Maxwell-Bloch Equations Research Articles

All-optical switching via spontaneously generated coherence, relative phase and incoherent pumping in a V-type three-level system

All-optical switching in a V-type three-level atomic medium is studied and controlled by spontaneously generated coherence (SGC), relative phase of applied fields and incoherent pumping rate. By simultaneously numerical solving the coupled Maxwell–Bloch equations for atom and field on a spatio-temporal grid, we have observed a continuous-wave probe field is switched ON or OFF by periodically modulating the coupling field intensity or the relative phase of applied fields. It also is shows that although the SGC can distort the switching probe pulse, however, under the SGC the response of the medium depends sensitively on the relative phase, and the probe field can be switched synchronous or anti-synchronous with the relative phase. On the other hand, in order to overcome the switching pulse distortion caused by SGC, we have included an incoherent pumping field. By choosing the appropriate phase and pumping rate, the fluctuations of switching probe pulse can be easily overcome. The proposed model can be useful for the design of optical switches and optical storage devices as well as possible experimental implementations.

Rogue waves in the three-level defocusing coupled Maxwell–Bloch equations

The coupled Maxwell–Bloch (CMB) system is a fundamental model describing the propagation of ultrashort laser pulses in a resonant medium with coherent three-level atomic transitions. In this paper, we consider an integrable generalization of the CMB equations with the defocusing case. The CMB hierarchy is derived with the aid of a 3 × 3 matrix eigenvalue problem and the Lenard recursion equation, from which the defocusing CMB model is proposed as a special reduction of the general CMB equations. The n -fold Darboux transformation as well as the multiparametric n th-order rogue wave solution of the defocusing CMB equations are put forward in terms of Schur polynomials. As an application, the explicit rogue wave solutions from first to second order are presented. Apart from the traditional dark rogue wave, bright rogue wave and four-petalled rogue wave, some novel rogue wave structures such as the dark four-peaked rogue wave and the double-ridged rogue wave are found. Moreover, the second-order rogue wave triplets which contain a fixed number of these rogue waves are shown.

Stimulated slowing of Yb atoms on the narrow 1S0 → 3P1 transition

We analyzed bichromatic and polychromatic stimulated forces for laser cooling and trapping of Yb atoms using only the narrow ${}^{1}{S}_{0}\phantom{\rule{4pt}{0ex}}\ensuremath{\rightarrow}{\phantom{\rule{4pt}{0ex}}}^{3}{P}_{1}$ transition. Our model is based on numerical solutions of optical Bloch equations for two-level atoms driven by multiple time-dependent fields combined with Monte Carlo simulations, which account for realistic experimental conditions such as atomic beam divergence, geometry, and Gaussian laser modes. Using 1 W of laser power, we predict a loading rate of $\ensuremath{\approx}\phantom{\rule{4pt}{0ex}}{10}^{8}$ atoms/s into a 556-nm magneto-optical trap (MOT) with a slowing force of $\ensuremath{\approx}60{F}_{\text{rad}}$. We show that a square-wave modulation can produce similar stimulated forces with almost twice the velocity range and improve the MOT loading rate of Yb atoms by up to 70%.

Multidimensional spectroscopy of magneto-excitons at high magnetic fields.

We perform two-dimensional Fourier transform spectroscopy on magneto-excitons in GaAs at magnetic fields and observe Zeeman splitting of the excitons. The Zeeman components are clearly resolved as separate peaks due to the two-dimensional nature of the spectra, leading to a more accurate measurement of the Zeeman splitting and the Landé g factors. Quantum coherent coupling between Zeeman components is observed using polarization dependent one-quantum two-dimensional spectroscopy. We use two-quantum two-dimensional spectroscopy to investigate higher four-particle correlations at high magnetic fields and reveal the role of the Zeeman splitting on the two-quantum transitions. The experimental two-dimensional spectra are simulated using the optical Bloch equations, where many-body effects are included phenomenologically.

Global and localised temporal structures in driven ring quantum cascade lasers

Starting from a full set of effective Maxwell-Bloch equations for a ring quantum cascade laser in the limit of fast material dynamics we derive a new set of equations which require a considerably lower numerical load because they evolve on the time scale of the electric field. With the further assumption of laser very close to threshold the equations take the form of the generalised Lugiato-Lefever equation. Using the latter, we study the formation and stability of multi-peaked localised structures which can be regarded as portions of a global pattern and exhibit a snaking structure.

Spaceless description of active optical media

The acclaimed Maxwell-Bloch (or Arecchi-Bonifacio) equations are a valid dynamical model, effectively describing wave propagation in nonlinear optical media: from the amplification in input-output devices to multimode instabilities arising in laser systems. However, the inherent spatial variability of the physical observables represents an obstacle to fast simulations and analysis, especially whenever networks of active elements have to be considered. In this paper, we propose an approach which, stripping the spatial dependence of its role as a generator of dynamical richness, allows for a compelling simple portrait. It leads to (a few) ordinary differential equations in input-output configurations, complemented by a time-delayed feedback in closed-loop setups. Such scheme reproduces accurately the dynamics, paving the way to a plain treatment of the wealth of phenomena described by the Maxwell-Bloch equations.

Analysis of a strange attractor of the laser type by means of the Maxwell-Bloch equations using the Lorentz equations

This paper will show how the theory of equations where at least one of its terms is a derivative can be solved from two points of view: The first one involves the quantitative analysis of a system and the second one involves the qualitative analysis of a system. In this manuscript, we will show the one that uses the qualitative analysis since we are interested more than the solution of the equation, in its behavior over time and the nature of its solution in terms of how stable it may or may not be. This will be possible through the analysis of the characteristics of the system since they provide us with information on the behavior of the solution, such is the case of a very important one: stability. This stability is desired to be preserved in the face of small variations that the system may present, contrary to what happens when the explicit solution is obtained. The path of the qualitative analysis will lead us to the chaos theory, for this we will show the model and analysis for a strange attractor which is obtained from the Maxwell Bloch model for the laser. This model will present a fork-type bifurcation. To show the solution methodology, due to its complexity, we will take its analogous as a simplified Lorenz model through a change of variable that allows its qualitative analysis.

Femtosecond laser-induced quantum-beat superfluorescence of atomic oxygen in a flame

Among different approaches to generate mirrorless lasing, resonant multiphoton pumping of gas constituents by deep-UV laser pulses exhibits so far the highest efficiency and produces measurable lasing energies, but the underlying mechanism was not yet fully settled. Here, we report lasing generation from atomic oxygen in a methane-air flame via femtosecond two-photon excitation. Temporal profiles of the lasing pulses were measured for varying concentrations of atomic oxygen, which shows that the peak intensity and time delay of the lasing pulse approximately scales as $N$ and $1/\sqrt{N}$, respectively, where $N$ represents the concentration. These scaling laws match well with the prediction of oscillatory superfluorescence (SF), indicating that the lasing we observed is essentially SF rather than amplified spontaneous emission. In addition, the quantum-beating effect was also observed in the time-resolved lasing pulse. A theoretical simulation based on nonadiabatic Maxwell-Bloch equations well reproduces the experimental observations of the temporal dynamics of the lasing pulses. These results on fundamentals should be beneficial for the better design and applications of lasing-based techniques.

Observation of photon recoil effects in single-beam absorption spectroscopy with an ultracold strontium gas

We report on observing photon recoil effects in the absorption of a single monochromatic light at 689 nm through an ultracold 88Sr gas, where the recoil frequency is comparable to natural linewidth of the narrow-line transition 5s2 1S0–5s5p 3P1 in strontium. In the regime of high-saturation, the absorption profile becomes asymmetric due to the photon-recoil shift, which is of the same order as the natural linewidth. The lineshape is described by an extension of the optical Bloch equations including the momentum transfers to atoms during emission and absorption of photons. Our work reveals the photon recoil effects in a simplest single-beam absorption setting, which is of significant relevance to other applications such as saturation spectroscopy, Ramsey interferometry, and absorption imaging.

A laser cooling scheme for precision measurements using fermionic barium monofluoride (137Ba19F) molecules

We theoretically investigate the laser cooling of fermionic barium monofluoride (137BaF) molecules, which are promising candidates for precision studies of weak parity violation and nuclear anapole moments. This molecular species features two nuclear spins, resulting in a hyperfine structure that is considerably more complicated than the one found in the usual laser-cooled diatomics. We use optical Bloch equations and rate equations to show that optical cycling, sub-Doppler cooling and bichromatic forces can all be realized under realistically achievable experimental conditions.

Solutions of the Maxwell–Bloch equations in (1,1)-d in the presence of a pulsed pump

Using the basis formed by the eigenfunctions of the Lienard–Wiechert Green function, we develop and test an algorithm for solving exactly the Maxwell–Bloch equations in (1-space, 1-time). We use this algorithm to search for patterns in the solutions of a pulse pumped slab of two-level systems when the slab’s thickness L=p+1/4λ,λ being the wavelength of the resonant radiation and p being an integer. Both non-linearity and saturation effects are manifest as the strength of the pump is increased. For thicknesses on this lattice, the behavior of the slab is dominated by a single mode in the spectrum of the Green’s function. The resulting patterns do not appear at other thicknesses, because of the interference between competing modes

Generalization of the Menegozzi and Lamb maser algorithm to the transient superradiance regime

ABSTRACT We investigate the application of the conventional quasi-steady state maser modelling algorithm of Menegozzi & Lamb (ML) to the high field transient regime of the one-dimensional Maxwell-Bloch (MB) equations for a velocity distribution of atoms or molecules. We quantify the performance of a first order perturbation approximation available within the ML framework when modelling regions of increasing electric field strength, and we show that the ML algorithm is unable to accurately describe the key transient features of R. H. Dicke’s superradiance (SR). We extend the existing approximation to one of variable fidelity, and we derive a generalization of the ML algorithm convergent in the transient SR regime by performing an integration on the MB equations prior to their Fourier representation. We obtain a manifestly unique integral Fourier representation of the MB equations which is $\mathcal {O}\left(N\right)$ complex in the number of velocity channels N and which is capable of simulating transient SR processes at varying degrees of fidelity. As a proof of operation, we demonstrate our algorithm’s accuracy against reference time domain simulations of the MB equations for transient SR responses to the sudden inversion of a sample possessing a velocity distribution of moderate width. We investigate the performance of our algorithm at varying degrees of approximation fidelity, and we prescribe fidelity requirements for future work simulating SR processes across wider velocity distributions.

High-order Accurate Schemes for Maxwell-Bloch Equations with Material Interfaces

Efficient and high-order accurate FDTD schemes are developed for nonlinear dispersive models in active multi-level media with material interfaces in 2D and 3D arbitrary geometry. The considered nonlinear systems are multi-level generalizations of 2-level optical Maxwell-Bloch equations. Arbitrary numbers of atomic levels and macroscopic polarization vectors can be accounted for in the multi-level atomic (MLA) system. The MLA models are suitable for modeling active media with various properties, such as lasing, saturable absorption, reverse saturable absorption, etc. Composite overlapping grids are employed to handle complex geometry with material interfaces, using locally conforming curvilinear grids for curved boundaries and interfaces and Cartesian grids for the rest. The developed high-order schemes allow compact stencils in time integration, and efficient point-wise update of the numerical solutions.

Mbsolve: An open-source solver tool for the Maxwell-Bloch equations

The Maxwell-Bloch equations are a valuable tool to model light-matter interaction, where the application examples range from the description of pulse propagation in two-level media to the elaborate simulation of optoelectronic devices, such as the quantum cascade laser (QCL). In this work, we present mbsolve, an open-source solver tool for the Maxwell-Bloch equations. Here, we consider the one-dimensional Maxwell's equations, which are coupled to the Lindblad equation. The resulting generalized Maxwell-Bloch equations are treated without invoking the rotating wave approximation (RWA). Since this full-wave treatment is computationally intensive, we provide a flexible framework to implement different numerical methods and/or parallelization techniques. On this basis, we offer two solver implementations that use OpenMP for parallelization. Program summaryProgram Title: mbsolveCPC Library link to program files:https://doi.org/10.17632/nvgnkfphj9.1Code Ocean capsule:https://codeocean.com/capsule/5425444/treeLicensing provisions: GPLv3Programming language: C++, Python bindingsNature of problem: The Maxwell-Bloch equations are frequently used to model light-matter interaction in general and the dynamics of optoelectronic devices, such as the quantum cascade laser, in particular. Especially when the rotating wave approximation is not invoked, the Maxwell-Bloch equations require numerical treatment. Here, efficient numerical methods and parallelization are necessary to solve the equations.Solution method: We present mbsolve, a flexible and open-source solver framework for the Maxwell-Bloch equations. The framework enables the specification of simulation setups, which can be subsequently used to evaluate and verify numerical methods and/or parallelization techniques. Additionally, mbsolve contains a finite-difference time-domain solver implementation based on OpenMP that is ready to be used, for example, in the simulation of quantum cascade laser dynamics.

Optical cycling of AlF molecules

Aluminium monofluoride (AlF) is a promising candidate for laser cooling and trapping at high densities. We show efficient production of AlF in a bright, pulsed cryogenic buffer gas beam, and demonstrate rapid optical cycling on the Q rotational lines of the A 1Π ↔ X 1Σ+ transition. We measure the brightness of the molecular beam to be >1012 molecules per steradian per pulse in a single rotational state and present a new method to determine its velocity distribution in a single shot. The photon scattering rate of the optical cycling scheme is measured using three different methods, and is compared to theoretical predictions of the optical Bloch equations and a simplified rate equation model. Despite the large number of Zeeman sublevels (up to 216 for the Q(4) transition) involved, a high scattering rate of at least 17(2) × 106 s−1 can be sustained using a single, fixed-frequency laser without the need to modulate the polarisation. We deflect the molecu-lar beam using the radiation pressure force and measure an acceleration of 8.7(1.5) × 105 m s−2. Losses from the optical cycle due to vibrational branching to X 1Σ+, v″ = 1 are addressed efficiently with a single repump laser. Further, we investigate two other loss channels, parity mixing by stray electric fields and photo-ionisation. The upper bounds for these effects are sufficiently low to allow loading into a magneto‐optical trap.

Rydberg blockade in an ultracold strontium gas revealed by two-photon excitation dynamics

We demonstrate the interaction-induced blockade effect in an ultracold $^{88}$Sr gas via studying the time dynamics of a two-photon excitation to the triplet Rydberg series $5\mathrm{s}n\mathrm{s}\, ^3\textrm{S}_1$ for five different principle quantum numbers $n$ ranging from 19 to 37. By using a multi-pulse excitation sequence to increase the detection sensitivity we could identify Rydberg-excitation-induced atom losses as low as $<1\%$. Based on an optical Bloch equation formalism, treating the Rydberg-Rydberg interaction on a mean-field level, the van der Waals coefficients are extracted from the observed dynamics, which agree fairly well with \emph{ab initio} calculations.

N-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert approach

We mainly study [Formula: see text]-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert (RH) approach in this paper. The relevant RH problem has been constructed by performing spectral analysis of Lax pair. Then the jump matrix of the Maxwell–Bloch equations has been obtained. Finally, we gain the exact solutions of the Maxwell–Bloch equations by solving the special RH problem with reflectionless case.

Two-dimensional optical gap solitons and vortices in a coherent atomic ensemble loaded on optical lattices

Coherent atomic systems, e.g., resonantly cold atomic gases within which electromagnetically induced transparency (EIT) operates, have recently received great attention, because of their remarkable scientific properties and pivotal implications. Light behavior in such systems governed by various potentials is a new and interesting research focus, but missing literally report on two-dimensional (2D) localized gap modes in coherent atomic systems loaded on optical lattices. We survey such issue in a coherent atomic gas inside where the EIT turns on, trapped by 2D optical lattices—constituted by counter-propagating far-detuned Stark laser fields, in the framework of nonlinear Schrödinger equation derived from Maxwell-Bloch equations. Using the linear stability analysis and direct perturbed evolution we address the formation, property, and stability of 2D localized gap modes of two types, gap solitons and gap vortices, in forbidden band gaps of the underlying linear Bloch-wave spectrum. The former mode is fundamental gap solitons, and the latter belongs to higher-order gap solitons with embedded topological charge. Our results are helpful not only for in-depth understanding of soliton dynamics in coherent atomic ensemble loaded on periodic potentials, but also for laying the groundwork for forthcoming applications in optical communications and quantum information processing.

Optical instabilities in Fabry-Perot resonators

We start from the Maxwell-Bloch equations that govern the dynamics of Fabry-Perot lasers and, considering a doubled cavity, generalize our traveling wave formalism to the case in which the two-level medium does not fill the cavity but occupies only a portion of it. We linearize these equations and focus on the adiabatic elimination of both atomic fluctuations. Next, we restrict our attention to free running lasers under resonant conditions and analyze amplitude and phase instabilities. The examination of the unstable domain leads to the conclusion that the multimode Fabry-Perot instability arises near threshold only when the ratio of the longitudinal to the transverse atomic relaxation rate is substantially smaller than unity. This result agrees with our previous study in the limit of adiabatic elimination of the atomic polarization fluctuations only. We describe the self-pulsing behavior that arises from the multimode instability, and exhibits hysteretic behavior when the pump parameter is swept forward and backward. Finally, we investigate the single-mode instability and show that in the Fabry-Perot case there is no longer the correspondence between single-mode and multimode instabilities that is well known in the case of ring lasers. We confirm that in the case of resonant Fabry-Perot lasers the single-mode amplitude instability arises very far from threshold.

Numerical Simulations of Laser Pulse Propagation in Quantum Active Media: Using a semiclassical model

Active media are materials that consist of quantum systems, such as atoms, impurities, or quantum dots, and are characterized by strong resonant absorption and re-emission of radiation. In this article, we present a general framework for the numerical simulation and analysis of time-dependent radiation-induced phenomena in active media. The formulation used is based on the solution of semiclassical Maxwell–Bloch equations describing the evolution of each quantum element under the effect of external and re-emitted radiation. Within these Maxwell–Bloch equations, the coupling of quantum systems to classical Maxwellian fields poses computational challenges due to the strong nonlinearities involved. In contrast to traditional mesh-based solvers, we adopt an electric field integral-operator approach that can reliably account for near-field effects—including self-radiation—and is scalable to large systems. We then focus on media based on large numbers of quantum dots and demonstrate various physical effects arising from the near-field coupling, including polarization modulations and superradiance.