B. M. Rodríguez-Lara

The center of mass motion of trapped ions and neutral atoms is suitable for approximation by a time-dependent driven quantum harmonic oscillator whose frequency and driving strength may be controlled with high precision. We show the time evolution for these systems with continuous differentiable time-dependent parameters in terms of the three basic operations provided by its underlying symmetry, rotation, displacement, and squeezing, using a Lie algebraic approach. Our factorization of the dynamics allows for the intuitive construction of protocols for state engineering, for example, creating and removing displacement and squeezing, as well as their combinations, optimizing squeezing, or more complex protocols that work for slow and fast rates of change in the oscillator parameters.

Deformed oscillators allow for a generalization of the standard fermions and bosons, namely, for the description of para-particles. Such particles, while indiscernible in nature, can represent good candidates for descriptions of physical phenomena like topological phases of matter. Here, we report the digital quantum simulation of para-particle oscillators by mapping para-particle states to the state of a qubit register, which allow us to identify the para-particle oscillator Hamiltonian as an $XY$ model, and further digitize the system onto a universal set of gates. In both instances, the gate depth grows polynomially with the number of qubits used. To establish the validity of our results, we experimentally simulate the dynamics of para-fermions and para-bosons, demonstrating full control of para-particle oscillators on a quantum computer. Furthermore, we compare the overall performance of the digital simulation of dynamics of the driven para-Fermi oscillator to a recent analog quantum simulation result.

Para-particles are fascinating because they are neither bosons nor fermions. While unlikely to be found in nature, they might represent accurate descriptions of physical phenomena like topological phases of matter. We report the quantum simulation of para-particle oscillators by tailoring the native couplings of two orthogonal motional modes of a trapped ion. Our system reproduces the dynamics of para-bosons and para-fermions of even order very accurately. These results represent the first experimental analogy of para-particle dynamics in any physical system and demonstrate full control of para-particle oscillators.

We propose a trapped-ion platform to simulate a reconfigurable spin-spin Hamiltonian related to quantum thermodynamic processes. Starting from an experimental model describing two trapped-ions under slightly off-resonant first sideband driving with individually controlled driving phases, we follow an operational quantum optics approach to show that it produces an effective model appearing in recent quantum thermodynamics proposals. We show that projection into the vibrational ground state manifold allows full analytic treatment. As a practical example, we take experimental data from a $^{187}$Yb$^{+}$ trap and numerically simulate the reversal of heat flow between two thermal spins controlled by their quantum correlations.

We propose an ideal scheme for preparing vibrational $\mathrm{SU(1,1)} \otimes \mathrm{SU(1,1)}$ states in a two-dimensional ion trap using red and blue second sideband resolved driving of two orthogonal vibrational modes. Symmetric and asymmetric driving provide two regimes to realize quantum state engineering of the vibrational modes. In one regime, we show that time evolution synthesizes so-called $\mathrm{SU}(1,1)$ Perelomov coherent states, that is separable squeezed states and their superposition too. The other regime allows engineering of lossless 50/50 $\mathrm{SU}(2)$ beam splitter states that are entangled states. These ideal dynamics are reversible, thus, the non-classical and entangled states produced by our schemes might be used as resources for interferometry.

Motivated by the proposal to simulate para-Bose oscillators in a trapped-ion setup [C. Huerta Alderete and B. M. Rodríguez-Lara, Phys. Rev. A 95, 013820 (2017)], we introduce an overcomplete, nonorthogonal basis for para-Bose Hilbert spaces. The states spanning these bases can be experimentally realized in the trapped-ion simulation via time evolution. The para-Bose states show both nonclassical and semiclassical statistics on their Fock state distribution, asymmetric field quadrature variances, and do not minimize the uncertainty relation for the field quadratures. These properties are analytically controlled by the para-Bose order and the evolution time; both parameters might be feasible for fine tuning in the trapped-ion quantum simulation.

Quantum mechanics allows paraparticles with mixed Bose-Fermi statistics that have not been experimentally confirmed. We propose a trapped-ion scheme whose effective dynamics are equivalent to a driven para-Bose oscillator of even order. Our mapping suggest highly entangled vibrational and internal ion states as the laboratory equivalent of quantum simulated parabosons. Furthermore, we show the generation and reconstruction of coherent oscillations and para-Bose analogs of Gilmore-Perelomov coherent states from population inversion measurements in the laboratory frame. Our proposal, apart from demonstrating an analog quantum simulator of para-Bose oscillators, provides a quantum state engineering tool that foreshadows the potential use of paraparticle dynamics in the design of quantum information systems.

We introduce the cross-cavity quantum Rabi model describing the interaction of a single two-level system with two orthogonal boson fields and propose its quantum simulation by two-dimensional, bichromatic, first-sideband driving of a single trapped ion. We provide an introductory survey of the model, including its diagonalization in the two-level system basis, numerical spectra and its characteristics in the weak, ultra strong and deep strong coupling regimes. We also show that the particular case of degenerate field frequencies and balanced couplings allows us to cast the model as two parity deformed oscillators in any given coupling regime.

We show that the two-photon Jaynes-Cummings model, feasible of experimental realization in cavity or ion-trap quantum electrodynamics, can approximately produce nonlinear coherent states of the field. We introduce these nonlinear coherent states of the field as $2m$-photon added or subtracted coherent states in terms of raising and lowering field operators, also known as London phase operators or Susskind-Glogower operators.

We present a two-dimensional array of nearest-neighbor coupled waveguides that is the optical analog of a quantum optomechanical system. We show that the quantum model predicts the appearance of effective column isolation, diagonal-coupling and other non-trivial couplings in the two-dimensional photonic lattice under a standard approximation from ion-trap cavity electrodynamics. We provide an approximate impulse function for the case of effective column isolation and compare it with exact numerical propagation in the photonic lattice.

We try to classify the spectrum of the two-qubit Dicke model by calculating two quantum information measures of its eigenstates: the Wooters concurrence and the mutual quantum information. We are able to detect four spectral sets in each parity subspace of the model: one set is regular and given by the product of a Fock state of the field times the singlet Bell state of the qubits; the rest are fairly regular and related to the triplet states of the Bell basis. The singlet states become trapping states when we couple the Dicke model to an environment of harmonic oscillators, making them candidates for generating maximally entangled states in experimental realizations of ion trap quantum electrodynamics (QED) and circuit QED. Furthermore, they are robust and survive the inclusion of driving and dipole-dipole interactions, pointing to their use for storing quantum correlations, and it is straightforward to provide a generalization of these trapping states to the Dicke model with even number of qubits.

The interaction of a two-level atom with a single-mode quantized field is one of the simplest models in quantum optics. Under the rotating wave approximation, it is known as the Jaynes-Cummings model and without it as the Rabi model. Real-world realizations of the Jaynes-Cummings model include cavity, ion trap and circuit quantum electrodynamics. The Rabi model can be realized in circuit quantum electrodynamics. As soon as nonlinear couplings are introduced, feasible experimental realizations in quantum systems are drastically reduced. We propose a set of two photonic lattices that classically simulates the interaction of a single two-level system with a quantized field under field nonlinearities and nonlinear couplings as long as the quantum optics model conserves parity. We describe how to reconstruct the mean value of quantum optics measurements, such as photon number and atomic energy excitation, from the intensity and from the field, such as von Neumann entropy and fidelity, at the output of the photonic lattices. We discuss how typical initial states involving coherent or displaced Fock fields can be engineered from recently discussed Glauber-Fock lattices. As an example, the Buck-Sukumar model, where the coupling depends on the intensity of the field, is classically simulated for separable and entangled initial states.

We introduce the term Majoranon to describe particles that obey the Majorana equation, which are different from the Majorana fermions widely studied in various physical systems. A general procedure to simulate the corresponding Majoranon dynamics, based on a decomposition of the Majorana equation into two Dirac equations, is described in detail. It allows the simulation of the two-component chiral spinors, the building blocks of modern gauge theories, in the laboratory with current technology. Specifically, a Majoranon in one spatial dimension can be simulated with a single qubit plus a continuous degree of freedom, for example a single trapped ion. Interestingly, the dynamics of a Majoranon deviates most clearly from that of a Dirac particle in the rest frame, in which the continuous variable is redundant, making a possible laboratory implementation feasible with existing set ups.

We propose a scheme to simulate the dynamics of neutrino oscillations using trapped ions. For neutrinos in 1+1 dimensions, our scheme is experimentally implementable with existing trapped ion technology. We show that the three generation neutrino oscillations can be realized with three ions for 1+3 and 1+1 dimensions where the latter case only requires experimentally proven two-ion interactions. For this case, we discuss two setups utilizing different types of spin-spin interactions. Our method can be readily applied to two generation neutrino oscillations requiring fewer ions and lasers. We give a brief outline of a possible experimental scenario.