Fluctuational and dynamical back action of the measuring device

During a continuous monitoring of a quantum object, the measuring device can act back on the object in general in two different ways: First, it can influence the evolution of the expectation value of the measured observables. This is called the “dynamical back action,” and it is a regular, predictable effect. (An example is the dynamical back action produced by the dynamical rigidity of a capacity sensor, which will be discussed in chapter X.) Second, the measuring device can perturb the observables in a random way, increasing their uncertainties, i.e. producing random deviations from their expectation values. A key feature of linear, continuous measurements, as discussed in the last chapter, is that for them these two types of back action are completely independent. The fluctuational back action is fundamental and irremovable, while the dynamical back action can be avoided entirely by a suitable construction of the measuring device.

For nonlinear systems the situation is completely different, as one can readily see from a typical example of a nonlinear, continuous measurement: a monitoring of an oscillator's energy. In accord with the uncertainty relation, the energy measurement produces perturbations of the oscillator's phase, and as the measurement proceeds continuously, these perturbations cause the phase to diffuse in a Brownian-motion type way.

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, …).

The background for this book was a shift of interest among experimental physicists that began approximately 20 years ago. The shift was from quantum measurements on large ensembles of microscopic quantum objects, to experiments of very high precision on single, macroscopic objects. This shift was accompanied by developments in the “culture” of such measurements and the invention and elaboration of new experimental methods. In parallel with these experimental developments, there was a noticeable increase of theoretical interest in the foundations and interpretation of quantum theory, and even in attempts to revise its foundational principles. Although these efforts have not produced any significant changes in the fundamental structure of quantum theory, they have revealed a rather large number of “blank spots” and unsolved problems, to which the founders of quantum physics had paid no attention. A reasonable number of the blank spots have been filled in during the past decade.

The reader has probably noticed that this book can be divided into two parts. In the first (chapters I―VII and XI), the authors have tried to present the modern approach to the quantum theory of measurement. This part can be regarded as a tutorial appendix to textbooks for advanced undergraduate or graduate-level courses in quantum mechanics, where little attention is usually paid to measurement theory.

The quantum theory of measurement has been widely regarded, since the 1940s, as an esoteric subject of little relevance to “real” physics. Quite the opposite was true in the 1920s and 1930s when Niels Bohr, John von Neumann, and others were struggling, by means of gedankin experiments, to develop an understanding of how measurements on single quantum objects were to be incorporated into quantum theory. However, the understanding they achieved turned out to be of little use in the quantum mechanical applications that commanded the attention of physicists between 1940 and 1980: the interaction of photons, atomic nuclei, and elementary particles, the theory of masers and lasers, the properties of matter (superfluidity, superconductivity, semiconductors), etc.

In each of these applications, although the phenomena studied were quantum mechanical at heart, the measurements used to probe them were so far removed from the quantum domain that the only feature of the quantum theory of measurement which entered into the experiments was the probability interpretation of squared amplitudes. There was no need to invoke what soon came to be regarded as an esoteric, problematic, dubious “collapse (reduction) of the wave function.”

Fundamentally, the reason for this irrelevance of the quantum theory of measurement was technological.

Review of methods of measurement

There are two different variants of the task of measuring electromagnetic energy: i) measuring the energy in a mode of an electromagnetic resonator, and ii) measuring the energy of a traveling electromagnetic wave. This chapter deals with the first variant; the next chapter, with the second.

The general principles underlying QND measurements of energy were formulated in chapter IV. Let us recall that the main principles are: i) the response of the measuring device must be directly proportional to the energy (and not, for example, to the strength of the electric field or the charge on the capacitor); and ii) the response must contain no information about the phase of the electromagnetic oscillations. Chapter IV analyzed the simplest example of such a device: a ponderomotive sensor that registers the electromagnetic pressure on the resonator's wall.

The weakness of the electromagnetic pressure produced by a small number of quanta makes it unlikely that this ponderomotive method can be realized in practice. Because of this, several other schemes for QND energy measurements have been proposed. Most are based on nonlinear effects in dielectrics. Their practical realization is a rather complicated task because, when the energy density is low, the nonlinear effects hardly work at all, and the response of the measuring device is correspondingly small.

The measurement process and the uncertainty relation

In the unstarred sections of previous chapters, the back action of the measuring device on the measured object was analyzed by simply invoking the Heisenberg uncertainty relation for the perturbation and the measurement error. Is this simple method of analysis justified?

The measurement process does not show up in the standard formulation of the uncertainty relations. Instead, the standard formulation, which follows from the fundamental postulates of quantum mechanics (section 2.1), couples the intrinsic rms uncertainties of observables as defined by the quantum state. Nevertheless, the examples of indirect measurements discussed above (the Heisenberg microscope and Doppler speed meter in chapter I, and the electron probe in chapter III) exhibit a tight coupling between the measurement process and the uncertainty relations. In particular, both the measurement error and the perturbation of the object can be traced to the uncertainty properties of the probe's initial state, and from the fact that this initial state is subject to the uncertainty relations, one can show that the measurement error and perturbation are also subject to it.

Is it also possible, starting from the uncertainty relations as properties of the object's quantum state, to obtain the uncertainty relations for the measurement process? To answer this question, let us discuss the following thought experiment: We suppose that a generalized coordinate of some quantum object is measured, and that before the measurement the object was in a state with a very well defined momentum.

The two main types of quantum measurements

von Neumann's postulate of the reduction of the wave function answers the question of what happens to the quantum object during the measurement. However, the reduction postulate does not answer the question of how the measuring device must be designed to realize the desired measurement. To answer this second question, one must understand the connection between the measuring device as an ordinary physical system, and the nature of the desired measurement (the quantity to be measured, the desired precision, and the choice of how the measurement is to perturb the quantum object).

The Schrödinger equation cannot tell us the connection between the design of the measuring device and the nature of the measurement, because the Schrödinger equation neither governs nor describes the process of measurement. More specifically: The evolution of the wave function as described by the Schrödinger equation has two key features: it is a reversible evolution (from the final state in principle one can always evolve back to the initial one), and it is a deterministic evolution (the final state is determined uniquely by the initial one). By contrast, the reduction of the wave function in a measurement is irreversible (once the measurement is finished and the information has been extracted from it, it is impossible to return to the pre-measurement state), and it is nondeterministic (one cannot predict, before the measurement, the post-measurement state of the measured object).

My association with Erwin Schrödinger was not a close one, although I spent the summer of 1927 in Zürich, with the stated purpose (stated in my letters to the John Simon Guggenheim Memorial Foundation) of working under his supervision. In fact, I spent most of my time in my room, trying to solve the Schrödinger equation for a system consisting of two helium atoms. I did not have very much success, except that, as was mentioned later by John C. Slater, I formulated a determinant of the several spin-orbital functions of the individual electrons as a way of ensuring that the wave function is antisymmetric. This was a device that Slater made much use of in discussing the electronic structure of atoms and also of molecules in 1929 and 1931.

Walter Heitler and Fritz London were also in Zürich that summer, also with the plan of working with Schrödinger. They told me that they had talked with Schrödinger several times about their work, while walking through the woods. I did not have even that much contact with him, because he was working so hard on his own problems.

It might be possible to put theoretical physicists on a scale ranging from one extreme, those who deal with ideas, to the other, those who deal with mathematics. Wolfgang Pauli is an example of a theoretical physicist near the mathematics end.

The prehistory to Schrödinger's activity in unifed field theory

Unification is one of those long-standing quests of science. A superior point of view allows one to recognize connections and to uncover common roots. In the theory of general relativity Einstein succeeded in achieving a superior point of view in an especially impressive manner. By extension of the theory of special relativity he was able to comprehend gravitation in the geometrization of the space-time-continuum. After the success of this process of geometrization, the inclusion and unification of the electromagnetic field and possibly other fields could be considered to be a particularly important goal.

A few years after the discovery of general relativity, Weyl (1918) had already tried a fundamental extension of the framework of the theory in order to include electromagnetism as well. His attempt was based on the idea of gauge in variance–a concept which was to emerge 30 years later as a cornerstone of the modern theories of unification. Nevertheless, no agreement with observed facts could be reached by his concept of the path-dependence of a displaced length.

Whereas Weyl's generalization consisted in placing a connection of lengths beside the connection of directions given by the metric, Eddington (1923) followed a different course in considering the connection of directions as an a priori property of the manifold. In this case the metric becomes a deduced quantity.

Introduction

Erwin Schrödinger and those in his scientific generation – men like Werner Heisenberg and P. A. M. Dirac – were interested in physics because it provides us with the basic understanding of the laws of nature. Thus, even though their work has provided us with the basis of most of modern high technology, particularly through the emergence of the quantum theory of the solid state, their pursuit of physics was not motivated by this. In this sense we, in our generation, look upon Erwin Schrödinger's situation and that of his colleagues with a degree of envy.

In keeping with the tradition of Erwin Schrödinger, I have pleasure in presenting this overview of particle physics as in early 1986. I am sure this overview will be outdated by the time it sees print – but that is the fate of anything one may write in so fast moving a subject.

Physics is an incredibly rich discipline: it not only provides us with the basic understnading of the laws of nature, it is also the basis of most of modern high technology. This remark is relevant to our developing countries. A fine example of this synthesis of a basic understanding of nature with high technology is provided by liquid-crystal physics which was worked out at Bangalore by Prof. S. Chandrasekhar and his group. In this context, one may note that, because of this connection with high technology and materials’ exploitation, physics is the ‘science of wealth creation’ par excellence.

Introduction

Having the privilege of writing this paper as a grandson of Boltzmann I apologize for not being a historian of science. A manifestation of Boltzmann's influence on Schrödinger is Schrödinger's enthusiastic quotation of Boltzmann's line of thought: ‘His line of thought may be called my first love in science. No other has ever thus enraptured me or will ever do so again.’ (Schrödinger, 1929; reprinted in Schrödinger, 1957, p. XII.) Even though Schrödinger had no personal contact with Boltzmann, his scientific education at the University of Vienna was in the tradition of Boltzmann. His thesis advisor and director of the second Institute for Experimental Physics at the University of Vienna, Franz Serafin Exner, was an ardent admirer of Boltzmann and so was Boltzmann's successor on the chair for theoretical physics, Friedrich Hasenöhrl. Schrödinger paid the most impressive tribute to his teacher Hasenöhrl when he received the Nobel Prize in 1933. He avowed that Hasenöhrl might have stood in his place had he not been killed in the first world war (Schrödinger, 1935, p. 87).

Later one of Schrödinger's main fields of interest was the application of Boltzmann's statistical methods which he called ‘natural statistics’ to various problems. And it was not an accident that a paper in this field ‘On Einstein's Gas Theory’ triggered the idea of wave mechanics (Schrödinger, 1926, 1984).