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arXiv:1601.01880 · Journal of Mathematical Physics

Wavefunction collapse through backaction of counting weakly interacting photons

Lee E. Harrell

Published Mar 1, 2016• 0 citations
arXivHTMLPDF

Abstract

We apply the formalism of quantum measurement theory to the idealized measurement of the position of a particle with an optical interferometer, finding that the backaction of counting entangled photons systematically collapses the particle’s wavefunction toward a narrow Gaussian wavepacket at the location xest determined by the measurement without appeal to environmental decoherence or other spontaneous collapse mechanism. Further, the variance in the particle’s position, as calculated from the post-measurement wavefunction, agrees precisely with shot-noise limited uncertainty of the measured xest. Both the identification of the absolute square of the particle’s initial wavefunction as the probability density for xest and the de Broglie hypothesis emerge as consequences of interpreting the intensity of the optical field as proportional to the probability of detecting a photon. Linear momentum information that is encoded in the particle’s initial wavefunction survives the measurement, and the pre-measurement expectation values are preserved in the ensemble average.

DOI
10.1063/1.4944454

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We apply the formalism of quantum measurement theory to the idealized measurement of the position of a particle with an optical interferometer, finding that the backaction of counting entangled photons systematically collapses the particle’s wavefunction toward a narrow Gaussian wavepacket at the location xest determined by the measurement without appeal to environmental decoherence or other spontaneous collapse mechanism. Further, the variance in the particle’s position, as calculated from the post-measurement wavefunction, agrees precisely with shot-noise limited uncertainty of the measured xest. Both the identification of the absolute square of the particle’s initial wavefunction as the probability density for xest and the de Broglie hypothesis emerge as consequences of interpreting the intensity of the optical field as proportional to the probability of detecting a photon. Linear momentum information that is encoded in the particle’s initial wavefunction survives the measurement, and the pre-measurement expectation values are preserved in the ensemble average.
A novel approach for measuring wavefunction collapse is proposed which, unlike interferometric measurements, is not affected by decoherence. A mirror of a Michelson interferometer is transferred into a quantum superposition, where the decay of the mirror superposition is measured by the fact that it affects the probability of detecting photons in the interferometer. The experiment can be realised with currently available technology for photon generation and detection and is suitable for testing collapse models, which is shown by calculating the expected outcomes for the gravity-based models of Diósi and Penrose.
Weak measurements of photon position can be used to obtain direct experimental evidence of the wavefunction of a photon between generation and ultimate detection. Significantly, these measurement results can also be understood as complex valued conditional probabilities of intermediate photon positions. It is therefore possible to interpret the quantum state as a complex valued probability distribution from which measurement probabilities can be derived according to Bayesian rules. The conventional measurement probabilities derived from squares of the wavefunction then describes the effects of measurement back-action, which originate from a non-classical relation between dynamics and statistics that is characteristic of quantum mechanics. It is pointed out that this relation can be used to derive the complete Hilbert space formalism directly from complex probabilities, without the axiomatic introduction of quantum states or operators.
The standard measurement postulate of quantum mechanics stipulates an instantaneous, non-local update of the global wavefunction upon measurement of an entangled subsystem. This postulate has been a persistent source of foundational tension, both with the locality structure of relativistic physics and with the dynamical character of the decoherence program. We separate the measurement postulate into three logically independent claims: definite outcome realization, Born-rule probability assignment, and instantaneous projection for the global wavefunction. We argue that the third claim can be eliminated at the axiomatic level and propose a modified measurement postulate, Postulate M, that replaces it. Under Postulate M, a single pointer outcome is realized locally when environmental decoherence has einselected a pointer basis at the measurement site; the corresponding selection acts on the global wavefunction as a conditional restriction to a branch rather than as a physical operation on spacelike-separated regions. We show that Postulate M recovers the standard measurement postulate as a limiting idealization for vanishing decoherence time, derives Lüders’ rule for sequential measurements, and yields no-signaling as an automatic structural consequence — not only at the level of measurement statistics but at the level of physical states themselves. Two explicit illustrations in the EPR scenario — with one and with both particles coupled to local spin environments — demonstrate the framework concretely and exhibit its frame-independent, manifestly local character. The framework is built on standard decoherence theory, compatible with multiple candidate mechanisms for single-outcome realization including gravitationally-induced collapse proposals, and naturally situated within a ψ-ontic reading of the quantum state. It represents, we argue, a reformulation of quantum mechanics as a local, state-realist theory — preserving Einstein’s aspiration for an objective physical world in the only form consistent with the empirical content of quantum mechanics, namely realism about the wavefunction rather than about pre-measurement observable values.
We study the evolution of a quantum particle in a harmonic potential whose position and momentum are repeatedly monitored. A back-action of measuring devices is accounted for. Our model utilizes a generalized measurement corresponding to the positive operator-valued measure. We assume that upon measurement, the particle's wavefunction is projected onto one of the possible detector states depending on the observed result. We chose these post-measurement states to be moving Gaussian wavepackets. The wavefunction quantum Monte Carlo formalism is used to simulate single quantum trajectories of the particle. We show how classical trajectories emerge in the course of observation and study in detail the dispersion of position and momentum of the particle.
It is demonstrated that the collapse of the wavefunction in the quantum measurement process is reflected in the two-particle correlation function for the electron spin, which can in principle be studied by a triple Stern-Gerlach experiment. The change in momentum of a Stern-Gerlach magnet is itself sufficient to allow the determination of the electron spin, subject to the uncertainty principle, if the coherence of the two components of the spin is destroyed by the magnetic field. Similar considerations apply to the two-particle correlation function for the photon polarisation, and two-photon atomic cascades can be used in practice to investigate the measurement process and determine the extension of a photon wavepacket. The irreversibility of quantum mechanical measurements and its relation to the uncertainty principle is briefly considered.
Quantum physics provides us with a formalism with which we can try to understand the microscopic world. However, the underpinning concepts of the basic principles of quantum mechanics itself are surprisingly ill understood. Even the ubiquitous concept of the wavefunction collapsing post measurement, lives outside the formal theory of quantum mechanics and must be taken as a postulate. The Schr\"odinger equation, which governs the evolution of quantum systems, does not describe the system during the measurement process. This is referred to as the measurement problem, and has been debated since the inception of quantum theory. Many different interpretations of quantum mechanics try to elucidate what happens during wavefunction collapse, such as the famous many-worlds interpretation. However, these interpretations either give rise to the same physical outcome or simply are not testable, and so cannot be physically differentiated from one another. We attempt to gain insight into the measurement problem through experiments involving sequential measurements, both in the weak and strong regime. We find surprising outcomes such as the persistence of time-ordering asymmetry in the weak measurement limit, and the survival of coherence in the strong measurement limit.