Abstract Previous calculations [1] yielding the frequency spectrum of a one‐mode radiation field interacting with a single atom are extended to the case of M atoms. Under the assumption M „N (N mean photon number) the total Hamiltonian is diagonalized, and the correlation function 〈𝔈̂(t1 𝔈̂(t2)〉 revealing the frequency spectrum of the interacting field calculated quite generally. The following special initial conditions are discussed in detail: a) Any atom is either in the upper or the lower state, and b) the electric dipole moment of the ensemble of atoms is oscillating in phase or with a phase difference of π with respect to the electric field strength. In case a) the initial monochromatic line is broadened, and in case b) a frequency shift appears.
Abstract Previous calculations [1] yielding the frequency spectrum of a one‐mode radiation field interacting with a single atom are extended to the case of M atoms. Under the assumption M „N (N mean photon number) the total Hamiltonian is diagonalized, and the correlation function 〈𝔈̂(t1 𝔈̂(t2)〉 revealing the frequency spectrum of the interacting field calculated quite generally. The following special initial conditions are discussed in detail: a) Any atom is either in the upper or the lower state, and b) the electric dipole moment of the ensemble of atoms is oscillating in phase or with a phase difference of π with respect to the electric field strength. In case a) the initial monochromatic line is broadened, and in case b) a frequency shift appears.
Abstract Making use of the eigenfunctions of the total Hamiltonian, the correlation function 〈𝔈̂(−) (t1) 𝔈̂(+) (t2)〉 is calculated explicitly for a one‐mode radiation field interacting incessantly with an atom. In the frequency spectrum following from this correlation function there appear frequency shifts corresponding, just as in [7], in one case, to a change of the effective dielectric constant within the cavity due to the atom, and, in the other case, to an amplitude modulation due to the alternating absorption and emission of a photon by the atom.
Abstract Die Wechselwirkung der elektromagnetischen Eigenschwingung eines Resonators mit einem „Zwei‐Niveau‐Atom”︁ führt zu einer symmetrischen Aufspaltung der ungestörten Resonanzfrequenz in zwei Frequenzpaare mit stark verschiedenen Linienabständen. Das innere Frequenzpaar entspricht der „Verstimmung”︁ der Eigenfrequenz des Resonators durch das hineingebrachte anharmonische Atom als „Dielektrikum”︁. Sein Frequenzabstand nimmt mit steigender Feldamplitude ab und verschwindet im Grenzfall sehr großer Lichtquantenzahl. Das äußere Frequenzpaar wird durch die Frequenz der Absorption und Reemission eines Lichtquants durch das Atom bestimmt und ist mit einer schwachen Amplitudenmodulation der Resonatorschwingung verbunden. Sein Frequenzabstand steigt linear mit der Feldamplitude an. Alle Linien weisen eine spezifisch quantentheoretisch bedingte Verbreiterung auf. welche von der quantentheoretischen Energieunschärfe des schwingenden Systems in Verbindung mit der Anharmonizität des Atoms herrührt. (Harmonische Atomoszillatoren würden keine Linienverbreiterung der hier betrachteten Art verursachen).
This paer is concerned with the spontaneous-emision rate of two two-level atoms which have different resonance frequencies and whose spatial separation is much smaller than the corresponding mean resonance wavelength. The total radiation rate of such an atom pair is calculated completely quantum mechanically as a function of time. Explicit results are given for various values of the frequeny difference and the following initial states of the atomic system: (a) both atoms in the excited state, (b) one taom in the excited state the other in the ground state, (c) a symmetric combination of those states, (d) an antisymmetric combination of those states, and product states where each atom is in an equal superposition of its ground and excited states, for which the expectation values of the dipole moments are nonzero and may be considered to be osicillating (e) in phase, (f) in opposite phae, and (g) in random phase. Oscillations are found to appear in the radiation rate in all cases when the frequency difference becomes larger than a certain critical value, viz., the mean half-width of the atomic resonance lines. The oscillations are especially pronounced for the initial atomic states (c), (d), (e), and (f), which exhibit superradiance and radiation trapping. For example, for atoms in the antisymmetric state (d), for which the radiation from two identical atoms is zero (completely self-trapped), a pair of nonidentical atoms begins radiating and even oscillates into the superradiant region at times.
Abstract With the help of the well‐known eigenfunctions for a two‐level system coupled to a single mode of the radiation field, the correlation functions for both the atomic dipole and the electric field oscillation are calculated, in rotating‐wave approximation, for an atom emitting spontaneously into an ideal cavity. Using the Eberly‐Wódkiewicz formalism, the corresponding spectra are evaluated in the long‐time, narrow‐band detection limit. The two spectra exhibit the same Rabi splitting, but the ratio of the strengths of the two lines is significantly different, in the off‐resonance case, for the dipole and the field oscillation.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1, A2, whose separation r21 can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only A1 inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1, O2. The atoms exhibit superradiant behavior, whereas O1, O2 tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that A2 and A2 radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly inhomogeneous line broadening due to such causes as doppler shift arising from translational velocity. The ratio of the number of molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width ∆, where E0 is the amplitude of the electric field in the incident radiation and is the dipole moment matrix element for the transition. Excitation functions obtained in a previous paper (I) for various values of the ratio γ/β, where γ is the half line-width in the absence of inhomogeneous broadening, are used to obtain the variation of N2/N with time under conditions of mixed broadening for various values of the ratio γ/∆ and γ/β, when the exciting radiation is in exact resonance with the central frequency of the transition.