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arXiv:1811.06517 · Bulletin of the American Physical Society

Limitations on the use of the Heisenberg picture in quantum optics

J. D. Franson, R. A. Brewster

Published Nov 15, 2018• 0 citations
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Abstract

The Schrodinger and Heisenberg pictures are equivalent formulations of quantum mechanics. Here we use the Schrodinger picture to calculate the decoherence that occurs when an optical Schrodinger cat state is passed through a beam splitter. We find that an exponentially large amount of decoherence can occur even when there is a negligible change in the quadrature operator x(t) in the Heisenberg picture. Similar results have been observed in the decoherence produced by an optical amplifier, and we suggest that the usual quadrature operators in the Heisenberg picture do not provide a complete description of the output of optical devices.

DOI
10.48550/arxiv.1811.06517

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The Schrodinger and Heisenberg pictures are equivalent formulations of quantum mechanics. Here we use the Schrodinger picture to calculate the decoherence that occurs when an optical Schrodinger cat state is passed through a beam splitter. We find that an exponentially large amount of decoherence can occur even when there is a negligible change in the quadrature operator x(t) in the Heisenberg picture. Similar results have been observed in the decoherence produced by an optical amplifier, and we suggest that the usual quadrature operators in the Heisenberg picture do not provide a complete description of the output of optical devices.
The study of decoherence properties improves our understanding of the fundamental principles of quantum mechanics and advances the study of quantum information processing. Herein, we report a wave-optical experiment that can simulate the decoherence process of a Schrodinger’s cat state (SCS) by photon loss. The method is based on an analogy between image rotation in wave optics and a beam splitter in quantum optics. Experimental results show that the SCS rapidly decays into a statistical mixture of two Gaussian states with approximately 10% photon loss. This behavior can be well described within the framework of quantum optics.
A quantum theory of the lossless beam splitter is given in terms of the quasi mode theory of macroscopic canonical quantization used to treat problems in cavity quantum electrodynamics and quantum optics. A Heisenberg picture approach to quantum scattering theory is applied, in which the input and output operators that are related via the scattering operator are linked to quantum optical measurements described via multi-time quantum correlation functions. In the application to the beam splitter the Heisenberg equations of motion for the input operators associated with the quasi mode annihilation operators are formally solved in a rotating picture to show that the unitary transform of the incident quasi mode annihilation operator (via the scattering operator) is just a linear combination of the incident and reflected quasi mode annihilation operators, in accordance with assumptions made in previous treatments of the beam splitter. The results depend on conservation of the transverse component of the wave vector, which follows from the form of the quasi mode-quasi mode coupling constants, and on conservation of the unperturbed energy, which follows from scattering theory. The applicability of quantum scattering theory to the beam splitter is justified in the usual situation where integrated one photon and two photon detection rates are finite for incident light field states of interest.
The description of the time evolution of systems is technically known as the "Schrodinger picture." An important alternative description is the "Heisenberg picture," which indeed was the one Heisenberg used when he first formulated quantum mechanics. The Schrodinger and Heisenberg pictures represent two different but equivalent sets of rules. In other words, when applied to a given physical situation they lead to the same observable predictions. In some respects, the Heisenberg picture presents definite technical advantages. It is particularly convenient, for instance, in quantum field theory. The chapter discusses work consistently with the equivalent Schrodinger set of rules. For this reason, it describes briefly the Heisenberg formulation. The chapter shows that the expressions used in the Schrodinger picture can be carried over to the Heisenberg picture. It implies, however, that the kets associated with the possible values that can be found upon measurement of an observable are themselves time-dependent kets.
Amplification of the Schrödinger-cat state (even coherent-state superposition) can be accompanied by its non-exponential decoherence and strong squeezing effect in the degenerate optical parametric amplifier. In the classical pump approximation and small thermal noise limit, the Wigner function behaviour illustrates that this non-exponential effect is induced by the phase-selective character of amplification. The non-exponential decoherence is much slower, in contrast with phase-unselective amplification exhibiting faster exponential decay. Thus, if one is able to produce the Schrödinger-cat state with at least a small amplitude, then it can be partially amplified with only a weak decoherence effect.