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Coherence theory and coherence phenomena in a closed spin 1/2 system

Dannenberg, O

Published May 18, 2006• 1 citation
OpenAlex

Abstract

We study a simplified Heisenberg spin model in order to clarify the idea of decoherence in closed quantum systems. For this purpose, we define a new concept: the coherence function $\\Xi(t)$, which describes the dynamics of coherence in the whole system, and which is linked with the total coarse-grained (von Neumann) entropy of all particles. We discuss in some detail a general coherence theory and its elementary results. For example, in the particular setup, decoherence diagonalises reduced density matrices in all possible basis sets. As expected, decoherence is understood as a statistical process that is caused by the dynamics of the system, similar to entropy. Moreover, the concept of decoherence time is applicable in closed systems. However, in most cases, the decay of off-diagonal elements differs from the usual $\\exp(-t/\\tau_d)$ behaviour. We have solved the form of decoherence time $\\tau_d$ in an infinite Heisenberg model with respect to density $\\rho$, spatial dimension $D$ and $\\epsilon$ in a $1/r^{\\epsilon}$ -type of potential.

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We study a simplified Heisenberg spin model in order to clarify the idea of decoherence in closed quantum systems. For this purpose, we define a new concept: the coherence function Xi(t)\\Xi(t)Xi(t), which describes the dynamics of coherence in the whole system, and which is linked with the total coarse-grained (von Neumann) entropy of all particles. We discuss in some detail a general coherence theory and its elementary results. For example, in the particular setup, decoherence diagonalises reduced density matrices in all possible basis sets. As expected, decoherence is understood as a statistical process that is caused by the dynamics of the system, similar to entropy. Moreover, the concept of decoherence time is applicable in closed systems. However, in most cases, the decay of off-diagonal elements differs from the usual exp(−t/taud)\\exp(-t/\\tau_d)exp(−t/taud​) behaviour. We have solved the form of decoherence time taud\\tau_dtaud​ in an infinite Heisenberg model with respect to density rho\\rhorho, spatial dimension DDD and epsilon\\epsilonepsilon in a 1/repsilon1/r^{\\epsilon}1/repsilon -type of potential.
Abstract A simplified Heisenberg spin model is studied in order to examine the idea of decoherence in closed quantum systems. For this purpose, we present a quantifiable definition to quantum coherence Ξ, and discuss in some detail a general coherence theory and its elementary results. As expected, decoherence is understood as a statistical process that is caused by the dynamics of the system, similar to the growth of entropy. It appears that coherence is an important measure that helps to understand quantum properties of a system, e.g., the decoherence time can be derived from the coherence function Ξ(t), but not from the entropy dynamics. Moreover, the concept of decoherence time is applicable in closed and finite systems. However, in most cases, the decay of off‐diagonal elements differs from the usual exp(‐t/τ d ) behaviour. For concreteness, we report the form of decoherence time τ d in a finite Heisenberg model with respect to the number of particles N, density n ρ , spatial dimension D and ϵ in a η/r ϵ ‐type of potential.
A simplied Heisenberg spin model is studied in order to examine the idea of decoherence in closed quantum systems.For this purpose, we present a quantiable denition to quantum coherence Ξ, and discuss in some detail a general coherence theory and its elementary results.As expected, decoherence is understood as a statistical process that is caused by the dynamics of the system, similar to the growth of entropy.It appears that coherence is an important measure that helps to understand quantum properties of a system, e.g., the decoherence time can be derived from the coherence function Ξ(t), but not from the entropy dynamics.Moreover, the concept of decoherence time is applicable in closed and nite systems.However, in most cases, the decay of o-diagonal elements diers from the usual exp(-t/τ d ) behaviour.For concreteness, we report the form of decoherence time τ d in a nite Heisenberg model with respect to the number of particles N , density n ρ , spatial dimension D and ǫ in a η/r ǫ -type of potential.
We study a simplified Heisenberg spin model in order to clarify the idea of decoherence in closed quantum systems. For this purpose, we define a new concept: the decoherence function Ξ(t), which describes the dynamics of decoherence in the whole system, and which is linked with the total (von Neumann) entropy of all particles. As expected, decoherence is understood both as a statistical process that is caused by the dynamics of the system, and also as a matter of entropy. Moreover, the concept of decoherence time is applicable in closed systems and we have solved its behaviour in the Heisenberg model with respect to particle number N, density ρand spatial dimension D in a 1/r -type of potential. We have also studied the Poincare recurrences occurring in these types of systems: in an N=1000 particle system the recurrence time is close to the order of the age of the universe. This encourages us to conclude that decoherence is the solution for quantum-classical problems not only in practice, but also in principle.
In the thesis I study various quantum coherence phenomena and create some of the foundations for a systematic coherence theory. So far, the approach to quantum coherence in science has been purely phenomenological. In my thesis I try to answer the question what quantum coherence is and how it should be approached within the framework of physics, the metatheory of physics and the terminology related to them. It is worth noticing that quantum coherence is a conserved quantity that can be exactly defined. I propose a way to define quantum coherence mathematically from the density matrix of the system. Degenerate quantum gases, i.e., Bose condensates and ultracold Fermi systems, form a good laboratory to study coherence, since their entropy is small and coherence is large, and thus they possess strong coherence phenomena. Concerning coherence phenomena in degenerate quantum gases, I concentrate in my thesis mainly on collective association from atoms to molecules, Rabi oscillations and decoherence. It appears that collective association and oscillations do not depend on the spin-statistics of particles. Moreover, I study the logical features of decoherence in closed systems via a simple spin-model. I argue that decoherence is a valid concept also in systems with a possibility to experience recoherence, i.e., Poincaré recurrences. Metatheoretically this is a remarkable result, since it justifies quantum cosmology: to study the whole universe (i.e., physical reality) purely quantum physically is meaningful and valid science, in which decoherence explains why the quantum physical universe appears to cosmologists and other scientists very classical-like. The study of the logical structure of closed systems also reveals that complex enough closed (physical) systems obey a principle that is similar to Gödel's incompleteness theorem of logic. According to the theorem it is impossible to describe completely a closed system within the system, and the inside and outside descriptions of the system can be remarkably different. Via understanding this feature it may be possible to comprehend coarse-graining better and to define uniquely the mutual entanglement of quantum systems.
Decoherence and relaxation processes in quantum systems are studied on the basis of an exactly solvable model. That is, a generalized Coleman-Hepp model is analyzed in a framework of a spin coherent state representation which enables us to determine time evolution of the whole system as successive applications of rotation operators. A density matrix of the whole system and reduced density matrices of constituent subsystems are obtained to yield quasi-probability densities and averages of observables of an incident particle and a detector (reservoir) system. Numerical evaluation of these quantities is performed to find decoherence like phenomena for larger values of N (a number of detector spins) and S l (magnitude of a detector spin). An appearance of a mixed state (i.e., vanishing of off-diagonal elements of a reduced matrix) is explicitly shown when N and/or S l become infinity. Namely, this occurs in the thermodynamic limit ( N →∞) and/or in the classical apparatus limit ( S l →∞).
In this review, we discuss the decoherence and thermalization of a quantum spin system interacting with a spin bath environment, by numerically solving the time-dependent Schrödinger equation of the whole system. The effects of the topologic structure and the initial state of the environment on the decoherence of the two-spin and many-spin system are discussed. The role of different spin-spin coupling is considered. We show under which conditions the environment drives the reduced density matrix of the system to a fully decoherent state, and how the diagonal elements of the reduced density matrix approach those expected for the system in the microcanonical or canonical ensemble, depending on the character of the additional integrals of motion. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the microcanonical distribution (in each eigenenergy subspace) and canonical ensemble are states that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics. Furthermore, our numerical results show that a fully decoherent quantum system prefers to stay in an equilibrium state with a maximum entropy, indicating the validity of the second law of thermodynamics in the decoherence process.