Discrete and continuous measurements All the examples of measurements so far discussed have in common the fact that the experimental output is either a single number or a finite set of numbers. However, in real measurements the output is often a record of some continuous function of time, from which one can get an understanding of the behavior of the measured quantity during some time interval. This type of measurement is called continuous. Correspondingly, measurements that give a discrete set of numbers can be called discrete. Let us note that measurements are sometimes called continuous if they are made not instantaneously, but over a finite duration of time. Evidently, this is not a reasonable viewpoint; all real measurements last for a finite time. The dividing line between continuous and discrete measurements is in fact the character of the output signal: does it give information about the measured quantity only at some chosen moment of time, or does it permit a continuous monitoring of the quantity's time evolution. The necessity to develop the quantum theory of continuous measurements arose in the 1960s, when devices for making continuous measurements were developed with sensitivities close to the quantum level (masers, parametric amplifiers, optical heterodyne receivers, …).
Discrete and continuous measurements All the examples of measurements so far discussed have in common the fact that the experimental output is either a single number or a finite set of numbers. However, in real measurements the output is often a record of some continuous function of time, from which one can get an understanding of the behavior of the measured quantity during some time interval. This type of measurement is called continuous. Correspondingly, measurements that give a discrete set of numbers can be called discrete. Let us note that measurements are sometimes called continuous if they are made not instantaneously, but over a finite duration of time. Evidently, this is not a reasonable viewpoint; all real measurements last for a finite time. The dividing line between continuous and discrete measurements is in fact the character of the output signal: does it give information about the measured quantity only at some chosen moment of time, or does it permit a continuous monitoring of the quantity's time evolution. The necessity to develop the quantum theory of continuous measurements arose in the 1960s, when devices for making continuous measurements were developed with sensitivities close to the quantum level (masers, parametric amplifiers, optical heterodyne receivers, …).
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.
The VIM3 defines a quantity as a “property of a phenomenon, body, or substance”, leaving the characteristics of the term ‘quantity’, related to the chosen characteristics of the relevant properties. The question is: does necessarily a property also necessarily refer to the possible granularity of a phenomenon, body, or substance? Take, for example, for the quantity “mass”: it does not always have to take into account whether or not a phenomenon, body, or substance is subdivided into discrete entities? It depends of the frame of the analysis and also on the chosen measurement unit. In other cases, like temperature, the macroscopic properties are related to the statistical properties of granular substances like atoms and molecules are, so the present meaning of ‘temperature’ is generally lost at the numerical level where the entity’s statistics become meaningless. Yet another case is quantum physics. The paper illustrates the issue and possible solutions under development. Keywords: continuous; granular; quantity; magnitude; quanta; integer number; real number; counting; function
If one continuously measures a decaying system, the system will appear to never decay that was called quantum Zeno effect. The continuous measurement is defined by a sequence of measurements whose time interval t between measurements approaches zero. Later many works chose the time interval t as finite (and greater than the Zeno time) which corresponds to making equal spaced measurements over a discrete time interval. With the discrete variable formulism one can derive the so-called Anti-Zeno effect. Our study is trying to contrast the results between continuous time interval measurement versus discrete time interval measurement. We demonstrate that we can obtain so-called “Zeno” and “Anti-Zeno” in a classical system if we apply the definition of non-ideal measurement.
In recent years advances in technology, especially in the field of quantum optics, have made possible repetitive quantum measurements on a single quantum object, an issue that remained an academic curiosity until 25 years ago. This book, first published in 1992, provides a tutorial introduction to this field. The book is coauthored by an experimentalist and a theorist: the former - Braginsky - was the first experimentalist after Weber to develop techniques for gravitational-wave detection; the latter - Khalili - contributed with Braginsky to develop the formal apparatus to describe the experimental devices for measurement schemes. After the first part, intended as a pedagogical appendix of a standard textbook, and devoted to the modern approach to quantum measurement, the second part is a review of some new methods of quantum measurements for macroscopic objects. The central problem is the detection of a small time-dependent classical force that acts on a quantum probe. This is the typical situation that occurs in gravitational-wave detection, where the probe is a macroscopic moving mass weakly coupled to a classical wave. The problem of gravitational-wave detection and the limitation to accuracy of measurement of position of a moving mass - the so-called standard quantum limit (SQL), which is a consequence of the random back-action of the measuring device on the mass - triggered the development of new quantum nondemolition (QND) measurements, i.e. measurements where the measured quantity is left unperturbed after the measurement. SQL and QND measurements (both Braginsky inventions) permeate the whole book as a transversal chapter. The aim is to give simple criteria and recipes to design a measuring device in order to optimize the accuracy of the measurement, and, at the same time, to minimize the perturbation that the device exerts on the measured object, so that it will not interfere with successive measurements on the same object. The continuous measurements are regarded as the limiting case of a sequence of repeated discrete measurements, when the interval between the measurements is much smaller than the characteristic time-scale on which the measured quantity changes. The analysis of continuous measurements give to the authors the opportunity to illustrate the watchdog effect and the quantum Zeno paradox (real effect), while deriving the master equation for the evolution of the object state. Then, an entire chapter is devoted to the analysis of devices for continuous measurement of small mechanical displacements. The book is easy to understand, and can serve as a complementary reading for graduate level courses in quantum mechanics. The material of the book can be useful for all scientists and engineers involved in researches related to high-sensitive quantum measurements, quantum information theory and quantum computation. The reader, however, should be aware that this book presents a very particular point of view, with no room left to very relevant concepts in quantum measurement theory, as, for example, squeezing, quantum amplification, joint-measurements of non-commuting observables and phase-measurements. More regrettably, the list of references is unsatisfactorily incomplete, with no mention of alternative developments, for example, regarding the SQL for measuring the position of a free mass. Here, schemes to breach the SQL have been proposed in the literature (Yuen H P 1983 Phys. Rev. Lett. 51 719, Ozawa M 1988 Phys. Rev. Lett. 60 385), whereas from the book one could conclude that such SQL would be unsurpassable. Also, a criticism is in order of the way of deriving some SQL's, based solely on the Heisenberg relation, even in consideration that the aim of such derivations is mostly pedagogical.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
This chapter investigates quantum continuous measurements from the point of view of their group-theoretical structure. There are two sources of a group-theoretical structure. One is that evolution develops continuously in time. Another source of group-theoretical structure is the equivalence of different measurement outputs. This equivalence can be described by the transverse group transforming alternative outputs into each other. To treat the group structure connected with time evolution, the notation has to be made more detailed, with explicit marking of the initial and final instants. A quantum system under continuous measurement exhibits evolution of a quite new type, having classical and quantum features simultaneously. The Galilean type semigroups are in fact very similar in the two cases, but the representations arising in describing the evolution must be more complicated in the presence of a measurement.