Atomic systems master equation

As an example we treat the decay process of IV.6 in terms of the master equation. The decay probability y per unit time is a property of the radioactive nucleus or the excited atom, and can, in principle, be computed by solving the Schrodinger equation for that system. To find the long-time evolution of a collection of emitters write P(n, t) for the probability that there are n surviving emitters at time t. The transition probability for a... 

Equation (5.1) described the vibrational response of a single particle to an applied forceF(t). In a (crystalline) system of many mobile particles (ensemble), the problem is analogous but the question now is how the whole system responds to an external force or perturbation Let us define the system s state (a) as a particular configuration of its particles and the probability of this state as pa. In a thermodynamic system, transitions from an a to a p configuration occur as thermally activated events. If the transition frequency a- /5 is copa and depends only on a and / (Markovian), the time evolution of the system is given by a master equation which links atomic and macroscopic parameters (dynamics and kinetics)... 

The results obtained for the stochastic model show that surface reactions are well-suited for a description in terms of the master equations. Since this infinite set of equations cannot be solved analytically, numerical methods must be used for solving it. In previous Sections we have studied the catalytic oxidation of CO over a metal surface with the help of a similar stochastic model. The results are in good agreement with MC and CA simulations. In this Section we have introduced a much more complex system which takes into account the state of catalyst sites and the diffusion of H atoms. Due to this complicated model, MC and in some respect CA simulations cannot be used to study this system in detail because of the tremendous amount of required computer time. However, the stochastic ansatz permits to study very complex systems including the distribution of special surface sites and correlated initial conditions for the surface and the coverages of particles. This model can be easily extended to more realistic models by introducing more aspects of the reaction mechanism. Moreover, other systems can be represented by this ansatz. Therefore, this stochastic model represents an elegant alternative to the simulation of surface reaction systems via MC or CA simulations. 

We start by considering the hydrogen atom, the simplest possible system, in which one electron interacts with a nucleus of unit positive charge. Only two terms are required from the master equation (3.161) in chapter 3, namely, those describing the kinetic energy of the electron and the electron-nuclear Coulomb potential energy. In the space-fixed axes system and SI units these terms are... 

The chemical master equation (CME) for a given system invokes the same rate constants as the associated deterministic kinetic model. Yet the CME is more fundamental than the deterministic kinetic view. Just as Schrodinger s equation is the fundamental equation for modeling motions of atomic and subatomic particle systems, the CME is the fundamental equation for reaction systems. Remember that Schrodinger s equation is not a model for a specific mechanical system. Rather, it is a theoretical framework upon which models for particular systems can be developed. In order to write down a model for an atomic system based on Schrodinger s equation, one needs to know how to write down the Hamiltonian a priori. Similarly, the CME is not a model for a specific biochemical reaction system it is a theoretical framework. To determine the CME model for a reaction system, one must know what are the possible elementary reactions and the associated rate constants. 

Abstract Interaction between a quantum system and its surroundings - be it another similar quantum system, a thermal reservoir, or a measurement device - breaks down the standard unitary evolution of the system alone and introduces open quantum system behaviour. Coupling to a fast-relaxing thermal reservoir is known to lead to an exponential decay of the quantum state, a process described by a Lindblad-type master equation. In modern quantum physics, however, near isolation of individual quantum objects, such as qubits, atoms, or ions, sometimes allow them only to interact with a slowly-relaxing near-environment, and the consequent decay of the atomic quantum state may become nonexponential and possibly even nonmonotonic. Here we consider different descriptions of non-Markovian evolutions and also hazards associated with them, as well as some physical situations in which the environment of a quantum system induces non-Markovian phenomena. 

There are several theoretical approaches that can be used to calculate the dynamics and correlation properties of two atoms interacting with the quantized electromagnetic held. One of the methods is the wavefunction approach in which the dynamics are given in terms of the probability amplitudes [9]. Another approach is the Heisenberg equation method, in which equations of motion for the atomic and held operators are found from the Hamiltonian of a given system [10], The most popular approach is the master equation method, in which the equation of motion is found for the density operator of an atomic system weakly coupled to a system regarded as a reservoir [7,8,41], There are many possible realizations of reservoirs. The typical reservoir to which atomic systems are coupled is the quantized three-dimensional multimode vacuum held. The major advantage of the master equation is that it allows us to consider the evolution of the atoms plus held system entirely in terms of atomic operators. 

The master equation involves the so-called reduced density operator p describing the system of two atoms, which is obtained from the total density operator pT by tracing over vacuum field (reservoir) states... 

On transforming Eq. (19) into the Schrodinger picture, the master equation of the two-atom system takes the form... 

The presence of the collective parameters Ti2 and ft 2 introduces off-diagonal terms in the Hamiltonian II and in the dissipative part of the master equation. This suggests that in the presence of the interaction between the atoms the bare atomic states are no longer the eigenstates of the two-atom system. We can diagonalize the Hamiltonian (32) with respect to the dipole-dipole interaction and find collective states of the two-atom system. 

The master equation (38) provides the simplest example of the effects introduced by the coherent interaction of atoms with the radiation field. These effects include the shifts of the energy levels of the system, produced by the dipole-dipole interaction, and the phenomena of enhanced (superradiant) and reduced (subradiant) spontaneous emission, which appear in the changed damping rates to (T + Ti2) and (T — Ti2), respectively. 

The choice of the collective states (40) as a basis leads to a complicated master equation whose physical properties are tractable only for very specific values of the parameters involved. A different choice of basis collective states is proposed here, which allows us to obtain a simple master equation of the system of two nonidentical atoms. Moreover, we will show that it is possible to create a maximally entangled state in the system of two nonidentical atoms that can be decoupled from the external environment and, at the same time, the state exhibits a strong coherent coupling with the remaining states. 

Here, we discuss an alternative scheme where the superposition state <1>) can be generated in two identical atoms driven in free space by a coherent laser field. This can happen when the atoms are in nonequivalent positions in the driving field, where the atoms experience different intensities and phases of the driving field. The populations of the collective states of the system can be found from the master equation (31). We use the set of the collective states (35) as an appropriate representation for the density operator... 

After transforming to the collective state basis, the master equation (31) leads to a closed system of 15 equations of motion for the density matrix elements [46]. However, for a specifically chosen geometry for the driving field, namely, that the field is propagated perpendicularly to the atomic axis (k rn = 0), the system of equations decouples into 9 equations for symmetric and 6 equations for antisymmetric combinations of the density matrix elements [45-50]. In this case, we can solve the system analytically, and find that the steady-state values of the populations are [45,46]... 

The dynamics of the collective two-atom system in a squeezed vacuum can be determined from the master equation of the density operator of the system or from the equations of motion for the transition probability amplitudes [22]. In Section II.B, we derived the master equation for the density operator of a two atom system interacting with the ordinary vacuum field. It is our purpose to extend the master equation to the case of a squeezed vacuum field. The method of derivation of the master equation is a straightforward extension of that presented in Section II.B. 

The system of vibrational master equations has been numerically integrated with the same initial condition described in Sect. 2.3.1. Figure 15 reports the N distribution obtained in JVE with allowance for recombination at different times. One can note that at a time of 10" 3 s the N distribution with recombination is very similar to that calculated without recombination. The only part affected by recombination and by the presence of atoms is the tail of N -distribution, which is strongly overestimated in JVE without recombination (see Fig. 8). This is due to the chemical deactivation, which increases the depopulation of higher vibrational levels. These considerations suggest that the kjj values (which depend on the tail of N distribution) are smaller than the corresponding values calculated without atoms. 

The model mechanism, discussed in the preceeding sections, requires the following refinements (a) A complete temporal coupling between the system of master equations and the relevant Boltzmann equation should always be made in order to obtain edf s and N distributions consistent with each other and with the presence of atomic species. The recombination of the atoms should be taken into account. 

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