Maxwell-Bloch Equations Research Articles

Broadband optical two-dimensional coherent spectroscopy of a rubidium atomic vapor

Optical two-dimensional coherent spectroscopy (2DCS) has become a powerful tool for studying energy level structure, dynamics, and coupling in many systems including atomic ensembles. Various types of two-dimensional (2D) spectra, including the so-called single-quantum, zero-quantum, and double-quantum 2D spectra, of both D lines (D$_1$ and D$_2$ transitions) of potassium (K) atoms have been reported previously. For rubidium (Rb), a major difference is that the D-lines are about 15 nm apart as opposed to only about 3 nm for K. Simultaneously exciting both D-lines of Rb atoms requires a broader laser bandwidth for the experiment. Here, we report a broadband optical 2DCS experiment in an Rb atomic vapor. A complete set of single-quantum, zero-quantum, and double-quantum 2D spectra including both D-lines of Rb atoms were obtained. The experimental spectra were reproduced by simulated 2D spectra based on the perturbation solutions to the optical Bloch equations. This work in Rb atoms complements previous 2DCS studies of K and Rb with a narrower bandwidth that covers two D-lines of K or only a single D-line of Rb. The broadband excitation enables the capability to perform double-quantum and multi-quantum 2DCS of both D-lines of Rb to study many-body interactions and correlations in comparison with K atoms.

Sensitivity Improvement and Determination of Rydberg Atom-Based Microwave Sensor

We present a theoretical and experimental investigation of the improvement and determination of the sensitivity of Rydberg atom-based microwave RF sensor. An optical Bloch equation has been set up based on the configuration that two-color cascading lasers exciting atom to highly Rydberg state and a microwave RF coupling this Rydberg state to its adjacent neighbor. The numerical simulation shows that the sensitivity of the atomic RF sensor is correlated with the amplitude strengths of the applied two lasers and the RF itself. It also depends on the frequency detuning of the coupling laser, which induces an asymmetrically optical splitting. The coupling laser frequency fixing at the shoulder of the stronger one is more favorable for a higher sensitivity. Accordingly, we perform an experimental demonstration for the optimization of all these parameters and the sensitivity is improved to 12.50(04) nVcm−1·Hz−1/2.

Exchange of orbital angular momentum of light via noise-induced coherence

The noise-induced coherence created via the quantum interference of incoherent radiation in atomic three-level systems of V and \ensuremath{\Lambda} types driven by a pair of weak laser pulses is shown to result in exchange of optical vortices. In the three- level V-type atom-light coupling the system is populated in its ground level, while in the \ensuremath{\Lambda} model the system is initially prepared in an electromagnetically induced transparency state. By solving the quantum optical Maxwell-Bloch equations and with quantum interference of incoherent radiation present, we show that the orbital angular momentum (OAM) of the vortex probe beam can be transferred to a generated signal field. While the exchange efficiency in the V configuration is higher, the losses are less in the \ensuremath{\Lambda} scheme when we consider such a noise-induced coherence. We further discuss the effects of phase mismatching and inhomogeneous broadening on the energy conversion between light beams carrying OAM.

Λ-enhanced gray molasses in a tetrahedral laser beam geometry.

We report the observation of sub-Doppler cooling of lithium using an irregular-tetrahedral laser beam arrangement, which is produced by a nanofabricated diffraction grating. We are able to capture 11(2)% of the lithium atoms from a grating magneto-optical trap into Λ-enhanced D1 gray molasses. The molasses cools the captured atoms to a radial temperature of 60(9) μK and an axial temperature of 23(3) μK. In contrast to results from conventional counterpropagating beam configurations, we do not observe cooling when our optical fields are detuned from Raman resonance. An optical Bloch equation simulation of the cooling dynamics agrees with our data. Our results show that grating magneto-optical traps can serve as a robust source of cold atoms for tweezer-array and atom-chip experiments, even when the atomic species is not amenable to sub-Doppler cooling in bright optical molasses.

Collective Resonance of D States in Rubidium Atoms Probed by Optical Two-Dimensional Coherent Spectroscopy

Collective resonance of interacting particles has important implications in many-body quantum systems and their applications. Strong interactions can lead to a blockade that prohibits the excitation of a collective resonance of two or more nearby atoms. However, a collective resonance can be excited with the presence of weak interaction and has been observed for atoms in the first excited state (P state). Here, we report the observation of collective resonance of rubidium atoms in a higher excited state (D state) in addition to the first excited state. The collective resonance is excited by a double-quantum four-pulse excitation sequence. The resulting double-quantum two-dimensional (2D) spectrum displays well-isolated peaks that can be attributed to collective resonances of atoms in P and D states. The experimental one-quantum and double-quantum 2D spectra can be reproduced by a simulation based on the perturbative solutions to the optical Bloch equations, confirming collective resonances as the origin of the measured spectra. The experimental technique provides a new approach for preparing and probing collective resonances of atoms in highly excited states.

Master equation for the quantum Rabi model in the adiabatic regime

A qubit--harmonic-oscillator open system coupled to thermal baths was considered and the master equation describing the evolution of the open system was deduced when the qubit transition frequency is $\ensuremath{\lesssim}0.1$ the oscillator frequency. The master equation is valid in all the qubit-oscillator state space and holds for all values of the qubit-oscillator coupling including the ultrastrong and deep strong coupling regimes. It only requires an oscillator frequency much larger than the relaxation rates. The qubit-oscillator coupling can enhance or decrease both the relaxation rates and the frequency shifts induced by the thermal baths. It was found that weak, sinusoidal qubit driving forces the qubit-oscillator open system to behave like a driven qubit whose evolution is governed by equations similar to those of the Bloch vector in the optical Bloch equations and whose transition frequency decreases with increasing qubit-oscillator coupling strength. Finally, it was shown how one can reach the adiabatic regime by using sinusoidal qubit driving with large driving frequency and the concepts of $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses were generalized to manipulate transitions between dressed states.

Dark-dark solitons, soliton molecules and elastic collisions in the mixed three-level coupled Maxwell–Bloch equations

We study the integrable three-level coupled Maxwell–Bloch equations with the mixed focusing-defocusing case. The n-fold Darboux transformation is constructed on basis of a linear 3×3 matrix eigenvalue problem. As an application, the n-dark-dark soliton solution in a simple determinant form is presented and the asymptotic behavior of the n-dark-dark soliton solution is rigorously given. In particular, under the resonant mechanism, the unusual dark-dark soliton molecules which represent the soliton bound states are analytically demonstrated. The double- and triple-dark-dark soliton molecules that are composed of two and three solitons propagating with identical velocities are graphically shown, respectively. The elastic collisions between a double- or triple-dark-dark soliton molecule and a usual dark-dark soliton, and the elastic collision of two double-dark-dark soliton molecules are discussed via the standard asymptotic analysis method.

Real-time multimode dynamics of terahertz quantum cascade lasers via intracavity self-detection: observation of self mode-locked population pulsations.

Mode-locking operation and multimode instabilities in Terahertz (THz) quantum cascade lasers (QCLs) have been intensively investigated during the last decade. These studies have unveiled a rich phenomenology, owing to the unique properties of these lasers, in particular their ultrafast gain medium. Thanks to this, in QCLs a modulation of the intracavity field intensity gives rise to a strong modulation of the population inversion, directly affecting the laser current. In this work we show that this property can be used to study in real-time the dynamics of multimode THz QCLs, using a self-detection technique combined with a 60GHz real-time oscilloscope. To demonstrate the potential of this technique we investigate a 4.2THz QCL operating in free-running, and observe a self-starting periodic modulation of the laser current, producing trains of regularly spaced, ∼100ps-long pulses. Depending on the drive current we find two distinct regimes of oscillation with dramatically different properties: a first regime at the fundamental cavity repetition rate, characterised by large amplitude and phase noise, with coherence times of a few tens of periods; a much more regular second-harmonic-comb regime, with typical coherence times of ∼105 oscillation periods. We interpret these measurements using a set of effective semiconductor Maxwell-Bloch equations that qualitatively reproduce the fundamental features of the laser dynamics, indicating that the observed carrier-density and optical pulses are in antiphase, and appear as a rather shallow modulation on top of a continuous wave background. Thanks to its simple implementation and versatility, the demonstrated broadband self-detection technique is a powerful tool for the study of ultrafast dynamics in THz QCLs.

Effects of neighboring transitions on the mechanisms of electromagnetically induced absorption and transparency in an open degenerate multilevel system

In this study, optical Bloch equations with and without neighboring hyperfine states near the degenerate two-level system (DTLS) in the challenging case of ^{85}Rb D2 transition, which involves the Doppler broadening effect, are solved. The calculated spectra agree well with the experimental results obtained based on the coupling-probe scheme with orthogonal linear polarizations of the coupling and probe fields. The mechanisms of electromagnetically induced absorption (electromagnetically induced transparency) for the open F_g=3 rightarrow F_e=2 and 3 transitions (open F_g=2 rightarrow F_e=2 and 3 transitions) are determined to be the effect of the strong closed F_g=3 rightarrow F_e=4 transition line (strong closed F_g=2 rightarrow F_e=1 transition line); this finding is based on a comparison between the calculated absorption profiles of the DTLS without neighboring states and those of all levels with neighboring states, depending on the coupling and probe power ratios. Furthermore, based on the aforementioned comparison, the crucial factors that enhance or reduce the coherence effects and lead to the transformation between electromagnetically induced absorption and electromagnetically induced transparency, are (1) the power ratios between the coupling and probe beams, (2) the openness of the excited state, and (3) effects of the neighboring states due to Doppler broadening in a real atomic system.

Dynamics in direct two-photon transition by frequency combs

Based on the proposed theoretical model of a three-level system, the optical Bloch equations including the direct two-photon transition (DTPT) process using the optical frequency comb (OFC) were derived and the population distribution of particles in the upper states varying with the velocity of the atoms was obtained. Comparing to the resonance two-photon transition process, that population was increased by a factor of 1.4 without the Doppler shift, which is consistent with our previous experimental results. Simultaneously, the relationships between momentum transfers, and atomic velocity and pulse number were analyzed. When applied to a multi-level system it was found that the population of particles in the excited states increased by a few percentages. The novel approach of DTPT using OFC improved the utilization of comb teeth and atoms, increased the momentum transfer path, reduced the reachable Doppler temperature limit, and encouraged us to use OFC to cool multiple elements simultaneously through the DTPT process. By analyzing the Doppler temperature of 133Cs and 87Rb in one dimension, it was found that this process can lower a temperature below 100 mK and generate dipolar molecules 133Cs87Rb via photoassociation, which provides us with a new tool to create dipolar molecules and to investigate their complex rovibrational spectra in ultra-cold chemistry.

Peculiaritie of Polarization Waves Behavior under Excitation of an Extended Resonant Medium by Overlapping Extremely Short Light Pulses

With the coherent interaction of a sequence of extremely short light pulses with a resonant medium (when the duration of the pulses and the delays between them are shorter than the relaxation times of the population difference T1 and polarization T2 of the medium), it is possible to create spatial electromagnetically induced gratings (EMIG) of polarization and population difference in the medium, as well as their ultrafast control. The dynamics of such gratings is of interest in connection with the possibility of ultrafast control over the properties of a medium at times of the order of the pulse duration and their influence on the stability of generation of extremely short pulses in a laser with a short linear cavity. In this paper, based on the numerical solution of the system of Maxwell-Bloch equations, an unusual and interesting possibility of the appearance of narrow and alternating sections in the medium is shown and studied in detail, in which harmonic polarization waves of the medium are formed, propagating in opposite directions relative to neighboring sections. This situation occurs when two extremely short attosecond pulses collide in a two-level resonant medium. Keywords: attosecond pulses, electromagnetically induced gratings, polarization waves.

Негармонические пространственные структуры разности населенностей, создаваемые униполярными прямоугольными импульсами в резонансной среде

In the case of coherent interaction with a medium of extremely short light pulses from a carrier of a harmonic form (when the pulse durations are shorter than the population relaxation times T_1 and polarization T_2 of the medium), electromagnetically induced gratings (EMIGs) of the population difference, which have a pronounced harmonic dependence on the coordinates, may appear in it. These structures can occur when pulses do not overlap at the same time or overlap in the medium. Recently, the possibility of obtaining unipolar electromagnetic pulses of the optical and adjacent ranges of non-harmonic shape, for example, rectangular and triangular, with a duration less than or comparable to the duration of the PCR in this range, has attracted interest. In this work, using the numerical solution of the system of Maxwell-Bloch equations, we study EMIG induced by rectangular attosecond pulses in a two-level resonant medium. The possibility of inducing an EMIG of a non-harmonic form in the form of light-induced channels, a kind of microresonators, with a size of the order of the wavelength of the resonant transition of the medium, whose parameters can be controlled, for example, by the amplitude of the incident pulses, is shown. It has been suggested that it is possible to create an EMIG of a predetermined non-harmonic form only in the general case of using unipolar pulses.

Поляритонный резонанс в автомодуляции асимметричного состояния сверхизлучающего лазера

Based on the numerical solution to the semiclassical Maxwell-Bloch equations in a one-dimensional model of a superradiant laser with a low-Q symmetric Fabry-Perot cavity, a comparative analysis of the spectrum of polariton modes of the linearized problem and the discrete spectrum of the of quasi-monochromatic generation with weak self-modulation under the condition of strong spontaneous mirror symmetry breaking of the counterpropagating electromagnetic waves, population inversion and polarization of an active medium is carried out. It is shown that the main frequencies of resonant self-modulation of a steady asymmetric superradiant phase state are close to the frequencies of polariton modes that are calculated in the linearized problem for the average level of the population inversion taken from the nonlinear problem and have small decay rates.

Особенности поведения волн поляризации при возбуждении протяженной резонансной среды перекрывающимися предельно короткими световыми импульсами

With the coherent interaction of a sequence of extremely short light pulses with a resonant medium (when the duration of the pulses and the delays between them are shorter than the relaxation times of the population difference T1 and polarization T2 of the medium), it is possible to create spatial electromagnetically induced gratings (EMIG) of polarization and population difference in the medium, as well as their ultrafast control. The dynamics of such gratings is of interest in connection with the possibility of ultrafast control over the properties of a medium at times of the order of the pulse duration and their influence on the stability of generation of extremely short pulses in a laser with a short linear cavity. In this paper, based on the numerical solution of the system of Maxwell-Bloch equations, an unusual and interesting possibility of the appearance of narrow and alternating sections in the medium is shown and studied in detail, in which harmonic polarization waves of the medium are formed, propagating in opposite directions relative to neighboring sections. This situation occurs when two extremely short attosecond pulses collide in a two-level resonant medium.

Nonharmonic Spatial Population Difference Structures Created by Unipolar Rectangular Pulses in a Resonant Medium

In the case of coherent interaction with a medium of extremely short light pulses (ESPs) having a carrier frequency and harmonic shape (when the pulse durations are shorter than the population relaxation times T1 and polarization relaxation time T2 of the medium), electromagnetically induced gratings (EMIGs) of the population difference, which have a pronounced harmonic dependence on the coordinates, may appear in it. These structures can occur when pulses do not overlap or overlap in the medium. Recently, the possibility of obtaining unipolar electromagnetic pulses in the optical and adjacent ranges of non-harmonic shape, for example, rectangular and triangular, with a duration less or comparable to the duration of the extremely-short pulse in this range, has attracted interest. In this work, using the numerical solution of the system of Maxwell-Bloch equations, we study EMIG formation by rectangular attosecond pulses in a two-level resonant medium. The possibility of inducing an EMIG of a non-harmonic shape in the form of light-induced channels microresonators, (microcavities) with a size of the order of the wavelength of the resonant transition of the medium, whose parameters can be controlled, for example, by the amplitude of the incident pulses, is shown. It has been suggested that it is possible to create an EMIG of a predetermined non-harmonic shape only in the general case of using unipolar pulses. Keywords: attosecond pulses, unipolar pulses, rectangular pulses, electromagnetically induced gratings, polarization waves, light-induced microresonators.

Spatio-temporal properties of pulse propagation in a graphene quantum system

In this paper we have theoretically studied the spatial-temporal evolution of electromagnetic light propagation through a four-level graphene quantum system by using density matrix method and perturbation theory. The four-level graphene quantum medium interacted by an elliptical polarized coupling and a weak probe lights, respectively. We present the analytical solution for solving the Maxwell–Bloch equations for graphene and electromagnetic field in space and time domains. Then, we have analyzed the dynamic control of pulse propagation and optical dual switching in such a laser-driven quantum system. Our theoretical findings show that by adjusting the optical parameters such as elliptical angle i.e. phase difference between right-and-left circularly polarized, one can easily control the absorption spectrum and pulse propagation of the probe light in graphene medium. Our results may have potential applications in designing the new quantum devices for usage in quantum information processing.

Transient electromagnetically induced transparency spectroscopy of 87Rb atoms in buffer gas

We study the transient response dynamics of 87Rb atomic vapor buffered in 8 torr Ne gas through an electromagnetically induced transparency configured in Λ-scheme. Experimentally, the temporal transmission spectra versus probe detuning by switching on and off the coupling one show complex structures. The transmitted probe light intensity drops to a minimum value when the coupling light turns off, showing a strong absorption. Even at the moment of turning on the coupling light at a subsequent delayed time, the atomic medium shows a fast transient response. To account for the transient switching feature, in the time-dependent optical Bloch equation, we must take the transverse relaxation dephasing process of atomic vapor into account, as well as the fluorescence relaxation along with the optical absorption. This work supplies a technique to quantify the transverse relaxation time scale and to sensitively monitor its variation along the environment by observing the transient dynamics of coherent medium, which is helpful in characterizing the coherent feature of the atomic medium.

Variational analysis of driven-dissipative bosonic fields

We present a method to perform a variational analysis of the quantum master equation for driven-disspative bosonic fields with arbitrary large occupation numbers. Our approach combines the P representation of the density matrix and the variational principle for open quantum system. We benchmark the method by comparing it to wave-function Monte-Carlo simulations and the solution of the Maxwell-Bloch equation for the Jaynes-Cummings model. Furthermore, we study a model describing Rydberg polaritons in a cavity field and introduce an additional set of variational paramaters to describe correlations between different modes.

Rogue waves, semirational rogue waves and W-shaped solitons in the three-level coupled Maxwell–Bloch equations

In this paper, the coupled Maxwell–Bloch equations which describe the propagation of two optical pulses in an optical medium with coherent three-level atoms are studied by Darboux transformation. Both the general nth-order rogue wave solution involving two different choices of multiple roots for the spectral characteristic equation and the multiparametric nth-order semirational solution are obtained in terms of Schur polynomials. The explicit rogue wave solutions and semirational solutions from first to second order are provided. In contrast to the bright, dark and four-petaled structures, some unusual patterns such as triple-hole, twisted-pair, composite four-petaled and composite dark rogue waves are put forward. Moreover, the interactions between dark–bright solitons and dark rogue waves and interactions between breathers and dark rogue waves are shown. Further, the higher-order nonlinear superposition modes which feature triple and quadruple temporal–spatial distributions are presented. Finally, the state transitions between rogue waves and W-shaped solitons are found where the modulation instability growth rate tends to zero under the low perturbation frequency. Particularly, the dark and double-peak W-shaped solitons are examined.

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.