Standard Master Equation Research Articles

Class of exact memory-kernel master equations

A well-known situation in which a non-Markovian dynamics of an open quantum system $S$ arises is when this is coherently coupled to an auxiliary system $M$ in contact with a Markovian bath. In such cases, while the joint dynamics of $S$-$M$ is Markovian and obeys a standard (bipartite) Lindblad-type master equation (ME), this is in general not true for the reduced dynamics of $S$. Furthermore, there are several instances (\eg the dissipative Jaynes-Cummings model) in which a {\it closed} ME for the $S$'s state {\it cannot} even be worked out. Here, we find a class of bipartite Lindblad-type MEs such that the reduced ME of $S$ can be derived exactly and in a closed form for any initial product state of $S$-$M$. We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models

Decoherence of odd compass states in the phase-sensitive amplifying/dissipating environment

We study the evolution of odd compass states (specific superpositions of four coherent states), governed by the standard master equation with phase-sensitive amplifying/attenuating terms, in the presence of a Hamiltonian describing a parametric degenerate linear amplifier. Explicit expressions for the time-dependent Wigner function are obtained. The time of disappearance of the so called “sub-Planck structures” is calculated using the negative value of the Wigner function at the origin of phase space. It is shown that this value rapidly decreases during a short “conventional interference degradation time” (CIDT), which is inversely proportional to the size of quantum superposition, provided the anti-Hermitian terms in the master equation are of the same order (or stronger) as the Hermitian ones (governing the parametric amplification). The CIDT is compared with the final positivization time (FPT), when the Wigner function becomes positive. It appears that the FPT does not depend on the size of superpositions, moreover, it can be much bigger in the amplifying media than in the attenuating ones. Paradoxically, strengthening the Hamiltonian part results in decreasing the CIDT, so that the CIDT almost does not depend on the size of superpositions in the asymptotical case of very weak reservoir coupling. We also analyze the evolution of the Mandel factor, showing that for some sets of parameters this factor remains significantly negative, even when the Wigner function becomes positive.

Reaction rates for a generalized reaction-diffusion master equation.

It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.

Decoherence of quantum excitation of even/odd coherent states in thermal environment

In this paper, we study the decoherence of quantum excitation (photon-added) even /odd coherent states, $((\hat a{^{\dagger }})^{m} \left | {\alpha _ \pm } \right \rangle )$ , in a thermal environment by investigating the variation of negative part of the Wigner quasidistribution function vs. the rescaled time. For this purpose, at first we obtain the time-dependent Wigner function corresponding to the mentioned states in the framework of standard master equation. Then, the time evolution of the Wigner function associated with photon-added even /odd coherent states, as well as the number of added photons m are analysed. It is shown that, in both states, the negative part of the Wigner function decreases with time. By deriving the threshold value of the rescaled time for single photon-added even /odd coherent states, it is also found that, if the rescaled time exceeds the threshold value, the associated Wigner function becomes positive, i.e., the decoherence occurs completely.

Prospects for observing dynamical and anti- dynamical Casimir effects in circuit QED due to fast modulation of qubit parameters

We consider the nonstationary circuit quantum electrodynamics (circuit QED) architecture, where a single artificial two-level atom interacts with a cavity field mode under external modulation of one or more system parameters. Two different approaches are employed to study the effects of Markovian dissipation on modulation-induced transitions between the atom–field dressed states: the standard master equation of quantum optics and the recently formulated dressed-picture master equation. We estimate the associated transition rates and show that photon generation from vacuum (‘dynamical Casimir effect’, DCE) and coherent photon annihilation from nonvacuum states (‘anti-DCE’) are possible with the current state-of-the-art parameters.

Improved Dyson series expansion for steady-state quantum transport beyond the weak coupling limit: divergences and resolution.

We present a general theory to calculate the steady-state heat and electronic currents for nonlinear systems using a perturbative expansion in the system-bath coupling. We explicitly demonstrate that using the truncated Dyson-series leads to divergences in the steady-state limit, thus making it impossible to be used for actual applications. In order to resolve the divergences, we propose a unique choice of initial condition for the reduced density matrix, which removes the divergences at each order. Our approach not only allows us to use the truncated Dyson-series, with a reasonable choice of initial condition, but also gives the expected result that the steady-state solutions should be independent of initial preparations. Using our improved Dyson series we evaluate the heat and electronic currents up to fourth-order in system-bath coupling, a considerable improvement over the standard quantum master equation techniques. We then numerically corroborate our theory for archetypal settings of linear systems using the exact nonequilibrium Green's function approach. Finally, to demonstrate the advantage of our approach, we deal with the nonlinear spin-boson model to evaluate heat current up to fourth-order and find signatures of cotunnelling process.

Dissipative Dynamic of Circuit QED in Ultrastrong Coupling

In this paper we study the dissipation dynamics process in circuit QED systems in the ultrastrong coupling regime. We numerically calculated the average photon number of cavity mode and population of the excited state of the two level system (TLS), based on both the standard master equation and the recently developed Lindblad type master equation in [1]. Our results reveal how the two treatments make significant differences with the interaction strength between cavity and TLS increasing.

The decoherence of the quantum excitation of even/odd coherent states in a photon-loss channel

In this paper, the decoherence of the quantum excitation (the photon-added case) of even/odd coherent states, in a photon-loss channel is studied via the evaluation of the negative part of the Wigner quasi-distribution function in terms of the rescaled time (). We first obtain the explicit form of the time-dependent Wigner functions of the states considered, for arbitrary values of the photon excitations. To achieve this purpose, an equivalent Fokker–Planck-like equation which describes the time evolution of the Wigner function in the framework of the standard master equation of the density matrix is used. Then, by analyzing the time variation of the corresponding Wigner functions, the loss of the nonclassicality behavior and, hence, the decay of the negative part of the Wigner functions as a result of decoherence are clearly shown. In particular, we derive the threshold value of the rescaled time for single-photon-added even and odd coherent states; through this it is observed that the associated Wigner functions become positive whenever the rescaled time exceeds that value, i.e., the decoherence has become complete. By virtue of numerical calculations, it is also illustrated that the negative volume of the corresponding Wigner functions decreases with increasing time. In addition, the states with larger values of the excitation number m generally (but not absolutely always) possess smaller values of the rescaled threshold time.

Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime

Optomechanical systems can exhibit self-sustained limit cycles where the quantum state of the mechanical resonator possesses nonclassical characteristics such as a strongly negative Wigner density, as was shown recently in a numerical study by Qian et al. [Physical Review Letters, 109, 253601 (2012)]. Here we derive a Fokker-Planck equation describing mechanical limit cycles in the quantum regime which correctly reproduces the numerically observed nonclassical features. The derivation starts from the standard optomechanical master equation, and is based on techniques borrowed from the laser theory due to Haake's and Lewenstein. We compare our analytical model with numerical solutions of the master equation based on Monte-Carlo simulations, and find very good agreement over a wide and so far unexplored regime of system parameters. As one main conclusion, we predict negative Wigner functions to be observable even for surprisingly classical parameters, i.e. outside the single-photon strong coupling regime, for strong cavity drive, and rather large limit cycle amplitudes. The general approach taken here provides a natural starting point for further studies of quantum effects in optomechanics.

Energy transfer between two aggregates in light-harvesting complexes

Energy transfer processes between two aggregates in a coupled chromophoric-pigment (protein) system are studied via the standard master equation approach. Each pigment of the two aggregates is modeled as a two-level system. The excitation energy is assumed to be transferred from the donor aggregate to the acceptor aggregate. The model can be used to theoretically simulate many aspects of light-harvesting complexes (LHCs). By applying the real bio-parameters of photosynthesis, we numerically investigate the efficiency of energy transfer (EET) between the two aggregates in terms of some factors, e.g., the initial coherence of the donor aggregate, the coupling strengthes between the two aggregates and between different pigments, and the effects of noise from the environment. Our results provide evidence for that the actual numbers of pigments in the chromophoric rings of LHCs should be the optimum parameters for a high EET. We also give a detailed analysis of the effects of noise on the EET.

Suppression of quantum-mechanical reflection by environmental decoherence

In quantum mechanics an incoming particle wave packet with sufficient energy will undergo both transmission and reflection when encountering a barrier of lower energy, but in classical mechanics there is no reflection, only transmission. In this paper we seek to explain the disappearance of quantum-mechanical reflection in the quasiclassical limit, using the standard machinery of decoherence through environmental interaction. We consider two models. In the first, the incoming particle is classicalized by coupling to an environment and modeled using a standard master equation of Lindblad form with Lindblad operator proportional to position (the simplest version of quantum Brownian motion). We find, however, that suppression of reflection is achieved only for environmental interaction so strong that large fluctuations in momentum are generated, which blurs the distinction between incoming and reflected wave packets. This negative conclusion also holds for a complex potential which has similar implications for attempts to understand the suppression of the Zeno effect using the same mechanism (discussed in more detail elsewhere). A different master equation with Lindblad operator proportional to momentum is shown to be successful in suppressing reflection without large fluctuations but such a master equation is unphysical. We consider a second model in which the barrier is modeled quantum mechanically by a massive target particle coupled to an environment to maintain it in a quasiclassical state. This avoids the fluctuations problem since the incoming particle is not coupled to the environment directly. We find that reflection is significantly suppressed as long as the decoherence time scale of the target particle is much smaller than certain characteristic scattering time scales of the incoming particle, or equivalently, as long as the velocity fluctuations in the target are larger than the velocity of the incoming particle.

Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations

We present a first-principles derivation of the Markovian semi-group master equation without invoking the rotating wave approximation (RWA). Instead we use a time coarse-graining approach which leaves us with a free timescale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a non-trivial effect on the dynamics of the populations. We show that the two different master equations can exhibit strong qualitative differences for the population of the energy eigenstates even for such a simple model. The RWA-based master equation misses an important feature which the coarse-graining based scheme does not. By optimizing the coarse-graining timescale the latter scheme can be made to approach the exact solution much more closely than the RWA-based master equation.

Dissipative dynamics of quantum correlations in the strong-coupling regime

The dynamics of entanglement and quantum discord between two identical qubits strongly interacting with a common single mode leaky cavity field have been investigated beyond the rotating wave approximation (RWA) by using recently derived Lindblad type quantum optical master equation [F. Beaudoin, J.M. Gambetta, A. Blais, Phys. Rev. A {\bf 84}, 043832 (2011)] that can describe the losses of the cavity field in a strong atom-field coupling regime. Contrary to previous investigations of the same model in the dissipative regime by using the standard Lindblad quantum optical master equation in a strong-coupling regime, the atom-field steady states are found to be cavity decay rate independent and have a very simple structure determined solely by the overlap of initial atomic state with the subradiant state which is valid for all coupling regimes. Non-RWA dynamics are found to have remarkable effects on the steady state quantum discord and entanglement that cannot be achieved under RWA conditions, for instance, they can induce steady state entanglement even for the initial states that have no overlap with the subradiant state. Moreover, the non-RWA dynamics are found to reverse the initial state dependence of steady state entanglement and quantum discord contrary to the RWA case.

Optimal antibunching in passive photonic devices based on coupled nonlinear resonators

We propose the use of weakly nonlinear passive materials for prospective applications in integrated quantum photonics. It is shown that strong enhancement of native optical nonlinearities by electromagnetic field confinement in photonic crystal resonators can lead to single-photon generation only exploiting the quantum interference of two coupled modes and the effect of photon blockade under resonant coherent driving. For realistic system parameters in state of the art microcavities, the efficiency of such a single-photon source is theoretically characterized by means of the second-order correlation function at zero-time delay as the main figure of merit, where major sources of loss and decoherence are taken into account within a standard master equation treatment. These results could stimulate the realization of integrated quantum photonic devices based on non-resonant material media, fully integrable with current semiconductor technology and matching the relevant telecom band operational wavelengths, as an alternative to single-photon nonlinear devices based on cavity quantum electrodynamics with artificial atoms or single atomic-like emitters.

Thermoelectric transport in the three terminal quantum dot

The thermoelectric transport in the system composed of a quantum dot in contact with superconducting, ferromagnetic and normal metal electrodes has been studied. Such a system can support pure spin current in the normal electrode. In the limit of a large superconducting gap and weak coupling between the dot and the electrodes we investigate the sub-gap charge and spin transport via Andreev mechanism using the standard master equation technique, which is known to be valid in the sequential tunnelling regime. The Zeeman splitting of the dot level induces pure spin current in the ferromagnetic electrode under an appropriate bias. This opens a novel possibility to switch the spin current between two electrodes by electric means. The calculated spin and charge thermopower coefficients attain very large values, of the order of a few hundreds μV K−1, and show similar dependences on the position of the on-dot energy level and temperature.

Master-equation approach to optomechanics with arbitrary dielectrics

We present a master equation describing the interaction of light with dielectric objects of arbitrary sizes and shapes. The quantum motion of the object, the quantum nature of light, as well as scattering processes to all orders in perturbation theory are taken into account. This formalism extends the standard master equation approach to the case where interactions among different modes of the environment are considered. It yields a genuine quantum description, including a renormalization of the couplings and decoherence terms. We apply this approach to analyze cavity cooling of the center-of-mass mode of large spheres. Furthermore, we derive an expression for the steady-state phonon numbers without relying on resolved-sideband or bad-cavity approximations.

Generalized Gibbs state with modified Redfield solution: Exact agreement up to second order

A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.

Scattering laser light on cold atoms: Toward multiple scattering signals from single-atom responses

We deduce the coherent backscattering signal from two distant laser-driven atoms using single-atom equations. In contrast to the standard master equation treatment, this new approach is suitable for the generalization to a large number of atomic scatterers.

Theory of spectroscopy in an optically pumped effusive vapor

We present a theoretical framework for studying spatially dependent absorption in a thermal vapor of multilevel atoms, of arbitrary optical thickness. The atomic state dynamics, governed by a standard atom-optical master equation, are self-consistently coupled to the axial evolution of the probe beam intensity and the effusive gas dynamics. We derive steady-state equations for the spatially varying distributions of atomic populations and the probe beam intensity. From the latter, absorption coefficients in both the saturated and unsaturated regimes can be calculated. We present solutions to the resulting equations at various levels of approximation, including an example of the full numerical solution of a saturated, optically thick vapor of three-level atoms, demonstrating a breakdown of Beer's law, among other measurable effects.

Competition between current-induced excitation and bath-induced decoherence in molecular junctions

A general framework is presented to describe a resonant inelastic current inducing dynamics in the nuclear degrees of freedom of a molecule embedded between two electrodes. This approach makes use of the scattering theory of density matrices to account for the interaction between the scattering charge and the molecular modes to all orders and reduces in appropriate limits to both the standard master equation treatment for vibrational heating and the Landauer formalism for purely elastic transport. While the method presented here is equivalent to these approaches in limiting cases, it also goes well beyond their restrictions by incorporating the full quantum dynamics in the vibrational subspace in the presence of tunneling current. By application to the Au-C(60)-Au junction, it is shown that inclusion of vibrational coherences, which were previously neglected, is crucial to accurately predict the dynamics induced by current in molecular devices. Interaction with a bath of phonon modes is incorporated within the Bloch model and the competition between the bath-induced relaxation processes and the current-induced excitation is studied in detail over a range of temperatures.