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Cambridge University Press eBooks

Weak measurement, the energy–momentum tensor and the Bohm approach

R. Flack, Basil J. Hiley

Published Nov 30, 2014• 6 citations
Publisher

Abstract

In this chapter we show how the weak values, are related to the T 0 µ ( x, t ) component of the energy–momentum tensor. This enables the local energy and momentum to be measured using weak measurement techniques. We also show how the Bohm energy and momentum are related to T 0 µ ( x, t ) and therefore it follows that these quantities can also be measured using the same methods. Thus the Bohm “trajectories” can be empirically determined, as was shown by Kocsis et al . (2011a) in the case of photons. Because of the difficulties with the notion of a photon trajectory, we argue the case for determining experimentally similar trajectories for atoms where a trajectory does not cause these particular difficulties. Introduction The notion of weak measurement introduced by Aharonov, Albert and Vaidman (1988) and Aharonov and Vaidman (1990) has opened up a radically new way of exploring quantum phenomena. In contrast to the strong measurement (von Neumann, 1955), which involves the collapse of the wave function, a weak measurement induces a more subtle phase change which does not involve any collapse. This phase change can then be amplified and revealed in a subsequent strong measurement of a complementary operator that does not commute with the operator being measured. This amplification explains why it is possible for the result of a weak spin measurement of a spin-1/2 atom to be magnified by a factor of 100 (Aharonov et al ., 1988; Duck, Stevenson and Sudarshan, 1989). A weak measurement, then, provides a means of amplifying small signals as well as allowing us to gain new, more subtle information about quantum systems. One of the new features that we will concentrate on in this chapter is the possible measurement of the T 0 µ ( x, t ) components of the energy–momentum tensor.

DOI
10.1017/cbo9781107706927.007

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In this chapter we show how the weak values, are related to the T 0 µ ( x, t ) component of the energy–momentum tensor. This enables the local energy and momentum to be measured using weak measurement techniques. We also show how the Bohm energy and momentum are related to T 0 µ ( x, t ) and therefore it follows that these quantities can also be measured using the same methods. Thus the Bohm “trajectories” can be empirically determined, as was shown by Kocsis et al . (2011a) in the case of photons. Because of the difficulties with the notion of a photon trajectory, we argue the case for determining experimentally similar trajectories for atoms where a trajectory does not cause these particular difficulties. Introduction The notion of weak measurement introduced by Aharonov, Albert and Vaidman (1988) and Aharonov and Vaidman (1990) has opened up a radically new way of exploring quantum phenomena. In contrast to the strong measurement (von Neumann, 1955), which involves the collapse of the wave function, a weak measurement induces a more subtle phase change which does not involve any collapse. This phase change can then be amplified and revealed in a subsequent strong measurement of a complementary operator that does not commute with the operator being measured. This amplification explains why it is possible for the result of a weak spin measurement of a spin-1/2 atom to be magnified by a factor of 100 (Aharonov et al ., 1988; Duck, Stevenson and Sudarshan, 1989). A weak measurement, then, provides a means of amplifying small signals as well as allowing us to gain new, more subtle information about quantum systems. One of the new features that we will concentrate on in this chapter is the possible measurement of the T 0 µ ( x, t ) components of the energy–momentum tensor.
The relationship between the real part of the weak value of the momentum operator at a post selected position is discussed and the meaning of the experimentally determined stream-lines in the Toronto experiment of Kocsis et al is re-examined. We argue against interpreting the energy flow lines as photon trajectories. The possibility of performing an analogous experiment using atoms is proposed in order that a direct comparison can be made with the trajectories calculated by Philippidis, Dewdney and Hiley using the Bohm approach.
In a recent paper Mahler {\em et al.} have argued that the experiments of Kocsis {\em et al.} provide experimental evidence for Bohmian mechanics. Unfortunately these experiments used relativistic, zero rest mass photons whereas Bohmian mechanics is based on non-relativistic Schrödinger particles having non-zero rest mass. The experimental results can be accounted for in terms of a different approach based on the electromagnetic field that was already outlined by Bohm in an appendix of the second of his 1952 papers. A subsequent development of this approach by Bohm, Hiley, Holland, Kaloyerou, Lam and Dewdney, shows in detail how this theory accounts for experimental quantum phenomena in general. Using this theory, we are led to the conclusion that the experiments have constructed mean momentum flow lines by measuring the real part of what we term the weak Poynting vector. These results support and clarify the analysis of Bliokh {\em et al}. The experimental flow lines can be constructed independently of the number of photons in the beam leading to the conclusion that flow lines cannot be interpreted as `photon trajectories'. We discuss exactly how the notion of a photon arises in the field approach in the light of weak values.
There has been a recent revival of interest in the notion of a 'trajectory' of a quantum particle. In this paper, we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm 'trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis et al.
Summary form only given. Measurements are the very basis of Physics, especially in Quantum Mechanics, where they assume even a more fundamental role because of wave function collapse after a “strong” (projective) measurement. Furthermore, measuring a quantum-mechanical observable completely erases the information on its conjugate one (e.g. measurement of position erases information on momentum, and vice-versa). Nevertheless, in Quantum Mechanics other kinds of measurement are possible. For example, wave function collapse can be overcome through Weak Measurements, i.e. measurements performed with an interaction sufficiently weak not to collapse the original state, featuring several interesting properties. An example of these are weak values [1-3], realised for the first time in [4-6], that have been used for addressing fundamental questions [7-12] as Contextuality and are also a tool for Quantum Metrology [13-19]. One of the most intriguing properties of Weak Measurements is that, since they are not affected by wave function collapse, they can allow gathering simultaneous information on non-commuting observables [20], impossible with the standard (projective) measurement protocols. A second example is offered by Protective Measurements [21], a new technique able to extract information on the expectation value of an observable even measuring a single particle. In this talk, after a general introduction to WMs, we present the first realisation of sequential weak value measurements [22], i.e. a joint measurement of the weak value of (incompatible) polarizations on a single photon. Then, we present an experiment addressed to explore the connection between anomalous weak values and Contextuality [23], showing a clear violation of the inequality proposed in [12] while satisfying all the additional theoretical requests, unequivocally demonstrating the contextual nature of weak values. Finally, we present and discuss the first implementation of Protective Measurements, showing unprecedented measurement capability and demonstrating how the expectation value of an observable can be obtained with a single measurement on a single particle.
Quantum weak measurement, measuring some observable quantities within the selected subensemble of the entire quantum ensemble, can produce many interesting results such as the superluminal phenomena. An outcome of such a measurement is the weak value which has been applied to amplify some weak signals of quantum interactions in lots of previous references. Here, we apply the weak measurement to the system of relativistic cold atoms. According to mass-energy equivalence, the internal energy of an atom will contribute its rest mass and consequently the external momentum of center of mass. This implies a weak coupling between the internal and external degrees of freedom of atoms moving in the free space. After a duration of this coupling, a weak value can be obtained by post-selecting an internal state of atoms. We show that the weak value can change the momentum uncertainty of atoms and consequently help us to experimentally measure the weak effects arising from mass-energy equivalence.
A weak measurement does not completely collapse the wave function into one of the components of a superposition. A subsequent projective measurement on the weakly measured wave function can result in an amplification of the result for the original weak measurement. This chapter illustrates the procedure by a measurement of the average trajectories taken by particles in the two-slit experiment, in which the local momentum of a photon is encoded into its polarization, by means of a birefringent calcite crystal. The particle trajectories show the interference pattern. A second experimental example finds the real part and the imaginary part of a particle’s complex wave function. The amplitude and phase of the wave function are expressed in terms of its real and imaginary parts.