Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Cooperative behavior among three-level systems: Transient effects of coherent optical pumping

C. M. Bowden, Chi C. Sung

Published Oct 1, 1978• 58 citations
Publisher

Abstract

The dynamical evolution of an excited-state population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the excited states by an externally applied ($c$-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field which is treated quantum mechanically. It is found that stimulated Raman processes can significantly influence the dynamical evolution of population inversion and macroscopic polarization by producing coherence effects among the populations. We examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration. It is shown, using the mean-field approximation, that for uniform pumping and conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective radiation time ${\ensuremath{\tau}}_{R}$, and the pump pulse duration ${\ensuremath{\tau}}_{P}$ satisfy the inequalities, ${\ensuremath{\omega}}_{R}>\frac{1}{{\ensuremath{\tau}}_{R}}>\ensuremath{\gamma}$ and ${\ensuremath{\tau}}_{P}\ensuremath{\gtrsim}{\ensuremath{\tau}}_{R}$ ($\ensuremath{\gamma}$ is a characteristic internal dephasing rate for the molecules), the system is left in a state with classical transverse polarization when the pump pulse terminates. The system can evolve collectively from such a state only as superradiant (classical) evolution. For superfluorescent evolution which requires a state of preparation of complete inversion (zero transverse polarization), it is shown that the pump pulse must be effectively of area $\ensuremath{\pi}$ and that ${\ensuremath{\tau}}_{P}<{\ensuremath{\tau}}_{R}$. Our results show that when the former conditions are satisfied the delay time ${\ensuremath{\tau}}_{D}$ for collective free pulse evolution is a function of both ${\ensuremath{\tau}}_{P}$ and ${\ensuremath{\tau}}_{R}$. It is shown further that the pump pulse shape as well as temporal duration significantly affect the final state of preparation. The results of this work are interpreted in connection with recent reported results of experiments in superfluorescence and superradiance.

DOI
10.1103/physreva.18.1558

Related papers

Transient effects of dephasing and relaxation in cooperative evolution among three-level systems: II

C. M. Bowden, Chi C. Sung

The effect of uniform dephasing and relaxation on the dynamical evolution of the population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the extreme excited state by an externally applied ($c$-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field, which is treated quantum mechanically. The authors examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration ${\ensuremath{\tau}}_{p}$. It is found that in the homogeneously broadened regime, in the mean-field limit, for uniform pumping, and under conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective relaxation time ${\ensuremath{\tau}}_{R}$ and ${T}_{1}$ and ${T}_{2}$ satisfy the inequalities ${\ensuremath{\omega}}_{R}{\ensuremath{\tau}}_{R}$, ${\ensuremath{\omega}}_{R}{T}_{1}$, ${\ensuremath{\omega}}_{R}{T}_{2}\ensuremath{\gg}1$, collective transverse polarization will evolve if $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}>1$, ${\ensuremath{\omega}}_{R}{\ensuremath{\tau}}_{p}\ensuremath{\approx}\ensuremath{\pi}$ ($\ensuremath{\gamma}$ is the uniform dephasing rate, i.e., $\ensuremath{\gamma}={T}_{2}^{\ensuremath{-}1}$). If, on the other hand, the above conditions are met, but $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}<1$, no collective transverse polarization will evolve during the pump-pulse duration ${\ensuremath{\tau}}_{p}$. In the first case super-radiant pulse evolution (classical collective evolution) may evolve after pump-pulse termination. In the latter case it is possible for collective spontaneous relaxation (superfluorescence) to take place if other conditions are satisfied. Further, a convenient algorithm is established whereby results for the super-radiance or superfluorescence model in the mean-field, small-tipping-angle limit are shown to be formally identical to corresponding results for the lethargic, asymptotic regime of swept-gain super-radiance. In the latter case, the equations of motion are with respect to the retarded time frame and the field equations are replaced by Maxwell's equations. It is shown that the mean-field approximation for the case of super-radiance or superfluorescence is formally equivalent to the asymptotic-limit approximation for swept-gain super-radiance. It is found that a solitary pulse will evolve if the gain-to-loss ratio $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}>1$; if $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}<1$, no solitary pulse will evolve.

Physical Review A•Nov 1, 1979•26 citations

Dynamics of coherent cooperative effects in extended systems under incoherent pumping

A. Sh. Aslaeva, R. F. Malikov

The dynamics of the pulse shape and spectra of superfluorescence, stimulated superradiance, and coherent amplification of ultrashort light pulses is studied for a system of four-level atoms under incoherent cw pumping taking into account homogeneous and inhomogeneous broadening of the luminescence lines. It is found that incoherent pumping at a high level of excitation may significantly change the shape of the emission pulse in cooperative coherent processes. Also, it is found that, depending on the ratio of parameters of the pump, nonradiative transitions, and relaxation, different regimes of superfluorescence and stimulated superradiance may be realized.

Optics and Spectroscopy•Nov 1, 2004•0 citations

Superfluorescence without inversion in coherently driven three-level systems

Victor Kozlov, Olga A. Kocharovskaya, Yuri V. Rostovtsev +1 more

A superfluorescence effect in a V-type medium with initial Raman inversion, driven by a strong coherent field on one of the transitions, is studied analytically and numerically. It is shown that the collective spontaneous emission evolves in the presence of a minor population of an upper state of a superfluorescent transition. In the limit of the Rabi frequency of a driving field large compared to that of a superfluorescent pulse, the system of two equations for the fields and density-matrix equations for three states of the medium is reduced to Maxwell-Bloch equations for an effective two-level system. The shape of a superfluorescent pulse is perfectly described in terms of a slowly varying envelope function (which is a solution of the Maxwell-Bloch equations), filled with fast oscillations associated with Rabi floppings on the driving transition. In the average, i.e., after integrating over a period of Rabi oscillations, the upper superfluorescent state is populated less than other states, such that the phenomenon may be called ``superfluorescence without inversion'' by analogy with lasing without inversion.

Physical Review A•Aug 1, 1999•17 citations

Coherent pump dynamics, propagation, transverse, and diffraction effects in three-level superfluorescence and control of light by light

F. P. Mattar, C. M. Bowden

A model is presented for the dynamical evolution of superfluorescence from an optically pumped three-level system. The full propagation, transverse, and diffraction effects are taken into account. With the use of a previously developed algorithm, a computational simulation was constructed from this model and results are presented and discussed. In particular, it is shown that the injected coherent pump-pulse initial characteristics, such as on-axis area, temporal and radial width and shape, can have significant deterministic effects on the superfluorescent pulse delay time, peak intensity, temporal width, and shape. Thus, by specifying certain initial properties of the injected pump pulse, the superfluorescent pulse can be shaped and altered. The results predict the conditions under which an injected light pulse of a given frequency can be used to generate, shape, and control a second light pulse of a different frequency via a nonlinear medium, thus demonstrating a new aspect of the phenomenon of light control by light.

Physical Review A•Jan 1, 1983•27 citations

Coherent cooperative effects dynamics of extended systems with incoherent pumping

A. Sh. Aslaeva, R. F. Malikov

Numerical researches of the three-level atomic system on supervision of pulse and spectrum dynamics of superfluorescence are carried out at incoherent pumping depending on nonradiation relaxations, on influence of phase relaxations are shown. Pulse and spectrum dynamics of induced superradiation depending on the area and duration of input pulse at incoherent pumping, with and without taking into account phase relaxations, are investigated. Depending on media parameters different modes of induced superradiation are researched.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 20, 2004•0 citations

Cooperative radiation processes in two-level systems: Superfluorescence

R. Bonifacio, Luigi A. Lugiato

We give a general nonperturbative treatment of cooperative emission in systems of $N$ two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to ${N}^{2}$, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length $L$ of the active volume is much larger than a suitable threshold length ${L}_{T}$ (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length $L$ is smaller or of the same order of a suitable cooperation length ${L}_{c}$ (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to $N$). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to $N$, so that superfluorescence effects disappear.

Physical Review A•May 1, 1975•623 citations
The dynamical evolution of an excited-state population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the excited states by an externally applied (ccc-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field which is treated quantum mechanically. It is found that stimulated Raman processes can significantly influence the dynamical evolution of population inversion and macroscopic polarization by producing coherence effects among the populations. We examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration. It is shown, using the mean-field approximation, that for uniform pumping and conditions such that the pump Rabi rate \ensuremathωR{\ensuremath{\omega}}_{R}\ensuremathωR​, the characteristic collective radiation time \ensuremathτR{\ensuremath{\tau}}_{R}\ensuremathτR​, and the pump pulse duration \ensuremathτP{\ensuremath{\tau}}_{P}\ensuremathτP​ satisfy the inequalities, \ensuremathωR>1\ensuremathτR>\ensuremathγ{\ensuremath{\omega}}_{R}>\frac{1}{{\ensuremath{\tau}}_{R}}>\ensuremath{\gamma}\ensuremathωR​>\ensuremathτR​1​>\ensuremathγ and \ensuremathτP\ensuremath≳\ensuremathτR{\ensuremath{\tau}}_{P}\ensuremath{\gtrsim}{\ensuremath{\tau}}_{R}\ensuremathτP​\ensuremath≳\ensuremathτR​ (\ensuremathγ\ensuremath{\gamma}\ensuremathγ is a characteristic internal dephasing rate for the molecules), the system is left in a state with classical transverse polarization when the pump pulse terminates. The system can evolve collectively from such a state only as superradiant (classical) evolution. For superfluorescent evolution which requires a state of preparation of complete inversion (zero transverse polarization), it is shown that the pump pulse must be effectively of area \ensuremathπ\ensuremath{\pi}\ensuremathπ and that \ensuremathτP<\ensuremathτR{\ensuremath{\tau}}_{P}<{\ensuremath{\tau}}_{R}\ensuremathτP​<\ensuremathτR​. Our results show that when the former conditions are satisfied the delay time \ensuremathτD{\ensuremath{\tau}}_{D}\ensuremathτD​ for collective free pulse evolution is a function of both \ensuremathτP{\ensuremath{\tau}}_{P}\ensuremathτP​ and \ensuremathτR{\ensuremath{\tau}}_{R}\ensuremathτR​. It is shown further that the pump pulse shape as well as temporal duration significantly affect the final state of preparation. The results of this work are interpreted in connection with recent reported results of experiments in superfluorescence and superradiance.
The effect of uniform dephasing and relaxation on the dynamical evolution of the population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the extreme excited state by an externally applied (ccc-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field, which is treated quantum mechanically. The authors examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration ${\ensuremath{\tau}}_{p}$. It is found that in the homogeneously broadened regime, in the mean-field limit, for uniform pumping, and under conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective relaxation time ${\ensuremath{\tau}}_{R}$ and T1{T}_{1}T1​ and T2{T}_{2}T2​ satisfy the inequalities ${\ensuremath{\omega}}_{R}{\ensuremath{\tau}}_{R}$, ${\ensuremath{\omega}}_{R}{T}_{1}$, ${\ensuremath{\omega}}_{R}{T}_{2}\ensuremath{\gg}1$, collective transverse polarization will evolve if $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}>1$, ${\ensuremath{\omega}}_{R}{\ensuremath{\tau}}_{p}\ensuremath{\approx}\ensuremath{\pi}$ ($\ensuremath{\gamma}$ is the uniform dephasing rate, i.e., $\ensuremath{\gamma}={T}_{2}^{\ensuremath{-}1}$). If, on the other hand, the above conditions are met, but $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}<1$, no collective transverse polarization will evolve during the pump-pulse duration ${\ensuremath{\tau}}_{p}$. In the first case super-radiant pulse evolution (classical collective evolution) may evolve after pump-pulse termination. In the latter case it is possible for collective spontaneous relaxation (superfluorescence) to take place if other conditions are satisfied. Further, a convenient algorithm is established whereby results for the super-radiance or superfluorescence model in the mean-field, small-tipping-angle limit are shown to be formally identical to corresponding results for the lethargic, asymptotic regime of swept-gain super-radiance. In the latter case, the equations of motion are with respect to the retarded time frame and the field equations are replaced by Maxwell's equations. It is shown that the mean-field approximation for the case of super-radiance or superfluorescence is formally equivalent to the asymptotic-limit approximation for swept-gain super-radiance. It is found that a solitary pulse will evolve if the gain-to-loss ratio $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}>1$; if $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}<1$, no solitary pulse will evolve.
The dynamics of the pulse shape and spectra of superfluorescence, stimulated superradiance, and coherent amplification of ultrashort light pulses is studied for a system of four-level atoms under incoherent cw pumping taking into account homogeneous and inhomogeneous broadening of the luminescence lines. It is found that incoherent pumping at a high level of excitation may significantly change the shape of the emission pulse in cooperative coherent processes. Also, it is found that, depending on the ratio of parameters of the pump, nonradiative transitions, and relaxation, different regimes of superfluorescence and stimulated superradiance may be realized.
A superfluorescence effect in a V-type medium with initial Raman inversion, driven by a strong coherent field on one of the transitions, is studied analytically and numerically. It is shown that the collective spontaneous emission evolves in the presence of a minor population of an upper state of a superfluorescent transition. In the limit of the Rabi frequency of a driving field large compared to that of a superfluorescent pulse, the system of two equations for the fields and density-matrix equations for three states of the medium is reduced to Maxwell-Bloch equations for an effective two-level system. The shape of a superfluorescent pulse is perfectly described in terms of a slowly varying envelope function (which is a solution of the Maxwell-Bloch equations), filled with fast oscillations associated with Rabi floppings on the driving transition. In the average, i.e., after integrating over a period of Rabi oscillations, the upper superfluorescent state is populated less than other states, such that the phenomenon may be called ``superfluorescence without inversion'' by analogy with lasing without inversion.
A model is presented for the dynamical evolution of superfluorescence from an optically pumped three-level system. The full propagation, transverse, and diffraction effects are taken into account. With the use of a previously developed algorithm, a computational simulation was constructed from this model and results are presented and discussed. In particular, it is shown that the injected coherent pump-pulse initial characteristics, such as on-axis area, temporal and radial width and shape, can have significant deterministic effects on the superfluorescent pulse delay time, peak intensity, temporal width, and shape. Thus, by specifying certain initial properties of the injected pump pulse, the superfluorescent pulse can be shaped and altered. The results predict the conditions under which an injected light pulse of a given frequency can be used to generate, shape, and control a second light pulse of a different frequency via a nonlinear medium, thus demonstrating a new aspect of the phenomenon of light control by light.
Numerical researches of the three-level atomic system on supervision of pulse and spectrum dynamics of superfluorescence are carried out at incoherent pumping depending on nonradiation relaxations, on influence of phase relaxations are shown. Pulse and spectrum dynamics of induced superradiation depending on the area and duration of input pulse at incoherent pumping, with and without taking into account phase relaxations, are investigated. Depending on media parameters different modes of induced superradiation are researched.
We give a general nonperturbative treatment of cooperative emission in systems of NNN two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to N2{N}^{2}N2, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length LLL of the active volume is much larger than a suitable threshold length LT{L}_{T}LT​ (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length LLL is smaller or of the same order of a suitable cooperation length Lc{L}_{c}Lc​ (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to NNN). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to NNN, so that superfluorescence effects disappear.