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

Europhysics Letters (EPL)

Pump operator for lasers with multi-level excitation

H. J. Briegel, Georg M. Meyer, Berthold‐Georg Englert

Published Mar 1, 1996• 12 citations
Publisher

Abstract

In a multi-level laser the pump process involves several intermediate levels. The details of the atomic excitation are often irrelevant so that an effective heat bath description and a two-level–laser model are applicable. The heat bath approximation may be too crude, however; it does not predict phenomena such as intensity-noise squeezing and sub-Poissonian field statistics. We show how, in the general situation, the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation where a rate matrix, rather than a single rate, accounts for the excitation process. The resulting dependence of the excitation rates on the state of the laser field constitutes a correction to the standard laser model. We hold it responsible for the dynamic noise reduction in multi-level lasers.

DOI
10.1209/epl/i1996-00371-6

Related papers

Correlated atomic excitation in multilevel lasers

Hans-Juergen Briegel, Georg M. Meyer, Berthold‐Georg Englert

In a multi-level laser the pump process usually involves several intermediate levels as well as spontaneous relaxations and an external pump field. These details of the atomic excitation are often irrelevant and can be summarized by an effective heat-bath description and a two-level- laser model instead. Under certain circumstances, however, the heat-bath approximation will be too crude and it cannot account for phenomena such as intensity-noise squeezing and sub- Poissonian field statistics. We show how in the general situation the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation. In steady state the atomic excitation is then described by a rate matrix rather than a single rate. The matrix or operator nature of the pump process gives rise to a correlation of the excitation rates with the statistical moments of the laser field. This correlation represents a correction to the standard laser model. In particular, it can be identified as the physical origin of the phenomenon of dynamic noise reduction in multi-level lasers.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 29, 1996•1 citation

Dynamic noise reduction in multilevel lasers: Nonlinear theory and the pump-operator approach

Hans‐Jürgen Briegel, Georg M. Meyer, Berthold‐Georg Englert

We present a general theory that connects the pump process in multilevel lasers with the statistics of the laser field. The key ingredient in our approach is the derivation of an effective master equation that involves only the two laser levels and that contains a non-Markovian pump term. This pump term gives rise to a narrowing of the photon-number distribution in steady state. The qualitative features of this dynamic noise reduction can be inferred from the eigenvalues of the pump term, which are known analytically. As the mechanism that is responsible for the noise reduction, we identify a correlated excitation process in which the effective excitation rate from the lower to the upper laser level depends on the photon number. This correlated excitation process is very different from an excitation at equidistant times, such as in lasers with periodic external injection. We stress that our approach is general and the treatment nonlinear because we do not resort to approximations such as an adiabatic elimination of atomic variables or the linearization of a Fokker-Planck equation. It is therefore particularly relevant for the description of systems where the active medium consists of a few atoms only and where the field losses are not negligible on the time scale of the atomic relaxations. This would be the case in an ion-trap laser. For such a situation our results are substantially different from those obtained by a linearized treatment. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Feb 1, 1996•32 citations

Dynamic quantum-noise reduction in multilevel-laser systems

Helmut Ritsch, P. Zoller

Based on the standard laser model of a large number N of model atoms resonantly coupled to a single lasing mode, we show that the nonlinear dynamics of the active atoms of the laser can lead to output-intensity fluctuations significantly reduced below the shot-noise level. We identify the multiple recycling of the active electron from the lower lasing level to the upper level through the pumping as the key process leading to this dynamic-pump-noise reduction. This process has been neglected in most of the standard treatments of the laser so far. We find that the results are closely related to recent calculations based on the assumption of an external regular pump. For the widely used four-level model of the active atoms, the intensity noise can be reduced 50% below the shot-noise level. Generalizing the model to an m-level system, we find a quantum-noise reduction by a factor of 1/2m/(m-1)(m\ensuremath{\ge}3), leading to perfect output-intensity noise reduction in the limit of a large number of intermediate steps in the recycling process of the active atoms. Finally, we demonstrate that the bandwidth of the noise reduction can be significantly enhanced using a nonlinear absorber in the cavity.

Physical Review A•Feb 1, 1992•54 citations

Quantum Theory of Laser Radiation. II. Statistical Aspects of Laser Light

J. A. Fleck

We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with $N$ stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.

Physical Review•Sep 9, 1966•26 citations

Role of pumping statistics and dynamics of atomic polarization in quantum fluctuations of laser sources

Mikhail I. Kolobov, L. Davidovich, E. Giacobino +1 more

We solve the problem of a laser with variable pumping statistics for any relative magnitude of the atomic- and cavity-decay constants, and obtain a different regime of sub-Poissonian light generation. We show that, even for Poissonian pumping, the noise in the amplitude quadrature outside the cavity can be reduced up to 50% below the shot-noise level when the polarization but not the populations can be adiabatically eliminated. Maximum noise reduction in this case is obtained when the lower level decays much faster than the upper one and occurs at a frequency given by the geometrical mean of the decay rates of the field and the lower-level population. Furthermore, the full consideration of atomic memory effects leads to a generalization of previous results on regularly pumped lasers. We find that, for regular pumping, maximum amplitude-noise reduction (up to complete quieting) still occurs at zero frequency in all cases. However, a long-living polarizaton leads to an increase in amplitude noise and may even eliminate the dip at zero frequency, at the same time leading to a significant quenching of the laser linewidth in the bad-cavity limit.

Physical Review A•Feb 1, 1993•128 citations

Quantum Theory of Radiation. II. Statistical Aspects of Laser Light

J. A. Fleck

We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with $N$ stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.

Physical Review•Mar 10, 1967•4 citations
In a multi-level laser the pump process involves several intermediate levels. The details of the atomic excitation are often irrelevant so that an effective heat bath description and a two-level–laser model are applicable. The heat bath approximation may be too crude, however; it does not predict phenomena such as intensity-noise squeezing and sub-Poissonian field statistics. We show how, in the general situation, the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation where a rate matrix, rather than a single rate, accounts for the excitation process. The resulting dependence of the excitation rates on the state of the laser field constitutes a correction to the standard laser model. We hold it responsible for the dynamic noise reduction in multi-level lasers.
In a multi-level laser the pump process usually involves several intermediate levels as well as spontaneous relaxations and an external pump field. These details of the atomic excitation are often irrelevant and can be summarized by an effective heat-bath description and a two-level- laser model instead. Under certain circumstances, however, the heat-bath approximation will be too crude and it cannot account for phenomena such as intensity-noise squeezing and sub- Poissonian field statistics. We show how in the general situation the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation. In steady state the atomic excitation is then described by a rate matrix rather than a single rate. The matrix or operator nature of the pump process gives rise to a correlation of the excitation rates with the statistical moments of the laser field. This correlation represents a correction to the standard laser model. In particular, it can be identified as the physical origin of the phenomenon of dynamic noise reduction in multi-level lasers.
We present a general theory that connects the pump process in multilevel lasers with the statistics of the laser field. The key ingredient in our approach is the derivation of an effective master equation that involves only the two laser levels and that contains a non-Markovian pump term. This pump term gives rise to a narrowing of the photon-number distribution in steady state. The qualitative features of this dynamic noise reduction can be inferred from the eigenvalues of the pump term, which are known analytically. As the mechanism that is responsible for the noise reduction, we identify a correlated excitation process in which the effective excitation rate from the lower to the upper laser level depends on the photon number. This correlated excitation process is very different from an excitation at equidistant times, such as in lasers with periodic external injection. We stress that our approach is general and the treatment nonlinear because we do not resort to approximations such as an adiabatic elimination of atomic variables or the linearization of a Fokker-Planck equation. It is therefore particularly relevant for the description of systems where the active medium consists of a few atoms only and where the field losses are not negligible on the time scale of the atomic relaxations. This would be the case in an ion-trap laser. For such a situation our results are substantially different from those obtained by a linearized treatment. \textcopyright{} 1996 The American Physical Society.
Based on the standard laser model of a large number N of model atoms resonantly coupled to a single lasing mode, we show that the nonlinear dynamics of the active atoms of the laser can lead to output-intensity fluctuations significantly reduced below the shot-noise level. We identify the multiple recycling of the active electron from the lower lasing level to the upper level through the pumping as the key process leading to this dynamic-pump-noise reduction. This process has been neglected in most of the standard treatments of the laser so far. We find that the results are closely related to recent calculations based on the assumption of an external regular pump. For the widely used four-level model of the active atoms, the intensity noise can be reduced 50% below the shot-noise level. Generalizing the model to an m-level system, we find a quantum-noise reduction by a factor of 1/2m/(m-1)(m\ensuremath{\ge}3), leading to perfect output-intensity noise reduction in the limit of a large number of intermediate steps in the recycling process of the active atoms. Finally, we demonstrate that the bandwidth of the noise reduction can be significantly enhanced using a nonlinear absorber in the cavity.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We solve the problem of a laser with variable pumping statistics for any relative magnitude of the atomic- and cavity-decay constants, and obtain a different regime of sub-Poissonian light generation. We show that, even for Poissonian pumping, the noise in the amplitude quadrature outside the cavity can be reduced up to 50% below the shot-noise level when the polarization but not the populations can be adiabatically eliminated. Maximum noise reduction in this case is obtained when the lower level decays much faster than the upper one and occurs at a frequency given by the geometrical mean of the decay rates of the field and the lower-level population. Furthermore, the full consideration of atomic memory effects leads to a generalization of previous results on regularly pumped lasers. We find that, for regular pumping, maximum amplitude-noise reduction (up to complete quieting) still occurs at zero frequency in all cases. However, a long-living polarizaton leads to an increase in amplitude noise and may even eliminate the dip at zero frequency, at the same time leading to a significant quenching of the laser linewidth in the bad-cavity limit.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.