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Journal of the Korean Physical Society

Magnetism of a system interacting with a common environment

Hyuk-jae Lee, Seok Lee, Jae Hun Kim

Published Mar 1, 2013• 0 citations
Publisher

Abstract

We consider a system and environment governed by the Markovian process. The density matrix (the reduced density matrix) of the system satisfies the Kossakowski-Lindblad equation. We calculate the z-direction magnetization in the system for various types of initial states and interactions between the system and the environment. In all cases, the values of the magnetization asymptotically approached the same value in spite of the different initial states. However, for the different types of interactions between the system and the environment, the asymptotic values differed with the initial state.

DOI
10.3938/jkps.62.857

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We consider a system and environment governed by the Markovian process. The density matrix (the reduced density matrix) of the system satisfies the Kossakowski-Lindblad equation. We calculate the z-direction magnetization in the system for various types of initial states and interactions between the system and the environment. In all cases, the values of the magnetization asymptotically approached the same value in spite of the different initial states. However, for the different types of interactions between the system and the environment, the asymptotic values differed with the initial state.
We derive a new perturbative quantum master equation for the reduced density matrix of a system interacting with an environment (with a dense spectrum of energy levels). The total system energy (system plus environment) is constant and finite. This equation takes into account the finite energy effects of the environment due to the total energy conservation. This equation is more general than the common perturbative equations used for describing a system in interaction with an environment (like the Redfield equation or the Cohen-Tannoudji one) because these last equations can be deduced from it in the limit of an infinitely large environment. We apply numerically this equation to the spin-GORM model. This model represents the interaction of a two-level system with an environment described by random matrices. We compare our equation with the exact von Neumann equation of the total system and show its superiority compared to the Redfield equation (in the Markovian and non-Markovian cases).
We know that the quantum system evolves to different physical phenomena according to initial states. However, the situation is very different in the open system. We consider the system and the environment governed by the Markovian process. The density matrix (the reduced density matrix) of the system satisfies the Kossakowski-Lindblad equation. We calculate the expectation value of the z-direction-magnetization in the system for various types of 1) initial states and 2) interactions between the system and the environment. In all the cases, the values of the magnetization approached asymptotically to the same point even though the different initial states and different interaction types between the system and the environment are applied. The paths to asymptotically approach the final values are different according to the initial states and the interactions. These facts show that the physical results of expectation value are independent of the initial states.
In dealing with systems whose eigenstates are known, such as ideal spin systems in a magnetic field, it is sometimes useful to introduce a formalism involving the density matrix. Knowledge of the density matrix operator permits the expectation values of other operators to be determined in a way that is straightforward in principle. In the presence of a magnetic field or some other polarizing mechanism, the form of the density matrix will change, and the expectation value, for example, may be nonzero. Although the situation that has been considered involves an electron beam, a related but somewhat more complicated approach is used for polarized light beams.
We consider a fermionic quantum system exchanging particles with an environment at a fixed temperature and study its reduced evolution by means of a Redfield-I equation with time-dependent (non-Markovian) coefficients. We find that the description can be efficiently reduced to a standard-form Redfield-II equation, however, with a obeying a universal law. At early times, after the system and environment start in a product state, the appears to be very high, yet eventually it settles down towards the true environment value. In this way, we obtain a time-local master equation, offering high accuracy at all times and preserving the crucial properties of the density matrix. It includes non-Markovian relaxation processes beyond the secular approximation and time-averaging methods and can be further applied to various types of Gorini-Kossakowski-Sudarshan-Lindblad equations. We derive the theory from first principles and discuss its application using a simple example of a single quantum dot.
By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the parametric representation with environmental coherent states) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini–Kossakowski–Sudarshan–Lindblad equation, but holds regardless of Markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We find that, instead of a single set of Lindblad operators, in the general (non-Markovian) case there one set of Lindblad-like operators for each and every point of a symplectic manifold associated to the environment. This intricacy disappears under some assumptions (which are related to Markovianity and the classical limit of the environment), under which it is possible to recover the usual master-equation formalism. Finally, we find such Lindblad-like operators for two different models of a qubit in a bosonic environment, and show that in the classical limit of the environment their renown master equations are recovered.