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Annual Meeting Optical Society of America

Filtered correlations in the Jaynes-Cummings resonance fluorescence

H. M. Castro-Beltrán, J. J. Sánchez-Mondragón

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Atomic inversion revivals are the first evidence of the quantum character of the radiation, predicted by the Jaynes–Cummings model, that are experimentally measurable in cavity quantum electrodynamics.1 We propose another test by studying the two time photon correlations between spectrally resolved components of resonance fluorescence spectrum of the coherent Jaynes–Cummings model. The spectrum raises from spontaneous emission between adjacent energy many folds, whose splitting is given by the Rabi frequency for every Fock state. This originates sideband discreteness that can be observed when the filter bandwidth is smaller than the sideband width (atom-cavity coupling). There is photon correlation only if these photons are from consecutive emissions in the dresssed lader, producing a frequency offset. Also, a photon from a sideband is more likely to be emitted before a photon from the other sideband when detuning is finite, in a fashion analogous to that studied by Aspect et al.2 for free space rf. Moreover, the decay is due solely to the filter bandwidth, reflecting lack of cavity losses. These correlations could be observed in presence of mixed fields, therefore are less dependant on the coherence of the experimental cavity fields than the revivals are.

DOI
10.1364/oam.1990.ws8

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Atomic inversion revivals are the first evidence of the quantum character of the radiation, predicted by the Jaynes–Cummings model, that are experimentally measurable in cavity quantum electrodynamics.1 We propose another test by studying the two time photon correlations between spectrally resolved components of resonance fluorescence spectrum of the coherent Jaynes–Cummings model. The spectrum raises from spontaneous emission between adjacent energy many folds, whose splitting is given by the Rabi frequency for every Fock state. This originates sideband discreteness that can be observed when the filter bandwidth is smaller than the sideband width (atom-cavity coupling). There is photon correlation only if these photons are from consecutive emissions in the dresssed lader, producing a frequency offset. Also, a photon from a sideband is more likely to be emitted before a photon from the other sideband when detuning is finite, in a fashion analogous to that studied by Aspect et al.2 for free space rf. Moreover, the decay is due solely to the filter bandwidth, reflecting lack of cavity losses. These correlations could be observed in presence of mixed fields, therefore are less dependant on the coherence of the experimental cavity fields than the revivals are.
The authors give a theoretical description of the spectrally resolved intensity correlations of an electromagnetic field, in analogy with the definition of the physical time-dependent spectrum. The results are applied to evaluate the time correlations between the three lines in the spectrum of resonance fluorescence of a two-level atom. The authors include the possibility of a finite bandwidth of the incident radiation, and they allow for collisions with perturber atoms. Photons in the central Rayleigh line are emitted in a fully random fashion, without any correlation with previous or subsequent emissions. Two photons from the same sideband display antibunching in time, whereas two photons from opposite sidebands tend to bunch with a strong asymmetry in time. The effect of collisions and of a finite bandwidth is to diminish this asymmetry.
A two-level atom interacting with a cavity field initialized in a coherent state exhibits the collapse and revival features in its population inversion dynamics under the Jaynes–Cummings interaction. When the cavity field is prepared in a number-state filtered coherent state, obtained by filtering the number state | m 〉 from a coherent state | α 〉 , we observe micro-Rabi oscillations in the collapse region instead of a perfect collapse. We find that the Rabi frequency of these micro oscillations depends only on the filtered photo-number m, whereas the amplitude depends on both m and α . This dependence enables one to infer the filtered number state from the coherent-state distribution by analyzing the time evolution of the atomic population inversion. Conversely, the amplitude and frequency of the micro-Rabi oscillations can be tuned by filtering an appropriate number state. Furthermore, we investigate the effect of filtering multiple number states from a coherent state on the atomic population inversion dynamics.
We analyze the time-dependent physical spectrum of a driven Jaynes-Cummings model in which both the two-level system and the quantized cavity mode are subject to coherent classical driving. The time-dependent Hamiltonian is mapped, via well-defined unitary transformations, onto an effective stationary Jaynes-Cummings form. Within this framework, we derive closed-form expressions for the two-time correlation functions of both the atomic and field operators. These correlation functions are subsequently used to evaluate the time-dependent physical spectrum according to the Eberly-Wódkiewicz definition, which properly accounts for finite spectral resolution and transient emission dynamics. We show that the external driving leads to substantial modifications of the atomic spectral response, including controllable frequency shifts and asymmetric line shapes. Importantly, we identify a regime in which the driving parameters are chosen such that the coherent displacement induced in the cavity field exactly cancels out the initial coherent amplitude. In this limit, the system dynamics reduce to that of an effectively vacuum-initialized Jaynes-Cummings model, and the standard vacuum Rabi splitting is recovered. This behavior provides a clear and physically transparent interpretation of the spectral features as arising from coherent field displacement rather than from modifications of the underlying atom-cavity coupling.
. The Jaynes-Cummings Model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probability for fields with sharply defined energy (or photon number). For fields having a statistical distribution of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive – only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further nonclassical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics.
The Jaynes-Cummings Model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probability for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive -- only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further nonclassical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schroedinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to exploremore » still further the relationship between classical and quantum physics. In tribute to E. D. Jaynes, who first recognized the importance of the JCM for clarifying the differences and similarities between quantum and classical physics, we here present an overview of the theory of the JCM and some of the many remarkable discoveries about it.« less
A variant Jaynes-Cummings model (JCM) where the two-state atom coupled to the cavity mode is driven by a strong external field reveals many interesting behaviors. On resonance, a large scale oscillatory behavior of the mean cavity photon number is predicted. It varies sinusoidally with the amplitude of 2(/spl epsi//g)/sup 2/, centered at the same value, where /spl epsi/ is the (real) driving field amplitude and g the vacuum Rabi frequency. The period of the oscillation was found to be exactly twice the "revival time" of the atomic inversion, i.e., T/sub 0/=(4/spl pi//g)(/spl epsi//g). Furthermore, the oscillation then shows another kind of "collapse-and-revival" type behavior as time proceeds. It is very similar to that of the atomic inversion, but the size of oscillation and the time scales are fundamentally different. The "revival" time of the oscillation, for instance, is found to be exactly one half of the so-called "fractional revival" time of the atomic inversion. Our current interest is directed to the radiative behavior of the driven JCM in a more general situation where dissipation as well as detuning is introduced.