Quantum Master Equation Approach Research Articles

Exciton size and quantum transport in nanoplatelets.

Two-dimensional nanoplatelets (NPLs) are an exciting class of materials with promising optical and energy transport properties. The possibility of efficient energy transport between nanoplatelets raises questions regarding the nature of energy transfer in these thin, laterally extended systems. A challenge in understanding exciton transport is the uncertainty regarding the size of the exciton. Depending on the material and defects in the nanoplatelet, an exciton could plausibly extend over an entire plate or localize to a small region. The variation in possible exciton sizes raises the question how exciton size impacts the efficiency of transport between nanoplatelet structures. Here, we explore this issue using a quantum master equation approach. This method goes beyond the assumptions of Förster theory to allow for quantum mechanical effects that could increase energy transfer efficiency. The model is extremely flexible in describing different systems, allowing us to test the effect of varying the spatial extent of the exciton. We first discuss qualitative aspects of the relationship between exciton size and transport and then conduct simulations of exciton transport between NPLs for a range of exciton sizes and environmental conditions. Our results reveal that exciton size has a strong effect on energy transfer efficiency and suggest that manipulation of exciton size may be useful in designing NPLs for energy transport.

Influence of Coherent Tunneling and Incoherent Hopping on the Charge Transfer Mechanism in Linear Donor-Bridge-Acceptor Systems.

The mechanism of charge transfer has been observed to change from tunneling to hopping with increasing numbers of DNA base pairs in polynucleotides and with the length of molecular wires. The aim of this paper is to investigate this transition by examining the population dynamics using a tight-binding Hamiltonian with model parameters to describe a linear donor-bridge-acceptor (D-B-A) system. The model includes a primary vibration and an electron-vibration coupling at each site. A further coupling of the primary vibration with a secondary phonon bath allows the system to dissipate energy to the environment and reach a steady state. We apply the quantum master equation (QME) approach, based on second-order perturbation theory in a quantum dissipative system, to examine the dynamical processes involved in charge-transfer and follow the population transfer rate at the acceptor, ka, to shed light on the transition from tunneling to hopping. With a small tunneling parameter, V, the on-site population tends to localize and form polarons, and the hopping mechanism dominates the transfer process. With increasing V, the population tends to be delocalized and the tunneling mechanism dominates. The competition between incoherent hopping and coherent tunneling governs the mechanism of charge transfer. By varying V and the total number of sites, we also examine the onset of the transition from tunneling to hopping with increasing length.

Interaction effect on adiabatic pump of charge and spin in quantum dot

We investigate the pumped charge and spin at zero-bias by adiabatic modulation of two control parameters using the full counting statistics with quantum master equation approach. First we study higher order effects of the pumping frequency in general Markov systems and show the equivalence between our approach and the real-time diagrammatic approach. An adiabatic modulation of the control parameters induces the Berry-Sinitsyn-Nemenman (BSN) phase. We show that the origin of the BSN phase is a non-adiabatic effect. The adiabatically pumped charge (spin) is given by a summation of (i) a time integral of the instantaneous steady charge (spin) current and (ii) a geometric surface integral of the BSN curvature, which results from the BSN phase. In quantum dots (QDs) weakly coupled to two leads, we show that (i) is usually dominant if the thermodynamic parameters are modulated although it is zero if the thermodynamic parameters are fixed to zero-bias. To observe the spin effects, we consider collinear magnetic fields, which relate to spins through the Zeeman effect, with different amplitudes applying to the QDs and the leads. For interacting one level QD, we calculate analytically the pumped charge and spin by modulating the magnetic fields and the coupling strengths to the leads in the weak and strong interacting limits. We show that the difference between these two limits appears through the averages of the numbers of the electron with up and down spin in the QD. For the quantum pump by the modulation of the magnetic fields of the QD and one lead, the energy-dependences of linewidth functions, which are usually neglected, are essential.

Markov Chain-Like Quantum Biological Modeling of Mutations, Aging, and Evolution.

Recent evidence suggests that quantum mechanics is relevant in photosynthesis, magnetoreception, enzymatic catalytic reactions, olfactory reception, photoreception, genetics, electron-transfer in proteins, and evolution; to mention few. In our recent paper published in Life, we have derived the operator-sum representation of a biological channel based on codon basekets, and determined the quantum channel model suitable for study of the quantum biological channel capacity. However, this model is essentially memoryless and it is not able to properly model the propagation of mutation errors in time, the process of aging, and evolution of genetic information through generations. To solve for these problems, we propose novel quantum mechanical models to accurately describe the process of creation spontaneous, induced, and adaptive mutations and their propagation in time. Different biological channel models with memory, proposed in this paper, include: (i) Markovian classical model, (ii) Markovian-like quantum model, and (iii) hybrid quantum-classical model. We then apply these models in a study of aging and evolution of quantum biological channel capacity through generations. We also discuss key differences of these models with respect to a multilevel symmetric channel-based Markovian model and a Kimura model-based Markovian process. These models are quite general and applicable to many open problems in biology, not only biological channel capacity, which is the main focus of the paper. We will show that the famous quantum Master equation approach, commonly used to describe different biological processes, is just the first-order approximation of the proposed quantum Markov chain-like model, when the observation interval tends to zero. One of the important implications of this model is that the aging phenotype becomes determined by different underlying transition probabilities in both programmed and random (damage) Markov chain-like models of aging, which are mutually coupled.

Nonequilibrium Energy Transfer at Nanoscale: A Unified Theory from Weak to Strong Coupling.

Unraveling the microscopic mechanism of quantum energy transfer across two-level systems provides crucial insights to the optimal design and potential applications of low-dimensional nanodevices. Here, we study the non-equilibrium spin-boson model as a minimal prototype and develop a fluctuation-decoupled quantum master equation approach that is valid ranging from the weak to the strong system-bath coupling regime. The exact expression of energy flux is analytically established, which dissects the energy transfer as multiple boson processes with even and odd parity. Our analysis provides a unified interpretation of several observations, including coherence-enhanced heat flux and negative differential thermal conductance. The results will have broad implications for the fine control of energy transfer in nano-structural devices.

Effects of electron-phonon interaction on thermal and electrical transport through molecular nano-conductors

The topic of this review is the effects of electron-phonon interaction (EPI) on the transport properties of molecular nano-conductors. A nano-conductor connects to two electron leads and two phonon leads, possibly at different temperatures or chemical potentials. The EPI appears only in the nano-conductor. We focus on its effects on charge and energy transport. We introduce three approaches. For weak EPI, we use the nonequilibrium Green’s function method to treat it perturbatively. We derive the expressions for the charge and heat currents. For weak system-lead couplings, we use the quantum master equation approach. In both cases, we use a simple single level model to study the effects of EPI on the system’s thermoelectric transport properties. It is also interesting to look at the effect of currents on the dynamics of the phonon system. For this, we derive a semi-classical generalized Langevin equation to describe the nano-conductor’s atomic dynamics, taking the nonequilibrium electron system, as well as the rest of the atomic degrees of freedom as effective baths. We show simple applications of this approach to the problem of energy transfer between electrons and phonons.

Efficiency fluctuations in quantum thermoelectric devices

We present a method, based on characterizing efficiency fluctuations, to asses the performance of nanoscale thermoelectric junctions. This method accounts for effects typically arising in small junctions, namely, stochasticity in the junction's performance, quantum effects, and nonequilibrium features preventing a linear response analysis. It is based on a nonequilibrium Green's function (NEGF) approach, which we use to derive the full counting statistics (FCS) for heat and work, and which in turn allows us to calculate the statistical properties of efficiency fluctuations. We simulate the latter for a variety of simple models where our method is exact. By analyzing the discrepancies with the semi-classical prediction of a quantum master equation (QME) approach, we emphasize the quantum nature of efficiency fluctuations for realistic junction parameters. We finally propose an approximate Gaussian method to express efficiency fluctuations in terms of nonequilibrium currents and noises which are experimentally measurable in molecular junctions.

Accurate nonadiabatic quantum dynamics on the cheap: making the most of mean field theory with master equations.

In this article, we show how Ehrenfest mean field theory can be made both a more accurate and efficient method to treat nonadiabatic quantum dynamics by combining it with the generalized quantum master equation framework. The resulting mean field generalized quantum master equation (MF-GQME) approach is a non-perturbative and non-Markovian theory to treat open quantum systems without any restrictions on the form of the Hamiltonian that it can be applied to. By studying relaxation dynamics in a wide range of dynamical regimes, typical of charge and energy transfer, we show that MF-GQME provides a much higher accuracy than a direct application of mean field theory. In addition, these increases in accuracy are accompanied by computational speed-ups of between one and two orders of magnitude that become larger as the system becomes more nonadiabatic. This combination of quantum-classical theory and master equation techniques thus makes it possible to obtain the accuracy of much more computationally expensive approaches at a cost lower than even mean field dynamics, providing the ability to treat the quantum dynamics of atomistic condensed phase systems for long times.

Full counting statistics of transport electrons through a two-level quantum dot with spin–orbit coupling

We study the full counting statistics of transport electrons through a semiconductor two-level quantum dot with Rashba spin–orbit (SO) coupling, which acts as a nonabelian gauge field and thus induces the electron transition between two levels along with the spin flip. By means of the quantum master equation approach, shot noise and skewness are obtained at finite temperature with two-body Coulomb interaction. We particularly demonstrate the crucial effect of SO coupling on the super-Poissonian fluctuation of transport electrons, in terms of which the SO coupling can be probed by the zero-frequency cumulants. While the charge currents are not sensitive to the SO coupling.

Influence of Superconducting Leads Energy Gap on Electron Transport Through Double Quantum Dot by Markovian Quantum Master Equation Approach

The superconducting reservoir effect on the current carrying transport of a double quantum dot in Markovian regime is investigated. For this purpose, a quantum master equation (QME) at finite temperature is derived for the many-body density matrix of an open quantum system. The dynamics and the steady-state properties of the double quantum dot system for arbitrary bias are studied. We will show that how the populations and coherencies of the system states are affected by superconducting leads. The energy parameter of system contains essentially four contributions due to dots system-electrodes coupling, intra dot coupling, two quantum dots inter coupling and superconducting gap. The coupling effect of each energy contribution is applied to currents and coherencies results. In addition, the effect of energy gap is studied by considering the amplitude and lifetime of coherencies to get more current through the system.

Decoherence and lead-induced interdot coupling in nonequilibrium electron transport through interacting quantum dots: A hierarchical quantum master equation approach

The interplay between interference effects and electron-electron interactions in electron transport through an interacting double quantum dot system is investigated using a hierarchical quantum master equation approach which becomes exact if carried to infinite order and converges well if the temperature is not too low. Decoherence due to electron-electron interactions is found to give rise to pronounced negative differential resistance, enhanced broadening of structures in current-voltage characteristics and an inversion of the electronic population. Dependence on gate voltage is shown to be a useful method of distinguishing decoherence-induced phenomena from effects induced by other mechanisms such as the presence of a blocking state. Comparison of results obtained by the hierarchical quantum master equation approach to those obtained from the Born-Markov approximation to the Nakajima-Zwanzig equation and from the non-crossing approximation to the nonequilibrium Green's function reveals the importance of an inter-dot coupling that originates from the energy dependence of the conduction bands in the leads and the need for a systematic perturbative expansion.

Hanbury Brown and Twiss correlations in quantum Hall systems

We study a Hanbury Brown and Twiss (HBT) interferometer formed with chiral edge channels of a quantum Hall system. HBT cross correlations are calculated for a device operating both in the integer and fractional quantum Hall regimes, the latter at Laughlin filling fractions. We find that in both cases, when the current is dominated by electron tunneling, current-current correlations show antibunching, characteristic of fermionic correlations. When the current-current correlations are dominated by quasiparticle tunneling, the correlations reveal bunching, characteristic of bosons. For electron tunneling, we use the Keldysh technique, and show that the result for fractional filling factors can be obtained in a simple way from the results of the integer case. It is shown that quasiparticle-dominated cross-current correlations can be analyzed by means of a quantum master-equation approach. We present here a detailed derivation of the results [Campagnano et al., Phys. Rev. Lett. 109, 106802 (2012)] and generalize them to all Laughlin fractions.

An analytical continuation approach for evaluating emission lineshapes of molecular aggregates and the adequacy of multichromophoric Förster theory

In large photosynthetic chromophore-protein complexes not all chromophores are coupled strongly, and thus the situation is well described by formation of delocalized states in certain domains of strongly coupled chromophores. In order to describe excitation energy transfer among different domains without performing extensive numerical calculations, one of the most popular techniques is a generalization of Förster theory to multichromophoric aggregates (generalized Förster theory) proposed by Sumi [J. Phys. Chem. B 103, 252 (1999)] and Scholes and Fleming [J. Phys. Chem. B 104, 1854 (2000)]. The aim of this paper is twofold. In the first place, by means of analytic continuation and a time convolutionless quantum master equation approach, a theory of emission lineshape of multichromophoric systems or molecular aggregates is proposed. In the second place, a comprehensive framework that allows for a clear, compact, and effective study of the multichromophoric approach in the full general version proposed by Jang, Newton, and Silbey [Phys. Rev. Lett. 92, 218301 (2004)] is developed. We apply the present theory to simple paradigmatic systems and we show on one hand the effectiveness of time-convolutionless techniques in deriving lineshape operators and on the other hand we show how the multichromophoric approach can give significant improvements in the determination of energy transfer rates in particular when the systems under study are not the purely Förster regime. The presented scheme allows for an effective implementation of the multichromophoric Förster approach which may be of use for simulating energy transfer dynamics in large photosynthetic aggregates, for which massive computational resources are usually required. Furthermore, our method allows for a systematic comparison of multichromophoric Föster and generalized Förster theories and for a clear understanding of their respective limits of validity.

Ecological optimisation of irreversible harmonic oscillator Carnot refrigerator

A model of a quantum harmonic oscillator Carnot refrigerator with heat resistance, internal irreversibility and heat leakage is established in this paper. Based on the quantum master equation and semigroup approach, equations of some important performance parameters, such as cooling load, coefficient of performance (COP), exergy output rate, entropy generation rate and ecological function (a criterion E = A/τ−T0σ representing the optimal compromise between exergy output rate A/τ and exergy loss rate T0σ), for the irreversible quantum Carnot refrigerator are derived. The optimal ecological performance of the refrigerator at high temperature limit is analysed using numerical examples. The effects of internal irreversibility and heat leakage on the optimal ecological performance are discussed. Performance comparisons of the quantum refrigerator under maximum ecological function, maximum COP and maximum cooling load conditions are also performed. Some special cases, i.e. endoreversible, frictionless and without heat leakage, are discussed. Taking ecological optimisation criterion as the objective makes larger improvement of the COP and the reduction in the exergy loss rate at the cost of smaller cooling load drop.

Dephasing of electrons in the Aharonov–Bohm interferometer with a single-molecular vibrational junction

Phase relaxation of electrons transferring through an electromechanical transistor is studied using the Aharonov–Bohm interferometer. Using the quantum master equation approach, the phase properties of an electron are numerically analyzed based on the interference fringes. The coherence of the electron is partially destroyed by its scattering on excited levels of the local nanomechanical oscillator. The transmission amplitudes with respect to two adjacent mechanical vibrational levels have a phase difference of π. The character of the π phase shift depends on the oscillator frequency only and is robust over a wide range of values of the applied voltage, tunneling length and damping rate of the mechanical oscillator.

Emission lineshapes of the B850 band of light-harvesting 2 (LH2) complex in purple bacteria: A second order time-nonlocal quantum master equation approach

The emission lineshape of the B850 band in the light harvesting complex 2 of purple bacteria is calculated by extending the approach of 2nd order time-nonlocal quantum master equation [S. Jang and R. J. Silbey, J. Chem. Phys. 118, 9312 (2003)]. The initial condition for the emission process corresponds to the stationary excited state density where exciton states are entangled with the bath modes in equilibrium. This exciton-bath coupling, which is not diagonal in either site excitation or exciton basis, results in a new inhomogeneous term that is absent in the expression for the absorption lineshape. Careful treatment of all the 2nd order terms are made, and explicit expressions are derived for both full 2nd order lineshape expression and the one based on secular approximation that neglects off-diagonal components in the exciton basis. Numerical results are presented for a few representative cases of disorder and temperature. Comparison of emission line shape with the absorption line shape is also made. It is shown that the inhomogeneous term coming from the entanglement of the system and bath degrees of freedom makes significant contributions to the lineshape. It is also found that the perturbative nature of the theory can result in negative portion of lineshape in some situations, which can be removed significantly by inclusion of the inhomogeneous term and completely by using the secular approximation. Comparison of the emission and absorption lineshapes at different temperatures demonstrates the role of thermal population of different exciton states and exciton-phonon couplings.

Single-electron transistor strongly coupled to vibrations: counting statistics and fluctuation theorem

Using a simple quantum master equation approach, we calculate the full counting statistics of a single-electron transistor strongly coupled to vibrations. The full counting statistics contains both the statistics of integrated particle and energy currents associated with the transferred electrons and phonons. A universal as well as an effective fluctuation theorem are derived for the general case where the various reservoir temperatures and chemical potentials are different. The first relates to the entropy production generated in the junction, while the second reveals internal information of the system. The model recovers the Franck–Condon blockade, and potential applications to non-invasive molecular spectroscopy are discussed.

Quantum Transport in Networks and Photosynthetic Complexes at the Steady State

Recently, several works have analysed the efficiency of photosynthetic complexes in a transient scenario and how that efficiency is affected by environmental noise. Here, following a quantum master equation approach, we study the energy and excitation transport in fully connected networks both in general and in the particular case of the Fenna–Matthew–Olson complex. The analysis is carried out for the steady state of the system where the excitation energy is constantly “flowing” through the system. Steady state transport scenarios are particularly relevant if the evolution of the quantum system is not conditioned on the arrival of individual excitations. By adding dephasing to the system, we analyse the possibility of noise-enhancement of the quantum transport.

Geometrical pumping in quantum transport: Quantum master equation approach

For an open quantum system, we investigate the pumped current induced by a slow modulation of control parameters on the basis of the quantum master equation and full counting statistics. We find that the average and the cumulant generating function of the pumped quantity are characterized by the geometrical Berry-phase-like quantities in the parameter space, which is associated with the generator of the master equation. From our formulation, we can discuss the geometrical pumping under the control of the chemical potentials and temperatures of reservoirs. We demonstrate the formulation by spinless electrons in coupled quantum dots. We show that the geometrical pumping is prohibited for the case of noninteracting electrons if we modulate only temperatures and chemical potentials of reservoirs, while the geometrical pumping occurs in the presence of an interaction between electrons.

Multistep Quantum Master Equation Theory for Response Functions in Four Wave Mixing Electronic Spectroscopy of Multichromophoric Macromolecules

This work provides an alternative derivation of third order response functions in four wave mixing spectroscopy of multichromophoric macromolecular systems considering only single exciton states. For the case of harmonic oscillator bath linearly and diagonally coupled to exciton states, closed form expressions showing all the explicit time dependences are derived. These expressions can provide more solid physical basis for understanding 2-dimensional electronic spectroscopy signals. For more general cases of system-bath coupling, the quantum master equation (QME) approach is employed for the derivation of multistep time evolution equations for Green function-like operators. Solution of these equations is feasible at the level of 2nd order non-Markovian QME, and the new approach can account for inter-exciton coupling, dephasing, relaxation, and non-Markovian effects in a consistent manner.