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

arXiv:quant-ph/0204074 · Physical Review A

Approximate master equations for atom optics

D. J. Atkins, Howard M. Wiseman, Prahlad Warszawski

Published Feb 6, 2003• 15 citations
arXivHTMLPDF

Abstract

In the field of atom optics, the basis of many experiments is a two-level atom coupled to a light field. The evolution of this system is governed by a master equation. The irreversible components of this master equation describe the spontaneous emission of photons from the atom. For many applications, it is necessary to minimize the effect of this irreversible evolution. This can be achieved by having a far detuned light field. The drawback of this regime is that making the detuning very large makes the time step required to solve the master equation very small, much smaller than the time scale of any significant evolution. This makes the problem very numerically intensive. For this reason, approximations are used to simulate the master equation, which are more numerically tractable to solve. This paper analyzes four approximations: The standard adiabatic approximation, a more sophisticated adiabatic approximation (not used before), a secular approximation, and a fully quantum dressed-state approximation. The advantages and disadvantages of each are investigated with respect to accuracy, complexity, and the resources required to simulate. In a parameter regime of particular experimental interest, only the sophisticated adiabatic and dressed-state approximations agree well with the exact evolution.

DOI
10.1103/physreva.67.023802

Related papers

Dressed-AtomModel

Z. Ficek, Mohamed Ridza Wahiddin

In this chapter, we present a powerful method of solving the Bloch equations or the master equation of a driven system called the dressed-atom model. The method is valid in situations where the Rabi frequency of the applied driving field is much larger than the spontaneous emission rate of the atom, . Under such conditions, one can make the secular approximation that consists of dropping terms oscillating in time with frequencies 2 and higher. These terms, if kept in the master equation, would make corrections to the dynamics of the system of the order of /, and thus are negligible. Although limited in the range of parameters for which it can be used, the dressed-atommodel provides a physical insight into the properties and dynamics of the system. Within this model, one can explicitly calculate energy states and transition rates between them in a relatively simple way. The knowledge of the energy states and transition rates is for most of the problems enough to fully understand the underlying physics. There are two mathematically different approaches to the dressed-atommodel, but giving the same results, depending on whether we treat the driving field classically or quantum mechanically. These are the semiclassical and quantum dress-atom models. In the following we explain in details these two dressed-atommodels.

•Apr 19, 2016•0 citations

Master-Equation Approach to Spontaneous Emission

G. S. Agarwal

Spontaneous emission from a system of $N$ identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state $|{z}_{0}〉$.

Physical Review A•Nov 1, 1970•232 citations

One-atom lasers

Yi Mu, Craig M. Savage

One-atom lasers are important because their governing equations can be solved exactly, even with a quantized field. We present a fully quantum-mechanical treatment of one-atom lasers modeled by quantum-optical master equations. These are solved numerically without any significant approximations. We show that laser action is possible with one atom, and that it might be achievable experimentally. Laser action is characterized by the dominance of stimulated emission over spontaneous emission. We use the one-atom laser model to investigate, without approximation, some interesting generic laser phenomena. Under certain conditions lasers produce intensity squeezed light, and then the laser linewidth increases with the pumping rate, in contrast with standard lasers. We also report ``self-quenching'' behavior: lasers with incoherent pumping out of the lower laser level turn off when the pumping is sufficiently fast because the coherence between the laser levels is destroyed.

Physical Review A•Nov 1, 1992•206 citations

Quantum optical master equations: The use of damping bases

Hans‐Jürgen Briegel, Berthold‐Georg Englert

We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.

Physical Review A•Apr 1, 1993•239 citations

Quantum optical master equations: The one-atom laser

Christian Ginzel, Hans‐Jürgen Briegel, Ullrich Martini +2 more

We present a detailed numerical study of the one-atom laser, that is, a single two-level atom interacting with one lasing mode, whereby both the atom and the photon field are coupled to reservoirs. The stationary as well as the dynamical properties of the model are calculated directly from the quantum master equation with the help of two numerical methods. These numerical methods do not need any quasi-probability representation and they do not require approximations. We find that the one-atom laser exhibits most of the typical features of a normal laser. In the region far below threshold some aspects, among them the linewidth, are changed due to eigenvalues of the master equation with imaginary parts. In this regime the complexity of the eigenvalues prominently enters the dynamical behavior.

Physical Review A•Jul 1, 1993•101 citations

Atom-only descriptions of the driven-dissipative Dicke model

François Damanet, Andrew J. Daley, Jonathan Keeling

We investigate how to describe the dissipative spin dynamics of the driven-dissipative Dicke model, describing $N$ two-level atoms coupled to a cavity mode, after adiabatic elimination of the cavity mode. To this end, we derive a Redfield master equation which goes beyond the standard secular approximation and large detuning limits. We show that the secular (or rotating wave) approximation and the large detuning approximation both lead to inadequate master equations, that fail to predict the Dicke transition or the damping rates of the atomic dynamics. In contrast, the full Redfield theory correctly predicts the phase transition and the effective atomic damping rates. Our work provides a reliable framework to study the full quantum dynamics of atoms in a multimode cavity, where a quantum description of the full model becomes intractable.

arXiv:1901.02473•Physical Review A•Mar 25, 2019•70 citations
In the field of atom optics, the basis of many experiments is a two-level atom coupled to a light field. The evolution of this system is governed by a master equation. The irreversible components of this master equation describe the spontaneous emission of photons from the atom. For many applications, it is necessary to minimize the effect of this irreversible evolution. This can be achieved by having a far detuned light field. The drawback of this regime is that making the detuning very large makes the time step required to solve the master equation very small, much smaller than the time scale of any significant evolution. This makes the problem very numerically intensive. For this reason, approximations are used to simulate the master equation, which are more numerically tractable to solve. This paper analyzes four approximations: The standard adiabatic approximation, a more sophisticated adiabatic approximation (not used before), a secular approximation, and a fully quantum dressed-state approximation. The advantages and disadvantages of each are investigated with respect to accuracy, complexity, and the resources required to simulate. In a parameter regime of particular experimental interest, only the sophisticated adiabatic and dressed-state approximations agree well with the exact evolution.
In this chapter, we present a powerful method of solving the Bloch equations or the master equation of a driven system called the dressed-atom model. The method is valid in situations where the Rabi frequency of the applied driving field is much larger than the spontaneous emission rate of the atom, . Under such conditions, one can make the secular approximation that consists of dropping terms oscillating in time with frequencies 2 and higher. These terms, if kept in the master equation, would make corrections to the dynamics of the system of the order of /, and thus are negligible. Although limited in the range of parameters for which it can be used, the dressed-atommodel provides a physical insight into the properties and dynamics of the system. Within this model, one can explicitly calculate energy states and transition rates between them in a relatively simple way. The knowledge of the energy states and transition rates is for most of the problems enough to fully understand the underlying physics. There are two mathematically different approaches to the dressed-atommodel, but giving the same results, depending on whether we treat the driving field classically or quantum mechanically. These are the semiclassical and quantum dress-atom models. In the following we explain in details these two dressed-atommodels.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
One-atom lasers are important because their governing equations can be solved exactly, even with a quantized field. We present a fully quantum-mechanical treatment of one-atom lasers modeled by quantum-optical master equations. These are solved numerically without any significant approximations. We show that laser action is possible with one atom, and that it might be achievable experimentally. Laser action is characterized by the dominance of stimulated emission over spontaneous emission. We use the one-atom laser model to investigate, without approximation, some interesting generic laser phenomena. Under certain conditions lasers produce intensity squeezed light, and then the laser linewidth increases with the pumping rate, in contrast with standard lasers. We also report ``self-quenching'' behavior: lasers with incoherent pumping out of the lower laser level turn off when the pumping is sufficiently fast because the coherence between the laser levels is destroyed.
We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.
We present a detailed numerical study of the one-atom laser, that is, a single two-level atom interacting with one lasing mode, whereby both the atom and the photon field are coupled to reservoirs. The stationary as well as the dynamical properties of the model are calculated directly from the quantum master equation with the help of two numerical methods. These numerical methods do not need any quasi-probability representation and they do not require approximations. We find that the one-atom laser exhibits most of the typical features of a normal laser. In the region far below threshold some aspects, among them the linewidth, are changed due to eigenvalues of the master equation with imaginary parts. In this regime the complexity of the eigenvalues prominently enters the dynamical behavior.
We investigate how to describe the dissipative spin dynamics of the driven-dissipative Dicke model, describing NNN two-level atoms coupled to a cavity mode, after adiabatic elimination of the cavity mode. To this end, we derive a Redfield master equation which goes beyond the standard secular approximation and large detuning limits. We show that the secular (or rotating wave) approximation and the large detuning approximation both lead to inadequate master equations, that fail to predict the Dicke transition or the damping rates of the atomic dynamics. In contrast, the full Redfield theory correctly predicts the phase transition and the effective atomic damping rates. Our work provides a reliable framework to study the full quantum dynamics of atoms in a multimode cavity, where a quantum description of the full model becomes intractable.