We present a derivation of the Redfield formalism for treating the dissipative dynamics of a time-dependent quantum system coupled to a classical environment. We compare such a formalism with the master equation approach where the environments are treated quantum mechanically. Focusing on a time-dependent spin-1/2 system we demonstrate the equivalence between both approaches by showing that they lead to the same Bloch equations and, as a consequence, to the same characteristic times $T_{1}$ and $T_{2}$ (associated with the longitudinal and transverse relaxations, respectively). These characteristic times are shown to be related to the operator-sum representation and the equivalent phenomenological-operator approach. Finally, we present a protocol to circumvent the decoherence processes due to the loss of energy (and thus, associated with $T_{1}$). To this end, we simply associate the time-dependence of the quantum system to an easily achieved modulated frequency. A possible implementation of the protocol is also proposed in the context of nuclear magnetic resonance.
In this paper we derive the Lindblad and Redfield forms with and without secular approximation from the Born-Markov master equation for open quantum systems. The spectral correlation tensor of bath (the Fourier transform of the bath correlation function) and then the coefficients in the two forms of the master equation are reevaluated according to the scheme in Ref.[Phys. Rev. A 99, 022118 (2019)]. They are complex numbers rather than the real numbers getting from traditional simplified methods. The dynamics of two models [one is an open three-level quantum system model, and the other is the m…
Heat transport in driven quantum systems: Comparison between the Floquet-Redfield equation and the master equation in the instantaneous eigenbasis
Aalto University School of Science（阿尔托大学科学学院）；Chalmers University of Technology（查尔姆斯理工大学）；Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）
Here we present a Lindblad master equation that approximates the Redfield equation, a well known master equation derived from first principles, without significantly compromising the range of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation only truncates terms in the Redfield equation that average out over a time-scale typical of the quantum system. The first step in this approximation is to properly renormalize the system Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling. In the second step, we swap…
We provide a comprehensive study of the energy transfer phenomenon -- populating a given energy level -- in 3- and 4-level quantum systems coupled to two thermal baths. In particular, we examine the effects of an external periodic driving and the coherence induced by the baths on the efficiency of the energy transfer. We consider the Floquet-Lindblad and the Floquet-Redfield scenarios, which both are in the Born-Markov, weak-coupling regime but differ in the treatment of the secular approximation, and for the latter, we develop an appropriate Floquet-type master equation by employing a partial…
Following the ideas N. N. Bogoliubov used to derive the classical and quantum nonlinear kinetic equations, we give an alternative derivation of the Redfield quantum linear master equation, which is widely used in the theory of open quantum systems, as well as higher-order corrections to it. This derivation naturally considers initially correlated system-reservoir states arising from the previous system-reservoir dynamics. It turns out that the Redfield equation does not require any modifications in this case. The expressions of higher-order corrections are simpler than those obtained by other …
Foerster and modified Redfield theories play one of the central roles in the description of excitation energy transfer in molecular systems. However, in the present state, these theories describe only the dynamics of populations of local electronic excitations or delocalized exciton eigenstates, respectively, i.e., the diagonal elements of the density matrix in the corresponding representation. They do not give prescription for propagating the off-diagonal elements of the density matrix (coherences). This is commonly accepted as a limitation of these theories. Here we derive formulas for the d…
We develop a more accurate secular form for the Redfield non-secular master equation, which is insensitive to the choice of the basis. This completely solves the ambiguity on which basis should be used when the secular approximation of the Redfield master equation is applied.
A comparison of different master equations for driven-dissipative dynamics in composite quantum systems: Dispersive readout in structured electromagnetic environments
Sorbonne Université（索邦大学）；Institut des NanoSciences de Paris（巴黎纳米科学研究所）；LPENS, Département de Physique, École Normale Supérieure, Centre Automatique et Systèmes (CAS), MINES ParisTech, Université PSL, Sorbonne Université, CNRS, Inria（物理与能源环境研究实验室，巴黎高等师范学院，自动系统与中央研究院，国立巴黎高等矿业学校，巴黎文理研究大学，索邦大学，法国国家科学研究中心，法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）
The reduced dynamics of an open quantum system obtained from an underlying microscopic Hamiltonian can in general only approximately be described by a time local master equation. The quality of that approximation depends primarily on the coupling strength and the structure of the environment. Various such master equations have been proposed with different aims. Choosing the most suitable one for a specific system is not straight forward. By focusing on the accuracy of the reduced dynamics we provide a thorough assessment for a selection of methods (Redfield Equation, Quantum Optical Master Equ…
Two different Master Equation approaches have been formally derived to address the dynamics of open quantum systems interacting with a thermal environment (such as sunlight). They have led to two different physical results: non-secular equations that show noise-induced (Fano) coherences and secular equations that do not. An experimental test for the appearance of non-secular terms is proposed using Ca atoms in magnetic fields excited by broadband incoherent radiation. Significantly different patterns of fluorescence are predicted, allowing for a clear test for the validity of the secular and n…
Modified Bloch-Redfield Master Equation for Incoherent Excitation of Multilevel Quantum Systems
提出了一种计算与非相干光弱相互作用的多能级量子系统的密度矩阵随时间演化的有效理论方法。该方法结合了布洛赫-雷德菲尔德理论与单光子相干性的部分长期近似，导致在一个主方程，明确暴露依赖于跃迁速率和跃迁偶极矩之间的角度的能量基础。修正的布洛赫-雷德菲尔德主方程允许在非相干光照射下量子相干与非相干能量转移之间的明确区分。完全非相干制度的特点是正交跃迁偶极矩的能量基础，导致由相干自由保利型主方程的动力学演化。相干区要求能量基中存在非正交跃迁偶极矩，并导致噪声诱导的量子相干性和粒子数-相干耦合的产生。作为第一个应用，我们考虑了单基态非相干光激发下的激发态相干态的动力学，观察到了在小激发态分裂的情况下粒子数到相干态的转移和非平衡准稳态的形成。早期推导出的V型系统的分析表达式[Phys. Rev. Lett. 113，113601（2014）]被发现提供了在小分子和大分子限制下的多级激发态布居和相干态的几乎定量的描述。
The Redfield equation describes the dynamics of a quantum system weakly coupled to one or more reservoirs and is widely used in theory of open quantum system. However, the assumption of weak system-reservoir coupling is often not fully adequate and higher-order corrections to the Redfield equation with respect to the system-bath coupling is required. Here we propose a general method of derivation of higher-order corrections to the Redfield quantum master equation based on the hierarchical equations of motion (HEOM). Also we derive conditions of validity of the Redfield equation as well as the …
Starting from a microscopic system-baths description, we derive the general conditions for a time-local quantum master equation (QME) to satisfy the first and second law of thermodynamics at the fluctuating level. Using counting statistics, we show that the fluctuating second law can be rephrased as a Generalized Quantum Detailed Balance condition (GQDB), i.e., a symmetry of the time-local generators which ensures the validity of the fluctuation theorem. When requiring in addition a strict system-bath energy conservation, the GQDB reduces to the usual notion of detailed balance which ensures Q…
The Lindblad form guarantees complete positivity of a Markovian quantum master equation (QME). However, its microscopic derivation for a quantum system weakly interacting with a thermal bath requires several approximations, which may result in inaccuracies in the QME. Recently, various Lindbladian QMEs were derived without resorting to the secular approximation from the Redfield equation which does not guarantee the complete positivity. Here we explicitly calculate, in a perturbative manner, the equilibrium steady states of these Lindbladian QMEs. We compare the results with the steady state o…
Exact solution of Markovian master equations for quadratic fermi systems: thermal baths, open XY spin chains, and non-equilibrium phase transition
本文将三次量子化方法推广到用4 n × 4 n矩阵对角化的方法统一精确地处理n个费米子二次系统的Redfield和Lindblad主方程。从Redfield方程出发，详细分析了非平衡热驱动。我们解释如何计算所有物理相关的量，如非平衡期望值的本地观测，各种熵或信息措施，或时间演化和性质的放松。我们还讨论了如何确切地处理显式时间相关的问题。然后，一般的形式主义，研究热驱动的开放XY自旋1/2链。我们发现，最近提出的非平衡量子相变在开放的XY链生存的Redfield模型内的热驱动。特别是，长程磁相关的相位可以通过非平衡稳态对外部（浴或体）参数的超敏性来表征。研究的热传输，我们发现足够强的热驱动的负热导，以及非单调依赖的热流浴耦合的强度。
The Hamiltonian of mean force is a widely used concept to describe the modification of the usual canonical Gibbs state for a quantum system whose coupling strength with the thermal bath is non-negligible. Here we perturbatively derive general approximate expressions for the Hamiltonians of mean force in the weak-coupling approximation and in the high-temperature one. We numerically analyse the accuracy of the corresponding expressions and show that the precision of the Bloch-Redfield equantum master equation can be improved if we replace the original system Hamiltonian by the Hamiltonian of me…
Understanding nonsecular dynamics in open quantum systems is addressed here, with emphasis on systems with large numbers of Bohr frequencies, zero temperature, and fast driving. We employ the master equation, which replaces arithmetic averages of the decay rates in the open system, with their geometric averages, and find that it can improve the second order perturbation theory, known as the Redfield equation, while enforcing complete positivity on quantum dynamics. The characteristic frequency scale that governs the approximation is the minimax frequency: the minimum of the maximum system osci…
We put forth a new class of quantum master equations that correctly reproduce the asymptotic state of an open quantum system beyond the infinitesimally weak system-bath coupling limit. Our method is based on incorporating the knowledge of the reduced steady state into its dynamics. The correction not only steers the reduced system towards a correct steady state but also improves the accuracy of the dynamics, thereby refining the archetypal Born-Markov weak-coupling second-order master equations. In case of equilibrium, since a closed form for the steady state exists in terms of a mean force Gi…
Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, w…
In the framework of theory of open quantum systems, we derive quantum master equations for the ultrastrong system-bath coupling regime and, more generally, the strong-decoherence regime. In this regime, the strong decoherence is complemented by slow relaxation processes. We use a generalization of the Foerster and modified Redfield peturbation theories known in theory of excitation energy transfer. Also, we show that the mean force Gibbs state in the corresponding limits are stationary for the derived master equations.
Master equations are a vital tool to model heat flow through nanoscale thermodynamic systems. Most practical devices are made up of interacting sub-system, and are often modelled using either local master equations (LMEs) or global master equations (GMEs). While the limiting cases in which either the LME or the GME breaks down are well understood, there exists a 'grey area' in which both equations capture steady-state heat currents reliably, but predict very different transient heat flows. In such cases, which one should we trust? Here, we show that, when it comes to dynamics, the local approa…
Here, we study a minimal model, the three-level V system coupled to two heat baths, and investigate the role of quantum coherences in heat transport in both the transient regime and in the nonequilibrium steady-state. In our model, energy is exchanged between the baths through two parallel pathways, which can be made distinct through the nondegeneracy of excited levels (energy splitting $Δ$) and a control parameter $α$, which adjusts the strength of one of the arms. Using a nonsecular quantum master equation of Redfield form, we succeed in deriving closed-form expressions for the quantum coher…
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a…
There is a remarkable characteristic of photosynthesis in nature, that is, the energy transfer efficiency is close to 100%. Recently, due to the rapid progress made in the experimental techniques, quantum coherent effects have been experimentally demonstrated. Traditionally, the incoherent theories are capable of calculating the energy transfer efficiency, e.g., (generalized) Förster theory and modified Redfield theory. However, in order to describe the quantum coherent effects in photosynthesis, the coherent theories have been developed, such as hierarchical equation of motion, quantum path i…
In a recent Letter, Guarnieri et al. [1] reported on the possibility of generating steady-state coherence (SSC) in the energy basis of a two-level system by the sole action of a single thermal bath also for weak system-bath coupling and due to a composite form of this interaction. The results are based on a time-local Bloch-Redfield master equation without secular approximation and the associated set of Bloch equations, all derived in the Supplemental Material (SM) of Ref. [1]. We comment that the results reported by the authors present both technical flaws and conceptual limitations that subs…
We address the structure of the Liouvillian superoperator for a broad class of bosonic and fermionic Markovian open systems interacting with stationary environments. We show that the accurate application of the partial secular approximation in the derivation of the Bloch-Redfield master equation naturally induces a symmetry on the superoperator level, which may greatly reduce the complexity of the master equation by decomposing the Liouvillian superoperator into independent blocks. Moreover, we prove that, if the steady state of the system is unique, one single block contains all the informati…
It is very common in the literature to write down a Markovian quantum master equation in Lindblad form to describe a system with multiple degrees of freedom and weakly connected to multiple thermal baths which can, in general, be at different temperatures and chemical potentials. However, the microscopically derived quantum master equation up to leading order in system-bath coupling is of the so-called Redfield form which is known to not preserve complete positivity in most cases. Additional approximations to the Redfield equation are required to obtain a Lindblad form. We lay down some fundam…
Solids can be cooled by driving impurity ions with lasers, allowing them to transfer heat from the lattice phonons to the electromagnetic surroundings. This exemplifies a quantum thermal machine, which uses a quantum system as a working medium to transfer heat between reservoirs. We review the derivation of the Bloch-Redfield equation for a quantum system coupled to a reservoir, and its extension, using counting fields, to calculate heat currents. We use the full form of this equation, which makes only the weak-coupling and Markovian approximations, to calculate the cooling power for a simple …
