup allows the implementation of other one-dimensional Dirac-type equations.

Quantum simulation of noncausal kinematic transformations

We propose the implementation of Galileo group symmetry operations or, in general, linear coordinate transformations in a quantum simulator. With an appropriate encoding, unitary gates applied to our quantum system give rise to Galilean boosts or spatial and time parity operations in the simulated dynamics. This framework provides us with a flexible toolbox that enhances the versatility of quantum simulation theory, allowing the direct access to dynamical quantities that would otherwise require full tomography. Furthermore, this method enables the study of noncausal kinematics and phenomena beyond special relativity in a quantum controllable system.

Digital quantum simulation of the Holstein model in trapped ions

We propose the implementation of the Holstein model by means of digital methods in a linear chain of trapped ions. We show how the simulation fidelity scales with the generation of phononic excitations. We propose a decomposition and a stepwise trapped-ion implementation of the Holstein Hamiltonian. Via numerical simulations, we study how the protocol is affected by realistic gates. Finally, we show how measurements of the size of the simulated polaron can be performed.

Quantum simulation of interacting fermion lattice models in trapped ions

We propose a method of simulating efficiently many-body interacting fermion lattice models in trapped ions, including highly nonlinear interactions in arbitrary spatial dimensions and for arbitrarily distant couplings. We map products of fermionic operators onto nonlocal spin operators and decompose the resulting dynamics in efficient steps with Trotter methods, yielding an overall protocol that employs only polynomial resources. The proposed scheme can be relevant in a variety of fields such as condensed-matter or high-energy physics, where quantum simulations may solve problems intractable for classical computers.

Zitterbewegung of electrons in quantum wells and dots in the presence of an in-plane magnetic field

We study the effect of an in-plane magnetic field on the zitterbewegung (ZB) of electrons in a semiconductor quantum well (QW) and in a quantum dot (QD) with the Rashba and Dresselhaus spin-orbit interactions (SOIs). We obtain a general expression of the time-evolution of the position vector and current of the electron in a semiconductor QW. The amplitude of the oscillatory motion is directly related to the Berry connection in momentum space. We find that in presence of the magnetic field the ZB in a QW does not vanish when the strengths of the Rashba and Dresselhaus SOIs are equal. The in-plane magnetic field helps to sustain the ZB in QWs even at a low value of k(0)d (where d is the width of the Gaussian wavepacket and k(0) is the initial wavevector). The trembling motion of an electron in a semiconductor QW with high Landé g-factor (e.g. InSb) is sustained over a long time, even at a low value of k(0)d. Further, we study the ZB of an electron in QDs within the two sub-band model numerically. The trembling motion persists in time even when the magnetic field is absent as well as when the strengths of the SOI are equal. The ZB in QDs is due to the superposition of oscillatory motions corresponding to all possible differences of the energy eigenvalues of the system. This is an another example of multi-frequency ZB phenomenon.

Experimental quantum simulations of many-body physics with trapped ions

Direct experimental access to some of the most intriguing quantum phenomena is not granted due to the lack of precise control of the relevant parameters in their naturally intricate environment. Their simulation on conventional computers is impossible, since quantum behaviour arising with superposition states or entanglement is not efficiently translatable into the classical language. However, one could gain deeper insight into complex quantum dynamics by experimentally simulating the quantum behaviour of interest in another quantum system, where the relevant parameters and interactions can be controlled and robust effects detected sufficiently well. Systems of trapped ions provide unique control of both the internal (electronic) and external (motional) degrees of freedom. The mutual Coulomb interaction between the ions allows for large interaction strengths at comparatively large mutual ion distances enabling individual control and readout. Systems of trapped ions therefore exhibit a prominent system in several physical disciplines, for example, quantum information processing or metrology. Here, we will give an overview of different trapping techniques of ions as well as implementations for coherent manipulation of their quantum states and discuss the related theoretical basics. We then report on the experimental and theoretical progress in simulating quantum many-body physics with trapped ions and present current approaches for scaling up to more ions and more-dimensional systems.

Quantum simulation of quantum field theories in trapped ions

We propose the quantum simulation of fermion and 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.

Klein tunneling of a quasirelativistic Bose-Einstein condensate in an optical lattice

A proof-of-principle experiment simulating effects predicted by relativistic wave equations with ultracold atoms in a bichromatic optical lattice that allows for a tailoring of the dispersion relation is reported. We observe the analog of Klein tunneling, the penetration of relativistic particles through a potential barrier without the exponential damping that is characteristic for nonrelativistic quantum tunneling. Both linear (relativistic) and quadratic (nonrelativistic) dispersion relations are investigated, and significant barrier transmission is observed only for the relativistic case.

Zitterbewegung (trembling motion) of electrons in semiconductors: a review

We review recent research on Zitterbewegung (ZB, trembling motion) of electrons in semiconductors. A brief history of the subject is presented, the trembling motion in semi-relativistic and spin systems is considered and its main features are emphasized. ZB of charge carriers in monolayer and bilayer graphene as well as in carbon nanotubes is elaborated in some detail. We describe the effects of an external magnetic field on ZB using monolayer graphene as an example. The nature of electron ZB in crystalline solids is explained. We also review various simulations of the trembling motion in a vacuum and in semiconductors, and mention ZB-like wave phenomena in sonic and photonic periodic structures. An attempt is made to quote all the relevant literature on the subject.

Quantum simulation of the Klein paradox with 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.

Classical simulation of relativistic Zitterbewegung in photonic lattices

We present the first experimental realization of an optical analog for relativistic quantum mechanics by simulating the Zitterbewegung (trembling motion) of a free Dirac electron in an optical superlattice. Our photonic setting enables a direct visualization of Zitterbewegung as a spatial oscillatory motion of an optical beam. Direct measurements of the wave packet expectation values in superlattices with tuned miniband gaps clearly show the transition from weak-relativistic to relativistic and far-relativistic regimes.

Quantum coherence of discrete kink solitons in ion traps

We propose to realize quantized discrete kinks with cold trapped ions. We show that long-lived solitonlike configurations are manifested as deformations of the zigzag structure in the linear Paul trap, and are topologically protected in a circular trap with an odd number of ions. We study the quantum-mechanical time evolution of a high-frequency, gap separated internal mode of a static kink and find long coherence times when the system is cooled to the Doppler limit. The spectral properties of the internal modes make them ideally suited for manipulation using current technology. This suggests that ion traps can be used to test quantum-mechanical effects with solitons and explore ideas for the utilization of the solitonic internal modes as carriers of quantum information.

Photonic analog of Zitterbewegung in binary waveguide arrays

An optical analog of Zitterbewegung (ZB), i.e., of the trembling motion of Dirac electrons caused by the interference between positive and negative energy states, is proposed for spatial beam propagation in binary waveguide arrays. In this optical system the ZB is simply observable as a quiver spatial oscillatory motion of the beam center of mass around its mean trajectory.

Delocalization of relativistic dirac particles in disordered one-dimensional systems and its implementation with cold atoms

We study theoretically the localization of relativistic particles in disordered one-dimensional chains. It is found that the relativistic particles tend to delocalization in comparison with the nonrelativistic particles with the same disorder strength. More intriguingly, we reveal that the massless Dirac particles are entirely delocalized for any energy due to the inherent chiral symmetry, leading to a well-known result that particles are always localized in one-dimensional systems for arbitrary weak disorders to break down. Furthermore, we propose a feasible scheme to detect the delocalization feature of the Dirac particles with cold atoms in a light-induced gauge field.

Confining stationary light: dirac dynamics and klein tunneling

We discuss the properties of 1D stationary pulses of light in an atomic ensemble with electromagnetically induced transparency in the limit of tight spatial confinement. When the size of the wave packet becomes comparable or smaller than the absorption length of the medium, it must be described by a two-component vector which obeys the one-dimensional two-component Dirac equation with an effective mass m;{*} and effective speed of light c;{*}. Then a fundamental lower limit to the spatial width in an external potential arises from Klein tunneling and is given by the effective Compton length lambda_{C}=variant Planck's over 2pi/(m;{*}c;{*}). Since c;{*} and m;{*} can be externally controlled and can be made small, it is possible to observe effects of the relativistic dispersion for rather low energies or correspondingly on macroscopic length scales.

Observing Zitterbewegung with ultracold atoms

We propose an optical lattice scheme which would permit the experimental observation of Zitterbewegung (ZB) with ultracold, neutral atoms. A four-level tripod variant of the setup for stimulated Raman adiabatic passage (STIRAP) has previously been proposed for generating non-Abelian gauge fields. Dirac-like Hamiltonians, which exhibit ZB, are simple examples of such non-Abelian gauge fields; we show how a variety of them can arise, and how ZB can be observed, in a tripod system. We predict that the ZB should occur at experimentally accessible frequencies and amplitudes.