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Quantum Optics Journal of the European Optical Society Part B

Incoherent excitation of the Jaynes-Cummings system

L. Tian, H. J. Carmichael

Published Apr 1, 1992• 28 citations
Publisher

Abstract

The authors calculate scattered, transmitted and reflected spectra for a coupled two-level atom and quantized electromagnetic cavity mode driven by broad-band chaotic light. They obtain analytical results in the secular approximation and compare these with exact numerical results. Under cavity QED conditions (strong dipole coupling) the spectra show multiple resonances at the incommensurate transition frequencies of the Jaynes-Cummings Hamiltonian. They obtain expressions for linewidths and transition amplitudes that help them to assess the prospects for an experimental observation of these Jaynes-Cummings resonances.

DOI
10.1088/0954-8998/4/2/007

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When a single atom interacts with one mode of an optical cavity and the coupling to the cavity mode is sufficiently large, the spectroscopic response of the composite system to a weak probe is a doublet, with peaks located at the eigenfrequencies of the one-quantum excited states of the Jaynes–Cummings Hamiltonian. So far as these one-quantum states are concerned the spectroscopy of the Jaynes–Cummings model is the same as that of coupled, damped harmonic oscillators. This equivalence has been used to observe a doublet spectrum (the so-called "vacuum" Rabi splitting) by exciting a collective dipole oscillator inside an optical cavity.1,2 At higher levels of excitation the spectroscopy of the Jaynes–Cummings model differs markedly from that of coupled harmonic oscillators. Coupled harmonic oscillators give a doublet at all excitation levels; because of its noncommensurate excited state energies the Jaynes–Cummings model gives a multitude of additional peaks. In this paper we show how the excited states of the Jaynes–Cummings Hamiltonian can be probed by driving a cavity containing a single atom with broadband chaotic light. For a sufficiently large dipole coupling strength the spectrum of the light scattered out of the cavity mode shows peaks at the many resonance frequencies of the Jaynes–Cummings Hamiltonian. A similar multipeaked spectrum is seen in transmission. The spectrum of the light reflected from the cavity shows corresponding absorption dips superimposed on the broadband input.

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Spectra of the atomic resonance fluorescence and of the cavity-transmitted light in a driven Jaynes-Cummings system are studied theoretically, where the cavity is driven by a classical field of arbitrary strength. In the regime of strong atom-cavity coupling, anomalous resonance peaks occur at $\ifmmode\pm\else\textpm\fi{}\sqrt{n}{g}^{\ensuremath{'}} (n=2,3,\dots{}),$ with the modified vacuum Rabi frequency ${g}^{\ensuremath{'}},$ both in the atomic fluorescence and the cavity transmission. These peaks as well as the observed shift of the vacuum Rabi splitting are explained in terms of the quasienergy levels induced dynamically by the strong driving field. For a stronger driving field, it is shown that the spectrum of the cavity transmission deviates from that of the atomic fluorescence. In the presence of significant damping, the spectrum of the cavity transmission approaches a singlet structure in the strong driving-field limit, and it is found additionally that a bimodal photon-number distribution, which results in a unique quintuplet spectrum of the atomic fluorescence, does not show up in the cavity transmission.

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The eigenstates of the Jaynes-Cummings Hamiltonian (the dressed atomic states) are widely used to describe the interaction between a single two-level atom and a single mode of the electromagnetic field. They provide a particularly useful basis when the secular approximation is justified. For an atom interacting fog with n photons in a cavity, the secular approximation requires that the n-photon Rabi frequency 2 n g be much larger than the atomic and cavity linewidths. Normally this condition is met when n is large and the coupling constant g is much smaller than the atomic and cavity linewidths. In this paper we allow g to be larger than the linewidths, and we allow the photon number to be small. We show that when the cavity is driven by a coherent field, the usual secular approximation cannot be used. The natural basis states become are now the eigenstates of the Jaynes-Cummings Hamiltonian plus the external field interaction. We solve this eigenvalue problem by separating the field and atomic parts of the Hamiltonian using a displacement and squeeze in the Hilbert-space of the field. We find that the Rabi frequencies are renormalized downwards as the strength of the driving field is increased. They eventually vanish at a critical field strength. At this critical field strength we observe a spontaneous symmetry breaking in the steady-state solution of the master equation for a single atom interacting with a driven cavity mode.

Annual Meeting Optical Society of America•Jan 1, 1990•0 citations
The authors calculate scattered, transmitted and reflected spectra for a coupled two-level atom and quantized electromagnetic cavity mode driven by broad-band chaotic light. They obtain analytical results in the secular approximation and compare these with exact numerical results. Under cavity QED conditions (strong dipole coupling) the spectra show multiple resonances at the incommensurate transition frequencies of the Jaynes-Cummings Hamiltonian. They obtain expressions for linewidths and transition amplitudes that help them to assess the prospects for an experimental observation of these Jaynes-Cummings resonances.
When a single atom interacts with one mode of an optical cavity and the coupling to the cavity mode is sufficiently large, the spectroscopic response of the composite system to a weak probe is a doublet, with peaks located at the eigenfrequencies of the one-quantum excited states of the Jaynes–Cummings Hamiltonian. So far as these one-quantum states are concerned the spectroscopy of the Jaynes–Cummings model is the same as that of coupled, damped harmonic oscillators. This equivalence has been used to observe a doublet spectrum (the so-called "vacuum" Rabi splitting) by exciting a collective dipole oscillator inside an optical cavity.1,2 At higher levels of excitation the spectroscopy of the Jaynes–Cummings model differs markedly from that of coupled harmonic oscillators. Coupled harmonic oscillators give a doublet at all excitation levels; because of its noncommensurate excited state energies the Jaynes–Cummings model gives a multitude of additional peaks. In this paper we show how the excited states of the Jaynes–Cummings Hamiltonian can be probed by driving a cavity containing a single atom with broadband chaotic light. For a sufficiently large dipole coupling strength the spectrum of the light scattered out of the cavity mode shows peaks at the many resonance frequencies of the Jaynes–Cummings Hamiltonian. A similar multipeaked spectrum is seen in transmission. The spectrum of the light reflected from the cavity shows corresponding absorption dips superimposed on the broadband input.
We explore how the Jaynes‐Cummings ladder transpires in the emitted spectra of a two‐level system in strong coupling with a single mode of light. We focus on the case of very strong coupling, that would be achieved with systems of exceedingly good quality (very long lifetimes for both the emitter and the cavity). We focus on the incoherent regime of excitation, that is realized with semiconductors quantum dots in microcavities, and discuss how reasonable is the understanding of the systems in terms of transitions between dressed states of the Jaynes‐Cummings Hamiltonian.
Abstract. We explore how the Jaynes-Cummings ladder transpires in the emitted spectra of a twolevel system in strong coupling with a single mode of light. We focus on the case of very strong coupling, that would be achieved with systems of exceedingly good quality (very long lifetimes for both the emitter and the cavity). We consider the incoherent regime of excitation, that is realized with semiconductors quantum dots in microcavities, and discuss how reasonable is the understanding of the systems in terms of transitions between dressed states of the Jaynes-Cummings Hamiltonian.
Spectra of the atomic resonance fluorescence and of the cavity-transmitted light in a driven Jaynes-Cummings system are studied theoretically, where the cavity is driven by a classical field of arbitrary strength. In the regime of strong atom-cavity coupling, anomalous resonance peaks occur at $\ifmmode\pm\else\textpm\fi{}\sqrt{n}{g}^{\ensuremath{'}} (n=2,3,\dots{}),$ with the modified vacuum Rabi frequency ${g}^{\ensuremath{'}},$ both in the atomic fluorescence and the cavity transmission. These peaks as well as the observed shift of the vacuum Rabi splitting are explained in terms of the quasienergy levels induced dynamically by the strong driving field. For a stronger driving field, it is shown that the spectrum of the cavity transmission deviates from that of the atomic fluorescence. In the presence of significant damping, the spectrum of the cavity transmission approaches a singlet structure in the strong driving-field limit, and it is found additionally that a bimodal photon-number distribution, which results in a unique quintuplet spectrum of the atomic fluorescence, does not show up in the cavity transmission.
In this dissertation we study the interaction of a single atom with a single mode of the electromagnetic field in an optical cavity. We emphasize strong coupling conditions, where the dipole coupling strength is larger than the dissipation rates of the cavity and the atom. By exciting the coupled atom and cavity mode with an external field, the excited-state spectroscopy of the coupled system is explored. Both quantum mechanical and semiclassical calculations are performed and their results are compared. In the limit of a weak driving field, both results show a two-peak ("vacuum" Rabi) spectrum. With increased strength of the driving field, differences are observed. The quantum calculation shows a multi-peak ( two "vacuum" Rabi peaks plus additional peaks) spectrum, while the semiclassical calculation does not. These additional peaks are produced by transitions between the excited states of the Jaynes-Cummings IV Hamiltonian indicative of the quantum nature of the coupled atom-cavity-mode system.
The eigenstates of the Jaynes-Cummings Hamiltonian (the dressed atomic states) are widely used to describe the interaction between a single two-level atom and a single mode of the electromagnetic field. They provide a particularly useful basis when the secular approximation is justified. For an atom interacting fog with n photons in a cavity, the secular approximation requires that the n-photon Rabi frequency 2 n g be much larger than the atomic and cavity linewidths. Normally this condition is met when n is large and the coupling constant g is much smaller than the atomic and cavity linewidths. In this paper we allow g to be larger than the linewidths, and we allow the photon number to be small. We show that when the cavity is driven by a coherent field, the usual secular approximation cannot be used. The natural basis states become are now the eigenstates of the Jaynes-Cummings Hamiltonian plus the external field interaction. We solve this eigenvalue problem by separating the field and atomic parts of the Hamiltonian using a displacement and squeeze in the Hilbert-space of the field. We find that the Rabi frequencies are renormalized downwards as the strength of the driving field is increased. They eventually vanish at a critical field strength. At this critical field strength we observe a spontaneous symmetry breaking in the steady-state solution of the master equation for a single atom interacting with a driven cavity mode.