Redfield Master Equation Research Articles

Photo-induced dynamics with continuous and discrete quantum baths.

The ultrafast quantum dynamics of photophysical processes in complex molecules is an extremely challenging computational problem with a broad variety of fascinating applications in quantum chemistry and biology. Inspired by recent developments in open quantum systems, we introduce a pure-state unraveled hybrid-bath method that describes a continuous environment via a set of discrete, effective bosonic degrees of freedom using a Markovian embedding. Our method is capable of describing both, a continuous spectral density and sharp peaks embedded into it. Thereby, we overcome the limitations of previous methods, which either capture long-time memory effects using the unitary dynamics of a set of discrete vibrational modes or use memoryless Markovian environments employing a Lindblad or Redfield master equation. We benchmark our method against two paradigmatic problems from quantum chemistry and biology. We demonstrate that compared to unitary descriptions, a significantly smaller number of bosonic modes suffices to describe the excitonic dynamics accurately, yielding a computational speed-up of nearly an order of magnitude. Furthermore, we take into account explicitly the effect of a δ-peak in the spectral density of a light-harvesting complex, demonstrating the strong impact of the long-time memory of the environment on the dynamics.

The role of anharmonicity in single-molecule spin-crossover

We exploit the system-bath paradigm to investigate vibrational-anharmonicity effects on spin-crossover in a single molecule. Focusing on weak coupling, we use the linear response approximation to deal with the nonlinear vibrational bath and propagate the Redfield master equation to obtain the equilibrium high spin fraction. We take both the anharmonicity in the bath potentials and the nonlinearity in the spin-vibration coupling into account and find a strong interplay between these two effects. Further, we show that the spin-crossover in a single molecule is always a gradual transition and the anharmonicity-induced phonon drag greatly affects the transition behavior.

Modeling Low- and High-Frequency Noise in Transmon Qubits with Resource-Efficient Measurement

Transmon qubits experience open-system effects that manifest as noise at a broad range of frequencies. We present a model of these effects using the Redfield master equation with a hybrid bath consisting of low- and high-frequency components. We use two-level fluctuators to simulate 1/f-like noise behavior, which is a dominant source of decoherence for superconducting qubits. By measuring quantum state fidelity under free evolution with and without dynamical decoupling (DD), we can fit the low- and high-frequency noise parameters in our model. We train and test our model using experiments on quantum devices available through IBM quantum experience. Our model accurately predicts the fidelity decay of random initial states, including the effect of DD pulse sequences. We compare our model with two simpler models and confirm the importance of including both high frequency and 1/f noise in order to accurately predict transmon behavior. Published by the American Physical Society 2024

NMR Relaxation by Redfield equation in a spin system [formula omitted

The Redfield master equation was solved analytically for a nuclear system with spin I=7/2. Using the irreducible tensor operator basis, the solutions of each density matrix element were computed. The experimental setup consisted of the 133Cs nuclei of the cesium-pentadecafluorooctanoate molecule in a lyotropic liquid crystal sample in the nematic phase at room temperature. Experimental longitudinal and transverse magnetization dynamics of the 133Cs nuclei were monitored, and the theoretical approach was used to generate valuable mathematical expressions with the highest accuracy through numerical procedures. This methodology can be extended to other nuclei with minimal difficulties.

Long-lived quantum coherent dynamics of a Λ-system driven by a thermal environment.

We present a theoretical study of quantum coherent dynamics of a three-level Λ-system driven by a thermal environment (such as blackbody radiation), which serves as an essential building block of photosynthetic light-harvesting models and quantum heat engines. By solving nonsecular Bloch-Redfield master equations, we obtain analytical results for the ground-state population and coherence dynamics and classify the dynamical regimes of the incoherently driven Λ-system as underdamped and overdamped depending on whether the ratio Δ/[rf(p)] is greater or less than one, where Δ is the ground-state energy splitting, r is the incoherent pumping rate, and f(p) is a function of the transition dipole alignment parameter p. In the underdamped regime, we observe long-lived coherent dynamics that lasts for τc ≃ 1/r, even though the initial state of the Λ-system contains no coherences in the energy basis. In the overdamped regime for p = 1, we observe the emergence of coherent quasi-steady states with the lifetime τc = 1.34(r/Δ2), which have a low von Neumann entropy compared to conventional thermal states. We propose an experimental scenario for observing noise-induced coherent dynamics in metastable He* atoms driven by x-polarized incoherent light. Our results suggest that thermal excitations can generate experimentally observable long-lived quantum coherent dynamics in the ground-state subspace of atomic and molecular Λ-systems in the absence of coherent driving.

Non-Markovian Quantum Dynamics in a Squeezed Reservoir.

We study non-Markovian dynamics of an open quantum system system interacting with a nonstationary squeezed bosonic reservoir. We derive exact and approximate descriptions for the open system dynamics. Focusing on the spin boson model, we compare exact dynamics with Redfield theory and a quantum optical master equation for both short and long time dynamics and in non-Markovian and Markov regimes. The squeezing of the bath results in asymptotic oscillations in the stationary state, which are captured faithfully by the Redfield master equation in the case of weak coupling. Furthermore, we find that the bath squeezing direction modifies the effective system–environment coupling strength and, thus, the strength of the dissipation.

Analytic expression for the optical exciton transition rates in the polaron frame

When an optical emitter is strongly coupled to a vibrational bath the polaron transformation is often used to permit an accurate second-order Redfield master equation. However, the optical transition rates in the polaron frame are not analytic and approximations typically need to be made which result in the loss of anything other than simple additive effects of the two baths. In this paper, we derive an intuitive analytic expression for the polaron frame optical transition rates by means of a finite mode truncation of the vibrational bath. Using this technique, calculations of the transition rates converge for only a few modes in the truncated spectral density, and capture non-additive effects such as population inversion of a two-level system.

Optimal Control for Quantum Optimization of Closed and Open Systems

We provide a rigorous analysis of the quantum optimal control problem in the setting of a linear combination $s(t)B+(1-s(t))C$ of two noncommuting Hamiltonians $B$ and $C$. This includes both quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA). The target is to minimize the energy of the final ``problem'' Hamiltonian $C$, for a time-dependent and bounded control schedule $s(t)\in [0,1]$ and $t\in \mc{I}:= [0,t_f]$. It was recently shown, in a purely closed system setting, that the optimal solution to this problem is a ``bang-anneal-bang'' schedule, with the bangs characterized by $s(t)= 0$ and $s(t)= 1$ in finite subintervals of $\mc{I}$, in particular $s(0)=0$ and $s(t_f)=1$, in contrast to the standard prescription $s(0)=1$ and $s(t_f)=0$ of quantum annealing. Here we extend this result to the open system setting, where the system is described by a density matrix rather than a pure state. This is the natural setting for experimental realizations of QA and QAOA. For finite-dimensional environments and without any approximations we identify sufficient conditions ensuring that either the bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, and recover the optimality of $s(0)=0$ and $s(t_f)=1$. However, for infinite-dimensional environments and a system described by an adiabatic Redfield master equation we do not recover the bang-type optimal solution. In fact we can only identify conditions under which $s(t_f)=1$, and even this result is not recovered in the fully Markovian limit. The analysis, which we carry out entirely within the geometric framework of Pontryagin Maximum Principle, simplifies using the density matrix formulation compared to the state vector formulation.

Validity of Born-Markov master equations for single- and two-qubit systems

The urgent need for reliable simulation tools to match the extreme accuracy needed to control tailored quantum devices highlights the importance of understanding open quantum systems and their modeling. To this end, we compare here the commonly used Redfield and Lindblad master equations against numerically exact results in the case of one and two resonant qubits transversely coupled at a single point to a Drude-cut ohmic bath. All the relevant parameters are varied over a broad range which allows us to give detailed predictions about the validity and physically meaningful applicability of the weak-coupling approaches. We characterize the accuracy of the approximate approaches by comparing the maximum difference of their system evolution superoperators with numerically exact results. After optimizing the parameters of the approximate models to minimize the difference, we also explore if and to what extent the weak-coupling equations can be applied at least as phenomenological models. Optimization may lead to an accurate reproduction of experimental data, but yet our results are important to estimate the reliability of the extracted parameter values such as the bath temperature. Our findings set general guidelines for the range of validity of the usual Born-Markov master equations and indicate that they fail to accurately describe the physics in surprisingly broad range of parameters, in particular at low temperatures. Since quantum-technological devices operate there their accurate modeling calls for a careful choice of methods.

A multifunctional quantum thermal device: With and without inner coupling

A three-level system attached to three thermal baths is manipulated to be a microscopic thermal device integrating a valve, a refrigerator, an amplifier, and a thermometer in the quantum regime, via tuning the inner coupling strength of the system and the temperatures of the external baths. We discuss the role of the inner coupling as well as the steady-state quantum coherence in these thermal functions using the Redfield master equation under a partial secular approximation. A high-sensitive thermometer for the low-temperature terminal can be established without the assistance from the inner coupling of the system. Our study of this multifunctional thermal device provides a deeper insight to the underlying quantum thermodynamics associated with the quantum coherence.

Transport in conductors and rectifiers: Mean-field Redfield equations and nonequilibrium Green's functions

We derive a closed equation of motion for the one particle density matrix of a quantum system coupled to multiple baths using the Redfield master equation combined with a mean-field approximation. The steady-state solution may be found analytically with perturbation theory. Application of the method to a one-dimensional noninteracting quantum wire yields an expression for the current that reproduces the celebrated Landauer's formula. Nonlinear rectification is found for the case of a mesoscopic three-dimensional semiconductor $p\ensuremath{-}n$ junction. The results are in good agreement with numerical simulations obtained using nonequilibrium Green's functions, supporting the validity of the Redfield equations for the description of transport.

Reservoir engineering of Cooper-pair-assisted transport with cold atoms

We show how Cooper-pair-assisted transport, which describes the stimulated transport of electrons in the presence of Cooper-pairs, can be engineered and controlled with cold atoms, in regimes that are difficult to access for condensed matter systems. Our model is a channel connecting two cold atomic gases, and the mechanism to generate such a transport relies on the coupling of the channel to a molecular BEC, with diatomic molecules of fermionic atoms. Our results are obtained using a Floquet–Redfield master equation that accounts for an exact treatment of the interaction between atoms in the channel. We explore, in particular, the impact of the coupling to the BEC and the interaction between atoms in the junction on its transport properties, revealing non-trivial dependence of the produced particle current. We also study the effects of finite temperatures of the reservoirs and the robustness of the current against additional dissipation acting on the junction. Our work is experimentally relevant and has potential applications to dissipation engineering of transport with cold atoms, studies of thermoelectric effects, quantum heat engines, or Floquet Majorana fermions.

Thermal rectification and heat amplification in a nonequilibrium V-type three-level system.

Thermal rectification and heat amplification are investigated in a nonequilibrium V-type three-level system with quantum interference. By applying the Redfield master equation combined with full counting statistics, we analyze the steady-state heat transport. The noise-induced interference is found to be able to rectify the heat current, which paves a new way to design quantum thermal rectifier. Within the three-reservoir setup, the heat amplification is clearly identified far from equilibrium, which is in absence of the negative differential thermal conductance.

Atom-only descriptions of the driven-dissipative Dicke model

We investigate how to describe the dissipative spin dynamics of the driven-dissipative Dicke model, describing $N$ two-level atoms coupled to a cavity mode, after adiabatic elimination of the cavity mode. To this end, we derive a Redfield master equation which goes beyond the standard secular approximation and large detuning limits. We show that the secular (or rotating wave) approximation and the large detuning approximation both lead to inadequate master equations, that fail to predict the Dicke transition or the damping rates of the atomic dynamics. In contrast, the full Redfield theory correctly predicts the phase transition and the effective atomic damping rates. Our work provides a reliable framework to study the full quantum dynamics of atoms in a multimode cavity, where a quantum description of the full model becomes intractable.

Spin transfer torque induced paramagnetic resonance

We show how the spin-transfer torque generated by an ac voltage may be used to excite a paramagnetic resonance of an atomic spin deposited on a metallic surface. This mechanism is independent of the environment of the atom and may explain the ubiquity of the paramagnetic resonance reported by Baumann $\textit{et al.}$ [$\href{http://dx.doi.org/10.1126/science.aac8703}{Science \textbf{350}, 417 (2015)}$]. The current and spin dynamics are modeled by a time-dependent Redfield master equation generalized to account for the periodic driven voltage. Our approach shows that the resonance effect is a consequence of the nonlinearity of the coupling between the magnetic moment and the spin-polarized current which generates a large second-harmonic amplitude that can be measured in the current signal.

Many-body open quantum systems beyond Lindblad master equations

Many-body quantum systems present a rich phenomenology which can be significantly altered when they are in contact with an environment. In order to study such setups, a number of approximations are usually performed, either concerning the system, the environment, or both. A typical approach for large quantum interacting systems is to use master equations which are local, Markovian, and in Lindblad form. Here, we present an implementation of the Redfield master equation using matrix product states and operators. We show that this allows us to explore parameter regimes of the many-body quantum system and the environment which could not be probed with previous approaches based on local Lindblad master equations. We also show the validity of our results by comparing with the numerical exact thermofield-based chain-mapping approach.

QmeQ 1.0: An open-source Python package for calculations of transport through quantum dot devices

QmeQ is an open-source Python package for numerical modeling of transport through quantum dot devices with strong electron–electron interactions using various approximate master equation approaches. The package provides a framework for calculating stationary particle or energy currents driven by differences in chemical potentials or temperatures between the leads which are tunnel coupled to the quantum dots. The electronic structures of the quantum dots are described by their single-particle states and the Coulomb matrix elements between the states. When transport is treated perturbatively to lowest order in the tunneling couplings, the possible approaches are Pauli (classical), first-order Redfield, and first-order von Neumann master equations, and a particular form of the Lindblad equation. When all processes involving two-particle excitations in the leads are of interest, the second-order von Neumann approach can be applied. All these approaches are implemented in QmeQ. We here give an overview of the basic structure of the package, give examples of transport calculations, and outline the range of applicability of the different approximate approaches. Program summaryProgram Title:QmeQProgram Files doi:http://dx.doi.org/10.17632/8687mrhgg9.1Licensing provisions: BSD 2-ClauseProgramming language:PythonExternal libraries:NumPy, SciPy, CythonNature of problem: Calculation of stationary state currents through quantum dots tunnel coupled to leads.Solution method: Exact diagonalization of the quantum dot Hamiltonian for a given set of single particle states and Coulomb matrix elements. Numerical solution of the stationary-state master equation for a given approximate approach.Restrictions: Depending on the approximate approach the temperature needs to be sufficiently large compared to the coupling strength for the approach to be valid.

Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle

We use Pontryagin's minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed Quantum Approximate Optimization Algorithm. Focusing on the Sherrington-Kirkpatrick spin-glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parameterization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. For the particular systems we study, we find numerically that the optimal nonadiabatic bang-bang protocols outperform conventional quantum annealing in the presence of weak white additive external noise and weak coupling to a thermal bath modeled with the Redfield master equation.

Finite coupling effects in double quantum dots near equilibrium

A weak coupling quantum master equation provides reliable steady-state results only in the van Hove limit, i.e., when the system-lead coupling approaches zero. Recently, J. Thingna et al. [Phys. Rev. E 88, 052127 (2013)] proposed an alternative approach, based on an analytic continuation of the Redfield solution, to evaluate the reduced density matrix up to second order in the system-bath coupling. The approach provides accurate results for harmonic oscillator and spin-bosonic systems. We apply this approach to study fermionic systems and the calculation on an exactly solvable double quantum dot system shows that the method is rigorously valid only near equilibrium, i.e., linear response regime. We further compare to the Redfield and the secular Redfield (Lindblad-type) master equations that are inaccurate in all parameter regimes. Lastly, we consider the non-trivial problem of strong Coulomb interaction and illustrate the interplay between system-lead coupling, inter-dot tunneling, and Coulomb strength that can be captured only via the analytic continuation method.

Violation of Onsager's theorem in approximate master equation approaches

The consistency with Onsager's theorem is examined for commonly used perturbative approaches, such as the Redfield and second-order von Neumann master equations, for thermoelectric transport through nanostructures. We study a double quantum dot, which requires coherences between states for a correct description, and we find that these perturbative approaches violate Onsager's theorem. We show that the deviations from the theorem scale with the lead-coupling strength in an order beyond the one considered systematically in the respective approach.