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arXiv:0811.3177 · Physical Review A

Theory versus experiment for vacuum Rabi oscillations in lossy cavities

Marcin Wilczewski, Marek Czachor

Published Mar 24, 2009• 16 citations
arXivHTMLPDF

Abstract

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 $Q$ 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 $Q$ factor we do not improve the coherence properties of the cavity.

DOI
10.1103/physreva.79.033836

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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×\else\texttimes\fi1010Q=3.31\ifmmode\times\else\texttimes\fi{}{10}^{10}Q=3.31\ifmmode×\else\texttimes\fi1010, whereas QQQ reported in the experiment was Q=7\ifmmode×\else\texttimes\fi107Q=7\ifmmode\times\else\texttimes\fi{}{10}^{7}Q=7\ifmmode×\else\texttimes\fi107. 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 \ensuremathΔt\ensuremath{\Delta}t\ensuremathΔt (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.
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 $\ensuremath{\varsigma}>400$.
There are only fistful of systems in physics that can be described by an exact theory from first principles and, at the same time, can be investigated under experimental conditions approaching the idealized theoretical ones. The one-atom-maser or micromaser provides such an example, making a detailed study of the fundamental properties of the atom-field interaction possible. The situation realized in a micromaser is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a cavity field. In the micromaser the atoms have a dual purpose of both pumping the field and also probing it via measurements of the outgoing atoms. Any other means of measuring the field inside the resonator has the detrimental effect of lowering its Q-factor and thereby the photon storage time. The long photon storage time of the resonator allows for the decay of the field to be negligible during the passage of an atom and its interaction with the field. The behavior of atoms in a cavity is governed by the oscillatory exchange of energy between the atoms and the field, which is called Rabi oscillation. For cavity fields in the vacuum and few photon number states (Fock states), Rabi oscillations have been measured in the past. To prevent thermal effects from polluting the pure number states these experiments were performed at low temperatures (below 1 K). However, the observed resolution of the Rabi oscillations measurements was quite disappointing (only about 2% of the predicted theoretical value) and precluded more ambitious experiments like atom-atom correlations or time resolved investigations of the maser field dynamics. The observed contrast of Rabi oscillations can be considered as a figure of merit for the experimental resolution of quantum features of a maser field. Therefore, a high resolution in the measurements of Rabi oscillations is crucial. This thesis describes the methods and improvements applied to enhance the resolution of Rabi oscillations by a factor of 8. The following upgrades and developments proved to be crucial in this work: 1) The base temperature of the cryogenic system was lowered to 0.3 K, i.e. one third of the original value. At the same time, the time during which the system remains at the base temperature was improved by almost of an order of magnitude (to more than 12 h). 2) For the promotion of the atoms to the Rydberg regime new three-step diode laser setup was constructed. The excitation efficiency compared to the old dye laser was increased by two orders. Using top-of-fringe stabilization with new spectroscopic methods, specially designed error detection and feedback schemes as well as state-of-the-art PID regulators the continuous locking of lasers is about 8 h (compared to about 20 min with the old dye laser system). 3) The flexibility of using the three step excitation allows to excite different maser states and for the first time to investigate pure maser transition. 4) Various improvements on the atomic oven and the channeltron detection unit led to the reliable and stable production and detection of the atomic beam. Observed distribution of the atomic beam statistics reaches to within 6% the Poisson limit which is expected in the absence of experimental imperfections. 5) New developed techniques in magnetic field compensation allow to perform such measurements with two orders of magnitude better accuracy compared to the previous experiments. To demonstrate the new capabilities of the apparatus and to apply its improved resolution of quantum fields we performed a previously impossible in-depth analysis of mean photon number and the atomic inversion produced by a micromaser for various detunings between the cavity- and atomic resonance frequency.
The coherent evolution of the atom–cavity state in bimodal (cavity) experiments has been analysed for a realistic time dependence in detuning the atomic transition frequency. Apart from a 'smooth switch' of the atomic resonance from one to the second mode of a bimodal cavity, we considered also an additional (effective) interaction between the field modes of the cavity, known as a 'communication channel'. Comparison of our model computations has been made especially with the measurements by Rauschenbeutel et al (2001 Phys. Rev. A 64 050301) who demonstrated for the first time the entanglement of the field modes in a bimodal cavity. It is shown that the agreement between the (theoretically) predicted and experimental phase shifts can be improved by allowing a 'communication' between the two field modes during a short but finite switch of the atomic transition frequency from one mode to the other. We therefore suggest that the details of the atom–cavity detuning should be taken into account for the future interpretation of bimodal cavity experiments.
One of the canonical questions in quantum optics is the nature of the radiative properties of an atom when the normal vacuum fluctuations of the electromagnetic reservoir are replaced by the asymmetric, reduced fluctuations of a squeezed vacuum. While the basic radiative linewidth-narrowing effect has been known for over a decade [C. W. Gardiner, Phys. Rev. Lett. 56, 1917 (1986)], experimental realizations with operationally definable definitive manifestations of the quantum nature of the squeezed reservoir have been largely lacking from subsequent investigations. This paper presents measurements on an experimentally realized atom--squeezed-light system, in which the squeezed-light output of a subthreshold optical parametric oscillator illuminates an atom strongly coupled to a high-finesse optical resonator. Transmission of a weak probe field incident on the atom-cavity system is investigated both theoretically and experimentally. Alteration of the transmitted probe spectrum has been observed, as has a transmission modulation that depends on the phase of the squeezed field relative to a saturating coherent field (displaced squeezing). In certain parameter regimes, properties unique to the quantum nature of the squeezed light have been identified in the theoretical treatment, but complications in the experiment prevent their unequivocal measure. It is found that the observed effects of the squeezed light are dramatically reduced relative to the predictions of an idealized theory. This is quantitatively attributed to the effects of atomic beam fluctuations and a simple modeling of the atomic beam as an additional loss mechanism in the theory leads to reasonable agreement with the data.
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.