Carlos Alexandre Brasil

Protecting a quantum state from environmental noise by an incompatible finite-time measurement

We show that measurements of finite duration performed on an open two-state system can protect the initial state from a phase-noisy environment, provided the measured observable does not commute with the perturbing interaction. When the measured observable commutes with the environmental interaction, the finite-duration measurement accelerates the rate of decoherence induced by the phase noise. For the description of the measurement of an observable that is incompatible with the interaction between system and environment, we have found an approximate analytical expression, valid at zero temperature and weak coupling with the measuring device. We have tested the validity of the analytical predictions against an exact numerical approach, based on the superoperator-splitting method, that confirms the protection of the initial state of the system. When the coupling between the system and the measuring apparatus increases beyond the range of validity of the analytical approximation, the initial state is still protected by the finite-time measurement, according with the exact numerical calculations.

A simple derivation of the Lindblad equation

We present a derivation of the Lindblad equation -an important tool for the treatment of nonunitary evolutions -that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings formato We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad formo The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.Apresentamos uma derivacao da equacao de Lindblad -uma ferramenta importante no tratamento de evolucoes nao-unitarias -acessivel a estudantes de graduacao em fisica ou matematica com nocoes basicas de mecânica quântica. Consideramos aqui um caso especifico, correspondente a uma situacao bem simples, onde o sistema principal interage com um banho de osciladores harmonicos a temperatura nula, com hamiltoniano de interacao que se assemelha ao modelo de Jaynes-Cummings. Iniciamos com a equacao de Born-Markov e, atraves do traco parcial dos graus de liberdade do banho, obtemos uma equacao na forma de Lindblad. Essa situacao especifica e bem instrutiva, pois permite verificar que os lindblads representam a contribuicao do sistema principal ao hamiltoniano de interacao com o banho, e que a aproximacao markoviana e vital para o surgimento do lindbladiano. A deducao formal da equacao de Lindblad para situacoes gerais requer o uso do formalismo de semigrupos dinâmicos quânticos e consideracoes mais abrangentes sobre o ambiente e a temperatura do que as utilizadas aqui.

How Much Time Does a Measurement Take

We consider the problem of measurement using the Lindblad equation, which allows the introduction of time in the interaction between the measured system and the measurement apparatus. We use analytic results, valid for weak system-environment coupling, obtained for a two-level system in contact with a measurer (Markovian interaction) and a thermal bath (non-Markovian interaction), where the measured observable may or may not commute with the system-environment interaction. Analysing the behavior of the coherence, which tends to a value asymptotically close to zero, we obtain an expression for the time of measurement which depends only on the system-measurer coupling, and which does not depend on whether the observable commutes with the system-bath interaction. The behavior of the coherences in the case of strong system-environment coupling, found numerically, indicates that an increase in this coupling decreases the measurement time, thus allowing our expression to be considered the upper limit for the duration of the process.

Elliptical orbits in the phase-space quantization

The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfelds original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.

Understanding the pointer states

In quantum mechanics, pointer states are eigenstates of the observable of the measurement apparatus that represent the possible positions of the display pointer of the equipment. The origin of this concept lies in attempts to fill the blanks in the Everetts relative-state interpretation, and to make it a fully valid description of physical reality. To achieve this, it was necessary to consider not only the main system interacting with the measurement apparatus (like von Neumann and Everett did) but also the role of the environment in eliminating correlations between different possible measurements when interacting with the measurement apparatus. The interaction of the environment with the main system (and the measurement apparatus) is the core of the decoherence theory, which followed Everetts thesis. In this article, we review the measurement process according to von Neumann, Everetts relative state interpretation, the purpose of decoherence and some of its follow-up until Wojciech Zureks primordial paper that consolidated the concept of pointer state, previously presented by Heinz Dieter Zeh. Employing a simple physical model consisting of a pair of two-level systems -- one representing the main system, the other the measurement apparatus -- and a thermal bath -- representing the environment -- we show how pointer states emerge, explaining its contributions to the question of measurement in quantum mechanics, as well as its limitations. Finally, we briefly show some of its consequences. This paper is accessible to readers with elementary knowledge about quantum mechanics, on the level of graduate courses.

The master equation for the reduced open-system dynamics, including a Lindbladian description of finite-duration measurement

Abstract.We consider the problem of the measurement of a system occurring during a finite-time interval, while environmentally induced noise decreases the system-state coherence. We assume a Markovian measuring device and, therefore, use a Lindbladian description for the measurement dynamics. For studying the case of noise produced by a non-Markovian environment, whose definition does not include the measuring apparatus, we use the Redfield approach to the interaction between system and environment. In the present hybrid theory, to trace out the environmental degrees of freedom, we introduce an analytic method based on superoperator algebra and Nakajima-Zwanzig projectors. The resulting master equation, describing the reduced system dynamics, is illustrated in the case of a qubit under phase noise during a finite-time measurement.

Analysis of the Jun Ishiwara’s “The universal meaning of the quantum of action”

Abstract Here, we present an analysis of the paper “Universelle Bedeutung des Wirkungsquantums” (The universal meaning of the quantum of action), published by Jun Ishiwara in German in the “Proceedings of Tokyo Mathematico-Physical Society 8 (1915) 106–116”. In his work, Ishiwara, established in the Sendai University, Japan, proposed – simultaneously with Arnold Sommerfeld, William Wilson and Niels Bohr in Europe – the phase-space-integral quantization, a rule that would be incorporated into the old-quantum-theory formalism. The discussions and analysis render this paper fully accessible to undergraduate students of physics with elementary knowledge of quantum mechanics.

Efficient finite-time measurements under thermal regimes

Contrary to conventional quantum mechanics, which treats measurement as instantaneous, here we explore a model for finite-time measurement. The main two-level system interacts with the measurement apparatus in a Markovian way described by the Lindblad equation, and with an environment, which does not include the measuring apparatus. To analyse the environmental effects on the final density operator, we use the Redfield approach, allowing us to consider a non-Markovian noise. In the present hybrid theory, to trace out the environmental degrees of freedom, we use a previously developed analytic method based on superoperator algebra and Nakajima-Zwanzig superoperators. Here, we analyse two types of system-environment interaction, phase and amplitude damping, which allows us to conclude that, in general, a finite-time quantum measurement performed during a certain period is more efficient than an instantaneous measurement performed at the end of it, because the rate of change of the populations is attenuated by the system-measurement apparatus interaction.

Weak values in collision theory

Weak measurements have an increasing number of applications in contemporary quantum mechanics. They were originally described as a weak interaction that slightly entangled the translational degrees of freedom of a particle to its spin, yielding surprising results after post-selection. That description often ignores the kinetic energy of the particle and its movement in three dimensions. Here, we include these elements and re-obtain the weak values within the context of collision theory by two different approaches, and prove that the results are compatible with each other and with the results from the traditional approach. To provide a more complete description of the Stern-Gerlach apparatus, we use weak vectors, a generalization of the weak values.