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arXiv:quant-ph/0507093

Quantum optics in different representations of the algebra of canonical commutation relations (I): Unexpected properties of Rabi oscillations -- theory and experiment

Marcin Wilczewski, Marek Czachor

Published Jul 11, 2005• 0 citations
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Abstract

We discuss the Jaynes-Cummings model in different representations of the algebra of canonical commutation relations. The first conclusion is that all the irreducible representations lead to equivalent physical predictions. However, the reducible representation recently introduced as a candidate for `QED without infinities' leads to new effects. We analyze from this perspective the experiments on Rabi oscillations performed by the Kastler-Brossel Laboratory group from Paris. Surprisingly, the results seem to support the reducible representation approach. We also discuss possibilities of more definitive tests of the new formalism.

DOI
10.48550/arxiv.quant-ph/0507093

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We discuss the Jaynes-Cummings model in different representations of the algebra of canonical commutation relations. The first conclusion is that all the irreducible representations lead to equivalent physical predictions. However, the reducible representation recently introduced as a candidate for `QED without infinities' leads to new effects. We analyze from this perspective the experiments on Rabi oscillations performed by the Kastler-Brossel Laboratory group from Paris. Surprisingly, the results seem to support the reducible representation approach. We also discuss possibilities of more definitive tests of the new formalism.
Rabi oscillations observed in the classic experiment of the Haroche group [M. Brune {\\it et al.}, Phys. Rev. Lett. {\\bf 76}, 1800 (1996)] exhibited unexpectedly strong damping. The damping had no theoretical explanation and was interpreted in terms of experimental noise of unclear origin (cavity relaxation time was Trmcav=220T_{\\rm cav}=220Trmcav​=220 mu\\mumus, but the data were fitted with Trmcavapprox50T_{\\rm cav}\\approx 50Trmcav​approx50 mu\\mumus). Here we show that stronger relaxation of Rabi oscillations is typical of electromagnetic fields quantized in reducible representations of the algebra of canonical commutation relations. The relaxation is then of fundamental origin, and is the first of a sequence of collapses and revivals that are predicted to occur even in exact vacuum. The degree of relaxation depends on parameters of the representation. We estimate which representations fit the data and show that with Trmcavapprox1T_{\\rm cav}\\approx 1Trmcav​approx1 ms one should observe a revival of the `decayed' vacuum Rabi oscillation at times roughly 10 times longer than those employed in the experiment. Confirmation of the revival would have to change the ways we think of quantum fields. Unfortunately, a negative result would not be conclusive.
A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials is tested on the example of quantum electrodynamic fields produced by a classical current. The scheme implies a modified but very physically looking Hamiltonian. We solve Heisenberg equations of motion and compute photon statistics. Poisson statistics naturally occurs and no infrared divergence is found even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincare covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to Renyi's α\alphaα. It is shown that the ``Shannon limit α→1\alpha\to 1α→1 plays here a role of correspondence principle with the standard regularized formalism.
The paper continues the analysis of vacuum Rabi oscillations we started in part I [Phys. Rev. A 79, 033836 (2009)]. Here we concentrate on experimental consequences for cavity QED of two different classes of representations of harmonic-oscillator Lie algebras. The zero-temperature master equation, derived in part I for irreducible representations of the algebra, is reformulated in a reducible representation that models electromagnetic fields by a gas of harmonic-oscillator wave packets. The representation is known to introduce automatic regularizations that in irreducible representations would have to be justified by ad hoc arguments. Predictions based on this representation are characterized in thermodynamic limit by a single parameter $\ensuremath{\varsigma}$, responsible for collapses and revivals of Rabi oscillations in exact vacuum. Collapses and revivals disappear in the limit $\ensuremath{\varsigma}\ensuremath{\rightarrow}\ensuremath{\infty}$. Observation of a finite $\ensuremath{\varsigma}$ would mean that cavity quantum fields are described by a non-Wightmanian theory, where vacuum states are zero-temperature Bose-Einstein condensates of a NNN-particle bosonic oscillator gas and, thus, are nonunique. The data collected in the experiment of Brune et al. [Phys. Rev. Lett. 76, 1800 (1996)] are consistent with any $\ensuremath{\varsigma}>400$.
We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.
A presentation of the canonical commutation relation of Bose fields is given in a way which is independent of the choice of the bases of the test functions and covariant with respect to the Euclidean transformation of the coordinate system. It is shown that the representation is characterized by an integral on the conjugate space L* of the space L of the test functions and a real function on Σ⊗L* where Σ is the group of the transformations f→u-1f+φ; fεL*, φεL and u is a Euclidean transformation of L*. The conditions for the irreducibility of a representation and the unitary equivalence of the representations and the existence of unique vacuum state are given. An example of the inequivalent Euclidean covariant irreducible representations containing unique vacuum state is given.
It often goes unnoticed that, even for a finite number of degrees of freedom, the canonical commutation relations have many inequivalent irreducible unitary representations; the free particle and a particle in a box provide examples that are both simple and well-known. The representations are unitarily inequivalent because the spectra of the position and momentum operators are different, and spectra are invariant under unitary transformations. The existence of these representations can have consequences that run from the merely unexpected to the barely conceivable. To start with, states of a single particle that belong to inequivalent representations will always be mutually orthogonal; they will never interfere with each other. This property, called superseparability elsewhere, is well-defined mathematically, but has not yet been observed. This article suggests two single-particle interference experiments that may reveal its existence. The existence of inequivalent irreducibile representations may be traced to the existence of different self-adjoint extensions of symmetric operators on infinite-dimensional Hilbert spaces. Analysis of the underlying mathematics reveals that some of these extensions can be interpreted in terms of topological, geometrical and physical quantities that can be controlled in the laboratory. The tests suggested are based on these interpretations. In conclusion, it is pointed out that mathematically rigorous many-worlds interpretations of quantum mechanics may be possible in a framework that admits superseparability.