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Direct test of representation of canonical commutation relations employed in field quantization

Marek Czachor, Marcin Wilczewski

Published Sep 18, 2005• 0 citations
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Abstract

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 $T_{\\rm cav}=220$ $\\mu$s, but the data were fitted with $T_{\\rm cav}\\approx 50$ $\\mu$s). 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 $T_{\\rm cav}\\approx 1$ 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.

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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.
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 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.
The 1996 experiment by Brune et al. [Phys. Rev. Lett. 76, 1800 (1996)] on vacuum Rabi oscillation is analyzed by means of alternative models of atom-reservoir interaction. Agreement with experimental Rabi oscillation data can be obtained if one defines jump operators in the dressed-state basis and takes into account thermal fluctuations between dressed states belonging to the same manifold. Such low-frequency transitions could be ignored in a closed cavity, but the cavity employed in the experiment was open, which justifies our assumption. The cavity quality factor corresponding to the data is approximately $Q=3.31\ifmmode\times\else\texttimes\fi{}{10}^{10}$, whereas QQQ reported in the experiment was $Q=7\ifmmode\times\else\texttimes\fi{}{10}^{7}$. The rate of decoherence arising from opening of the cavity can be of the same order as an analogous correction coming from finite time resolution $\ensuremath{\Delta}t$ (formally equivalent to collisional decoherence). Peres-Horodecki separability criterion shows that the rate at which the atom-field state approaches a separable state is controlled by fluctuations between dressed states from the same manifold and not by the rate of transitions toward the ground state. In consequence, improving the QQQ factor we do not improve the coherence properties of the cavity.
Using an echo technique proposed by Morigi et al., we have time-reversed the atom-field interaction in a cavity quantum electrodynamics experiment. The collapse of the atomic Rabi oscillation in a coherent field is reversed, resulting in an induced revival signal. The amplitude of this "echo" is sensitive to nonunitary decoherence processes. Its observation demonstrates the existence of a mesoscopic quantum superposition of field states in the cavity between the collapse and the revival times.
Quantum Rabi oscillation in cavity quantum electrodynamics is at the very heart of the interaction of an atomic system with a single electromagnetic field in a resonator. It has proved a powerful tool to entangle the atom with the field in the case of an empty cavity (vacuum Rabi oscillation). On the other side, as soon as the field is macroscopic, its state seems unaffected and one observes the well-known phenomenon of classical Rabi oscillation which only affects the atomic populations. We propose to experimentally investigate the frontier between those two behaviours, when one has to take into account the fluctations of the photon number. In order to do so, we let interact circular Rydberg atoms with a microwave resonator in which a mesocopic field is injected (a few tens of photons). We observe a strong correlation between specific atomic states and the phase of the injected field, which is measured by a homodyne technique. Those correlations explain in an easy way all the standard features of quantum Rabi oscillations (especially the collapses and revivals). Moreover they allow for the preparation of atom-field entangled state as well as non classical Schroedinger cat like states for the field.