Maxwell-Bloch Equations Research Articles

Theoretical study of the control of absorption, transparency, and amplification in a microwave- and RF-driven four-level (Δ + ∇)-type closed-contour interaction system

This paper describes a theoretical study of the optical responses of an atomic medium to the probe field in a four-level (Δ + ∇)-type closed-contour interaction system driven by two pump laser fields (P1, P2), one probe laser field (Pr) and two microwave or radio frequency (RF) coupling fields (L, U). Here, L and U are considered to connect the forbidden electric dipole transitions, i.e. two hyperfine ground states of the Δ subsystem and two hyperfine excited states of the ∇ subsystem, respectively. Since each subsystem is a closed-loop system, relative phases are thought to be present between the optical fields (pump and probe fields), and the microwave or RF fields. The manipulation of the quantum coherence of the system by the strengths of coupling fields along with the relative phases is demonstrated in terms of the changes occurring in the absorption of the probe field by the medium. The probe absorption and dispersion profiles are extracted by numerically solving the optical Bloch equations for the system under steady-state conditions. In this study, we start by illuminating the system with the probe only, and then, by applying the pump fields and the coupling fields one by one, the changes in the absorptive and dispersive probe line profiles are investigated for two distinct cases that depend on the relative strengths of the pump fields, e.g. when P2 is stronger than P1 and vice versa. Furthermore, the individual effects of the strengths and phases of the coupling fields on the probe absorption, transparency and amplification are also explored for both cases. All the changes that appear in the probe signal as a result of the application of different fields are explained by the modified interaction fields or the corresponding effective Rabi frequencies obtained from the partial dressed-state analysis.

Self-starting harmonic comb emission in THz quantum cascade lasers

Harmonic comb states have proven to be ubiquitous in mid-IR quantum cascade lasers. We report here on robust, pure, self-starting harmonic mode locking in Copper-based double-metal THz quantum cascade lasers. Different harmonic orders can be excited in the same laser cavity depending on the pumping condition, and stable harmonic combs spanning more than 600 GHz at 80 K are reported. Such devices can be RF injected, and the free running coherence is assessed by means of a self-mixing technique performed at 50 GHz. A theoretical model based on Maxwell-Bloch equations including an asymmetry in the gain profile is used to interpret the data.

Darboux transformations and solutions of nonlocal Hirota and Maxwell–Bloch equations

AbstractIn this paper, based on the Hirota and Maxwell–Bloch (H‐MB) system and its application in the theory of the femtosecond pulse propagation through an erbium doped fiber, we define two kinds of nonlocal Hirota and Maxwell–Bloch (NH‐MB) systems, namely, ‐symmetric NH‐MB system and reverse space‐time NH‐MB system. Then, we construct the Darboux transformations of these NH‐MB systems. Meanwhile, we derive the explicit solutions by the Darboux transformations.

Modeling the time-dependent electron dynamics in dielectric materials induced by two-color femtosecond laser pulses: Applications to material modifications

Controlling the electron dynamics during laser-matter interactions is a key factor to control the energy deposition and subsequent material modifications induced by femtosecond laser pulses. One way to achieve this goal is to use two-color femtosecond laser pulses. In this paper, the electron dynamics in dielectric materials induced by two-color femtosecond laser pulses is studied by solving dedicated optical Bloch equations. This model includes photo- and impact ionization, the laser heating of conduction electrons, their recombination to the valence band, and their collisions with phonons. The influence of photon energies, laser intensities, and pulse-to-pulse delay is analyzed. Depending on the interaction process, colors cooperate to excite electrons or drive them independently. For the given laser parameters, an optimal pulse-to-pulse delay is found which enhances significantly the energy deposition into the material, in agreement with experimental observations.

Solitons in a Chiral Medium

The dynamics of a field in a thin waveguide surrounded by helically arranged two-level atoms is studied. The interaction of the induced polarizations of atoms with the field propagating inside the waveguide is described by a system of reduced Maxwell–Bloch equations (RMBEs) in the approximation of unidirectional propagation of the field. Nonlocal dipole–dipole interaction (DDI) of the polarizations of atoms in the helix is described in the approximation of the interaction of nearest neighbors in a curvilinear medium. Soliton solutions of an integrable reduction of the system of equations describing the asymmetric propagation of field pulses in the waveguide in the forward and backward directions are found by the method of Riemann problem with zeros. It is shown that, depending on the chirality sign or the propagation direction, a field pulse in the waveguide can have either the shape of a sharp peak or a nearly rectangular shape. Solutions describing the evolution of field pulses on a nonzero pedestal show that the shape and amplitude of the field pulses can be controlled by the external pumping parameters.

Synthesizing optical spectra using computer-generated holography techniques

Experimental control and detection of atoms and molecules often rely on optical transitions between different electronic states. In many cases, substructure such as hyperfine or spin-rotation structure leads to the need for multiple optical frequencies spaced by MHz to GHz. The task of creating multiple optical frequencies—optical spectral engineering—becomes challenging when the number of frequencies becomes large, a situation that one could encounter in complex molecules and atoms in large magnetic fields. In this work, we point out a novel mapping between computer-generated holography (CGH) and spectrum generation via phase modulation, which is an established technique. Using this mapping, we synthesize arbitrary optical spectra by modulating a monochromatic light field with a time-dependent phase generated through CGH techniques. Our method is compatible with non-linear optical processes such as sum frequency generation and second harmonic generation. Additional requirements that arise from the finite lifetimes of excited states can also be satisfied in our approach. As a proof-of-principle demonstration, we generate spectra suitable for cycling photons on the X–B transition in CaF, and verify via optical Bloch equation simulations that one can achieve high photon scattering rates, which are important for fluorescent detection and laser cooling. Our method could offer significant simplifications in future experiments that would otherwise be prohibitively complex.

Second-harmonic generation in nonlinear plasmonic lattices enhanced by quantum emitter gain medium.

We report on a theoretical study of second-harmonic generation (SHG) in plasmonic nanostructures interacting with two-level quantum emitters (QEs) under incoherent energy pump. We generalize the driven-dissipative Tavis-Cummings model by introducing the anharmonic surface plasmon-polariton (SPP) mode coupled to QEs and examine physical properties of corresponding SPP-QE polariton states. Our calculations of the SHG efficiency for strong QE-SPP coupling demonstrate orders of magnitude enhancement facilitated by the polariton gain. We further discuss time-domain numerical simulations of SHG in a square lattice comprising Ag nanopillars coupled to QEs utilizing a fully vectorial nonperturbative nonlinear hydrodynamic model for conduction electrons coupled to Maxwell-Bloch equations for QEs. The simulations support the idea of gain enhanced SHG and show orders of magnitude increase in the SHG efficiency as the QEs are tuned in resonance with the lattice plasmon mode and brought above the population inversion threshold by incoherent pumping. By varying pump frequency and tuning QEs to a localized plasmon mode, we demonstrate further enhancement of the SHG efficiency facilitated by strong local electric fields. The incident light polarization dependence of the SHG is examined and related to the symmetries of participating plasmon modes.

Rogue wave management for the generalized inhomogeneous nonlinear Schrödinger Maxwell–Bloch equation with external potential

We have revealed the interesting characteristic features of rogue waves for the various choices of potentials as they propagate through an inhomogeneous erbium-doped fiber system. We have demonstrated analytically that the intensity and the trajectories of the rogue waves can be influenced through the suitable choices of the modulating parameters. Additionally, the behavioral change in the rogue waves in the absence of potential has also been discussed. Obtained results confirm that the external potentials strongly influence the dynamics of rogue waves. We anticipate that our results can be useful to improve signal transmission through optical rogue waves.

The analysis of coherent phenomena for both linear and non-linear interactions in a four-level ladder (Ξ)-type configuration using density matrix formalism in dressed state representation

In this paper, we present a theoretical study of Autler-Townes (AT) splitting, electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) phenomena in a four-level ladder (Ξ)-type configuration formed by two coupling and one probe laser fields using density matrix formalism in dressed state representation (DSR). The DSR is followed by the analytical solution of the optical Bloch equations (OBEs) under density matrix formalism in bare state representation (BSR). The density matrix elements are presented in both linear and non-linear interaction regimes under DSR. The concept of transfer of population (TOP) mechanism has been shown by calculating the different transition probabilities in DSR. It has been found that the population in a particular dressed state is greatly enhanced by increasing the strength of coupling field and played a key role to manipulate the induced coherence between different energy levels. The present theoretical model of DSR offers an excellent interpretation of the formation of EIT, EIA and conversion from EIT to EIA in both linear and third order non-linear probe absorption profiles. Besides, the dispersion profiles in both linear and non-linear interaction regimes have also been studied and the conversion of normal to anomalous dispersion profile which drastically modifies the optical responses of the medium has been discussed. We have found that the amplitudes of the coherent signals are more sensitive to the third order non-linear interaction compared to first order linear interaction.

Localized gap modes of coherently trapped atoms in an optical lattice.

We theoretically investigate one-dimensional localized gap modes in a coherent atomic gas where an optical lattice is formed by a pair of counterpropagating far-detuned Stark laser fields. The atomic ensembles under study emerge as Λ-type three-level configuration accompanying the effect of electromagnetically induced transparency (EIT). Based on Maxwell-Bloch equations and the multiple scales method, we derive a nonlinear equation governing the spatial-temporal evolution of the probe-field envelope. We then uncover the formation and properties of optical localized gap modes of two kinds, such as the fundamental gap solitons and dipole gap modes. Furthermore, we confirm the (in)stability regions of both localized gap modes in the respective band-gap spectrum with systematic numerical simulations relying on linear-stability analysis and direct perturbed propagation. The predicted results may enrich the nonlinear horizon to the realm of coherent atomic gases and open up a new door for optical communication and information processing.

Hardware-neutral tools for the exploration of optical phenomena in near-resonant atomic systems

In the last three decades, a lot of research has been devoted to the optical response of an atomic media in near-to-resonant conditions and to how nonlinear optical properties are enhanced in these systems. However, as current research turns its attention towards multi-level and multidimensional systems interacting with several electromagnetic fields, the ever-increasing complexity of these problems makes it difficult to treat the semiclassical model of the Maxwell–Bloch equations analytically without any strongly-limiting approximations. Thus, numerical methods and particularly robust and fast computational tools, capable of addressing such class of modern and future problems in photonics, are mandatory. In this paper, we describe the development and implementation of a Maxwell–Bloch numerical solver that exploits the massive parallelism of the GPUs to tackle efficiently problems in multidimensional settings or featuring Doppler broadening effects. This constitutes a simulation tool that is capable of addressing a vast class of problems with considerable reduction of simulation time, featuring speedups up to 15 compared with the same codes running on a CPU.

Зависимость спектра генерации и синхронизации мод от ширины запрещенной фотонной зоны в гетеролазерах класса С с распределенной обратной связью волн в резонаторе Фабри-Перо

Based on the numerical solution to the nonlinear Maxwell-Bloch equations, the joint implementation of active (parametric) and passive (coherent) mode locking for a class C heterolaser with a low-Q combined Fabry-Perot cavity in the presence of an effective distributed feedback of counter-propagating waves is demonstrated. It is shown that the multiplicity of the spectral width of the photonic band gap of the cavity to the intermode spacing far from this zone contributes to the effective phasing of quasi-monochromatic quasi-equidistant modes, thus making it possible a phenomenon of parametric coherent self-mode-locking. A number of examples are used to demonstrate typical properties of a quasi-periodic sequence of soliton-like pulses generated in the found regime, which does not require external modulation of the pump or laser parameters, or the placement of a fast-saturating absorber in it.

Modelling of the Fluctuation and Coherent Dynamics in Active Metamaterial Devices

Active metamaterials, i.e., coupled systems consist of metamaterials and active or gain media. Modelling the interaction between unconventional electromagnetic response in metamolecules and quantum dynamics of active media in active metamaterials requires the development of a unified and efficient semiclassical model. Here, we propose a semiclassical model based on Langevin Maxwell-Bloch equations and simplified rate equations for a multi-level quantum system. Using the semiclassical model and the finite-difference time-domain (FDTD) algorithm, a methodology for the self-consistent descriptions of both the coherent lasing process and the spatio-temporal spontaneous noise dynamics in active metamaterials is developed. To validate the proposed method, we numerically study the luminescence of QDs hybridized with plasmonic and all-dielectric metasurfaces and compare with the experimental results. The proposed semiclassical method reveals a new approach for developing new types of nanodevices based on active metasurfaces.

Resonant forward-scattered field in the high-saturation regime: Elastic and inelastic contributions

We measure the resonant forward scattering of light by a highly saturated atomic medium through the flashes emitted immediately after an abrupt extinction of the probe beam. The experiment is done in a dilute regime where the phenomena are well captured using the independent scattering approximation. Comparing our measurements to a model based on Maxwell-Bloch equations, our experimental results are consistent with contributions from only the elastic component, whereas the attenuation of the coherent transmission power is linked to the elastic and inelastic scatterings. In the large saturation regime and at the vicinity of the atomic resonance, we derive an asymptotic expression relating the elastic scattering power to the forward-scattered power.

Intensity Noise of Semiconductor Lasers Induced by Inter-Mode Beating

Beating between the lasing main mode and the non-lasing side modes is one of the intensity noise sources of single longitudinal mode lasers. Its characteristics were investigated experimentally recent years. This paper presents a theoretical study of mode-beating noises in semiconductor lasers, and intends to explain the experimentally observed phenomena. The effects of coherent population oscillation (CPO) induced by mode beating on gain coefficient and on reflective index are analyzed based on Maxwell-Bloch equation and rate equations. The power ratio of the side mode over its conjugation mode generated by four wave mixing (FWM) is discussed; formulas of amplitude and frequency of mode beating intensity noise are given with relations to pump levels and mode orders. The obtained results are suitable for the edge emitting diode lasers and the external cavity lasers. The line shape of mode beating intensity noises is discussed based on CPO-induced phase noise, giving explanations of the experimentally observed phenomenon of multiple sub-peak spectra. The roles of intensity noise reduction by nonlinear optical absorber (NLA) inserted in the laser cavity are described. More factors related to the mode beating noise are discussed qualitatively, such as spectral hole burning of gain (SHB), the effect of external cavity, and higher order FWM.

Attosecond-pulse formation in the water-window range by an optically dressed hydrogen-like plasma-based C5+ x-ray laser

In this paper, we present the analytical theory of attosecond pulse formation via optical modulation of an active medium of the hydrogen-like C5+ plasma-based X-ray laser at 3.4 nm wavelength in the "water window" range, taking into account a variation of the population inversion caused by radiative decay of the upper lasing states. We derive an analytical solution for the X-ray field amplified by an X-ray laser with time-dependent population inversion, which is simultaneously irradiated by a strong optical laser field, and use it to find the optimal conditions for the attosecond pulse formation from a narrowband seeding X-ray field. We show that the shape of pulses can be improved at the cost of reduced pulse peak intensity (i) via external attenuation of the resonant spectral component of the amplified X-ray field or (ii) by using a resonantly absorbing medium (the active medium of the X-ray laser after the change of sign of the population inversion) for the pulse formation. The results of the analytical theory are in a good agreement with the numerical solutions of the Maxwell-Bloch equations which account for the nonlinearity, as well as the amplified spontaneous emission, of the active medium. Both analytically and numerically we show the possibility to produce a train of attosecond pulses with sub-200 as duration and the peak intensity exceeding 10^12 W/cm^2 at the carrier wavelength 3.4 nm in the "water window" range, which makes them attractive for the biological and medical applications.

Propagation of electric field generated by periodic pumping in a stable medium of two-level atoms of the Maxwell–Bloch model

We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann–Hilbert (RH) problem, using simultaneous spectral analysis of both the Lax equations. The proposed matrix RH problem solves the problem of the propagation of a sinusoidal signal in an unperturbed stable medium (attenuator). It is proved that this RH problem provides the causality principle for the region t < x, and for the region of the light cone, 0 < x < t allows us to find the asymptotics of the transmitted signal. First, we study the asymptotics of the RH problem for large times, and then, we obtain asymptotic formulas for the mixed problem solution of the Maxwell–Bloch equations when the attenuator is long enough. Three sectors are obtained in the light cone where the asymptotics have essentially different behaviors.

Study of multi-window electromagnetically induced transparency (EIT) and related dispersive signals in V-type systems in the Zeeman sublevels of hyperfine states of 87Rb-D2 line

We have theoretically studied the multi-window electromagnetically induced transparency (EIT) and the related dispersive signals in multi-level V-type systems which are formed by the co-propagating circularly polarized pump (σ −) and probe (σ +) laser fields in the Zeeman sublevels of hyperfine states of 87Rb-D2 line. The optical Bloch equations (OBEs) are derived and numerically solved under the steady-state condition to obtain the probe absorption and dispersion spectra. We present the study of a single-window EIT for the three-level, double-window EIT for the four-level and triple-window EIT for the five-level V-type systems under both Doppler-free and Doppler-broadened conditions. The dispersive signals corresponding to the multi EIT windows for each system are also observed separately. From the knowledge of the slopes of the probe dispersion signals, we have calculated group index (n g) of the medium considering three different V-type schemes. The effect of temperature on the EIT, dispersion and group index profiles is also studied in detail. It is found that the amplitudes of EIT and dispersion signals and also the positive and negative values of n g are decreased as the temperature of the medium increases. To understand the Doppler-broadening effect on the group index profiles, we have considered three different velocity groups of atoms in the calculation of group index profiles. Moreover, the effects of probe intensity variation on the Doppler-broadened EIT and dispersive signals are studied extensively, and the conversions of EIT to EIA and dispersive positive slope to negative slope are observed under strong probe regime. Also, we have investigated the variation of dispersive slope and group index with the variation of probe intensity and probe frequency, respectively, at the multi-frequency position of the multi-window EIT. Finally, the switching between subluminal (slow light) to superluminal (fast light) propagation of light with the variation of both probe intensity and probe frequency is discussed in detail.

Floquet states in dissipative open quantum systems

We theoretically investigate basic properties of nonequilibrium steady states of periodically-driven open quantum systems based on the full solution of the Maxwell–Bloch equation. In a resonant driving condition, we find that the transverse relaxation, also known as decoherence, significantly destructs the formation of Floquet states while the longitudinal relaxation does not directly affect it. Furthermore, by evaluating the quasienergy spectrum of the nonequilibrium steady states, we demonstrate that Rabi splitting can be observed as long as the decoherence time is as short as one third of the Rabi-cycle. Moreover, we find that Floquet states can be formed even under significant dissipation when the decoherence time is substantially shorter than the cycle of driving, once the driving field strength becomes strong enough. In an off-resonant condition, we demonstrate that the Floquet states can be realized even in weak field regimes because the system is not excited and the decoherence mechanism is not activated. Once the field strength becomes strong enough, the system can be excited by multi-photon absorption and the decoherence process becomes active. As a result, the Floquet states are significantly disturbed by the environment even in the off-resonant condition. Thus, we show here that the suppression of energy transfer from light to matter is a key condition for the realization of Floquet states in both on- and off-resonant conditions not only because it prevents material damage but also because it contributes to preserving coherence.

Line-shape analysis of double-quantum multidimensional coherent spectra

Double-quantum two-dimensional coherent spectroscopy (MDCS) is a powerful optical method that is used to study optical properties of atomic and complex molecular systems and semiconductor materials. Double-quantum 2D spectra and particularly the peak lineshapes on the spectra can also provide information about many-body interactions. We model 2D spectra by solving the optical Bloch equations and show the effects of correlation between coupled resonances, which also explains the discrepancies between the experimental results reported by multiple groups.