Quantum Master Equation Approach Research Articles

From Liouville to Landauer: Electron transport and the bath assumptions made along the way

A generalized quantum master equation approach is introduced to describe electron transfer in molecular junctions that spans both the off-resonant (tunneling) and resonant (hopping) transport regimes. The model builds on prior insights from scattering theory but is not limited to a certain parameter range with regard to the strength of the molecule–electrode coupling. The framework is used to study the simplest case of energy and charge transfer between the molecule and the electrodes for a single site noninteracting Anderson model in the limit of symmetric and asymmetric coupling between the molecule and the electrodes. In the limit of elastic transport, the Landauer result is recovered for the current by invoking a single active electron Ansatz and a binary collision approximation for the memory kernel. Inelastic transport is considered by allowing the excitation of electron–hole pairs in the electrodes in tandem with charge transport. In the case of low bias voltages where the Fermi levels of the electrodes remain below the molecular state, it is shown that the current arises from tunneling and the molecule remains neutral. However, once the threshold is reached for aligning the fermi level of one electrode with the molecular orbital, a small amount of charge transfer occurs with a negligible amount of hopping current. While inelasticity in the current has a minimal impact on the shape of the current–voltage curve in the case of symmetric electrode coupling, the results for a slight asymmetry in coupling demonstrate complete charge transfer and a significant drop in current. These results provide encouraging confirmation that the framework can describe charge transport across a wide range of electrode–molecule coupling and provide a unique perspective for developing new master equation treatments for energy and charge transport in molecular junctions. An extension of this work to account for inelastic scattering from electron–vibrational coupling at the molecule is straightforward and will be the subject of subsequent work.

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra.

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light-matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori-Nakajima-Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method's capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

Simulating Open Quantum System Dynamics on NISQ Computers with Generalized Quantum Master Equations.

We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.

Theoretical Study on Thermal Structural Fluctuation Effects of Intermolecular Configurations on Singlet Fission in Pentacene Crystal Models.

Singlet fission (SF) occurs as a result of complex excited state relaxation dynamics in molecular aggregates, where a singlet exciton (FE) state is converted into a double-triplet exciton (TT) state through the interactions with several other degrees of freedom, such as nuclear motions. In this study, we combined quantum dynamics simulation based on the quantum master equation approach with all-atom-based classical molecular mechanics/molecular dynamics to examine the thermal structural fluctuation (i.e., static disorder) effects of intermolecular configuration on SF in pentacene crystal models. In particular, we considered two types of static-disordered models, in which excited states are assumed to interact with nuclear motions of intermolecular modes in the classical mechanical/statistical manner. We found that the introduction of static disorder effects leads to a faster decay of coherence between the FE and charge transfer (CT) states in the early stage of SF, contributing to the accelerations of several FE → TT relaxation pathways. Such acceleration in these models is shown to be attributed to fluctuations in the energies and electronic coupling of the CT states based on relative relaxation factor analysis. The present study is expected to contribute to further development of bottom-up materials design for efficient SF in condensed phases where the exitonic system interacts with nuclear motions in various coupling strengths.

Coherence crossover dynamics in the strong coupling regime

The transient dynamics of quantum coherence of Gaussian states is investigated. The state is coupled to an external environment which can be described by a Fano–Anderson type Hamiltonian. Solving the quantum Langevin equation, we obtain the Greens functions which are used to compute the time evolved first and second moments of the quadrature operators. From the quadrature operator moments, we construct the covariance matrix which is used to measure the coherence in the system. The coherence is estimated using the relative entropy of coherence measure. We consider three different environment spectral densities in our analysis viz, the Ohmic, the sub-Ohmic, and the super-Ohmic densities. In our work, we study the dynamics of the coherent state, squeezed state, and displaced squeezed state. For all these states we observe that when the coupling with the system and the environment is weak, the coherence monotonically decreases and eventually vanishes in the long time limit. Thus all the states exhibit Markovian evolution in the weak coupling limit. In the strong coupling limit, the dynamics for the initial period is Markovian and after a certain period, it becomes non-Markovian where we observe an environmental backaction on the system. To confirm the non-Markovianity of the evolution we use a detection method based on Bures distance formula. Thus in the strong coupling limit, we observe a dynamical crossover from Markovian nature to non-Markovian behavior. This crossover is very abrupt under some environmental conditions and for some parameters of the quantum state. Using a quantum master equation approach we verify the crossover from the dynamics of the dissipation and fluctuation coefficients and the results endorse those obtained from coherence dynamics.

Tensor-Train Thermo-Field Memory Kernels for Generalized Quantum Master Equations.

The generalized quantum master equation (GQME) approach provides a rigorous framework for deriving the exact equation of motion for any subset of electronic reduced density matrix elements (e.g., the diagonal elements). In the context of electronic dynamics, the memory kernel and inhomogeneous term of the GQME introduce the implicit coupling to nuclear motion and dynamics of electronic density matrix elements that are projected out (e.g., the off-diagonal elements), allowing for efficient quantum dynamics simulations. Here, we focus on benchmark quantum simulations of electronic dynamics in a spin-boson model system described by various types of GQMEs. Exact memory kernels and inhomogeneous terms are obtained from short-time quantum-mechanically exact tensor-train thermo-field dynamics (TT-TFD) simulations and are compared with those obtained from an approximate linearized semiclassical method, allowing for assessment of the accuracy of these approximate memory kernels and inhomogeneous terms. Moreover, we have analyzed the computational cost of the full and reduced-dimensionality GQMEs. The scaling of the computational cost is dependent on several factors, sometimes with opposite scaling trends. The TT-TFD memory kernels can provide insights on the main sources of inaccuracies of GQME approaches when combined with approximate input methods and pave the road for the development of quantum circuits that implement GQMEs on digital quantum computers.

Heavy Quarkonium Dynamics in the QGP with a Quantum Master Equation Approach

Heavy Quarkonium Dynamics in the QGP with a Quantum Master Equation Approach

Nonadiabatically driven open quantum systems under out-of-equilibrium conditions: Effect of electron-phonon interaction

In this paper we explore the effects of nonadiabatic external driving on the dynamics of an electronic system coupled to two electronic leads and to a phonon mode, with and without damping. In the limit of slow driving, we establish nonadiabatic corrections to thermodynamic and transport quantities. In particular, we study the first-order correction to the work done by the driving, the charge current, and the vibrational excitation using a perturbative expansion. We then compare the results to the numerically exact hierarchical equations of motion (HEOM) approach. Furthermore, the HEOM analysis spans both the weak and strong system-bath coupling regime and the slow- and fast-driving limits. We show that the electronic friction and the nonadiabatic corrections to the charge current provide a clear indicator for the Franck-Condon effect and for nonresonant tunneling processes. We also discuss the validity of the approximate quantum master equation approach and the benefits of using HEOM to study nonadiabatically driven open quantum systems out of equilibrium.

Lyapunov equation in open quantum systems and non-Hermitian physics

The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation of motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master equation approaches.

Quantum master equation approach to heat transport in dielectrics and semiconductors

We report on the derivation of the heat transport equation for nonmetals using a quantum Markovian master equation in Lindblad form. We first establish the equations of motion describing the time variation of the on-site energy of atoms in a one dimensional periodic chain that is coupled to a heat reservoir. In the continuum limit, the Fourier law of heat conduction naturally emerges, and the heat conductivity is explicitly obtained. It is found that the effect of the heat reservoir on the lattice is described by a heat source density that depends on the diffusion coefficients of the atoms. We show that the Markovian dynamics is equivalent to the long wavelength approximation for phonons, which is typical for the case of elastic solids. The high temperature limit is shown to reproduce the classical heat conduction equation.

Exciton transport in amorphous polymers and the role of morphology and thermalisation

Understanding the transport mechanism of electronic excitations in conjugated polymers is key to advancing organic optoelectronic applications, such as solar cells, organic light-emitting diodes and flexible electronics. While crystalline polymers can be studied using solid-state techniques based on lattice periodicity, the characterisation of amorphous polymers is hindered by an intermediate regime of disorder and the associated lack of symmetries. To overcome these hurdles we have developed a reduced state quantum master equation approach based on the Merrifield exciton formalism. This new approach allows us to study the dynamics of excitons’ centre of mass and charge separation (CS), going beyond the standard model of charge-neutral Frenkel excitons. Using this model we study exciton transport in conjugated polymers and its dependence on morphology and temperature. Exciton dynamics consists of a thermalisation process, whose features depend on the relative strength of thermal energy, electronic couplings and disorder, resulting in remarkably different transport regimes. By applying this method to representative systems based on poly(p-phenylene vinylene) (PPV) we obtain insight into the role of temperature and disorder on localisation, CS, non-equilibrium dynamics, and experimental accessibility of thermal equilibrium states of excitons in amorphous polymers.

Unraveling current-induced dissociation mechanisms in single-molecule junctions.

Understanding current-induced bond rupture in single-molecule junctions is both of fundamental interest and a prerequisite for the design of molecular junctions, which are stable at higher-bias voltages. In this work, we use a fully quantum mechanical method based on the hierarchical quantum master equation approach to analyze the dissociation mechanisms in molecular junctions. Considering a wide range of transport regimes, from off-resonant to resonant, non-adiabatic to adiabatic transport, and weak to strong vibronic coupling, our systematic study identifies three dissociation mechanisms. In the weak and intermediate vibronic coupling regime, the dominant dissociation mechanism is stepwise vibrational ladder climbing. For strong vibronic coupling, dissociation is induced via multi-quantum vibrational excitations triggered either by a single electronic transition at high bias voltages or by multiple electronic transitions at low biases. Furthermore, the influence of vibrational relaxation on the dissociation dynamics is analyzed and strategies for improving the stability of molecular junctions are discussed.

Nonlinear electric conductivity and THz-induced charge transport in graphene

Based on the quantum master equation approach, the nonlinear electric conductivity of graphene is investigated under static electric fields for various chemical potential shifts. The simulation results show that, as the field strength increases, the effective conductivity is firstly suppressed, reflecting the depletion of effective carriers due to the large displacement in the Brillouin zone caused by the strong field. Then, as the field strength exceeds 1 MV m−1, the effective conductivity increases, overcoming the carrier depletion via the Landau–Zener tunneling process. Based on the nonlinear behavior of the conductivity, the charge transport induced by few-cycle THz pulses is studied to elucidate the ultrafast control of electric current in matter.

Strong coupling effects in quantum thermal transport with the reaction coordinate method

We present a semi-analytical approach for studying quantum thermal energy transport at the nanoscale. Our method, which is based on the reaction coordinate method, reveals the role of strong system-bath coupling effects in quantum energy transport. Considering as a case study the nonequilibrium spin-boson model, a collective coordinate is extracted from each thermal environment and added into the system to construct an enlarged system (ES). After performing additional Hamiltonian’s truncation and transformation, we obtain an effective two-level system with renormalized parameters, resulting from the strong system-bath coupling. The ES is weakly coupled to its environments, thus can be simulated using a perturbative Markovian quantum master equation approach. We compare the heat current characteristics of the effective two-state model to other techniques, and demonstrate that we properly capture strong system-bath signatures such as the turnover behavior of the heat current as a function of system-bath coupling strength. We further investigate the thermal diode effect and demonstrate that strong couplings moderately improve the rectification ratio relative to the weak coupling limit. The effective Hamiltonian method that we developed here offers fundamental insight into the strong coupling behavior, and is computationally economic. Applications of the method toward studying multi-level quantum thermal machines are anticipated.

Suppression of shot noise in a spin-orbit coupled quantum dot.

We study theoretically the transport properties of electrons in a quantum dot system with spin–orbit coupling. By using the quantum master equation approach, the shot noise and skewness of the transport electrons are calculated. We obtain super-Poisson noise behaviour by investigating the full counting statistics of the transport system. We discover super-Poisson behaviour is more obvious with the spin polarization increasing. More importantly, we discover the suppression of shot noise induced by spin–orbit coupling. The value of shot noise is gradually decreasing when spin–orbit coupling strength increases.

Controllable phase-dependent Wigner-function negativity at steady state via parametric driving and feedback

Generating the negative Wigner functions where the corresponding Wigner states are nonclassical has been recognized as a powerful tool for successfully performing quantum information and computing protocols beyond the scope of classical computers. Here, we present the possibility to generate and engineer the negative Wigner function at a steady state using parametric (two-photon) driving and homodyne-based feedback in a quantum van der Pol (vdP) oscillator. Specifically, we employ a quantum master equation approach for calculating the Wigner function of the vdP oscillator field in phase space and, furthermore, quantifying its negativity content. We clearly show that the negative-value magnitudes, regions, and shapes of the Wigner function can be effectively tuned by the parametric driving phase and the parametric driving amplitude, as well as the feedback coefficient within a large range. We identify different contributions of these involved parameters to the Wigner-function negativity. In the present scheme, more complex quantum coherence and interference phenomena are introduced via the parametric driving and feedback, which stabilizes the phase of the vdP oscillator field and renders the capability to generate the negative Wigner function. Therefore, the enhanced Wigner-function negativity can be achieved under these optimized system parameters. Our in-depth study provides insight into the formation and in situ control of the desirable Wigner nonclassical states. The obtained results are not limited to the vdP oscillator systems and should be generally applicable to other coherent coupled systems within the reach of modern experimental facilities.

High-order harmonic generation in graphene: Nonlinear coupling of intraband and interband transitions

We investigate high-order harmonic generation (HHG) in graphene with a quantum master equation approach. The simulations reproduce the observed enhancement in HHG in graphene under elliptically polarized light [N. Yoshikawa et al, Science 356, 736 (2017)]. On the basis of a microscopic decomposition of the emitted high-order harmonics, we find that the enhancement in HHG originates from an intricate nonlinear coupling between the intraband and interband transitions that are respectively induced by perpendicular electric field components of the elliptically polarized light. Furthermore, we reveal that contributions from different excitation channels destructively interfere with each other. This finding suggests a path to potentially enhance the HHG by blocking a part of the channels and canceling the destructive interference through band-gap or chemical potential manipulation.

Realistic nonlocal refrigeration engine based on Coulomb-coupled systems.

We investigate, in detail, a triple quantum dot system that exploits Coulomb coupling to achieve nonlocal refrigeration. The system under investigation is a derivative of the nonlocal thermodynamic engine, originally proposed by Sánchez and Büttiker [Phys. Rev. B 83, 085428 (2011)PRBMDO1098-012110.1103/PhysRevB.83.085428], that employs quadruple quantum dots to attain efficient nonlocal heat harvesting. Investigating the cooling performance and operating regime using the quantum master equation approach, we point out some crucial aspects of the refrigeration engine. In particular, we demonstrate that the maximum cooling power for the setup is limited to about 70% of the optimal design. Proceeding further, we point out that to achieve a target reservoir temperature lower than the average temperature of the current path, the applied voltage must be greater than a given threshold voltage V_{TH} that increases with the decrease in the target reservoir temperature. In addition, we demonstrate that the maximum cooling power, as well as the coefficient of performance, deteriorates as one approaches a lower target reservoir temperature. The triple quantum dot system, investigated in this paper, combines fabrication simplicity along with descent cooling power and may pave the way towards the practical realization of efficient nonlocal cryogenic refrigeration systems.

Berry-phase-like effect of thermo-phonon transport in optomechanics

We investigate thermo-phonon transport and its nontrivial Berry-phase-like effect in an optomechanical system with a squeezed vacuum injection. By taking the cumulant generation function approach the exact expressions of the thermo-phonon flux and optomechanical Berry phase are derived analytically. Further, the quantum master equation approach is invoked to verify the analytical results of the transport properties. It is shown that the steady-state thermo-phonon flux can be modulated by varying optically the nonequilibrium characteristics of the system via the squeezed vacuum. In particular, an adiabatic modulation of squeezing parameters induces an optomechanical Berry-phase-like effect and as a result provides an additional geometric phonon response across the macroscopic mechanical motion near the quantum regime, which can also be seen as a consequence of the asymmetric jumping probability between transition associated with phonon absorption and emission in a thermal bath. The present method and results are general and can be straightforwardly extended to any multimode oscillator systems and therefore pave the way to the thermal noise energy harvesting and rectification in coupled oscillator systems with inertial terms.

Density matrix to quantum master equation (QME) model for arrays of Coulomb coupled quantum dots in the sequential tunneling regime

Coulomb coupled quantum dot arrays with staircase ground state configuration have been proposed in literature for enhancing heat-harvesting and refrigeration performance (Erdman et al 2018 Phys. Rev. B 98, 045433; Walldorf et al 2017 Phys. Rev. B 96, 115415; Daré 2019 Phys. Rev. B 100 195427; Zhang and Chen 2019 Physica E 114, 113635; Daré and Lombardo 2017 Phys. Rev. B 96, 115414; Zhang et al 2016 Energy 95, 593; Sánchez and Büttiker 2011 Phys. Rev. B 83 085428; Singha 2018 Phys. Lett. A 382, 3026). Due to their mutual Coulomb interaction, a performance analysis of such systems remains complicated and necessitates consideration of microscopic physics using density matrix formulation. However the path of transport analysis starting from the system Hamiltonian to density matrix formulation is complicated and lacks the simplicity and intuitive aspect of sequential electron transport conveyed by the quantum master equation (QME) approach. In this paper, starting from the system Hamiltonian and employing the density matrix formulation, I derive the QME of a system of three quantum dots, two of which are electro-statically coupled. The framework elaborated in this paper can be further extended to derive QME of systems with higher number of Coulomb coupled quantum dots. Hence, the formulation developed in this paper can pave the way towards an intuitive analysis of transport physics for an array of Coulomb coupled quantum dots in the sequential tunneling regime.