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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

Single dressed-atom laser: lasing out of a driven Jaynes-Cummings system

Young-Tak Chough, Jung Bog Kim, Kyungwon An

Published Jan 20, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1109/cleopr.1999.817879

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Collapse and revival dynamics in a driven Jaynes-Cummings system

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A single atom coupled to a quantized mode of an electromagnetic cavity, driven by an external coherent field, is described by the Hamiltonian H djc = ig(a†σ− − aσ+) + iε(a† − a), where (a†, a) and (σ+, σ) are the raising and lowering operators for the cavity mode and atom, respectively. This is the single atom version of the Hamiltonian used in the study of optical bistability. It describes the evolution of the atom-cavity system on time scales much smaller than the inverse spontaneous emission and cavity decay rates. For 2ε/g < 1, HDJC possesses a discrete set of interaction eigenstates1 with eigenvalues E n ± = ± g n [ 1 − ( 2 ε / g ) 2 ] 3 / 4 . These form a renormalized version of the usual Jaynes-Cummings eigenstates2, which correctly take into account the effect of the driving field to all orders of magnitude. We investigate the collapse and revival nature of the dynamical evolution of the single atom in a cavity, with and without dissipation and above and below the threshold 2ε/g = 1 of the driven Jaynes-Cummings Hamiltonian.

Annual Meeting Optical Society of America•Jan 1, 1992•0 citations
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.
The driven two-photon Jaynes-Cummings model (JCM) exhibits a collapse and revival phenomenon in its mean photon number on a time scale much larger than the periodic revival time for two-state inversion in the usual two-photon JCM. The time scale of revivals for these oscillations is much larger than the corresponding time scale for the driven one-photon JCM and can be explained with the Hamiltonian of type ${H}_{\ensuremath{\eta}}=\ifmmode\pm\else\textpm\fi{}\sqrt{{(J}_{\ensuremath{\eta}}^{\ifmmode\dagger\else\textdagger\fi{}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}{(J}_{\ensuremath{\eta}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}}$ constructed explicitly for this purpose. The effect of Stark shifts is also studied and it is observed that the dynamics is strongly influenced by such Stark shifts both at early stages as well as at late stages of the evolution but in different ways, and thus Stark shifts play a significant role in determining the dynamical evolution of the system.
We study the dynamics of the driven Jaynes-Cummings model where an atom interacts with a single-mode cavity field but is also directly driven by an external radiation field. We show that the dynamic variables can be described in terms of ordinary Jaynes-Cummings dynamic variables with a different initial condition. We investigate the special case of an initial coherent state. We find collapses and revivals in the average number of photons at a much larger time scale than their Jaynes-Cummings model counterparts. We show that these ``super-revivals'' are related to the large-time-scale revivals that occur in the average electric field in the ordinary Jaynes-Cummings model. We also examine the photon statistics in the driven Jaynes-Cummings model and show that the cavity field is sub-Poissonian for certain interaction times.
Collapse and revivals in the Jaynes-Cummings model are studied within the context of the Mollow transformation for a single-mode coherent-state cavity field. The Mollow-transformed Jaynes-Cummings Hamiltonian has two atom-field terms, one corresponding to a classical field and the other to a quantized field driving the atomic transition. As such it maps onto the complementary problem of an atom in an optical cavity, whose field is initially in the vacuum state, driven by an external classical field. Both problems have the same atomic-state dynamics. It is shown that the revivals can be associated with two distinct properties of the quantized field of the transformed Hamiltonian. Revivals occurring at even (odd) multiples of the revival time are correlated with field states that are close to the initial field state and for which the average energy in the field is a minimum (maximum). Using semiclassical dressed states, we are able to map the problem for the field states onto one involving two uncoupled oscillators driven by off-resonant fields, provided effects related to dispersion are neglected. Analytic expressions for the photon number distribution of the transformed cavity field, including dispersion, are obtained for the even revival times.
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
The dynamics of the driven Jaynes-Cummings model where a two-level atom is coupled to a single mode of cavity fields and is driven by an external coherent field is investigated. We find that in this model quantum collapse and revivals of the atomic population inversion and squeezing of the cavity field can be generated from a thermal photon state. We also notice that the squeezing is very sensitive to the mean number of thermal photons initially in the cavity, but the quantum collapse and revivals are not. The influence of the amplitude of the driving field and various detuning parameters on the nonclassical effects is discussed. The result shows that one can use the external field to control nonclassical effects in the Jaynes-Cummings model.
A single atom coupled to a quantized mode of an electromagnetic cavity, driven by an external coherent field, is described by the Hamiltonian H djc = ig(a†σ− − aσ+) + iε(a† − a), where (a†, a) and (σ+, σ) are the raising and lowering operators for the cavity mode and atom, respectively. This is the single atom version of the Hamiltonian used in the study of optical bistability. It describes the evolution of the atom-cavity system on time scales much smaller than the inverse spontaneous emission and cavity decay rates. For 2ε/g < 1, HDJC possesses a discrete set of interaction eigenstates1 with eigenvalues E n ± = ± g n [ 1 − ( 2 ε / g ) 2 ] 3 / 4 . These form a renormalized version of the usual Jaynes-Cummings eigenstates2, which correctly take into account the effect of the driving field to all orders of magnitude. We investigate the collapse and revival nature of the dynamical evolution of the single atom in a cavity, with and without dissipation and above and below the threshold 2ε/g = 1 of the driven Jaynes-Cummings Hamiltonian.