Basic Properties of a Mean Field Laser Equation

arXiv: Mathematical Physics, 2018

We study the dynamics of the solution of a non-linear quantum master equation describing a simple laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and a set of two level atoms that are coupled to two reservoirs. First, we establish the existence of a unique regular stationary state for the non-linear evolution equation under consideration. Second, we examine the free interaction solutions, i.e., the solutions to the non-linear quantum master equation that coincide with unitary evolutions generated by the Hamiltonian resulting from neglecting the interactions between the laser mode, atoms and the bath. We obtain that a family of non-constant free interaction solutions is born at the regular stationary state as a relevant parameter, which is denoted by $C_\\mathfrak{b} $, passes through the critical value $1$. These free interaction solutions yield the periodic solutions of the Maxwell Bloch equations modeling our physical system in...

2020

We study the dynamics of a non-linear Gorini-Kossakowski-Sudarshan-Lindblad equation with mean-field Hamiltonian that describe a simple laser under the mean field approximation. The open quantum system is formed by a single mode optical cavity and a set of two level atoms, each coupled to a reservoir. We prove that the mean field quantum master equation has a unique regular stationary solution. In case a relevant parameter C_b, i.e., the cavity cooperative parameter, is less than 1, we prove that any regular solution converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. We obtain that a locally exponential stable family of periodic solutions is born at the regular stationary state as C_b passes through the critical value 1. Then, the mean-field laser equation has a Hopf bifurcation at C_b =1 of supercritical-like type. Thus, we establish the mechanism by which the full quantum dynamics yields th...

Annales Henri Poincaré, 2018

We deal with the dynamical system properties of a Gorini–Kossakowski–Sudarshan–Lindblad equation with mean-field Hamiltonian that models a simple laser by applying a mean-field approximation to a quantum system describing a single-mode optical cavity and a set of two-level atoms, each coupled to a reservoir. We prove that the mean-field quantum master equation has a unique regular stationary solution. In case a relevant parameter $$C_\mathfrak {b} $$ C b , i.e., the cavity cooperative parameter, is less than 1, we prove that any regular solution converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. We obtain that a locally exponential stable limit cycle is born at the regular stationary state as $$C_\mathfrak {b} $$ C b passes through the critical value 1. Then, the mean-field laser equation has a Poincaré–Andronov–Hopf bifurcation at $$C_\mathfrak {b} =1 $$ C b = 1 of supercritical-like type. N...

1994

Physical Review A, 1993

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.

1999

Quantum Optics has had an impressive development during the last two decades. Laser-based devices are invading our everyday life: printers, pointers, a new surgical technology replacing the old scalpel, transport of information, among many other applications which give the idea of a well established theory. Even though, various theoretical problems remain open. In particular Quantum Optics continues to inspire mathematicians in their own research. This survey is aimed at giving a taste of some mathematical problems raised by Quantum Optics within the field of Stochastic Analysis. Especially, we want to show the interplay between classical and non commutative stochastic differential equations associated to the so called master equations of Quantum Optics. Contents 1. Introduction 2 1.1. Notations 3 1.2. Deriving the master equation 4 2. The associated Quantum Stochastic Differential Equation 7 2.1. Quantum noises in continuous time 8 2.2. The master equation for the quantum cocycle 9 2.3. The Quantum Markov Flow 10 3. Unravelling: classical stochastic processes related to quantum flows 12 3.1. The algebra generated by the number operator 12 3.2. Position, momentum and their algebras 13 3.3. The evolution on A N , A q , A p. 15 4. Stationary state and the convergence towards the equilibrium 16 References 18 Research partially supported by FONDECYT grant 1960917 and the "Cátedra Presidencial en Análisis Estocástico y Física Matemática".

Physica D: Nonlinear Phenomena, 2000

A statistical model of self-organization in a generic class of one-dimensional nonlinear Schrödinger (NLS) equations on a bounded interval is developed. The main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine-scale, random fluctuations, or radiation. The model is derived from equilibrium statistical mechanics by using a mean-field approximation of the conserved Hamiltonian and particle number (L 2 norm squared) for finite-dimensional spectral truncations of the NLS dynamics. The continuum limits of these approximated statistical equilibrium ensembles on finite-dimensional phase spaces are analyzed, holding the energy and particle number at fixed, finite values. The analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the NLS ground state equation, and that the remaining energy resides in Gaussian fluctuations equipartitioned over wavenumbers. Some results of direct numerical integration of the NLS equation are included to validate empirically these properties of the most probable states for the statistical model. Moreover, a theoretical justification of the mean-field approximation is given, in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit.

Physical Review E, 2013

The recently developed technique combining the weak-coupling limit with the Floquet formalism is applied to a model of a two-level atom driven by a strong laser field and weakly coupled to heat baths. First, the case of a single electromagnetic bath at zero temperature is discussed and the formula for resonance fluorescence is derived. The expression describes the well-known Mollow triplet, but its details differ from the standard ones based on additional simplifying assumptions. The second example describes the case of two thermal reservoirs: an electromagnetic one at finite temperature and the second dephasing one, which can be realized as a phononic or buffer gas reservoir. It is shown using the developed thermodynamical approach that the latter system can work in two regimes depending on the detuning sign: a heat pump transporting heat from the dephasing reservoir to an electromagnetic bath or heating both, always at the expense of work supplied by the laser field.

Physical Review Letters, 2013

We introduce a versatile method to compute electronic steady-state properties of strongly correlated extended quantum systems out of equilibrium. The approach is based on dynamical mean-field theory (DMFT), in which the original system is mapped onto an auxiliary nonequilibrium impurity problem imbedded in a Markovian environment. The steady-state Green's function of the auxiliary system is solved by full diagonalization of the corresponding Lindblad equation. The approach can be regarded as the nontrivial extension of the exact-diagonalization-based DMFT to the nonequilibrium case. As a first application, we consider an interacting Hubbard layer attached to two metallic leads and present results for the steady-state current and the nonequilibrium density of states.

Rendiconti del Seminario Matematico e Fisico di Milano, 1996

Stochastic differential equations for processes with values in Hilbert spaces are now largely used in the quantum theory of open systems. In this work we present a class of such equations and discuss their main properties; moreover, we explain how they are derived from purely quantum mechanical models, where the dynamics is represented by a unitary evolution in a Hilbert space, and how they are related to the theory of continual measurements. An essential tool is an isomorphism between the bosonic Fock space and the Wiener space, which allows to connect certain quantum objects with probabilistic ones. * To appear in Rendiconti del Seminario Matematico e Fisico di Milano