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In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of the method arising from the Sampling Theorem, which defines the critical Nyquist frequency and the aliasing effect. We then define the Fourier Transform (FT) and the FFT in a way that can be implemented in one and more dimensions. Finally, we show how to apply the FFT in the solution of PDEs related to problems involving two spatial dimensions, specifically the Poisson equation, the diffusion equation and the wave equation for elliptic, parabolic and hyperbolic cases respectively., Comment: Prepared for educational purposes, 9 pages, 9 figures. Accepted for publication in the educative section of Revista Mexicana de Fisica

We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral properties of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bands, with a gap proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, whereas non-degenerate spectra produce flat distributions. By using the eigenstate participation ratio, we predict the behavior of a walker under static phase noise in the coin and sites, showing a correlation between low participation ratios and localization, and high ratios with delocalization. Our results show that the average eigenstate participation ratio provides insights equivalent to computationally intensive mean squared displacement calculations. We observe a transition from super-diffusive to sub-diffusive behavior for uniformly distributed noise within the range $-\pi/3$ to $\pi/3$ and saturation of the mean square distance when the number of steps exceeds the graph size by an order of magnitude. Finally, we propose a quantum circuit implementation of our model., Comment: 16 pages, 12 figures

The representation of Spanish-speaking students in STEM identity literature, particularly in physics identity, is conspicuously minimal. This study addresses this gap with a two-pronged approach. First, a physics identity instrument was adapted for Spanish-speaking STEM students to promote inclusivity in educational research. Data from 334 Mexican STEM students was collected and analysed for validation through factor analyses and Cronbach's alpha, confirming the structural integrity and reliability of the adapted instrument. Second, the instrument was employed to examine the correlation between physics identity and academic performance in a separate dataset from 200 engineering students, using multivariate linear and logistic regressions. The findings underscore gender's impact on physics course grades. Notably, variations in physics identity among engineering students indicated a potential link with their chosen field. Despite the importance of physics in engineering, physics identity was lower than expected, and performance in physics courses did not consistently correlate with a strong physics identity. However, the `interest in physics topics` subconstruct did correlate with physics course grades. This research fills a critical void by providing an adapted instrument for assessing physics identity in Spanish-speaking students and underscores the need for pedagogical shifts to enhance physics identity and STEM outcomes for Latino students., Comment: 34 pages, 3 figures, 6 tables

We explore a family of paraxial beams constructed by the linear superposition of Laguerre-Gaussian beams, representing an optical analogue to generalized $SU(2)$ Lie group coherent states. A single complex parameter controls a smooth transition between Laguerre-Gaussian and Hermite-Gaussian beams, with intermediate beams that merge characteristics of both families. Our beams exhibit propagation-invariant properties, up to a scaling factor, a highly desirable feature for optical applications, validated via holographic experimental results., Comment: 12 pages, 6 figures

We explore Majorana stellar representation on the Riemann sphere for optical scalar beams and vector fields, leveraging the spin and orbital angular momentum of light. For scalar beams, the star constellations for the Laguerre-Gaussian beam basis reside at the poles of the sphere. For vector fields, these constellations represent well-defined circular polarization. The underlying $su(2)$ symmetry of angular momenta enables populating the sphere through generalized unitary rotations of the basis elements, resulting in optical analogues of Bloch and generalized $su(2)$ coherent states for both scalar beams and vector fields. Constellations for scalar Hermite-Gaussian beams and vector Hermite-Gaussian partial Poincar\'e fields, restricted to linear polarization, occupy the equator of the sphere as a result of these rotations. Additionally, we investigate scalar and vector analogues of cat codes and kings of quantumness, providing insights into the visualization and characterization of complex optical beams and fields with potential applications in optical communications., Comment: 24 pages, 11 figures

We use the spatial degree of freedom of light modes to construct optical analogues of generalized quantum coherent states for Hermite- and Laguerre-Gauss modes. Our optical analogues preserve the statistical properties of their quantum counterparts, encoded in their amplitude and phase distributions. We explore three basic symmetries that provide generalized displaced, rotated, and squeezed coherent states. Given the substantial interest in squeezed states for probing matter, we believe that the optical analogues introduced here have significant implications for optical sensing. Specifically, the single-particle nature of our spatial modes makes them robust candidates for sensing photosensitive materials. Overall, our approach opens the door to optical metrology and sensing protocols that mimic those already existing in the quantum realm, and facilitates further exploration of the quantum state zoo through classical optical analogues., Comment: 42 pages, 14 figures

We revisit the Jaynes-Cummings and anti-Jaynes-Cummings model through the lens of Lie theory, aiming to highlight the efficacy of an operator-based approach for diagonalization. We focus on explicitly delineating the steps from an underlying abstract supersymmetry, provided by the $u(1 \vert 1)$ superalgebra, into concrete proper states and energies in the laboratory frame. Additionally, we explore the anisotropic Rabi model possessing an underlying supersymmetry, provided by the $osp(2 \vert 2)$ superalgebra, in a squeezed reference frame, where it is possible to approximate its spectral characteristics by an effective Jaynes--Cummings model. Finally, we identify a regime for a factorizable anisotropic Rabi model, exhibiting an equally spaced, double degenerate energy spectrum with a unique ground state energy. Our work aims to merge mathematical physics with practical quantum optics, underscoring the critical role of Lie theory., Comment: 21 pages, 4 figures

We report electromagnetic field modes solving the inhomogeneous Maxwell equations for parabolic gradient index fibers in the low refractive index contrast approximation. The first family comprises accelerating fields characterized by an intensity distribution center tracing a circular trajectory transverse to the fiber optical axis. These fields maintain an invariant shape for both their intensity and phase distributions while rotating around their center. The second family comprises breathing fields characterized by an intensity distribution center aligned with the fiber optical axis. These fields exhibit intensity distribution scaling along propagation, while their phase swirls and rotates around the optical axis without changing their intensity distribution shape and topological charge., Comment: 11 pages, 4 figures

Large quantum photonic systems hold promise for surpassing classical computational limits, yet their state preparation remains a challenge. We propose an alternative approach to study multiparticle dynamics by mapping the excitation mode of these systems to physical properties of the Laguerre-Gauss modes. We construct coherent states establishing a direct link between excitation number dynamics and the evolution of the Laguerre-Gauss mode. This highlights the photon transverse spatial degree of freedom as a versatile platform for testing fundamental aspects of quantum multiparticle systems., Comment: 14 pages, 7 figures

We present a Hamiltonian model describing two pairs of mechanical and optical modes under standard optomechanical interaction. The vibrational modes are mechanically isolated from each other and the optical modes couple evanescently. We recover the ranges for variables of interest, such as mechanical and optical resonant frequencies and naked coupling strengths, using a finite element model for a standard experimental realization. We show that the quantum model, under this parameter range and external optical driving, may be approximated into parametric interaction models for all involved modes. As an example, we study the effect of detuning in the optical resonant frequencies modes and optical driving resolved to mechanical sidebands and show an optical beam splitter with interaction strength dressed by the mechanical excitation number, a mechanical bidirectional coupler, and a two-mode mechanical squeezer where the optical state mediates the interaction strength between the mechanical modes., Comment: 18 pages, 6 figures

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., Comment: 19 pages, 9 figures

We present a Lie algebraic approach to a Hamiltonian class covering driven, parametric quantum harmonic oscillators where the parameter set -- mass, frequency, driving strength, and parametric pumping -- is time-dependent. Our unitary-transformation-based approach provides a solution to our general quadratic time-dependent quantum harmonic model. As an example, we show an analytic solution to the periodically driven quantum harmonic oscillator without the rotating wave approximation; it works for any given detuning and coupling strength regime. For the sake of validation, we provide an analytic solution to the historical Caldirola--Kanai quantum harmonic oscillator and show that there exists a unitary transformation within our framework that takes a generalized version of it onto the Paul trap Hamiltonian. In addition, we show how our approach provides the dynamics of generalized models whose Schr\"odinger equation becomes numerically unstable in the laboratory frame., Comment: 26 pages, 7 figures

We demonstrate scattering control of Gaussian-like wave packets propagating with constant envelope velocity and invariant waist through coupled resonator optical waveguides (CROW) via an external resonator coupled to multiple sites of the CROW. We calculate the analytical reflectance and transmittance using standard scattering methods from waveguide quantum electrodynamics and show it is possible to approximate them for an external resonator detuned to the CROW. Our analytical and approximate results are in good agreement with numerical simulations. We engineer various configurations using an external resonator coupled to two sites of a CROW to show light trapping with effective exponential decay between the coupling sites, wave packet splitting into two pairs of identical Gaussian-like wave packets, and a non-local Mach-Zehnder interferometer., Comment: 20 pages, 7 figures

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., Comment: 7 pages, 5 figures, 1 table

We study the dynamics of a class-B semiconductor microring laser coupled to a neutral microring resonator for common fabrication parameters. For zero detuning between the resonators, we identify five dynamical regions controlled by the normalized pump rate of the laser and coupling strength between the resonators. These regions show stable lasing with phase locking for either sufficiently small or large coupling strength parameter values. Multistability and chaos arise in between these stable regions where at least one fixed point is attractive. The transition from stable to unstable lasing regions in the parameter space phase diagram is a crossover., Comment: 16 pages, 5 figures

We study an optomechhanical device supporting at least three optical modes in the infrared telecommunication band and three mechanical vibration modes. We model the coherent driving of each optical mode, independently of each other, to obtain an effective Hamiltonian showing the different types of parametric processes allowed in the device. We propose a bichromatic driving scheme, in the lossy optical cavity regime, under a mean field approximation, that provides the quantum simulation of a parametric oscillator with optical control of its parameters., Comment: 12 pages, 2 figures, Quantum Fest 2021

We use a normal mode approach to show full and partial state transfer in a class of coupled resonator networks with underlying $su(2)$ symmetry that includes the so-called $J_{x}$ photonic lattice. Our approach defines an auxiliary Hermitian coupling matrix describing the network that yields the normal modes of the system and its time evolution in terms of orthogonal polynomials. These results provide insight on the full quantum state reconstruction time in a general $su(2)$ network of any size and the full quantum transfer time in the $J_{x}$ network of size $4 n + 1$ with $n=1,2,3,\ldots$ In the latter, our approach shows that the Fock state probability distribution of the initial state is conserved but the amplitudes suffer a phase shift proportional to $\pi/2$ that results in partial quantum state transfer for any other network size., Comment: 13 pages, 1 figure

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., Comment: 11 pages, 7 figures, 1 table

The dynamical control of energy transfer between interacting systems is fundamental in diverse applications related to physical, electronic and chemical processes. Recent developments show that noise may enhance or suppress power transfer in systems described by Coupled-Mode Theory. We show a semi-analytic approach to utilize dynamical noise to produce routing in coupled mode devices. We present results for an optical mode-coupling device with induced noise in the refractive index of the material. However, our approach is valid for networks of coupled oscillators of any type as long as their dynamics are described by Coupled-Mode Theory., Comment: 15 pages, 4 figures

There are diverse teaching methodologies to promote both collaborative and individual work in undergraduate physics courses. However, few educational studies seek to understand how students learn and apply new knowledge through open-ended activities that require mathematical modeling and experimentation focused on environmental problems. In this work, we propose a novel home experiment to simulate the dynamics of a particulate under temperature inversion and model it as damped harmonic motion. Twenty six first year students enrolled in STEM majors answered six qualitative questions after designing and developing the experiment. These questions helped analyze the students epistemological beliefs about their learning process of physics topics and its applications. Results showed that this type of open-ended experiments could facilitate the students understanding of physics phenomena. In addition, this experiment showed that it could help physics professors to promote students epistemological development by giving their students the opportunity to search for different sources of knowledge and becoming self-learners instead of looking at the professor as the epistemological authority. At the end, students described this activity as a positive experience that helped them realize alternative ways to apply physics topics in different contexts of their environment., Comment: 23 pages, 5 figures

We propose a generalized Jaynes-Cummings model that includes but is not limited to an extensive collection of experimental and theoretical proposals from the literature. It covers nonlinear boson terms, nonlinear dispersive and multi-boson exchange interaction. Our model features an underlying Lie graded algebra symmetry reminiscent to supersymmetric quantum mechanics. This allows us to propose a diagonalization scheme and calculate its analytic time evolution. In consequence, we are able to construct closed forms for relevant observables and explore the dynamics of particular realizations of our model independent of their complexity. As an practical example, we show the evolution of the population inversion and the boson quadratures for an initial state consisting of the qubit in the ground state interacting with a coherent field for a selection of cases including the standard JC model with Stark shift, Kerr-like terms, intensity dependent coupling, multi-boson exchange and algebraic deformations., Comment: 20 pages, 8 figures

We discuss the decay of a two-level system into an engineered reservoir of coupled harmonic oscillators in the single-excitation manifold and propose its optical simulation with an homogeneous chain of coupled waveguides where individual elements couple to an external waveguide. We use two approaches to study the decay of the optical analogue for the probability amplitude of the two-level system being in the excited state. A Born approximation allows us to provide analytic closed-form amplitudes valid for small propagation distances. A Fourier-Laplace approach allows us to estimate an effective decay rate valid for long propagation distances. In general, our two analytic approximations match our numerical simulations using coupled mode theory and show non-Markovian decay into the engineered reservoir. In particular, we focus on two examples that provide enhancement or suppression of the decay decay rate using flat-top or Gaussian coupling distributions., Comment: 17 pages, 8 figures

Cholesky factorization provides photonic lattices that are the isospectral partners or the square root of other arrays of coupled waveguides. The procedure is similar to that used in supersymmetric quantum mechanics. However, Cholesky decomposition requires initial positive definite mode coupling matrices and the resulting supersymmetry is always broken. That is, the isospectral partner has the same range than the initial mode coupling matrix. It is possible to force a decomposition where the range of the partner is reduced but the characteristic supersymmetric intertwining is lost. As an example, we construct the Cholesky isospectral partner and the square root of a waveguide necklace with cyclic symmetry. We use experimental parameters from telecommunication C-band to construct a finite element model of these Cholesky photonics lattices to good agreement with our analytic prediction., Comment: 15 pages, 4 figures

We demonstrate lasing in arrays of microring resonators with underlying para-Fermi symmetry. The properties of the algebra allow the analytic prediction of propagation constants and normal modes of an array of passive resonators that show the optical analogue of a zero-energy mode that is its own chiral pair. The rest of the normal modes in the linear model form chiral pairs that are evenly distributed around the pseudo zero-energy mode. We use this information to construct a class-B laser model where even and odd sites are driven with different strength in a pattern following para-Fermi algebra chirality to demonstrate lasing with enhancement or suppression of the zero-energy mode. Zero-energy mode lasing is stable to initial conditions for a large set of driving ratios. Lasing with zero-energy mode suppression is unstable and more susceptible to initial field configurations., Comment: 14 pages, 5 figures

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.

Estimating the quality of a single-photon source is crucial for its use in quantum technologies. The standard test for semiconductor sources is a value of the second-order correlation function of the emitted field below $1/2$ at zero time-delay. This criterion alone provides no information regarding the amplitude of the single-photon contribution for general quantum states. Addressing this question requires the knowledge of additional observables. We derive an effective second-order correlation function, strongly connected to the Mandel-$Q$ parameter and given in terms of both the second-order correlation and the average photon number, that provides a lower bound on the single-to-multi-photon projection ratio. Using both observables individually allows for lower and upper bounds for the single-photon projection. Comparing the tightness of our bounds with those in the literature, we find that relative bounds may be better described using the average photon number, while absolute bounds for low excitation states are tighter using the vacuum projection. Our results show that estimating the quality of a single-photon source based on additional information is very much dependent on what aspect of the quantum state of light one is interested in., Comment: 8 pages (format changed to published two-column), 9 Figures

We consider a coupled mode system where the effective propagation constants of localized modes are amenable to modulation. Starting from an unmodulated system where power transfer is heavily suppressed, we demonstrate that small, periodic modulation of the propagation constants enhance power transfer using a slowly varying envelope approximation for the field mode amplitudes. We calculate an approximate modulation frequency enabling complete transfer between otherwise negligibly coupled elements. The ability to control these modulations by electrical or thermal effects allows for reconfigurable multiport switching. We use an array of coupled silica waveguides and the thermo-optic effect to test our predictions in the telecom C-band. However, this requires a refractive index modulation with period of the order $~10^{-3}~\mathrm{m}$ and yields total power transfer with a propagation distance of the order of $10^{-1}~\mathrm{m}$, which might make it unattractive for integrated photonic applications. Nevertheless, our results are valid for devices described by an equivalent coupled mode matrix for space or time propagation; for example, arrays of microring or terahertz resonators, microwave cavities, radio frequency antennas, or RLC circuits., Comment: 15 pages, 4 figures

Spectral collapse, the transition from discrete to continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model in optical phase space and find that, in the so-called degenerate qubit regime, the collapse is similar to that happening in the transition from a quantum harmonic to an inverted quadratic potential with the free-partical potential as transition point. In this regime, it is possible to construct Dirac-normalizable eigenfunctions for the model that show well defined parity. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs., Comment: 11 pages, 3 figures

We present a proposal for the optical simulation of para-Fermi oscillators in arrays of coupled waveguides. We use a representation that arises as a deformation of the $su(2)$ algebra. This provides us with a set of chiral and a zero-energy-like normal modes. The latter is its own chiral pair and suggest the addition of controlled losses/gains following a pattern defined by parity. In these non-Hermitian para-Fermi oscillators, the analog of the zero-energy mode presents the largest effective loss/gain and it is possible to tune the system to show sequences of exceptional points and varying effective losses/gains. These arrays can be used for mode suppression or enhancement depending on the use of loss or gain, in that order. We compare our coupled mode theory predictions with finite element method simulations to good agreement., Comment: 12 pages, 4 figures

Recent work highlights that cooperation in the one-shot Prisoner's dilemma (PD) is primarily driven by moral preferences for doing the right thing, rather than social preferences for equity or efficiency. By contrast, little is known on what motivates cooperation in the Stag-Hunt Game (SHG). Cooperation in the SHG fundamentally differs from cooperation in the PD in that it is not costly, but risky: players have no temptation to deviate from the cooperative outcome, but cooperation only pays off if the other player cooperates. Here, we provide data from a large (N=436), pre-registered, experiment. Contrary to what has been observed for the PD, we find that SHG cooperation is primarily driven by preferences for efficiency, rather than preferences for doing the right thing.

The optomechanical state transfer protocol provides effective, lossy, quantum beam-splitter-like dynamics where the strength of the coupling between the electromagnetic and mechanical modes is controlled by the optical steady-state amplitude. By restricting to a subspace with no losses, we argue that the transition from mode-hybridization in the strong coupling regime to the damped-dynamics in the weak coupling regime, is a signature of the passive parity-time ($\mathcal{PT}$) symmetry breaking transition in the underlying non-Hermitian quantum dimer. We compare the dynamics generated by the quantum open system (Langevin or Lindblad) approach to that of the $\mathcal{PT}$-symmetric Hamiltonian, to characterize the cases where the two are identical. Additionally, we numerically explore the evolution of separable and correlated number states at zero temperature as well as thermal initial state evolution at room temperature. Our results provide a pathway for realizing non-Hermitian Hamiltonians in optomechanical systems at a quantum level., Comment: 14 pages, 6 figures

We study propagation in a cyclic symmetric multicore fiber where the core radii randomly fluctuate along the propagation direction. We propose a hybrid analytic-numerical method to optimize the amplitude and frequency of the fluctuations that suppress power transfer between outer and inner cores. Our predictions are confirmed by numerical experiments using finite difference beam propagation methods for realistic C-band fibers., Comment: 11 pages, 4 figures

Exceptional points (EPs) are degeneracies of non-Hermitian operators where, in addition to the eigenvalues, corresponding eigenmodes become degenerate. Classical and quantum photonic systems with EPs have attracted tremendous attention due to their unusual properties, topological features, and an enhanced sensitivity that depends on the order of the EP, i.e. the number of degenerate eigenmodes. Yet, experimentally engineering higher-order EPs in classical or quantum domains remains an open challenge due to the stringent symmetry constraints that are required for the coalescence of multiple eigenmodes. Here we analytically show that the number-resolved dynamics of a single, lossy, waveguide beamsplitter, excited by $N$ indistinguishable photons and post-selected to the $N$-photon subspace, will exhibit an EP of order $N+1$. By using the well-established mapping between a beamsplitter Hamiltonian and the perfect state transfer model in the photon-number space, we analytically obtain the time evolution of a general $N$-photon state, and numerically simulate the system's evolution in the post-selected manifold. Our results pave the way towards realizing robust, arbitrary-order EPs on demand in a single device., Comment: 5 pages, 2 figures

The quantum Rabi model accepts analytical solutions in the so-called degenerate qubit and relativistic regimes with discrete and continuous spectrum, in that order.We show that solutions in the laboratory frame are the superposition of even and odd displaced number states, in the former, and infinitely squeezed coherent states, in the latter, of the boson field correlated to the internal states of the qubit. We propose a single parameter model that interpolates between these discrete and continuous spectrum regimes to study the spectral statistics for first and second neighbor differences before the so-called spectral collapse. We find two central first neighbor differences that interweave and fluctuate keeping a constant second neighbor separation., Comment: 15 pages, 5 figures

Nearest-neighbor coupled-mode theory is a powerful framework to describe electromagnetic-wave propagation in multicore fibers but it lacks precision as the separation between cores decreases. We use abstract symmetries to study a ring of evenly distributed identical cores around a central core, a common configuration used in telecommunications and sensing. We find its normal modes and their effective propagation constants while including the effect of all high-order inter-core couplings. Finite-element simulations support our results to good agreement. Only two of these effective modes involve fields in all the cores. These two modes display opposite-sign phase configurations between the fields in the external cores and the central core. These two modes still appear in the limit where the external cores become a continuous ring. Our results might help improve predictions for crosstalk in telecommunications or precision in sensing applications., Comment: 13 pages, 3 figures

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., Comment: 13 pages, 4 figures

We study the effect of finite size and misalignment on a fundamental optomechanical setup: a Fabry-P\'erot cavity with one fixed and one moveable mirror. We describe in detail light confinement under these real world imperfections and compare the behaviour of the intracavity and output fields to the well-known ideal case. In particular, we show that it is possible to trace the motion of the movable mirror itself by measuring intensity changes in the output field even in the presence of fabrication shortcomings and thermal noise. Our result might be relevant to the transition from high precision research experiments to everyday commercial applications of optomechanics; such as high-precission stepmotor or actuator positioning., Comment: 14 pages, 4 figures

Spectral characterization is a fundamental step in the development of useful quantum technology platforms. Here, we study an ensemble of interacting qubits coupled to a single quantized field mode, an extended Dicke model that might be at the heart of Bose-Einstein condensate in a cavity or circuit-QED experiments for large and small ensemble sizes, respectively. We present a semi-classical and quantum analysis of the model. In the semi-classical regime, we show analytic results that reveal the existence of a third regime, in addition of the two characteristic of the standard Dicke model, characterized by one logarithmic and two jump discontinuities in the derivative of the density of states. We show that the finite quantum system shows two different types of clustering at the jump discontinuities, signaling a precursor of two excited quantum phase transitions. These are confirmed using Peres lattices where unexpected order arises around the new precursor. Interestingly, Peres conjecture regarding the relation between spectral characteristics of the quantum model and the onset of chaos in its semi-classical equivalent is valid in this model as a revival of order in the semi-classical dynamics occurs around the new phase transition., Comment: 15 pages, 6 figures

We propose a technique for robust optomechanical state transfer using phase-tailored composite pulse driving with constant amplitude. Our proposal is inspired by coherent control techniques in lossless driven qubits. We demonstrate that there exist optimal phases for maximally robust excitation exchange in lossy strongly-driven optomechanical state transfer. In addition, our proposed composite phase driving also protects against random variations in the parameters of the system. However, this driving can take the system out of its steady state. For this reason, we use the ideal optimal phases to produce smooth sequences that both maintain the system close to its steady state and optimize the robustness of optomechanical state transfer., Comment: 16 pages, 7 figures, 52 references

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., Comment: 12 pages, 7 figures

Group theory (GT) provides a rigorous framework for studying symmetries in various disciplines in physics ranging from quantum field theories and the standard model to fluid mechanics and chaos theory. To date, the application of such a powerful tool in optical physics remains limited. Over the past few years however, several quantum-inspired symmetry principles (such as parity-time invariance and supersymmetry) have been introduced in optics and photonics for the first time. Despite the intense activities in these new research directions, only few works utilized the power of group theory. Motivated by this status quo, here we present a brief overview of the application of GT in optics, deliberately choosing examples that illustrate the power of this tool in both continuous and discrete setups. We hope that this review will stimulate further research that exploits the full potential of GT for investigating various symmetry paradigms in optics, eventually leading to new photonic devices., Comment: 20 page, 5 figures

We study light propagation through cyclic arrays, composed by copies of a given $\mathcal{PT}$-symmetric dimer, using a group theoretical approach and finite element modeling. The theoretical mode-coupling analysis suggest the use of these devices as output port replicators where the dynamics are controlled by the impinging light field. This is confirmed to good agreement with finite element propagation in an experimentally feasible necklace of passive $\mathcal{PT}$-symmetric dimers constructed from lossy and lossless waveguides., Comment: 10 pages, 4 figures

We use the microscopic derivation of the Jaynes-Cummings model master equation under field losses to study the dynamics of initial field states beyond the single-excitation manifold. We show that field-qubit detuning, as well as finite temperature, modify the effective decay rate in the model using entropy measures, like qubit-field purity and von Neumann entropy of the field, for initial Fock states. For initial semi-classical states of the field, we show that the microscopic approach, in phase space, provides an evolution to thermal equilibrium that is smoother than the one provided by the standard phenomenological approach., Comment: 18 pages, 14 figures, 2 movies