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CRC Press eBooks

Introduction to Continuous Quantum Measurements

Michael B. Mensky

Published Oct 19, 2017• 0 citations
Publisher

Abstract

This chapter explores the path-integral theory of continuous quantum measurements. It explains the principal ideas of this theory on a qualitative level with a minimum of mathematical apparatus. Quantum mechanics appeared as a theory of microscopic bodies when it had been proved that the motion of microscopic systems cannot be described in the framework of classical physics. However, quantum effects may be important even for macroscopic bodies. From a certain point of view the main object in quantum mechanics is probability amplitude because it expresses the principal difference between quantum and classical theory. Quantum mechanics differs in that not probabilities but probability amplitudes should be summed up for a quantum system. The uncertainty principle is good for expressing features of instantaneous quantum measurements but it is inconvenient for continuous measurements. The variance of outputs of a precise measurement may be considered to be quantum measurement noise.

DOI
10.1201/9781315140292-1

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This chapter explores the path-integral theory of continuous quantum measurements. It explains the principal ideas of this theory on a qualitative level with a minimum of mathematical apparatus. Quantum mechanics appeared as a theory of microscopic bodies when it had been proved that the motion of microscopic systems cannot be described in the framework of classical physics. However, quantum effects may be important even for macroscopic bodies. From a certain point of view the main object in quantum mechanics is probability amplitude because it expresses the principal difference between quantum and classical theory. Quantum mechanics differs in that not probabilities but probability amplitudes should be summed up for a quantum system. The uncertainty principle is good for expressing features of instantaneous quantum measurements but it is inconvenient for continuous measurements. The variance of outputs of a precise measurement may be considered to be quantum measurement noise.
Advances in technology are taking the accuracy of macroscopic as well as microscopic measurements close to the quantum limit, for example, in the attempts to detect gravitational waves. Interest in continuous quantum measurements has therefore grown considerably in recent years. Continuous Quantum Measurements and Path Integrals examines these measurements using Feynman path integrals. The path integral theory is developed to provide formulae for concrete physical effects. The main conclusion drawn from the theory is that an uncertainty principle exists for processes, in addition to the familiar one for states. This implies that a continuous measurement has an optimal accuracy-a balance between inefficient error and large quantum fluctuations (quantum noise). A well-known expert in the field, the author concentrates on the physical and conceptual side of the subject rather than the mathematical.
This chapter reviews the main points in the theory of instantaneous measurements. A description of continuous measurements as limits of series of instantaneous measurements turns out to be very cumbersome and in fact inadequate. A characteristic feature of quantum systems is that measurements in them unavoidably affect their dynamics. The well known manifestation of this back reaction is the uncertainty principle, discussion of which can be found in any textbook on quantum mechanics. The operator plays the role of a propagator for a quantum system under continuous measurement. The quantum Zeno effect is an example of a continuous measurement giving a trivial result. However, this is a consequence of the very special character of the measurement procedure described by a discrete set of orthogonal projectors each of which is a bounded operator. The description of approximate measurements was derived for the case when precise measurement is a discrete-spectrum measurement.
This chapter explores accurate formulation of the approach, including a description of the evolution of the measured system and the generalized unitarity condition for this evolution. It argues that evolution under continuous measurement has both quantum and classical features. The chapter deals with the evolution of the system subject to continuous measurement. It demonstrates in general form and considers the specific example of a free particle scattered because its position is measured. The quantum theory of measurements has been discussed not only because of its conceptual interest but also for practical purposes. The chapter discusses the application of the measurement amplitude for estimation of probabilities of different measurement outputs. However, the same amplitude can be used to describe the evolution of the system subject to the measurement. The chapter shows the continuous measurement changes the evolution of the quantum system undergoing the measurement.
This chapter aims to express the deviation, the variance of outputs, in the simpler and more transparent form of a specific type of action uncertainty principle (AUP). It considers two simple examples of the application of AUP in quantum mechanics: position monitoring of a free particle and of a harmonic oscillator. The chapter considers only the case of a one-dimensional particle and oscillator. Both rougher and finer measurements are disadvantageous as compared with the optimal measurement, the former because of classical and the latter because of quantum measurement noise. The variance of the measurement outputs corresponding to the optimal level of measurement puts an absolute quantum limit on the given type of measurement. The path-integral approach has already been applied to measurements of quantum fields: an electromagnetic field and a gravitational field. Measurements of this type can be called continual because they are prolonged in time and protracted in space.
Preface INTRODUCTION TO CONTINUOUS QUANTUM MEASUREMENTS Quantum and Classical Systems Amplitudes and Alternatives Paths and Continuous Measurements The Action Uncertainty Principle INSTANTANEOUS AND SEQUENTIAL MEASUREMENTS Measurement of a Quantum System Quantum Zeno Paradox Approximate and Sequential Measurements TECHNIQUE OF PATH INTEGRALS Propagators and Path Integrals Definition of a Path Integral The Path Integral for an Oscillator Gaussian Path Integrals CONTINUOUS MEASUREMENT AND EVOLUTION OF THE MEASURED SYSTEM The Measurement Amplitude Effective Lagrangian Evolution of a Quantum System Subject to Continuous Measurement Scattering by the Measuring Medium CONTINUOUS MEASUREMENTS OF OSCILLATORS Position Monitoring (Path measurement) Estimation of Force Acting on an Oscillator Spectral Measurements of an Oscillator Evolution of a Harmonic Oscillator Subject to Spectral Measurement Measurement of a Path of a Nonlinear Oscillator CONTINUOUS QUANTUM NONDEMOLITION MEASUREMENTS Quantum Nondemolition Measurements Monitoring of Linear Momentum Velocity versus Momentum QND Measurements of Oscillators MEASUREMENT OF AN ELECTROMAGNETIC FIELD Exposition of the Method Measurement Amplitude Analysis of the Results Another Definition of the Field Strength Explicit Model of a Meter TIME IN QUANTUM COSMOLOGY Statement of the Problem Measurements in Quantum Cosmology Minisuperspace without Measurement Minisuperspace under Measurement Analysis of the results Concluding Remarks THE ACTION UNCERTAINTY PRINCIPLE Uncertainties in Continuous Measurements Classical Regime of Measurements Quantum Regime and AUP AUP and Equations of Motion Examples from Quantum Mechanics Quantum Nondemolition Measurements AUP in Quantum Gravity Concluding Remarks GROUP-THEORETICAL STRUCTURE OF QUANTUM CONTINUOUS MEASUREMENTS Evolution under Measurement Transverse Group for Alternatives Longitudinal Semigroup for Evolution Unification of Transverse and Longitudinal Structures Concluding Remarks PATHS AND MEASUREMENTS: FURTHER DEVELOPMENT Quantum and Classical Features in Continuous Measurement The Two-Slit Experiment Wave-Particle Dualism Conceptual Problems References Index
Aiming at the quantum mechanics wave function description probability feature, statistical interpretation, randomicity, uncertainty, nonlocality and agnosticism etc fundamental principles and concepts, based on the mass-space interaction, mass-space space-time view, microscopic mass-energy systems correlating with space-time model etc principles and theories, and the quantum entanglement particle experiments etc facts, present the matter-ordering-energy conservation laws and the Dong Pei innovating group interpretation of quantum mechanics. Simultaneously indicate: In the universe there exists the ordering- and disordering-energy matter, together known as matter. Existing the absolute space and absolute time. The absolute space-time is continuous. Matter and the absolute space-time together create and compose the objective world. The objective world is a harmonious and unified whole. Physical laws are completely unified. The objective world and matter are all finite. Matter exists, evolves and moves in the absolute space-time. Interactions are the most essential and most root basic attribute of matter. The interaction among matter is the mass-space interaction. The mass-space interaction is continuous and stable, is not random and intermittent. Macroscopic and microscopic regions all observe the identical physical laws. Elementary particles and they compose the microscopic mass-energy systems are all the ordering-energy matter, their mass-energy space-time is a interdependent whole, their feature, state, structure and motion laws are all entirely determined by the mass-space interaction, correspond with the determinism and causality, the motion state and feature, motion orbitals, position and speed of particles in the absolute space-time are all absolutely determined and could be measured exactly, their quantum states could be approximately described by the wave function of quantum mechanics. The wave functions are the simplified mathematical function approximate expression of quantum system states of adopted interaction, can be used to calculate the correlative physical quantities. The wave functions could approximately describe the state of the mass-energy space-time whole of a microscopic mass-energy system with the mass-space interaction acting and continuously moving. An eigenstate just is the state making the system energy minimum and in rather stable existence. Scientific theories are the abstract simplified approximate expression for the essential of natural phenomena. Physical theory is the abstract simplified approximate representation for the matter structure and its motion laws. Whereas the mathematical formulation then is the further simplified approximate representation for physical contents. All of theories, include quantum mechanics and relativity, only when the assumed approximate condition is satisfied, just are the applied correct scientific theory, could describe the objective world good enough but not exactly. Develop scientific theory, correctly cognize and reasonably interpret the objective world, without the end.