title: Aspects and applications of two-time averages in open quantum Markoffian systems

authors: Carmichael, Howard J.

abstract: The use of a quantum-mechanical operator master equation for the description of an open Markoffian system is discussed in two particular aspects. In both the evaluation of operator averages at two times is a central feature. We deal first with the concept of detailed balance, so familiar from classical statistical mechanics, and hope to bring the various usages of this term under the common perspective provided by the formal generalisation of classical results.

The application of detailed balancing principles to quantum systems obeying Markoffian master equations is discussed and the formal generalisation of the classical principle of detailed balance introduced. A view of detailed balance as the macroscopic consequence of microreversibility is taken. Here the close association between two-time averages and detailed balance requires that the various properties of the former be developed as a basis to our discussions. Both operator and phase-space formulations for evaluating averages are therefore discussed, and from them the respective operator and phase-space expressions for quantum detailed balance constructed. From here the classical limit with ℏ → o is shown to correspond to the introduction of detailed balance to open quantum Markoffian systems via a literal interpretation of “quantum-mechanical Fokker-Plank processes”. The relationship of full quantum detailed balance to the detailed balancing of transitions within the energy representation, such as is encountered in the Pauli master equation, is established through introducing the concept of diagonal master equations. For these utilisation of the energy representation allows a direct identification with the Pauli situation. Finally, a generalised master equation accounting for a variety of reservoir couplings is derived, and studied in relation to both the operator detailed balance and the diagonal master equations for thermal equilibrium. It is shown that the presence of certain features in the reservoir interactions generally ensure the failure of full detailed balance, and simultaneously provides deviations from the diagonal form. These observations are to be associated with an inadequacy in the approximations of the master equation method.

A second section of this work tackles the theoretical description of a particular physical system within the framework of the master equation formalism. The concept of the open Markoffian system is applied in relation to resonance fluorescence and the properties of two-time averages used in the determination of field correlation functions. Very intense illumination, which gives rise to the dynamical Stark effect, is studied with particular interest.
 
Within the context of a quantum-mechanical master equation for incident field plus atomic scattering centre we develop a full description of the resonance fluorescence phenomenon. From a stationary autocorrelation function fluorescence spectra corresponding to arbitrary incident field strengths may be evaluated for the conditions pertaining under atomic saturation. Spectra are also available for the nonstationary field associated with the transient region of atomic dynamics. Both semiclassical and one-photon approximations are illuminated in a simple fashion. Particular interest is taken in deriving the second-order correlation function for the scattered light. This is readily available from our formalism and exhibits the interesting feature of photon-antibunching at weak incident intensities. The region of weak scattering is familiar in its description by the Lorentzian electron oscillator model and this purely quantum feature provides a possible check for Q.E.D. versus recent improved semiclassical theories. The second-order correlation function is also discussed in relation to its potential for the experimental study of fluorescence spectra, particularly those in the region of the dynamical Stark effect. Possibilities for the extraction of spectral detail presently unattainable are clearly evident.

text: Aspects and applications of two-time averages in open quantum Markoffian systems The use of a quantum-mechanical operator master equation for the description of an open Markoffian system is discussed in two particular aspects. In both the evaluation of operator averages at two times is a central feature. We deal first with the concept of detailed balance, so familiar from classical statistical mechanics, and hope to bring the various usages of this term under the common perspective provided by the formal generalisation of classical results.

The application of detailed balancing principles to quantum systems obeying Markoffian master equations is discussed and the formal generalisation of the classical principle of detailed balance introduced. A view of detailed balance as the macroscopic consequence of microreversibility is taken. Here the close association between two-time averages and detailed balance requires that the various properties of the former be developed as a basis to our discussions. Both operator and phase-space formulations for evaluating averages are therefore discussed, and from them the respective operator and phase-space expressions for quantum detailed balance constructed. From here the classical limit with ℏ → o is shown to correspond to the introduction of detailed balance to open quantum Markoffian systems via a literal interpretation of “quantum-mechanical Fokker-Plank processes”. The relationship of full quantum detailed balance to the detailed balancing of transitions within the energy representation, such as is encountered in the Pauli master equation, is established through introducing the concept of diagonal master equations. For these utilisation of the energy representation allows a direct identification with the Pauli situation. Finally, a generalised master equation accounting for a variety of reservoir couplings is derived, and studied in relation to both the operator detailed balance and the diagonal master equations for thermal equilibrium. It is shown that the presence of certain features in the reservoir interactions generally ensure the failure of full detailed balance, and simultaneously provides deviations from the diagonal form. These observations are to be associated with an inadequacy in the approximations of the master equation method.

A second section of this work tackles the theoretical description of a particular physical system within the framework of the master equation formalism. The concept of the open Markoffian system is applied in relation to resonance fluorescence and the properties of two-time averages used in the determination of field correlation functions. Very intense illumination, which gives rise to the dynamical Stark effect, is studied with particular interest.
 
Within the context of a quantum-mechanical master equation for incident field plus atomic scattering centre we develop a full description of the resonance fluorescence phenomenon. From a stationary autocorrelation function fluorescence spectra corresponding to arbitrary incident field strengths may be evaluated for the conditions pertaining under atomic saturation. Spectra are also available for the nonstationary field associated with the transient region of atomic dynamics. Both semiclassical and one-photon approximations are illuminated in a simple fashion. Particular interest is taken in deriving the second-order correlation function for the scattered light. This is readily available from our formalism and exhibits the interesting feature of photon-antibunching at weak incident intensities. The region of weak scattering is familiar in its description by the Lorentzian electron oscillator model and this purely quantum feature provides a possible check for Q.E.D. versus recent improved semiclassical theories. The second-order correlation function is also discussed in relation to its potential for the experimental study of fluorescence spectra, particularly those in the region of the dynamical Stark effect. Possibilities for the extraction of spectral detail presently unattainable are clearly evident.

year: 1975

title: Odd multiperfect numbers of abundancy 4

authors: Broughan, Kevin A.

subjects: mathematics

abstract: Euler's structure theorem for any odd perfect number is extended to odd multiperfect numbers of abundancy power of 2. In addition, conditions are found for classes of odd numbers not to be 4-perfect: some types of cube, some numbers divisible by 9 as the maximum power of 3, and numbers where 2 is the maximum even prime power.

text: Odd multiperfect numbers of abundancy 4 Euler's structure theorem for any odd perfect number is extended to odd multiperfect numbers of abundancy power of 2. In addition, conditions are found for classes of odd numbers not to be 4-perfect: some types of cube, some numbers divisible by 9 as the maximum power of 3, and numbers where 2 is the maximum even prime power.

year: 2007

title: Cultural tattoos: meanings, descriptors, and attributions

authors: Nikora, Linda Waimarie

subjects: Maori

abstract: Body piercing and tattoo/ta moko were initially seen to be practiced by sailors,
criminals, specific cultural groups (e.g., Māori), or sub-cultural groups (e.g., bikers,
gang members, adolescents). In recent times, these practices have become part of
mainstream popular culture, and are enjoyed by a wide range of people. In this
study, we set out to explore patterns of body modifying behaviour engaged in, or
commented on, by a sample of university students. We invited undergraduate
psychology students from two courses to complete an ‘online’ questionnaire.
Students logged on to a web site, were presented with an information sheet, and
invited to respond. In this paper, we present the reasons why people in this sample
decided to obtain a tattoo and the meanings they ascribe to their modifications. We
will also consider the observations that people make of those who have culturally
inspired tattoos.

text: Cultural tattoos: meanings, descriptors, and attributions Body piercing and tattoo/ta moko were initially seen to be practiced by sailors,
criminals, specific cultural groups (e.g., Māori), or sub-cultural groups (e.g., bikers,
gang members, adolescents). In recent times, these practices have become part of
mainstream popular culture, and are enjoyed by a wide range of people. In this
study, we set out to explore patterns of body modifying behaviour engaged in, or
commented on, by a sample of university students. We invited undergraduate
psychology students from two courses to complete an ‘online’ questionnaire.
Students logged on to a web site, were presented with an information sheet, and
invited to respond. In this paper, we present the reasons why people in this sample
decided to obtain a tattoo and the meanings they ascribe to their modifications. We
will also consider the observations that people make of those who have culturally
inspired tattoos.

year: 2003