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Physical Review A

Theory of collective resonance fluorescence in strong driving fields

G. Compagno, Franco Persico

Published Jun 1, 1982• 35 citations
Publisher

Abstract

An S-conserving theory of collective resonance fluorescence by $N$ two-level atoms driven by a uniform strong field is presented. The detuning between field and atoms can be varied. A canonical transformation to dress the atoms by the driving field and Langevin equations for dressed operators are used to investigate the behavior of the system. For sufficiently large detuning the dressed atoms are shown to reach a steady state after a transient super-radiant-like decay in times short, compared with the long decay time which is attained on resonance and which is the same as the single-atom spontaneous-decay time. In the transient regime the parameters of the quasi-super-radiant decay are shown to vary with detuning and with the driving-field amplitude. Both the amplitude spectrum and the integrated intensity of the fluorescence field, emitted off resonance by the dressed atoms during the transient, are shown to bear pulselike traces of the quasi-super-radiant decay. The elastic components of the off-resonance amplitude spectrum at the end of the super-radiant-like transient are broadened with respect to resonance, and the integrated intensity exhibits a sudden transient decrease in an appropriate range of detuning, which has been given the name "superdarkening" and which disappears on resonance. After the steady-state regime is attained, the resonance intensity spectrum is shown to coincide with that obtained by other authors, while off resonance the inelastic lines are broadened in a super-radiant-like fashion.

DOI
10.1103/physreva.25.3138

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We study the transient fluorescence spectra of two-level atoms driven by a strong optical field when the atoms are initially prepared in pure dressed states of the atom-field Hamiltonian. Spectra obtained here differ qualitatively from those observed in the steady state. In the case of steady-state resonant excitation, a three-peaked spectrum symmetric about the laser frequency is obtained; with pure-dressed-state preparation, the symmetry of the spectrum about the driving field is broken---one side peak is initially enhanced and the other is initially suppressed. Furthermore, the temporal oscillations of fluorescence intensity observed in previous calculations of transient fluorescence spectra appropriate to atoms prepared in a superposition of dressed states [J. H. Eberly et al., J. Phys. B 13, 217 (1980)] are suppressed when pure-dressed-state preparation is considered.

Physical Review A•Jul 1, 1986•22 citations
An S-conserving theory of collective resonance fluorescence by NNN two-level atoms driven by a uniform strong field is presented. The detuning between field and atoms can be varied. A canonical transformation to dress the atoms by the driving field and Langevin equations for dressed operators are used to investigate the behavior of the system. For sufficiently large detuning the dressed atoms are shown to reach a steady state after a transient super-radiant-like decay in times short, compared with the long decay time which is attained on resonance and which is the same as the single-atom spontaneous-decay time. In the transient regime the parameters of the quasi-super-radiant decay are shown to vary with detuning and with the driving-field amplitude. Both the amplitude spectrum and the integrated intensity of the fluorescence field, emitted off resonance by the dressed atoms during the transient, are shown to bear pulselike traces of the quasi-super-radiant decay. The elastic components of the off-resonance amplitude spectrum at the end of the super-radiant-like transient are broadened with respect to resonance, and the integrated intensity exhibits a sudden transient decrease in an appropriate range of detuning, which has been given the name "superdarkening" and which disappears on resonance. After the steady-state regime is attained, the resonance intensity spectrum is shown to coincide with that obtained by other authors, while off resonance the inelastic lines are broadened in a super-radiant-like fashion.
A canonical transformation, which diagonalizes the single-mode radiation Hamiltonian in the rotating wave approximation, is exploited to dress a two-level atom by the strong monochromatic driving field in the resonance fluorescence phenomenon. Langevin equations of motion are obtained for dressed atomic operators, and their solutions yield the amplitude of waves scattered into the fluorescence field. The power spectrum of the scattered field is obtained from the first-order correlation functions between dressed atomic operators. The results of the present treatment are compared with others, and advantages and disadvantages of the present method are discussed.
We investigate the steady-state spontaneous emission spectrum of a three-level atom driven by two coherent fields and the absorption spectrum of a weak probe passing through a collection of such driven atoms. We find that the fluorescence spectrum is strongly affected by the decay rates of all the levels involved in the atomic evolution and not just by the decay parameters of the specific transition whose emission spectrum is being monitored. In particular, the spectral components can acquire very different widths and peak heights relative to the case of the standard resonance fluorescence in which a two-level system is driven by a single near-resonant field. An external probe signal passing through the gas of three-level atoms may be absorbed or amplified, as in the standard two-level case, but under specific operating conditions, amplification (or absorption) occurs over the entire range where the atomic response is appreciable. In this case the existence of amplification or absorption is controlled solely by the population difference between two dressed states of the system. We provide numerical results in support of our arguments for arbitrary values of the atomic and field parameters and also develop an analytic description in the limit of strong driving fields that leads to explicit line-shape and linewidth formulas.
We examine the radiative properties of a two-level atom driven by a strong bichromatic field with frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{0}$-${\mathrm{\ensuremath{\delta}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{0}$+${\mathrm{\ensuremath{\delta}}}_{2}$, which can be asymmetrically placed about the atomic transition frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ and can have different Rabi frequencies ${\mathrm{\ensuremath{\Omega}}}_{1}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$. Applying the optical Bloch equations for the bichromatic excitation, we derive an infinite set of equations of motion for the time evolution of the atomic variables. The equations are solved numerically by matrix inversion and Laplace transforms. Using the quantum regression theorem, we then solve for the steady-state total fluorescence intensity, and the resonance-fluorescence and absorption spectra. The spectra are found to depend on the frequency difference 2\ensuremath{\delta}=${\mathrm{\ensuremath{\delta}}}_{1}$+${\mathrm{\ensuremath{\delta}}}_{2}$, the average detuning \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{0}$-1/2(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$), and the Rabi frequencies of the driving fields. For ${\mathrm{\ensuremath{\Omega}}}_{1}$=${\mathrm{\ensuremath{\Omega}}}_{2}$=\ensuremath{\Omega}, the total fluorescent intensity displays a series of maxima for \ensuremath{\Delta}=\ifmmode\pm\else\textpm\fi{}n\ensuremath{\delta}\ensuremath{\mp}${\mathrm{\ensuremath{\Omega}}}^{2}$/4\ensuremath{\delta}, where the n are the odd integers. The intensity-dependent shift from the resonances n\ensuremath{\delta} is explained as an analog of the generalized Bloch-Siegert shift. The resonance-fluorescence spectrum for \ensuremath{\Delta}\ensuremath{\ne}0 appears to contain more peaks than that for \ensuremath{\Delta}=0.This is due to the splitting of the central peak and the even sidebands into doublets. The absorption spectrum of a weak-probe beam for \ensuremath{\Delta}=0 and equal Rabi frequencies consists of a symmetric series of dispersionlike sidebands separated from ${\mathrm{\ensuremath{\omega}}}_{0}$ by integer multiples of \ensuremath{\delta}, together with a central absorption peak at ${\mathrm{\ensuremath{\omega}}}_{0}$, whose amplitude oscillates with \ensuremath{\Omega}. For \ensuremath{\Delta}\ensuremath{\ne}0 and/or ${\mathrm{\ensuremath{\Omega}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\Omega}}}_{2}$, the odd sidebands remain dispersionlike, while the central peak and the even sidebands split into absorption-emission doublets. A simple physical interpretation of the spectral features is given in terms of the dressed-atom model.
The dependence of squeezing on the number of atoms is discussed in the time-dependent collective resonance fluorescence of N two-level atoms, coherently driven by a strong resonant laser field. It is shown that, for strong driving field, squeezing appears only in the transient regime of resonance fluorescence. The location of squeezing in time is independent of the number of atoms and the initial atomic populations. It is found that despite the reduction in the amount of squeezing per single atom, squeezing increases in the whole resonance fluorescence field as the number of atoms increases, and attains its maximal value for moderate N. Moreover, the maximal value of squeezing is dependent on the intensity of the driving field. As the latter increases, the maximum of squeezing increases and shifts towards larger numbers of atoms. For a very large number of atoms, however, squeezing is altogether removed.
We study the transient fluorescence spectra of two-level atoms driven by a strong optical field when the atoms are initially prepared in pure dressed states of the atom-field Hamiltonian. Spectra obtained here differ qualitatively from those observed in the steady state. In the case of steady-state resonant excitation, a three-peaked spectrum symmetric about the laser frequency is obtained; with pure-dressed-state preparation, the symmetry of the spectrum about the driving field is broken---one side peak is initially enhanced and the other is initially suppressed. Furthermore, the temporal oscillations of fluorescence intensity observed in previous calculations of transient fluorescence spectra appropriate to atoms prepared in a superposition of dressed states [J. H. Eberly et al., J. Phys. B 13, 217 (1980)] are suppressed when pure-dressed-state preparation is considered.