Quantum Master Equation Research Articles

Quantum theory of a potential biological magnetic field sensor: Radical pair mechanism in flavin adenine dinucleotide biradicals

Recent studies in vitro and in vivo suggest that flavin adenine dinucleotide (FAD) on its own might be able to act as a biological magnetic field sensor. Motivated by these observations, in this study, we develop a detailed quantum theoretical model for the radical pair mechanism (RPM) for the flavin adenine biradical within the FAD molecule. We use the results of existing molecular dynamics simulations to determine the time-varying distance between the radicals on FAD, which we then feed into a quantum master equation treatment of the RPM. In contrast to previous semi-classical models, which are limited to the low-field and high-field cases, our quantum model can predict the full magnetic field dependence of the transient absorption signal. Our model's predictions are consistent with experiments at physiological pH values.

Simulating Non-Markovian Dynamics in Multidimensional Electronic Spectroscopy via Quantum Algorithm.

Including the effect of the molecular environment in the numerical modeling of time-resolved electronic spectroscopy remains an important challenge in computational spectroscopy. In this contribution, we present a general approach for the simulation of the optical response of multichromophore systems in a structured environment and its implementation as a quantum algorithm. A key step of the procedure is the pseudomode embedding of the system-environment problem resulting in a finite set of quantum states evolving according to a Markovian quantum master equation. This formulation is then solved by a collision model integrated into a quantum algorithm designed to simulate linear and nonlinear response functions. The workflow is validated by simulating spectra for the prototypical excitonic dimer interacting with fast (memoryless) and finite-memory environments. The results demonstrate, on the one hand, the potential of the pseudomode embedding for simulating the dynamical features of nonlinear spectroscopy, including lineshape, spectral diffusion, and relaxations along delay times. On the other hand, the explicit synthesis of quantum circuits provides a fully quantum simulation protocol of nonlinear spectroscopy harnessing the efficient quantum simulation of many-body dynamics promised by the future generation of fault-tolerant quantum computers.

Differentiable master equation solver for quantum device characterization

Differentiable models of physical systems provide a powerful platform for gradient-based algorithms, with particular impact on parameter estimation and optimal control. Quantum systems present a particular challenge for such characterization and control, owing to their inherently stochastic nature and sensitivity to environmental parameters. To address this challenge, we present a versatile differentiable quantum master equation solver facilitating direct computation of steady-state solutions, and incorporate this solver into a framework for device characterization capable of dealing with additional nondifferentiable parameters. Our approach utilizes gradient-based optimization and Bayesian inference to provide estimates and uncertainties in quantum device parameters. To showcase our approach, we consider steady-state charge transport through electrostatically defined quantum dots. Using simulated data, we demonstrate efficient estimation of parameters for a single quantum dot, and model selection as well as the capability of our solver to compute time evolution for a double quantum dot system. Our differentiable solver stands to widen the impact of physics-aware machine learning algorithms on quantum devices for characterization and control. Published by the American Physical Society 2024

Skewed-Gaussian model of small-photon-number coherent Ising machines

A Gaussian quantum theory of bosonic modes has been widely used to describe quantum optical systems, including coherent Ising machines (CIMs) that consist of χ(2) degenerate optical parametric oscillators (DOPOs) as nonlinear elements. However, Gaussian models have been thought to be invalid in the extremely strong-gain-saturation limit. Here, we develop an extended Gaussian model including two third-order fluctuation products, 〈δX̂3〉 and 〈δX̂δP̂2〉, which we call self-skewness and cross-skewness, respectively. This new model which we call the skew-Gaussian model more precisely replicates the success probability predicted by the quantum master equation (QME), relative to Gaussian models. We also discuss the impact of skew variables on the performance of CIMs. Published by the American Physical Society 2024

How Sophisticated Are Neural Networks Needed to Predict Long-Term Nonadiabatic Dynamics?

Nonadiabatic dynamics is key for understanding solar energy conversion and photochemical processes in condensed phases. This often involves the non-Markovian dynamics of the reduced density matrix in open quantum systems, where knowledge of the system's prior states is necessary to predict its future behavior. In this study, we explore time-series machine learning methods for predicting long-time nonadiabatic dynamics based on short-time input data, comparing these methods with the physics-based transfer tensor method (TTM). To understand the impact of memory time on these approaches, we demonstrate that non-Markovian dynamics can be represented as a linear map within the Nakajima-Zwanzig generalized quantum master equation framework. We further propose a practical method to estimate the effective memory time, within a given tolerance, for reduced density matrix propagation. Our predictive models are applied to various physical systems, including spin-boson models, multistate harmonic (MSH) models with Ohmic spectral densities and for a realistic organic photovoltaic system composed of a carotenoid-porphyrin-fullerene triad dissolved in tetrahydrofuran. Results indicate that the simple linear-mapping fully connected neural network (FCN) outperforms the more complicated nonlinear-mapping networks including the gated recurrent unit (GRU) and the convolutional neural network/long short-term memory (CNN-LSTM) in systems with short memory times, such as spin-boson and MSH models. Conversely, the nonlinear CNN-LSTM and GRU models yield higher accuracy in the triad MSH systems characterized by long memory times. These findings offer valuable insights into the role of effective memory time in non-Markovian quantum dynamics, providing practical guidance for the application of time-series machine learning models to complex chemical systems.

Complete positivity and thermal relaxation in quadratic quantum master equations

Complete positivity and thermal relaxation in quadratic quantum master equations

Reduced density matrix dynamics in multistate harmonic models via time-convolution and time-convolutionless quantum master equations with quantum-mechanical and semiclassical kernels.

In this work, we explore the electronic reduced density matrix (RDM) dynamics using time-convolution (TC) and time-convolutionless (TCL) quantum master equations (QMEs) that are based on perturbative electronic couplings within the framework of multistate harmonic (MSH) models. The MSH model Hamiltonian consistently incorporates the electronic-vibrational correlations between all pairs of states by satisfying the pairwise reorganization energies directly obtained from all-atom simulations, representing the globally heterogeneous environments that couple to the multiple states differently. We derive the exact quantum-mechanical and a hierarchy of semiclassical approximate expressions for the kernels in TC and TCL QMEs that project the full RDM for general shifted harmonic systems, including the MSH model. These QMEs are applied to simulate RDM dynamics of photoinduced charge transfer (PICT) in organic photovoltaic carotenoid-porphyrin-fullerene triad solvated in tetrahydrofuran solution and the excitation energy transfer (EET) dynamics in photosynthetic Fenna-Matthews-Olson complexes from C. tepidum and P. aestuarii. Our results show that while both TC and TCL QMEs capture similar phenomena in PICT and EET processes, TC QME generally provides more accurate results than TCL QME, particularly in the initial oscillation of EET population dynamics. This study highlights the effectiveness of the TC and TCL QMEs in modeling RDM dynamics of nonadiabatic processes, offering insights for realistic condensed phase systems.

Fast and robust production of quantum superposition states by the fractional shortcut to adiabaticity

The fractional shortcut to adiabaticity (f-STA) for the production of quantum superposition states is proposed firstly via a three-level system with a Λ-type linkage pattern and a four-level system with a tripod structure. The fast and robust production of the coherent superposition states is studied by comparing the populations for the f-STA and the fractional stimulated Raman adiabatic passage (f-STIRAP). The states with equal proportions can be produced by fixing the controllable parameters of the driving pulses at the final moment of the whole process. The effects of the pulse intensity and the time delay of the pulses on the production process are discussed by monitoring the populations on all of the quantum states. In particular, the spontaneous emission arising from the intermediate state is investigated by the quantum master equation. The result reveals that the f-STA exhibits superior advantages over the f-STIRAP in producing the superposition states.

Continuous-variable electromechanical quantum thermal transistors

Abstract We present a scheme to realize quantum thermal transistor effects in a continuous-variable electromechanical system including two microwave cavities and one mechanical oscillator. The thermal noise fluxes between the quantum system and its baths are evaluated by quantum master equation. It is shown that the thermal noise flux at one microwave cavity as an emitter can be dissipated into the other as a collector by combining the heating Stokes and cooling anti-Stokes processes. The indirect energy transfers between the two microwave modes can be significantly amplified by small energy changes at the mechanical oscillator as the base. The extremely high amplification depends sensitively on the detunings of the two microwave modes, which provides a new tool for precision measurements. This study opens the door for constructing quantum thermal transistors using various continuous-variable systems and is well accessible based on current experimental techniques.

Exact finite-time correlation functions for multiterminal setups: Connecting theoretical frameworks for quantum transport and thermodynamics

Transport in open quantum systems can be explored through various theoretical frameworks, including the quantum master equation, scattering matrix, and Heisenberg equation of motion. The choice of framework depends on factors such as the presence of interactions, the coupling strength between the system and environment, and whether the focus is on steady-state or transient regimes. Existing literature mainly treats these frameworks independently. Our work establishes connections between them by clarifying the role and status of these approaches using two paradigmatic models for single and multipartite quantum systems in a two-terminal setup under voltage and temperature biases. We derive analytical solutions in both steady-state and transient regimes for the populations, currents, and current correlation functions. Exact results from the Heisenberg equation are shown to align with scattering matrix and master equation approaches within their respective validity regimes. Crucially, we establish a protocol for the weak-coupling limit, bridging the applicability of master equations at weak-coupling with Heisenberg or scattering matrix approaches at arbitrary coupling strength. Published by the American Physical Society 2024

Dynamics of a small quantum system open to a bath with thermostat.

We investigate dynamics of a small quantum system open to a bath with thermostat. We introduce another bath, called a superbath, weakly coupled with the bath to provide it with a thermostat, which has either the Lindblad or Redfield type. We treat the interaction between the system and bath via a rigorous perturbation theory. Due to the thermostat, the bath behaves dissipative and stochastic, for which the usual Born-Markov assumption is not needed. We consider a specific example of a harmonic oscillator system of interest and a photonic bath in a large container, and a superbath of the Caldeira-Legget oscillators distributed on the inner surface of the container. After taking the trace over the superbath states, we use the P representation for the total harmonic system of the system and bath. We derive the reduced time-evolution equationfor the system by explicitly finding the correlation between the system and bath beyond the product state that was not obtainable in the previous theory for the system and bath isolated from environment, and marginalizing bath degrees of freedom. Remarkably, the associated dynamic equationfor the system density matrix is of the same form as the Redfield master equationwith different coefficients depending on thermostat used. We find the steady state does not depend on the thermostat but the time-dependent state does, which agrees with common expectation. We expect to apply our theory to general systems. Unlike the usual quantum master equations, our reduced dynamics allows investigation for time-dependent protocols, and nonequilibrium quantum stochastic dynamics will be investigated in the future.

Dynamics of high-dimensional quantum systems coupled to a harmonic bath. General theory and implementation via multiconfigurational wave packets and truncated hierarchical equations for the mean-fields.

Modeling the dynamics of a quantum system coupled to a dissipative environment becomes particularly challenging when the system's dimensionality is too high to permit the computation of its eigenstates. This problem is addressed by introducing an eigenstate-free formalism, where the open quantum system is represented as a mixture of high-dimensional, time-dependent wave packets governed by coupled Schrödinger equations, while the environment is described by a multi-component quantum master equation. An efficient computational implementation of this formalism is presented, employing a variational mixed Gaussian/multiconfigurational time-dependent Hartree (G-MCTDH) ansatz for the wave packets and propagating the environment dynamics via hierarchical equations, truncated at the first or second level of the hierarchy. The effectiveness of the proposed methodology is demonstrated on a 61-dimensional model of phonon-driven vibrational relaxation of an adsorbate. G-MCTDH calculations on 4- and 10-dimensional reduced models, combined with truncated hierarchical equations for the mean fields, nearly quantitatively replicate the full-dimensional quantum dynamical results on vibrational relaxation while significantly reducing the computational time. This approach thus offers a promising quantum dynamical method for modeling complex system-bath interactions, where a large number of degrees of freedom must be explicitly considered.

Connected sum for modular operads and Beilinson–Drinfeld algebras

Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. Motivated by this example, we introduce a notion of connected sum for general modular operads. We show that a connected sum induces a commutative product on the space of functions associated to the modular operad. Moreover, we combine this product with Barannikov’s non-commutative Batalin–Vilkovisky structure present on this space of functions, obtaining a Beilinson–Drinfeld algebra. Finally, we study the quantum master equation using the exponential defined using this commutative product.

BRST covariant phase space and holographic Ward identities

This paper develops an enlarged BRST framework to treat the large gauge transformations of a given quantum field theory. It determines the associated infinitely many Noether charges stemming from a gauge fixed and BRST invariant Lagrangian, a result that cannot be obtained from Noether’s second theorem. The geometrical significance of this result is highlighted by the construction of a trigraded BRST covariant phase space, allowing a BRST invariant gauge fixing procedure. This provides an appropriate framework for determining the conserved BRST Noether current of the global BRST symmetry and the associated global Noether charges. The latter are found to be equivalent with the usual classical corner charges of large gauge transformations. It allows one to prove the gauge independence of their physical effects at the perturbative quantum level. In particular, the underlying BRST fundamental canonical relation provides the same graded symplectic brackets as in the classical covariant phase space. A unified Lagrangian Ward identity for small and large gauge transformations is built. It consistently decouples into a bulk part for small gauge transformations, which is the standard BRST-BV quantum master equation, and a boundary part for large gauge transformations. The boundary part provides a perturbation theory origin for the invariance of the Hamiltonian physical -matrix under asymptotic symmetries. Holographic anomalies for the boundary Ward identity are studied and found to be solutions of a codimension one Wess-Zumino consistency condition. Such solutions are studied in the context of extended BMS symmetry. Their existence clarifies the status of the 1-loop correction to the subleading soft graviton theorem.

Microscopic analysis of relaxation behavior in nonlinear optical conductivity of graphene

We produce a general formalism to study the interband dynamical optical conductivity in the nonlinear regime of graphene in the presence of a quantum bath comprising phonons and electrons. When a quantum solid of graphene is subjected to an intense electric field in the optical frequency range, the observation of a nonlinear response is facilitated by formulating a quantum master equation of the density operator associated with the Hamiltonian encapsulated in a spin-boson model of dissipative quantum statistical mechanics. Our results reveal the nonlinear steady-state regime’s population inversion and decoherence. The present method enables us to investigate further the nonlinear interband optical conductivity of pristine and gapped graphene characterized by a single dimensionless parameter at finite temperatures. Different bath spectra’ effects on phonons and electrons are examined in detail. The temperature dependence of conductivity reveals that changing temperature can enable us to make a transition from the linear to the nonlinear regime for fixed optical field parameters. Interestingly, a fascinating switching-like behavior is observed for the low-temperature optical conductivity of the gapped graphene while we vary the energy gap as well as the frequency of the externally applied field. Although our general formulation can address a variety of nonequilibrium responses of the two-band system, it also facilitates a connection with the phenomenological modeling of nonlinear optical conductivity.

Quantum control of polariton emission in a microcavity-quantum well system under magnetic field

In this work, a quantum dissipative model is employed to investigate the influence of a perpendicular magnetic field on the photoluminescence (PL) spectrum of a quantum well embedded within a microcavity. This model incorporates both the exact electron-hole interaction within the semiconductor and the light-matter coupling between the fundamental photonic mode and the fermionic particles. The loss and pumping mechanisms are described using the quantum master equation, and the PL spectrum is determined via the quantum regression theorem. Our findings demonstrate that the magnetic field acts as a control mechanism in the polariton emission energy, the emission linewidth and the intensity distribution along the emission line. Finally, it is observed that the magnetic field can redistribute the density matrix occupations leading to modifications in the average number of polaritons in the system.

Extended perturbative approach including Redfield and Förster limits for qualitative analysis of exciton dynamics in any photosynthetic light harvesting and reaction center

According to many reports, the various structures of photosynthetic light-harvesting/reaction-center complexes and their molecular-dynamics simulations necessitate a numerically efficient and quality-conserved theory of excitation energy transfer and exciton relaxation in large pigment systems. Although exciton dynamics depend on various parameters, such as exciton coupling strength, exciton–phonon coupling, site energy values for each pigment, and temperature, classifying the transition mechanism for any Hamiltonian into perturbatively delocalized or localized theories is challenging. In this study, perturbative quantum master equations of a reduced density matrix for any orthogonal transformation similar to the coherent modified Redfield theory are derived. Our approach qualitatively conserves the dynamics of relevant perturbative approximations in each limiting case. As an application, any orthogonal transformation of a relevant system is optimized using the average of the square of interactions between orthogonal state transitions. The numerical results for two pigment systems are compared with the limiting formalisms of the modified Redfield and Förster theory.

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

Non-equilibrium rate theory for polariton relaxation dynamics.

We derive an analytic expression of the non-equilibrium Fermi's golden rule (NE-FGR) expression for a Holstein-Tavis-Cumming Hamiltonian, a universal model for many molecules collectively coupled to the optical cavity. These NE-FGR expressions capture the full-time-dependent behavior of the rate constant for transitions from polariton states to dark states. The rate is shown to be reduced to the well-known frequency domain-based equilibrium Fermi's golden rule (E-FGR) expression in the equilibrium and collective limit and is shown to retain the same scaling with the number of sites in non-equilibrium and non-collective cases. We use these NE-FGR to perform population dynamics with a time-non-local and time-local quantum master equation and obtain accurate population dynamics from the initially occupied upper or lower polariton states. Furthermore, NE-FGR significantly improves the accuracy of the population dynamics when starting from the lower polariton compared to the E-FGR theory, highlighting the importance of the non-Markovian behavior and the short-time transient behavior in the transition rate constant.

On the existence of heterotic-string and type-II-superstring field theory vertices

We consider the problem of the existence of heterotic-string and type-II-superstring field theory vertices in the product of spaces of bordered surfaces parameterizing the left- and right-moving sectors of these theories. It turns out that this problem can be solved by proving the existence of a solution to the BV quantum master equation in moduli spaces of bordered spin-Riemann surfaces. We first prove that for arbitrary genus ▪, ▪ Neveu–Schwarz boundary components, and ▪ Ramond boundary components such solutions exist. We also prove that these solutions are unique up to homotopy in the category of BV algebras. Furthermore, we prove that there exists a map in this category under which these solutions are mapped to fundamental classes of Deligne-Mumford stacks of associated punctured spin-Riemann surfaces. These results generalize the work of Costello on the existence of a solution to the BV quantum master equations in moduli spaces of bordered Riemann surfaces which, through the work of Sen and Zwiebach, are related to the existence of bosonic-string vertices, and their relation to fundamental classes of Deligne-Mumford stacks of associated punctured Riemann surfaces. Using the existence of solutions to the BV quantum master equation in moduli spaces of spin-Riemann surfaces, we prove that heterotic-string and type-II-superstring field theory vertices, for arbitrary genus ▪ and an arbitrary number of any type of boundary components, exist. Furthermore, we prove the existence of a solution to the BV quantum master equation in spaces of bordered N=1 super-Riemann surfaces for arbitrary genus ▪, ▪ Neveu–Schwarz boundary components, and ▪ Ramond boundary components.