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

Journal of Physics B Atomic and Molecular Physics

Effects of long-lived incoherences on coherent population trapping

G. S. Agarwal, N. Nayak

Published Oct 28, 1986• 11 citations
Publisher

Abstract

The effects of the non-Markov character of several types of relaxation on population trapping are examined. Different types of relaxation can arise as a result of (i) intermediate-state interactions and (ii) the non-Lorentzian character of the lineshapes of the pump and probe beams used for the study of the trapping. In each case the dynamical equations for the appropriately defined elements of the density matrix will have multiplicative colour noise, for which the correlation time tau c , need not be short. A very interesting regime of physical behavior is obtained if the correlation time and of the Rabi frequencies, G i , of the transitions are such that G i tau approximately=1. The relevant mathematical equations for the two mechanisms of relaxation, (i) and (ii), are related using methods based on Fokker-Planck equations and matrix continued fractions; the authors calculate physical quantities such as excited-state populations and coherences for a range of parameters including detunings, correlation times and pump and probe intensities. Under conditions of population trapping, Raman signals have the same form as Fano profiles.

DOI
10.1088/0022-3700/19/20/021

Related papers

Some features of the optical manifestation of relaxation

Pavel A Apanasevich, Alexander P. Nizovtsev

It is shown that, in general, the interaction between relaxing quantum systems and radiation is described by non-Markov transport equations for the density matrix ρ p and the relaxation matrix depends on the properties of the radiation. The usual Markov-type equations for ρ p are obtained from these equations if γτ c ≪1 and (∊ 2 +4 V 2 ) 1/2 τ c ≪1 (γ is the width of the investigated transition in a quantum system, rc is the correlation time of the perturbations causing relaxation, e is the frequency detuning, and V is the energy of the interaction between the system and the radiation field). When the second inequality is disobeyed, the relaxation process deviates strongly from the Markov type. In the case of small values of V , this results in a non-Lorentzian profile of the absorption lines characterized by a steep descent in the wings. In the range of high values of V , when 2 V τ c 1, the whole relaxation process becomes greatly modified: the dependence of the relaxation coefficients on V becomes strong and a coupling appears between the relaxations of the diagonal and the nondiagonal elements of the density matrix. This is manifested, in particular, in the saturation effect: the term describing saturation in the expression for the polarizability changes considerably and the expression for the line width loses the component due to the adiabatic (elastic) perturbation.

Soviet Journal of Quantum Electronics•Aug 31, 1975•2 citations

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

Quantum-noise reduction in Raman lasers: Effects of collisions, population trapping, and Doppler shifts

K. J. Schernthanner, Helmut Ritsch

We investigate the effect of collisional broadening, of population trapping and repumping, and of detuning of the atomic levels on the quantum-statistical properties of Raman laser radiation. We find that under suitably chosen operating conditions the effect of such additional incoherent relaxation mechanisms for the atomic polarization and populations can be made very small. One obtains highly nonclassical values for the Mandel Q parameter and the output-intensity fluctuation spectrum S(\ensuremath{\omega}) for experimentally reasonable parameters. Including detuning of the laser and the pump light from the atomic levels in certain cases can lead to further narrowing of the intracavity photon number distribution, as compared to the resonant case, down to a Mandel Q parameter close to the optimum possible value of Q=-1. A subsequent preliminary analysis prevails that inhomogeneous Doppler broadening of the atomic lines, both for collinear and for counterpropagating pump and laser light, is not detrimental to the nonclassical intracavity state.

Physical Review A•May 1, 1994•16 citations

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

C. M. Bowden, Chi C. Sung

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.

Physical Review A•Oct 1, 1978•58 citations

Quantum dynamics of incoherently driven V-type systems: Analytic solutions beyond the secular approximation

Amro Dodin, Timur V. Tscherbul, Paul Brumer

Closed-form analytic solutions to non-secular Bloch-Redfield master equations for quantum dynamics of a V-type system driven by weak coupling to a thermal bath, relevant to light harvesting processes, are obtained and discussed. We focus on noise-induced Fano coherences among the excited states induced by incoherent driving of the V-system initially in the ground state. For suddenly turned-on incoherent driving, the time evolution of the coherences is determined by the damping parameter ζ=12(γ1+γ2)/Δp, where γi are the radiative decay rates of the excited levels i = 1, 2, and Δp=Δ(2)+(1-p(2))γ1γ2 depends on the excited-state level splitting Δ > 0 and the angle between the transition dipole moments in the energy basis. The coherences oscillate as a function of time in the underdamped limit (ζ ≫ 1), approach a long-lived quasi-steady state in the overdamped limit (ζ ≪ 1), and display an intermediate behavior at critical damping (ζ = 1). The sudden incoherent turn-on is shown to generate a mixture of excited eigenstates |e1〉 and |e2〉 and their in-phase coherent superposition |ϕ+〉=1r1+r2(r1|e1〉+r2|e2〉), which is remarkably long-lived in the overdamped limit (where r1 and r2 are the incoherent pumping rates). Formation of this coherent superposition enhances the decay rate from the excited states to the ground state. In the strongly asymmetric V-system where the coupling strengths between the ground state and the excited states differ significantly, additional asymptotic quasistationary coherences are identified, which arise due to slow equilibration of one of the excited states. Finally, we demonstrate that noise-induced Fano coherences are maximized with respect to populations when r1 = r2 and the transition dipole moments are fully aligned.

arXiv:1509.03399•The Journal of Chemical Physics•Jun 27, 2016•47 citations

Quantum jumps in a coherently driven Raman system

G. S. Agarwal, S. V. Lawande, Richard D’Souza

Early results of Agarwal and Jha [Z. Phys. B 35, 391 (1979)] are used to examine the question of quantum jumps in a Raman (\ensuremath{\Lambda}) system driven by two coherent fields. We discuss both short-time (${\ensuremath{\gamma}}_{1}$t\ensuremath{\lesssim}1) and long-time (${\ensuremath{\gamma}}_{2}$t\ensuremath{\approxeq}1) behavior of the intensity correlations assuming that the Stokes transition is very weak, ${\ensuremath{\gamma}}_{2}$\ensuremath{\ll}${\ensuremath{\gamma}}_{1}$. We calculate the counting statistics of the dark periods using the higher-order intensity correlations. The results differ from those obtained using arguments based on population trapping which predict only a very rare ``jump'' event. The intermittence in fluorescence produced by a coherently driven \ensuremath{\Lambda} system is interpreted in terms of an equivalent two-state Markov process.

Physical Review A•Jan 1, 1988•22 citations
The effects of the non-Markov character of several types of relaxation on population trapping are examined. Different types of relaxation can arise as a result of (i) intermediate-state interactions and (ii) the non-Lorentzian character of the lineshapes of the pump and probe beams used for the study of the trapping. In each case the dynamical equations for the appropriately defined elements of the density matrix will have multiplicative colour noise, for which the correlation time tau c , need not be short. A very interesting regime of physical behavior is obtained if the correlation time and of the Rabi frequencies, G i , of the transitions are such that G i tau approximately=1. The relevant mathematical equations for the two mechanisms of relaxation, (i) and (ii), are related using methods based on Fokker-Planck equations and matrix continued fractions; the authors calculate physical quantities such as excited-state populations and coherences for a range of parameters including detunings, correlation times and pump and probe intensities. Under conditions of population trapping, Raman signals have the same form as Fano profiles.
It is shown that, in general, the interaction between relaxing quantum systems and radiation is described by non-Markov transport equations for the density matrix ρ p and the relaxation matrix depends on the properties of the radiation. The usual Markov-type equations for ρ p are obtained from these equations if γτ c ≪1 and (∊ 2 +4 V 2 ) 1/2 τ c ≪1 (γ is the width of the investigated transition in a quantum system, rc is the correlation time of the perturbations causing relaxation, e is the frequency detuning, and V is the energy of the interaction between the system and the radiation field). When the second inequality is disobeyed, the relaxation process deviates strongly from the Markov type. In the case of small values of V , this results in a non-Lorentzian profile of the absorption lines characterized by a steep descent in the wings. In the range of high values of V , when 2 V τ c 1, the whole relaxation process becomes greatly modified: the dependence of the relaxation coefficients on V becomes strong and a coupling appears between the relaxations of the diagonal and the nondiagonal elements of the density matrix. This is manifested, in particular, in the saturation effect: the term describing saturation in the expression for the polarizability changes considerably and the expression for the line width loses the component due to the adiabatic (elastic) perturbation.
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.
We investigate the effect of collisional broadening, of population trapping and repumping, and of detuning of the atomic levels on the quantum-statistical properties of Raman laser radiation. We find that under suitably chosen operating conditions the effect of such additional incoherent relaxation mechanisms for the atomic polarization and populations can be made very small. One obtains highly nonclassical values for the Mandel Q parameter and the output-intensity fluctuation spectrum S(\ensuremath{\omega}) for experimentally reasonable parameters. Including detuning of the laser and the pump light from the atomic levels in certain cases can lead to further narrowing of the intracavity photon number distribution, as compared to the resonant case, down to a Mandel Q parameter close to the optimum possible value of Q=-1. A subsequent preliminary analysis prevails that inhomogeneous Doppler broadening of the atomic lines, both for collinear and for counterpropagating pump and laser light, is not detrimental to the nonclassical intracavity state.
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 ${\ensuremath{\omega}}_{R}$, 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.
Closed-form analytic solutions to non-secular Bloch-Redfield master equations for quantum dynamics of a V-type system driven by weak coupling to a thermal bath, relevant to light harvesting processes, are obtained and discussed. We focus on noise-induced Fano coherences among the excited states induced by incoherent driving of the V-system initially in the ground state. For suddenly turned-on incoherent driving, the time evolution of the coherences is determined by the damping parameter ζ=12(γ1+γ2)/Δp, where γi are the radiative decay rates of the excited levels i = 1, 2, and Δp=Δ(2)+(1-p(2))γ1γ2 depends on the excited-state level splitting Δ > 0 and the angle between the transition dipole moments in the energy basis. The coherences oscillate as a function of time in the underdamped limit (ζ ≫ 1), approach a long-lived quasi-steady state in the overdamped limit (ζ ≪ 1), and display an intermediate behavior at critical damping (ζ = 1). The sudden incoherent turn-on is shown to generate a mixture of excited eigenstates |e1〉 and |e2〉 and their in-phase coherent superposition |ϕ+〉=1r1+r2(r1|e1〉+r2|e2〉), which is remarkably long-lived in the overdamped limit (where r1 and r2 are the incoherent pumping rates). Formation of this coherent superposition enhances the decay rate from the excited states to the ground state. In the strongly asymmetric V-system where the coupling strengths between the ground state and the excited states differ significantly, additional asymptotic quasistationary coherences are identified, which arise due to slow equilibration of one of the excited states. Finally, we demonstrate that noise-induced Fano coherences are maximized with respect to populations when r1 = r2 and the transition dipole moments are fully aligned.
Early results of Agarwal and Jha [Z. Phys. B 35, 391 (1979)] are used to examine the question of quantum jumps in a Raman (\ensuremath{\Lambda}) system driven by two coherent fields. We discuss both short-time (${\ensuremath{\gamma}}_{1}$t\ensuremath{\lesssim}1) and long-time (${\ensuremath{\gamma}}_{2}$t\ensuremath{\approxeq}1) behavior of the intensity correlations assuming that the Stokes transition is very weak, ${\ensuremath{\gamma}}_{2}$\ensuremath{\ll}${\ensuremath{\gamma}}_{1}$. We calculate the counting statistics of the dark periods using the higher-order intensity correlations. The results differ from those obtained using arguments based on population trapping which predict only a very rare ``jump'' event. The intermittence in fluorescence produced by a coherently driven \ensuremath{\Lambda} system is interpreted in terms of an equivalent two-state Markov process.