Showing papers on "Master equation published in 2008"
TL;DR: An open quantum system, the time evolution of which is governed by a master equation, can be driven into a given pure quantum state by an appropriate design of the coupling between the system and t...
Abstract: An open quantum system, the time evolution of which is governed by a master equation, can be driven into a given pure quantum state by an appropriate design of the coupling between the system and t ...
969 citations
TL;DR: From a snapshot of a quantum evolution, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which a computable measure of "Markovianity" is introduced.
Abstract: We investigate what a snapshot of a quantum evolution--a quantum channel reflecting open system dynamics--reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of "Markovianity" is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results also clarify the geometry of the set of quantum channels with respect to being solutions of time (in)dependent master equations.
512 citations
TL;DR: The coarse master equations for peptide folding dynamics are constructed from atomistic molecular dynamics simulations and give access to the slow conformational dynamics but also shed light on the molecular mechanisms of the helix-coil transition.
Abstract: We construct coarse master equations for peptide folding dynamics from atomistic molecular dynamics simulations. A maximum-likelihood propagator-based method allows us to extract accurate rates for the transitions between the different conformational states of the small helix-forming peptide Ala5. Assigning the conformational states by using transition paths instead of instantaneous molecular coordinates suppresses the effects of fast non-Markovian dynamics. The resulting master equations are validated by comparing their analytical correlation functions with those obtained directly from the molecular dynamics simulations. We find that the master equations properly capture the character and relaxation times of the entire spectrum of conformational relaxation processes. By using the eigenvectors of the transition rate matrix, we are able to systematically coarse-grain the system. We find that a two-state description, with a folded and an unfolded state, roughly captures the slow conformational dynamics. A f...
485 citations
TL;DR: In this paper, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories.
Abstract: Stochastic thermodynamics provides a framework for describing small systems like colloids or biomolecules driven out of equilibrium but still in contact with a heat bath. Both, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories. Various exact relations involving the distribution of such quantities like integral and detailed fluctuation theorems for total entropy production and the Jarzynski relation follow from such an approach based on Langevin dynamics. Analogues of these relations can be proven for any system obeying a stochastic master equation like, in particular, (bio)chemically driven enzyms or whole reaction networks. The perspective of investigating such relations for stochastic field equations like the Kardar-Parisi-Zhang equation is sketched as well.
462 citations
TL;DR: The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.
Abstract: A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantu...
361 citations
TL;DR: Vidal et al. as mentioned in this paper extended the infinite time-evolving block decimation algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as (translationally invariant) tensor networks.
Abstract: The infinite time-evolving block decimation algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional (1D) quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a (translationally invariant) tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open (i.e., master equation) dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional (2D) classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.
359 citations
TL;DR: Numerical tests show that quantum coherence can cause significant changes in steady state donor/acceptor populations from those predicted by the FD theory and illustrate delicate cooperation of nonequilibrium and quantum coherent effects on the transient population dynamics.
Abstract: A theory of coherent resonance energy transfer is developed combining the polaron transformation and a time-local quantum master equation formulation, which is valid for arbitrary spectral densities including common modes. The theory contains inhomogeneous terms accounting for nonequilibrium initial preparation effects and elucidates how quantum coherence and nonequilibrium effects manifest themselves in the coherent energy transfer dynamics beyond the weak resonance coupling limit of the Forster and Dexter (FD) theory. Numerical tests show that quantum coherence can cause significant changes in steady state donor/acceptor populations from those predicted by the FD theory and illustrate delicate cooperation of nonequilibrium and quantum coherence effects on the transient population dynamics.
296 citations
TL;DR: In this paper, the Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n?4n matrix, provided that all bath operators are linear in the fermionic variables.
Abstract: The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n?4n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states (NESS) and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbour Heisenberg XY spin-1/2 chain in a transverse magnetic field.
289 citations
TL;DR: In this article, the authors investigated the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps.
Abstract: We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.
289 citations
TL;DR: In this article, the Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n x 4n matrix, provided that all bath operators are linear in the fermionic variables.
Abstract: The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n x 4n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbor Heisenberg XY spin 1/2 chain in a transverse magnetic field.
276 citations
TL;DR: In this article, the authors obtained general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q = 1−p to the left.
Abstract: In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice $${\mathbb{Z}}$$
with nearest neighbor hopping rates p to the right and q = 1−p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the $$N\to\infty$$
limit. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schutz for the case N = 2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m
th left-most particle is at x at time t. In one of these formulas we can take the $$N\to\infty$$
limit, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.
TL;DR: In this paper, it is shown that the Wangsness-bloch-Redfield master equation is equivalent to the master equation derived by K\"onig et al. [Phys. Rev. 76, 1715 (1996); Phys. Lett. B 54, 16820 (1996)].
Abstract: An important class of approaches to the description of electronic transport through molecules and quantum dots is based on the master equation. We discuss various formalisms for deriving a master equation and their interrelations. It is shown that the master equation derived by K\"onig et al. [Phys. Rev. Lett. 76, 1715 (1996); Phys. Rev. B 54, 16820 (1996)] is equivalent to the Wangsness-Bloch-Redfield master equation. The roles of the large-reservoir and Markov approximations are clarified. At low temperatures, when the quasiparticle lifetime becomes large, the Markov approximation can be derived from the assumption of weak tunneling under certain conditions. Interactions in the leads are shown to be irrelevant for the transport in the case of momentum-independent tunneling. It is explained why the $T$-matrix formalism gives incomplete results except for diagonal density operators to second order in the tunneling amplitudes. The time-convolutionless master equation is adapted to tunneling problems and a diagrammatic scheme for generating arbitrary orders in the tunneling amplitudes is developed.
TL;DR: In this paper, a nonperturbation theory for describing decoherence dynamics of electron charges in a double quantum dot gated by electrodes is developed. But the authors do not consider the back-reaction of the reservoirs being fully taken into account.
Abstract: In this paper, we develop a nonperturbation theory for describing decoherence dynamics of electron charges in a double quantum dot gated by electrodes. We extend the Feynman-Vernon influence functional theory to fermionic environments and derive an exact master equation for the reduced density matrix of electrons in the double dot for a general spectral density at arbitrary temperature and bias. We then investigate the decoherence dynamics of the double-dot charge qubit with backreaction of the reservoirs being fully taken into account. Time-dependent fluctuations and leakage effects induced from the dot-reservoir coupling are explicitly explored. The charge qubit dynamics from the Markovian to non-Markovian regime is systematically studied under various manipulating conditions. The decay behavior of charge qubit coherence and the corresponding relaxation time ${T}_{1}$ and dephasing time ${T}_{2}$ are analyzed in detail.
TL;DR: The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques, and explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
Abstract: We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
TL;DR: It is demonstrated that the controlled generation of Fock states with up to 15 photons in a microwave coplanar waveguide resonator coupled to a superconducting phase qubit can be described by a master equation where the lifetime of the n-photon Fock state scales as 1/n, in agreement with theory.
Abstract: We demonstrate the controlled generation of Fock states with up to 15 photons in a microwave coplanar waveguide resonator coupled to a superconducting phase qubit. The subsequent decay of the Fock states, due to dissipation, is then monitored by varying the time delay between preparing the state and performing a number-state analysis. We find that the decay dynamics can be described by a master equation where the lifetime of the n-photon Fock state scales as 1/n, in agreement with theory. We have also generated a coherent state in the microwave resonator, and monitored its decay process. We demonstrate that the coherent state maintains a Poisson distribution as it decays, with an average photon number that decreases with the same characteristic decay time as the one-photon Fock state.
TL;DR: A rigorous statistical method is proposed to approximate the complete statistical distribution of any observable of an MD simulation provided that one can identify conformational substates such that the transition process between them may be modeled with a memoryless jump process.
Abstract: Molecular dynamics (MD) simulations can be used to estimate transition rates between conformational substates of the simulated molecule. Such an estimation is associated with statistical uncertainty, which depends on the number of observed transitions. In turn, it induces uncertainties in any property computed from the simulation, such as free energy differences or the time scales involved in the system’s kinetics. Assessing these uncertainties is essential for testing the reliability of a given observation and also to plan further simulations in such a way that the most serious uncertainties will be reduced with minimal effort. Here, a rigorous statistical method is proposed to approximate the complete statistical distribution of any observable of an MD simulation provided that one can identify conformational substates such that the transition process between them may be modeled with a memoryless jump process, i.e., Markov or Master equation dynamics. The method is based on sampling the statistical distr...
TL;DR: An extensive set of equilibrium and kinetic data is presented and analyzed for an ultrafast folding protein—the villin subdomain and the theoretical model provides a detailed picture of the free-energy surface and a residue-by-residue description of the evolution of the folded structure, yet contains many fewer adjustable parameters than either the chemical- or physical-kinetics models.
Abstract: An extensive set of equilibrium and kinetic data is presented and analyzed for an ultrafast folding protein—the villin subdomain. The equilibrium data consist of the excess heat capacity, tryptophan fluorescence quantum yield, and natural circular-dichroism spectrum as a function of temperature, and the kinetic data consist of time courses of the quantum yield from nanosecond-laser temperature-jump experiments. The data are well fit with three kinds of models—a three-state chemical-kinetics model, a physical-kinetics model, and an Ising-like theoretical model that considers 105 possible conformations (microstates). In both the physical-kinetics and theoretical models, folding is described as diffusion on a one-dimensional free-energy surface. In the physical-kinetics model the reaction coordinate is unspecified, whereas in the theoretical model, order parameters, either the fraction of native contacts or the number of native residues, are used as reaction coordinates. The validity of these two reaction coordinates is demonstrated from calculation of the splitting probability from the rate matrix of the master equation for all 105 microstates. The analysis of the data on site-directed mutants using the chemical-kinetics model provides information on the structure of the transition-state ensemble; the physical-kinetics model allows an estimate of the height of the free-energy barrier separating the folded and unfolded states; and the theoretical model provides a detailed picture of the free-energy surface and a residue-by-residue description of the evolution of the folded structure, yet contains many fewer adjustable parameters than either the chemical- or physical-kinetics models.
TL;DR: In this paper, the authors examine the methodology from a rigorous point of view, discussing where it can be expected to work, and where it should fail, where the Kohn-Sham levels are misaligned.
Abstract: Density functional calculations for the electronic conductance of single molecules are now common. We examine the methodology from a rigorous point of view, discussing where it can be expected to work, and where it should fail. When molecules are weakly coupled to leads, local and gradient-corrected approximations fail, as the Kohn–Sham levels are misaligned. In the weak bias regime, exchange–correlation corrections to the current are missed by the standard methodology. For finite bias, a new methodology for performing calculations can be rigorously derived using an extension of time-dependent current density functional theory from the Schrodinger equation to a master equation.
TL;DR: In this article, an ultracold gas of neutral atoms subject to the periodic optical potential generated by a high-Q cavity mode is studied, where different routes to derive approximative multiparticle Hamiltonians in Bose-Hubbard form with rescaled or even dynamical parameters are derived.
Abstract: We study an ultracold gas of neutral atoms subject to the periodic optical potential generated by a high-Q cavity mode. In the limit of very low temperatures, cavity field and atomic dynamics require a quantum description. Starting from a cavity QED single atom Hamiltonian we use different routes to derive approximative multiparticle Hamiltonians in Bose-Hubbard form with rescaled or even dynamical parameters. In the limit of large enough cavity damping the different models agree. Compared to free space optical lattices, quantum uncertainties of the potential and the possibility of atom-field entanglement lead to modified phase transition characteristics, the appearance of new phases or even quantum superpositions of different phases. Using a corresponding effective master equation, which can be numerically solved for few particles, we can study time evolution including dissipation. As an example we exhibit the microscopic processes behind the transition dynamics from a Mott insulator like state to a self-ordered superradiant state of the atoms, which appears as steady state for transverse atomic pumping.
TL;DR: The exact master equation for two, and its generalization to N, harmonic oscillators interacting with a general environment are expected to be useful for the analysis of quantum coherence, entanglement, fluctuations, and dissipation of mesoscopic objects toward the construction of a theoretical framework for macroscopic quantum phenomena.
Abstract: In this paper we derive an exact master equation for two coupled quantum harmonic oscillators interacting via bilinear coupling with a common environment at arbitrary temperature made up of many harmonic oscillators with a general spectral density function. We first show a simple derivation based on the observation that the two harmonic oscillator model can be effectively mapped into that of a single harmonic oscillator in a general environment plus a free harmonic oscillator. Since the exact one harmonic oscillator master equation is available [B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992)], the exact master equation with all its coefficients for this two harmonic oscillator model can be easily deduced from the known results of the single harmonic oscillator case. In the second part we give an influence functional treatment of this model and provide explicit expressions for the evolutionary operator of the reduced density matrix which are useful for the study of decoherence and disentanglement issues. We show three applications of this master equation: on the decoherence and disentanglement of two harmonic oscillators due to their interaction with a common environment under Markovian approximation, and a derivation of the uncertainty principle at finite temperature for a composite object, modeled by two interacting harmonic oscillators. The exact master equation for two, and its generalization to N, harmonic oscillators interacting with a general environment are expected to be useful for the analysis of quantum coherence, entanglement, fluctuations, and dissipation of mesoscopic objects toward the construction of a theoretical framework for macroscopic quantum phenomena.
TL;DR: In this paper, a non-Boltzmann model of the radiating atomic and molecular electronic states present in lunar-return shock-layers is presented, and a novel approach of curve-fitting the non Boltzmann populations of radiating atoms and molecules is developed.
Abstract: This paper investigates the non-Boltzmann modeling of the radiating atomic and molecular electronic states present in lunar-return shock-layers. The Master Equation is derived for a general atom or molecule while accounting for a variety of excitation and de-excitation mechanisms. A new set of electronic-impact excitation rates is compiled for N, O, and N2+, which are the main radiating species for most lunar-return shock-layers. Based on these new rates, a novel approach of curve-fitting the non-Boltzmann populations of the radiating atomic and molecular states is developed. This new approach provides a simple and accurate method for calculating the atomic and molecular non-Boltzmann populations while avoiding the matrix inversion procedure required for the detailed solution of the Master Equation. The radiative flux values predicted by the present detailed non-Boltzmann model and the approximate curve-fitting approach are shown to agree within 5% for the Fire 1634 s case.
TL;DR: Dramatic improvements are reported by applying the quasi-steady-state approximation of the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and to the finite state projection algorithm.
Abstract: Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis–Menten enzyme kinetics, double phosphorylation, the Goldbeter–Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.
TL;DR: The partially reflected process is defined as a limit of the Markovian jump process generated by the Euler scheme for the underlying Ito dynamics with partial boundary reflection.
Abstract: The radiation (reactive or Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary; however, the relation between the reaction constant in the Robin boundary condition and the reflection probability is not well defined. In this paper we define the partially reflected process as a limit of the Markovian jump process generated by the Euler scheme for the underlying Ito dynamics with partial boundary reflection. Trajectories that cross the boundary are terminated with probability $P\sqrt{\Delta t}$ and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the Fokker–Planck equation in a half-space, wi...
TL;DR: In this article, the authors study numerically the dynamics of the Rabi Hamiltonian, which describes the interaction of a single cavity mode and a two-level atom without the rotating wave approximation.
Abstract: We study numerically the dynamics of the Rabi Hamiltonian, which describes the interaction of a single cavity mode and a two-level atom without the rotating wave approximation. We analyze this system subjected to damping and dephasing reservoirs, included via the usual Lindblad superoperators in the master equation. We show that the combination of the antirotating term and the atomic dephasing leads to linear asymptotic photon generation from the vacuum. We reveal the origins of the phenomenon and estimate its importance in realistic situations.
TL;DR: In this article, the authors analyzed the quantum regime of the dynamical backaction cooling of a mechanical resonator assisted by a driven harmonic oscillator (cavity) and derived the corresponding motional master equation using the Nakajima-Zwanzig formalism.
Abstract: We analyze the quantum regime of the dynamical backaction cooling of a mechanical resonator assisted by a driven harmonic oscillator (cavity). Our treatment applies to both optomechanical and electromechanical realizations and includes the effect of thermal noise in the driven oscillator. In the perturbative case, we derive the corresponding motional master equation using the Nakajima-Zwanzig formalism and calculate the corresponding output spectrum for the optomechanical case. Then we analyze the strong optomechanical coupling regime in the limit of small cavity linewidth. Finally, we consider the steady state covariance matrix of the two coupled oscillators for arbitrary input power and obtain an analytical expression for the final mechanical occupancy. This is used to optimize the drive's detuning and input power for an experimentally relevant range of parameters that includes the resolved-sideband-limit ground state cooling regime.
TL;DR: In this paper, the authors compare different quantum master equations for the time evolution of the reduced density matrix and propose a coarse-graining approach with a dynamically adapted coarsegraining time scale.
Abstract: We compare different quantum master equations for the time evolution of the reduced density matrix. The widely applied secular approximation (rotating wave approximation) applied in combination with the Born-Markov approximation generates a Lindblad-type master equation ensuring for completely positive and stable evolution and is typically well applicable for optical baths. For phonon baths however, the secular approximation is expected to be invalid. The usual Markovian master equation does not generally preserve positivity of the density matrix. As a solution we propose a coarse-graining approach with a dynamically adapted coarse-graining time scale. For some simple examples we demonstrate that this preserves the accuracy of the integro-differential Born equation. For large times we analytically show that the secular approximation master equation is recovered. The method can in principle be extended to systems with a dynamically changing system Hamiltonian, which is of special interest for adiabatic quantum computation. We give some numerical examples for the spin-boson model of cases where a spin system thermalizes rapidly, and other examples where thermalization is not reached.
TL;DR: In this article, the authors analyzed the quantum regime of the dynamical backaction cooling of a mechanical resonator assisted by a driven harmonic oscillator (cavity) and derived the corresponding motional master equation using the Nakajima-Zwanzig formalism.
Abstract: We analyze the quantum regime of the dynamical backaction cooling of a mechanical resonator assisted by a driven harmonic oscillator (cavity). Our treatment applies to both optomechanical and electromechanical realizations and includes the effect of thermal noise in the driven oscillator. In the perturbative case, we derive the corresponding motional master equation using the Nakajima-Zwanzig formalism and calculate the corresponding output spectrum for the optomechanical case. Then we analyze the strong optomechanical coupling regime in the limit of small cavity linewidth. Finally we consider the steady state covariance matrix of the two coupled oscillators for arbitrary input power and obtain an analytical expression for the final mechanical occupancy. This is used to optimize the drive's detuning and input power for an experimentally relevant range of parameters that includes the "ground state cooling" regime.
TL;DR: In this article, it is shown how a noise-filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations.
Abstract: Life processes in single cells and at the molecular level are inherently stochastic. Quantifying the noise is, however, far from trivial, as a major contribution comes from intrinsic fluctuations, arising from the randomness in the times between discrete jumps. It is shown in this paper how a noise-filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations. Such interpretation naturally exists in linear noise approximations, but it also provides an exact description of the jump process when the transition rates are linear. As an important example, it is shown in this paper how, by addressing the proximity of the underlying dynamics in an appropriate topology, a sequence of coupled birth-death processes, which can be relevant in gene expression, tends to a pure delay; this implies important limitations in noise suppression capabilities. Despite the exactness, in a linear regime, of the analysis of noise in conjunction with the network dynamics, we emphasize in this paper the importance of also analyzing dynamic behavior when transition rates are highly nonlinear; otherwise, steady-state solutions can be misinterpreted. The examples are taken from systems with macroscopic models leading to bistability. It is discussed that bistability in the deterministic mass action kinetics and bimodality in the steady-state solution of the master equation neither always imply one another nor do they necessarily lead to efficient switching behaviours: the underlying dynamics need to be taken into account. Finally, we explore some of these issues in relation to a model of the lac operation.
TL;DR: A formalism based on the master equation is adopted and it is shown how the probability density for the position of a molecular motor at a given time can be solved exactly in Fourier-Laplace space.
Abstract: Dynamic biological processes such as enzyme catalysis, molecular motor translocation, and protein and nucleic acid conformational dynamics are inherently stochastic processes. However, when such processes are studied on a nonsynchronized ensemble, the inherent fluctuations are lost, and only the average rate of the process can be measured. With the recent development of methods of single-molecule manipulation and detection, it is now possible to follow the progress of an individual molecule, measuring not just the average rate but the fluctuations in this rate as well. These fluctuations can provide a great deal of detail about the underlying kinetic cycle that governs the dynamical behavior of the system. However, extracting this information from experiments requires the ability to calculate the general properties of arbitrarily complex theoretical kinetic schemes. We present here a general technique that determines the exact analytical solution for the mean velocity and for measures of the fluctuations. We adopt a formalism based on the master equation and show how the probability density for the position of a molecular motor at a given time can be solved exactly in Fourier-Laplace space. With this analytic solution, we can then calculate the mean velocity and fluctuation-related parameters, such as the randomness parameter (a dimensionless ratio of the diffusion constant and the velocity) and the dwell time distributions, which fully characterize the fluctuations of the system, both commonly used kinetic parameters in single-molecule measurements. Furthermore, we show that this formalism allows calculation of these parameters for a much wider class of general kinetic models than demonstrated with previous methods.
TL;DR: In this article, the authors derived the Kraus operators from the point of view of the quantum channel and proved the normalization conditions of the corresponding Kraus operator from the perspective of the phase sensitive process.
Abstract: We solve various master equations to obtain density operators' infinite operator-sum representation via a new approach, i.e., by virtue of the thermo-entangled state representation that has a fictitious mode as a counterpart mode of the system mode. The corresponding Kraus operators from the point of view of quantum channel are derived, whose normalization conditions are proved. Miscellaneous characters possessed by different quantum channels, such as decoherence, phase diffusion, damping, and amplification, can be shown explicitly in the entangled state representation of the density operators. Squeezing transformation is applied to the density operator for decoherence to generate a master equation for describing the phase sensitive process. Partial trace method for deriving new density operators is also introduced. Throughout our discussion, the technique of integration within an ordered product (IWOP) of operators is fully used.