Equation Of Motion For Density Matrix Research Articles

Algorithm for solving a pump-probe model for an arbitrary number of energy levels.

We describe a generalized algorithm for evaluating the steady-state solution of the density matrix equation of motion, for the pump-probe scheme, when two fields oscillating at different frequencies couple the same set of atomic transitions involving an arbitrary number of energy levels, to an arbitrary order of the harmonics of the pump-probe frequency difference. We developed a numerical approach and a symbolic approach for this algorithm. We have verified that both approaches yield the same result for all cases studied, but require different computation time. The results are further validated by comparing them with the analytical solution of a two-level system to first order. We have also used both models to produce results up to the third order in the pump-probe frequency difference, for two-level systems, and up to first order for three- and four-level systems. In addition, we have used this model to determine accurately, the gain profile for a self-pumped Raman laser, for a system involving 16 Zeeman sublevels in the D1 manifold of ^{87}Rb atoms. We have also used this model to determine the behavior of a single-pumped superluminal laser. In many situations involving the applications of multiple laser fields to atoms with many energy levels, one often makes the approximation that each field couples only one transition, because of the difficulty encountered in accounting for the effect of another field coupling the same transition but with a large detuning. The use of the algorithm presented here would eliminate the need for making such approximations, thus improving the accuracy of numerical calculations for such schemes.

Quantum kinetic theory of nonlinear optical currents: Finite Fermi surface and Fermi sea contributions

The quantum kinetic framework provides a versatile method for investigating the dynamical optical and transport currents of crystalline solids. In this paper, starting from the density-matrix equations of motion, we present a general theoretical path to obtain the nonlinear optical response in an elegant and transparent manner. We devise an extensive kinetic theory that can be applied to materials with arbitrary band structures and captures intraband and interband coherence effects, finite Fermi surfaces, and disorder effects. We present a classification of the nonlinear optical currents arising from the interference of the interband and intraband components of the density matrix with distinct symmetry and quantum geometrical origin for each contribution. In this context, we report the following four primary findings: (i) The Fermi golden rule approach is insufficient to derive the correct expression for the injection current, a shortcoming that we remedy in our theory while associating the injection current with the intraband-interband contribution to the second-order density matrix. (ii) The interband-intraband contribution yields a resonant current that survives irrespective of any symmetry constraint in addition to the well-known anomalous nonlinear current (nonresonant), which requires time-reversal symmetry. (iii) Quite generally, the nonlinear current is significantly enhanced by contributions arising from the finite Fermi surface. (iv) The finite Fermi surface and Fermi sea additionally lead to sizable novel nonlinear effects via contributions we term double resonant and higher-order pole. We investigate such effects in sum frequency and difference frequency generation. As an illustration, we compute the nonlinear response of the topological antiferromagnet CuMnAs and thin film tilted Weyl semimetals as model systems dominated by interband coherence contributions. We find that the nonlinear response of CuMnAs is responsive to the direction of the finite magnetization field and the response of Weyl semimetal to the tilt. In addition, the choice of the polarization angle of the beam is crucial to have a nonlinear current in CuMnAs, while it is not the case for Weyl semimetals.

Transient evolution of optical magnetic resonance in rubidium vapor.

We observe the transient evolution of the laser pumped magnetic resonance in a paraffin-coated rubidium vapor cell and analyze the phenomena numerically by using the four-level density matrix. With the increased radio frequency (RF) sweep rate, the traditional Lorentz signal turns to an asymmetric shape at low sweep rate and starts to oscillate at a high sweep rate. The transient oscillation's features, including frequency and the decay time, are studied by suddenly turning on the RF field to the resonance Larmor precession frequency under the different RF field and light field parameters. The experimental result reveals the transient signals' strong dependence on the RF field power and light power. Especially, the transient oscillation frequency primarily depends on the RF field's power and whatever power the laser light is. When the laser power is lower, the transient oscillation frequency is proportional to the RF field's amplitude. These transient evolution signals are also confirmed with the numerical calculations based on the density-matrix equation of motion.

First-principles study of electron dynamics with explicit treatment of momentum dispersion on Si nanowires along different directions

ABSTRACTIn this research, ground-state electronic structure and optical properties along with photoinduced electron dynamics of Si nanowires oriented in various directions are reviewed. These nanowires are significant functional units of future nano-electronic devices. All observables are computed for a distribution of wave vectors at ambient temperature. Optical properties are computed under the approximation of momentum conservation. The total absorption is composed of partial contributions from fixed values of momentum. The on-the-fly non-adiabatic couplings obtained along the ab initio molecular dynamics nuclear trajectories are used as parameters for Redfield density matrix equation of motion. The main outcomes of this study are transition energies, light absorption spectra, electron and hole relaxation rates, and electron transport properties. The results of these calculations would contribute to the understanding of the mechanism of electron transfer process on the Si nanowires for optoelectronic applications.

First-Principles Study of Charge Carrier Dynamics with Explicit Treatment of Momentum Dispersion on Si Nanowires along <211> crystallographic Directions

ABSTRACTThe ground state structure, optical properties and charge carrier dynamics of silicon nanowire (SiNW) grown in <211> crystallographic direction is studied as a function of wavevector using density functional theory. This nanowire can be used as fundamental unit of nanoelectronic devices. The optical properties are computed under assumption of momentum conservation$\Delta \vec{k} = 0$. The on-the-fly non-adiabatic couplings for electronic degrees of freedom are obtained along the ab initio molecular dynamics nuclear trajectories, which are used as parameters for Redfield density matrix equation of motion. By investigating the photo-induced process on this nanowire, it is shown that high-energy photoexcitation relaxes to the band gap edge within 75 ps. The results of these calculations help to understand the mechanism of electron transfer process on the Si nanowire.

Millicharged fermion vacuum polarization in a cosmic magnetic field and generation of CMB elliptic polarization

The contribution of one loop milli-charged fermion vacuum polarization in cosmic magnetic field to the cosmic microwave background (CMB) polarization is considered. Exact and perturbative solution of the density matrix equations of motion in terms of the Stokes parameters are presented. For linearly polarized CMB at decoupling time, it is shown that propagation of CMB photons in cosmic magnetic field would generate elliptic polarization (circular and linear) of the CMB due to milli-charged fermion vacuum polarization. Analytic expressions for the degree of circular polarization and rotation angle of polarization plane of the CMB are presented. Depending on the ratio of milli-charged fermion relative charge to mass, $\epsilon/m_\epsilon$, and CMB observation frequency, it is shown that the acquired CMB degree of circular polarization could be of the order of magnitude $P_C(T_0)\sim 10^{-10}- 10^{-6}$ in the best scenario. The effect studied generates CMB polarization even in the case when the CMB is initially in thermal equilibrium. Limits on the magnetic field amplitude due to prior decoupling CMB polarization are presented.

Interband coherence response to electric fields in crystals: Berry-phase contributions and disorder effects

In solid state conductors, linear response to a steady electric field is normally dominated by Bloch state occupation number changes that are correlated with group velocity and lead to a steady state current. However, for a number of important physical observables, the most important response even in conductors can be electric-field induced coherence between Bloch states in different bands, such as that responsible for screening in dielectrics. Examples include the anomalous and spin-Hall effects, spin torques, minimum conductivities and the chiral anomaly. In this paper we present a general quantum kinetic theory of linear response to an electric field which can be applied to solids with arbitrarily complicated band structures and includes the inter-band coherence response and the Bloch-state repopulation responses on an equal footing. We aim to enable extensive transport theory applications using computational packages based on maximally localized Wannier functions. The formalism is based on density-matrix equations of motion, on a Born approximation treatment of disorder, and on an expansion in scattering rate to leading non-trivial order. The quasiparticle bands are treated in a completely general manner that allows for arbitrary forms of the spin-orbit interaction and for the broken time reversal symmetry of magnetic conductors. We demonstrate that the inter-band response in conductors consists primarily of two terms: an intrinsic contribution due to the entire Fermi sea that captures the Berry curvature contribution to wave-packet dynamics, and an anomalous contribution caused by scattering that is sensitive to the presence of the Fermi surface. We solve for some electric-field response properties of simple model systems that are known to be dominated by interband coherence contributions and discuss an extensive list of complicated problems that cannot be solved analytically.

Effects of Hydrostatic Pressure and Electric Field on the Electron-Related Optical Properties in GaAs Multiple Quantum Well.

The properties of the electronic structure of a finite-barrier semiconductor multiple quantum well are investigated taking into account the effects of the application of a static electric field and hydrostatic pressure. With the information of the allowed quasi-stationary energy states, the coefficients of linear and nonlinear optical absorption and of the relative refractive index change associated to transitions between allowed subbands are calculated with the use of a two-level scheme for the density matrix equation of motion and the rotating wave approximation. It is noticed that the hydrostatic pressure enhances the amplitude of the nonlinear contribution to the optical response of the multiple quantum well, whilst the linear one becomes reduced. Besides, the calculated coefficients are blueshifted due to the increasing of the applied electric field, and shows systematically dependence upon the hydrostatic pressure. The comparison of these results with those related with the consideration of a stationary spectrum of states in the heterostructure-obtained by placing infinite confining barriers at a conveniently far distance-shows essential differences in the pressure-induced effects in the sense of resonant frequency shifting as well as in the variation of the amplitudes of the optical responses.

Effects of applied electric fields on optical responses in elliptical quantum nanosystems

The optical absorption response of an elliptical quantum-dot-type nanostructure is investigated by numerically determining the electron states within the effective mass and parabolic approximations via the finite element method, taking into account the effect of an externally applied electric field. Validation of the numerical procedure is discussed through the comparison with analytical results of previous works for zero external field conditions. For finite field, the comparison is made with the outcome of degenerate perturbation theory. The calculated electron energies and wavefunctions are used to evaluate the linear and nonlinear light absorption and refractive index change coefficients in the system. Their corresponding expressions are those typically derived within a perturbative approach for solving the density matrix equation of motion. It is shown how the electric field and the change in the elliptical geometry affect the optical response of the considered structures. The extension of the treatment to consider the properties of elliptic nanoparticles under external field influences is briefly discussed.

Charge Transfer Mechanism in Titanium-Doped Microporous Silica for Photocatalytic Water-Splitting Applications

Solar energy conversion into chemical form is possible using artificial means. One example of a highly-efficient fuel is solar energy used to split water into oxygen and hydrogen. Efficient photocatalytic water-splitting remains an open challenge for researchers across the globe. Despite significant progress, several aspects of the reaction, including the charge transfer mechanism, are not fully clear. Density functional theory combined with density matrix equations of motion were used to identify and characterize the charge transfer mechanism involved in the dissociation of water. A simulated porous silica substrate, using periodic boundary conditions, with Ti4+ ions embedded on the inner pore wall was found to contain electron and hole trap states that could facilitate a chemical reaction. A trap state was located within the silica substrate that lengthened relaxation time, which may favor a chemical reaction. A chemical reaction would have to occur within the window of photoexcitation; therefore, the existence of a trapping state may encourage a chemical reaction. This provides evidence that the silica substrate plays an integral part in the electron/hole dynamics of the system, leading to the conclusion that both components (photoactive materials and support) of heterogeneous catalytic systems are important in optimization of catalytic efficiency.

Protocol for detection of nonsecular conversion through coherent nanooptical spectroscopy

The theoretical description of dynamics in open quantum systems becomes very demanding upon inclusion of non-Markovian effects. To simplify the computational implementation for density matrix equations of motion, the secular approximation is often applied. An experimental verification of its validity though remains difficult due to uncertainties in the system parameters and the absence of qualitatively distinct features. In this paper, we present the proposal for an experimental detection protocol sensitive to nonsecular processes neglected in the secular approximation. The protocol uses a combination of multidimensional coherent spectroscopy and nanoplasmonics. It allows for studies of nonsecular processes in various systems and provides a tool to experimentally verify the validity of the secular approximation. We apply this protocol to a system of CdSe/ZnS nanostructures and discuss the particular features originating from nonsecular processes on the resulting two-dimensional spectra.

Atom–photon entanglement beyond the multi-photon resonance condition

Atom---photon entanglement between the dressed atom and its spontaneous emission is studied in a near-degenerate three-level V-type atomic system in multi-photon resonance condition and beyond it. Taking into account the quantum interference due to the spontaneous emission, the density matrix equations of motion are numerically calculated in two-photon resonance condition and beyond it. The dynamical behavior of these two subsystems is investigated by using the von Neumann entropy. We apply the Floquet decomposition to the equations of motion to solve this time-dependent problem and identify the contribution of the different scattering processes to the atom---photon entanglement. In addition, the impact of the various nonlinear effects on the atom---photon entanglement is introduced in two-photon resonance condition. It is shown that the degree of entanglement (DEM) can be controlled via the intensity and the detuning of the coupling field as well as the quantum interference induced by spontaneous emission. We find that vacuum-induced interference has a major role in phase sensitivity of the DEM; however, beyond the two-photon resonance condition the DEM does not depend on the relative phase of the applied fields. Our results can be used for quantum information processing via entanglement.

Manipulating femtosecond spin-orbit torques with laser pulse sequences to control magnetic memory states and ringing

Femtosecond (fs) coherent control of collective order parameters is important for non--equilibrium phase dynamics in correlated materials. Here we propose a possible scheme for fs control of a ferromagnetic order parameter based on non--adiabatic optical manipulation of electron--hole ($e$--$h$) photoexcitations between spin--orbit--coupled bands that are exchange--split by magnetic interaction with local spins. We photoexcite fs carrier spin--pulses with controllable direction and time profile without using circularly--polarized light, via time--reversal symmetry--breaking by non--perturbative interplay between spin--orbit and magnetic exchange coupling of coherent photocarriers. We manipulate photoexcited {\em fs spin--orbit torques} to control complex switching pathways of the magnetization between multiple magnetic memory states. We calculate the photoinduced fs magnetic anisotropy in the time domain by using density matrix equations of motion rather than the quasi--equilibrium free energy. By comparing to pump--probe experiments, we identify a "sudden" magnetization canting induced by laser excitation, which displays magnetic hysteresis absent in static magneto--optical measurements and agrees with switchings measured by Hall magnetoresistivity. The fs magnetization canting switches direction with magnetic state and laser frequency, which distinguishes it from nonlinear optical and demagnetization longitudinal effects. By shaping two--color laser--pulse sequences analogous to multi--dimensional Nuclear Magnetic Resonance (NMR) spectroscopy, we show that sequences of clockwise or counter--clockwise fs spin--orbit torques can enhance or suppress magnetic ringing and switching rotation at any desired time. We propose protocols that can provide controlled access to four magnetic states via consequative 90$^{o}$ switchings.

Large Kerr coefficient induced by THz driven to intersubband of a GaN/AlGaN asymmetric quantum well nanostructure

In this article, a large Kerr nonlinearity induced by THz driven to intersubband of a GaN/AlGaN asymmetric quantum well nanostructure is investigated. To obtain the Kerr nonlinearity of the nanostructure, potential profiles, energy eigenvalues and wavefunctions, and overlap integrals are obtained by solving the Schrödinger and Poisson equations self-consistently. Knowing energy levels and related wavefunctions, one can investigate the nonlinear response of the structure by solving the density matrix equations of motion in the rotating wave approximation. The effect of the THz field, the control field, and the effect of the THz field detuning on the Kerr nonlinearity and the nonlinear absorption are observed. Our obtained results show that these parameters variations can cause a large Kerr nonlinearity and nonlinear absorption in GaN/AlGaN asymmetric quantum well nanostructure.

Electron dynamics in charged wet TiO2 anatase (001) surface functionalised by ruthenium ions

The surface of charged wet TiO2 anatase (001) functionalised by ruthenium ion at ambient temperatures is studied by computational modelling. Response of this model to photoexcitations at ambient temperatures is explored with the Redfield density matrix equation of motion on the basis of Kohn–Sham orbitals. The parameters of the Redfield equation are on-the-fly non-adiabatic couplings for electronic degrees of freedom obtained along the ab initio molecular dynamics nuclear trajectories. The main results in this study are the following: (1) optical properties of the doped models such as light absorption intensity and transition energies can be tuned by modifying total charge; (2) electron and hole relaxation rates depend on the initial excitation; and (3) in the doped model, excitations of lower energy provide quicker relaxation. Results of computational modelling would benefit understanding of the mechanism of electron transfer processes on the surface of ruthenium-doped TiO2.

Effects of magnetic field on the terahertz nonlinear optical properties in donor‐doped GaAs/AlGaAs quantum wells

AbstractEffects of the magnetic field on nonlinear optical properties at THz range in GaAs/AlGaAs quantum wells doped with donor atoms are investigated. Expressions for the third‐order nonlinear optical susceptibilities are obtained through the solution of the density matrix equations of motion within the rotating wave approximation. Donor binding energies are calculated variationally by means of an iterative shooting algorithm. Magnetic field has strong effect on the nonlinear susceptibility: it removes the degeneracy in energies of 2p± impurity states and increases the absolute value of the nonlinearity. It is also shown that a large and tunable optical nonlinear figure of merit is possible with the magnetic field applied in the growth direction. The nonlinear optical quantities are also calculated for donor distributions with different full width at half maximum values in the absence of magnetic field and the observed features at low energy part are attributed to the increasing homogeneity in the donor distribution.

Analytical model of transit time broadening for two-photon excitation in a three-level ladder and its experimental validation

We revisit transit time broadening for one of the typical experiment designs in molecular spectroscopy, that of a collimated supersonic beam of particles crossing a focused Gaussian laser beam. In particular, we consider a Doppler-free arrangement of a collimated supersonic beam of Na${}_{2}$ molecules crossing two counterpropagating laser beams that excite a two-photon transition in a three-level ladder scheme. We propose an analytical two-level model with a virtual intermediate level to show that the excitation line shape is described by a Voigt profile and provide the validity range of this model with respect to significant experimental parameters. The model also shows that line broadening due to the curvature of laser field wave fronts on the particle beam path is exactly compensated by increased transit time of particles farther away from the beam axis, such that the broadening is determined solely by the size of the laser beam waist. The analytical model is validated by comparing it with numerical simulations of density-matrix equations of motion using a split propagation technique and with experimental results.

Femtosecond all-optical modulation of collective spin in the (Ga,Mn)As ferromagnet

Using a many-body theory, we discuss some fundamental issues of femtomagnetism in magnetic electronic systems. We address the question of how spin may couple to transient optical coherence during time scales shorter than the photoexcitation duration and the characteristic times of interaction with the lattice. We also discuss the role of the competition between magnetic exchange and spin–orbit interactions in the nonthermal temporal evolution regime. Using density matrix equations of motion, we predict a femtosecond collective spin tilt leading to nonthermal magnetization modulation and all-optical ultrafast switching between different metastable magnetic states of (Ga,Mn)As ferromagnets. This spin dynamics is triggered by carrier coherences and by nonthermal populations photoexcited along the {111} equivalent directions of the Brillouin zone, which can be controlled by tuning the laser frequency/intensity and by using a small magnetic field. We present femtosecond magneto-optical spectroscopy experimental results that agree with our predictions. Our results indicate the possibility of reading the (Ga,Mn)As magnetic memory at THz speeds limited only by the pulse duration.

Quantum kinetic equation in phase-space textured multiband systems

Starting from the density-matrix equation of motion, we derive a semiclassical kinetic equation for a general two-band electronic Hamiltonian, systematically including quantum-mechanical corrections up to second order in space-time gradients. We find, in addition to band-projected corrections to the single-particle equation of motion due to phase-space Berry curvature, interband terms that we attribute to the nonorthorgonality of the projected Hilbert spaces. As examples, we apply our kinetic equation to electronic systems in the presence of spatially inhomogeneous and dynamical spin textures stemming from electromagnetic gauge potentials, and specifically to the electromagnetic response of massive 2D Dirac fermions and 3D Weyl fermions.

All-optical four-state magnetization reversal in (Ga,Mn)As ferromagnetic semiconductors

Using density matrix equations of motion and a tight-binding band calculation, we predict all-optical switching between four metastable magnetic states of (III,Mn)As ferromagnets. This switching is initiated non-thermally within 100 fs, during nonlinear coherent photoexcitation. For a single optical pulse, magnetization reversal is completed after ∼100 ps and controlled by the coherent femtosecond photoexcitation. Our predicted switching comes from magnetic nonlinearities triggered by a femtosecond magnetization tilt that is sensitive to un-adiabatic light-induced spin interactions.