Steady-state Master Equation Research Articles

Coherent and incoherent pumping of a three-level laser

This article investigates the quantum and semi-classical aspects of a three-level atom-cavity system within the context of cavity quantum electrodynamics. The study examines the behavior of the system through a quantum perspective and a semi-classical approximation. The steady-state master equation is solved in the atom-cavity basis, resulting in a closed set of equations describing the atom’s level occupancies and the cavity’s photon number. The accuracy of the semi-classical approximation is assessed by comparing it with quantum simulations. The research analyzes the system’s behavior near the laser threshold, highlighting the interplay between semi-classical and quantum behaviors. Additionally, the conversion of the three-level atom to a two-level atom is explored under specific conditions, enabling an investigation into the weak driving limit. Quantum simulation results are used to validate the proposed approximations. This work contributes to the understanding of atom-cavity interactions and provides insights into the transition from semi-classical to quantum behavior in such systems.

Phonon Blockade in Parametrically Pumped Acoustic Cavity at Finite Temperature

In this study, we investigated the phonon blockade effect in a parametrically driven and dissipative acoustic cavity at finite temperature. With the approximated analytical results based on the steady-state density-matrix master equation, we found that a quantum-interference-induced phonon blockade exists at finite temperature. We found a crossover between the quantum and thermal regimes on the curve of the second-order correlation function of the acoustic mode as the temperature increases. This phenomenon implies an asymmetry about the quantum and classic regimes. We also numerically simulated the single-phonon emission using the Monte Carlo wave function method. The results showed that a wide and deep dip around the zero time delay exists on the curve of the time-delayed second-order correlation function, which implies the possibility of observing a strong phonon blockade with pulse driving. Our study outlines a potential candidate for a efficient single-phonon source and applications in quantum information and phononic quantum networks.

Comprehensive Model for the Thermoelectric Properties of Two-Dimensional Carbon Nanotube Networks

Networks of semiconducting single-walled carbon nanotubes (SWCNTs) are interesting thermoelectric materials due to the interplay between CNT and network properties. Here, we present a unified model to explain charge and energy transport in SWCNT networks. We use the steady-state master equation for the random resistor network containing both intra- and intertube resistances, as defined through their one-dimensional density of states that are modulated by static Gaussian disorder. The tube-resistance dependence on the carrier density and disorder is described through the Landauer formalism. Electrical and thermoelectric properties of the network are obtained by solving Kirchhoff's laws through a modified nodal analysis, where we use the Boltzmann-transport formalism to obtain the conductivity, Seebeck coefficient, and electronic contribution to the thermal conductivity. The model provides a consistent description of a wide range of previously published experimental data for temperature and charge-carrier-density-dependent conductivities and Seebeck coefficients, with energetic disorder being the main factor to explain the experimentally observed mobility upswing with carrier concentration. Moreover, we show that, for lower disorder energies, the Lorentz factor obtained from the simulation is in accordance with the Wiedemann-Franz law for degenerate-band semiconductors. At higher disorder, deviations from simple band behavior are found. Suppressed disorder energy and lattice thermal conductivity can be the key to higher thermoelectric figures of merit, zT, in SWCNT networks, possibly approaching or even exceeding zT = 1. The general understanding of transport phenomena will help the selection of chirality, composition, and charge-carrier density of SWCNT networks to improve their efficiency of thermoelectric energy conversion.

Open quantum systems beyond Fermi's golden rule: Diagrammatic expansion of the steady-state time-convolutionless master equations

Steady-state observables, such as occupation numbers and currents, are crucial experimental signatures in open quantum systems. The time-convolutionless (TCL) master equation, which is both exact and time-local, is an ideal candidate for the perturbative computation of such observables. We develop a diagrammatic approach to evaluate the steady-state TCL generator based on operators rather than superoperators. We obtain the steady-state occupation numbers, extend our formulation to the calculation of currents, and provide a simple physical interpretation of the diagrams. We further benchmark our method on a single non-interacting level coupled to Fermi reservoirs, where we recover the exact expansion to next-to-leading order. The low number of diagrams appearing in our formulation makes the extension to higher orders accessible. Combined, these properties make the steady-state time-convolutionless master equation an effective tool for the calculation of steady-state properties in open quantum systems.

Unimolecular Decay of the Dimethyl-Substituted Criegee Intermediate in Alkene Ozonolysis: Decay Time Scales and the Importance of Tunneling.

We used the steady-state master equation to model unimolecular decay of the Criegee intermediate formed from ozonolysis of 2,3-dimethyl-2-butene (tetramethylethylene, TME). Our results show the relative importance and time scales for both the prompt and thermal unimolecular decay of the dimethyl-substituted Criegee intermediate, (CH3)2COO. Calculated reactive fluxes show the importance of quantum mechanical tunneling for both prompt and thermal decay to OH radical products. We constrained the initial energy distribution of chemically activated (CH3)2COO formed in TME ozonolysis by combining microcanonical rates k(E) measured experimentally under collision-free conditions and modeled using semiclassical transition-state theory (SCTST) with pressure-dependent yields of stabilized Criegee intermediates measured with scavengers in flow-tube experiments. Thermal decay rates under atmospheric conditions k(298 K, 1 atm) increase by more than 1 order of magnitude when tunneling is included. Accounting for tunneling has important consequences for interpreting pressure dependent yields of stabilized Criegee intermediates, particularly with regard to the fraction of Criegee intermediates formed in the zero-pressure limit.

The improvement of lasing without inversion in the presence of multi-photon transition in a three-level atom confined in an optical cavity in the steady-state regime

A three-level atom in \( \Lambda\) configuration trapped in a single-mode Fabry-Perot optical cavity is investigated. It is assumed that the transition between levels 2 and 3 in the atom is accomplished through the number of q photons. To solve the steady-state master equation for the atom-cavity system, the matrix continued-fraction method for applied physical quantities is used. The results show that by raising the number of transitions, the curves of the population inversion finally overlap with one another and the output mean photon number from the optical cavity rises and moreover the laser effect appears in the system. On the other hand, the outcomes reveal that by increasing the number of transitions despite the negativity of the population inversion, the number of output photons from the cavity grows and lasing interval becomes larger. Eventually the steps of the transformation from the three-level atom to two-level one in the case of multi-photon transition under several specific conditions have been studied. The obtained results of the simulations confirm the accuracy of the used approximations in the two-level pattern appropriately. In summary, the lasing process without population inversion along with the multi-photon transition in the three-level atom is discussed.

Exact Length Distribution of Filamentous Structures Assembled from a Finite Pool of Subunits.

Self-assembling filamentous structures made of protein subunits are ubiquitous in cell biology. These structures are often highly dynamic, with subunits in a continuous state of flux, binding to and falling off of filaments. In spite of this constant turnover of their molecular parts, many cellular structures seem to maintain a well-defined size over time, which is often required for their proper functioning. One widely discussed mechanism of size regulation involves the cell maintaining a finite pool of protein subunits available for assembly. This finite pool mechanism can control the length of a single filament by having assembly proceed until the pool of free subunits is depleted to the point when assembly and disassembly are balanced. Still, this leaves open the question of whether the same mechanism can provide size control for multiple filamentous structures that are assembled from a common pool of protein subunits, as is often the case in cells. We address this question by solving the steady-state master equation governing the stochastic assembly and disassembly of multifilament structures made from a shared finite pool of subunits. We find that, while the total number of subunits within a multifilament structure is well-defined, individual filaments within the structure have a wide, power-law distribution of lengths. We also compute the phase diagram for two multifilament structures competing for the same pool of subunits and identify conditions for coexistence when both have a well-defined size. These predictions can be tested in cell experiments in which the size of the subunit pool or the number of filament nucleators is tuned.

Computational Study of the Rate Coefficients for the Reactions of NO2 with CH3NHNH, CH3NNH2, and CH2NHNH2

The reactions of NO2 with cis-/trans-CH3NHNH, CH3NNH2 and CH2NHNH2 have been studied theoretically by quantum chemical calculations and steady-state unimolecular master equation analysis based on RRKM theory. The barrier heights for the roaming transition states between nitro (RNO2) and nitrite (RONO) isomerization reactions and those for the concerted HONO and HNO2 elimination reactions from RNO2 and RONO, affect the pressure dependences of the product-specific rate coefficients. At ambient temperature and pressure, the dominant product of the reactions of NO2 with cis-/trans-CH3NHNH and CH2NHNH2 would be expected to be HONO with trans-CH3NNH and CH2NNH2, respectively, whereas it is CH3N(NH2)NO2 for CH3NNH2 + NO2. The product-specific rate coefficients for the titled and related reactions on the same potential energy surfaces were proposed for kinetics modeling.

The Influence of Morphology on the Charge Transport in Two-Phase Disordered Organic Systems

ABSTRACTIn this work we use a three-dimensional Pauli master equation to investigate the charge carrier mobility of a two-phase system, which can mimic donor-acceptor and amorphous-crystalline bulk heterojunctions. Our approach can be separated into two parts: the morphology generation and the charge transport modeling in the generated blend. The morphology part is based on a Monte Carlo simulation of binary mixtures (donor/acceptor). The second part is carried out by numerically solving the steady-state Pauli master equation. By taking the energetic disorder of each phase, their energy offset and domain morphology into consideration, we show that the carrier mobility can have a significant different behavior when compared to a one-phase system. When the energy offset is non-zero, we show that the mobility electric field dependence switches from negative to positive at a threshold field proportional to the energy offset. Additionally, the influence of morphology, through the domain size and the interfacial roughness parameters, on the transport was also investigated.

Observation of Autler–Townes effect in a dispersively dressed Jaynes–Cummings system

We report on the spectrum of a superconducting transmon device coupled to a planar superconducting resonator in the strong dispersive limit where discrete peaks, each corresponding to a different number of photons, are resolved. A thermal population of 5.474 GHz photons at an effective resonator temperature of T = 120 mK results in a weak n = 1 photon peak along with the n = 0 photon peak in the qubit spectrum in the absence of a coherent drive on the resonator. Two-tone spectroscopy using independent coupler and probe tones reveals an Autler–Townes splitting in the thermal n = 1 photon peak. The observed effect is explained accurately using the four lowest levels of the dispersively dressed qubit–resonator system and compared to results from numerical simulations of the steady-state master equation for the coupled system.

Steady-state fluctuations of a genetic feedback loop: An exact solution

Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution [J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Phys. Rev. E 72, 051907 (2005), and subsequent studies]. We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.

Reduced Master Equation for Modeling of Ferromagnetic Single-Electron Transistor

In this paper, the reduced master equation which is a fast simulation method of spin dependent transport in ferromagnetic single electron transistors is presented, for first time. This simulation method follows steady state master equation in which all charge states of the system are considered, whereas charge states are decreased in reduced master equation. This method is based on two degrees of electron freedom which are charge and spin. This is applied in the condition that orthodox tunneling theory is applicable to calculate the tunneling rate of electrons through barriers. The comparison between the I-V characteristics of a ferromagnetic single-electron transistor by following the reduced and full master equation methods for different main parameters of these transistors show that the results are exactly the same at low bias voltages. Consequently, the reduced master equation method is not only more simplified and improves the speed of numerical simulation, but also the modeling results are as accurate as the results of the full maser equation method at low bias conditions.

Kinetics and Mechanisms of the Allyl + Allyl and Allyl + Propargyl Recombination Reactions

The kinetics and mechanisms of the self-reaction of allyl radicals and the cross-reaction between allyl and propargyl radicals were studied both experimentally and theoretically. The experiments were carried out over the temperature range 295-800 K and the pressure range 20-200 Torr (maintained by He or N(2)). The allyl and propargyl radicals were generated by the pulsed laser photolysis of respective precursors, 1,5-hexadiene and propargyl chloride, and were probed by using a cavity ring-down spectroscopy technique. The temperature-dependent absorption cross sections of the radicals were measured relative to that of the HCO radical. The rate constants have been determined to be k(C(3)H(5) + C(3)H(5)) = 1.40 × 10(-8)T(-0.933) exp(-225/T) cm(3) molecule(-1) s(-1) (Δ log(10)k = ± 0.088) and k(C(3)H(5) + C(3)H(3)) = 1.71 × 10(-7)T(-1.182) exp(-255/T) cm(3) molecule(-1) s(-1) (Δ log(10)k = ± 0.069) with 2σ uncertainty limits. The potential energy surfaces for both reactions were calculated with the CBS-QB3 and CASPT2 quantum chemical methods, and the product channels have been investigated by the steady-state master equation analyses based on the Rice-Ramsperger-Kassel-Marcus theory. The results indicated that the reaction between allyl and propargyl radicals produces five-membered ring compounds in combustion conditions, while the formations of the cyclic species are unlikely in the self-reaction of allyl radicals. The temperature- and pressure-dependent rate constant expressions for the important reaction pathways are presented for kinetic modeling.

Reduced Master Equation as a Novel and Fast Method for Modeling of Spin Transport in Ferromagnetic Single Electron Transistors

In this paper, we formulate and investigate the reduced master equation as a novel and fast simulation method of spin dependent transport in ferromagnetic single electron transistors (FMSET) which so far this method for modeling of FMSET has not been mentioned in no paper. This simulation method follows the steady state master equation in which all charge states of the system are considered, whereas the charge states decrease in reduced master equation. This method is based on two degrees of electron freedom which are charge and spin. This is applied in the condition that the orthodox tunneling theory is applicable to calculate the tunneling rate of electrons through barriers. A comparison between current-bias voltage and current-gate voltage characteristics of different ferromagnetic single electron transistors following the reduced and full master equation methods shows that the results at low bias voltages are the same. Consequently, the reduced master equation method for simulation of current-bias/gate voltage of FMSETs is more simplified and improves the speed of numerical simulation, and also the modeling results are as accurate as the results of the full master equation method at low bias conditions in current scale.

Theoretical study the trap and carrier-density dependent electron mobility in pentacene ab-plane by the steady master equation

The steady state master equation coupled Marcus–Hush electron transfer theory is developed to calculate the two-dimensional electron and hole mobilities in pentacene ab-plane. In this paper, we numerically investigated the influences of the random distributed traps and charge-carrier densities on the 2D mobilities. The study showed that shallow traps (<7.5 kT) give few effects on the mobilities for various carrier densities, while deep traps (>10 kT) remarkably decrease mobilities, in particularly, when the carriers density cannot quenched them. However, when the traps are fully quenched due to large carrier densities, the mobility will be nearly back to their intrinsic mobility values.

The Energetics of a Flaring Solar Active Region and Observed Flare Statistics

A stochastic model for the energy of a flaring solar active region is presented, generalizing and extending the approach of Wheatland and Glukhov. The probability distribution for the free energy of an active region is described by the solution to a master equation involving deterministic energy input and random jump transitions downward in energy (solar flares). It is shown how two observable distributions, the flare frequency-energy distribution and the flare waiting-time distribution, may be derived from the steady state solution to the master equation, for given choices for the energy input and for the rates of flare transitions. An efficient method of numerical solution of the steady state master equation is presented. Solutions appropriate for flaring, involving a constant rate of energy input and power-law distributed jump transition rates, are numerically investigated. The flarelike solutions exhibit power-law flare frequency-energy distributions below a high-energy rollover, set by the largest energy the active region is likely to have. The solutions also exhibit approximately exponential (i.e., Poisson) waiting-time distributions, despite the rate of flaring depending on the free energy of the system.

Steady-state master equation methods

This paper investigates the application of the steady-state master equation (SSME) to complex unimolecular reactions. It is shown how the steady-state approximation and the Boltzmann reservoir approximation can be applied, both together or separately, to improve the efficiency and the numerical robustness of simple master equation calculations. The case of a two-well isomerization is considered in detail, and two versions of the SSME are derived for it, one of which shows very clearly and analytically the relationship between the rate coefficients and the flux coefficients. It is shown for a reversible second order recombination reaction that the second order phenomenological equation can be regained in either version of the SSME. It is also shown how, under steady state conditions, the SSME may be used to determine rate coefficients for each reaction in a multiple well scheme.

Compact analytical model for room-temperature-operating silicon single-electron transistors with discrete quantum energy levels

A compact and analytical model for silicon single-electron transistors (SETs) considering the discrete quantum energy levels and the parabolic tunneling barriers is proposed. The model is based on a steady-state master equation that considers only the three most probable states derived from ground level and the first excited level for each number of electrons in the dot to reduce the complexity while accounting for the quantum-level spacing and multiple peaks in Coulomb oscillation. Negative differential conductance (NDC) characteristics and aperiodic Coulomb oscillations due to nonuniform quantum-level spacings can be reproduced in this model. The model was compared with measurements, and good agreement was obtained. Simulations of some basic circuits that utilize NDC are successfully carried out by applying our model to the HSPICE circuit simulation. Our model can provide suitable environments for designing CMOS-combined room-temperature-operating highly functional SET circuits.

A compact analytical model for asymmetric single-electron tunneling transistors

Analytical model for asymmetric single-electron tunneling transistors (SETTs), in which resistance and capacitance parameters of source/drain junctions are not equal, has been developed. The model is based on the steady-state master equation, takes only the two most-probable charging states into account, and is therefore very simple. Even so, it can accurately reproduce the peculiar behaviors of an asymmetric SETT, such as the skew in the drain current-gate voltage characteristics and the Coulomb staircase in the drain current-drain voltage characteristic. Analytical expressions for the charge in the Coulomb island and the capacitance components of the SETT are also derived according to the same scheme, and it is demonstrated that the model can precisely describe the various aspects of the SETT behavior.

Analytical Single-Electron Transistor (SET) Model for Design and Analysis of Realistic SET Circuits

In this work, we propose a compact, physically based, analytical single-electron transistor (SET) model suitable for the design and analysis of realistic SET circuits. The model is derived on the basis of the “orthodox” theory of correlated single-electron tunneling and the steady-state master equation method. The SET inverter characteristics are successfully calculated using the model implemented in the simulation program with integrated circuit emphasis (SPICE). The hybrid circuit of SETs with metal-oxide-semiconductor field-effect transistors (MOSFETs) is also successfully simulated. By utilizing the model, it is clarified that the drain-voltage-induced shift of the gate voltage dependence of SET current reaches one-half of the drain voltage in the case of a completely symmetric SET.