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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Exact solution of the Jaynes-Cummings model with relaxation

Alexander V. Gorokhov, Ilya E. Sinaiski

Published Jul 13, 2004• 1 citation
Publisher

Abstract

Exact analytical expression for the density matrix of atom + field system is obtained taking into account of the photons relaxation and frequency detuning (Jaynes-Cummings model with m-quantum transitions and dissipation). The photon mean number, atom population inversion and Fano Q - factor depending on initial states of the field and parameters of the one-atom maser are investigated.

DOI
10.1117/12.578911

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Exact analytical expression for the density matrix of atom + field system is obtained taking into account of the photons relaxation and frequency detuning (Jaynes-Cummings model with m-quantum transitions and dissipation). The photon mean number, atom population inversion and Fano Q - factor depending on initial states of the field and parameters of the one-atom maser are investigated.
The Jaynes-Cummings model (JCM) of two-level atom interacting with the photon mode in ideal cavity plays an essential role in modern quantum optics. In previous papers 1,2 an exact form of density matrix of the JCM with fixed atom position and photons dissipation was found. Here, taking into account the classical motion of the atom through the cavity, it is considered a case of nonideal cavity with zero temperature. We have obtained an exact expression for density matrix and calculated photon spectra and spectra of the mean number of photons in a cavity and and time dependencies of some values relevant for the one-atom maser theory.
An expression for the population inversion in the Jaynes-Cummings model and the most general initial state of the atom-field system is presented. States that maintain the same level of population inversion when the atom and the field are initially decoupled are found. When the atom is prepared in a coherent superposition of its states and the field is in an eigenstate of the Susskind-Glogower phase operator, for a certain choice of the relative phase between the atomic dipole and the field, coherent trapping occurs. The evolution of the squeezing of the field for these states is given, and the dependence on the phase and on the mean photon number of the field is studied.
Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density operator that belongs to the Jaynes-Cummings model with cavity damping. The detuning parameter is set equal to zero and the optical resonator does not contain any thermal photons. It is shown that the master equation for the density operator can be replaced by two algebraic recursion relations for vectors of dimension 2 and 4. These vectors are built up from suitably chosen matrix elements of the density operator. By performing an iterative procedure, the exact solution for each matrix element is found in the form of an infinite series. We demonstrate that all series are convergent and discuss how they can be truncated when carrying out numerical work. With the help of techniques from function theory, it is proved that our solutions respect the following conditions on the density operator: conservation of trace, Hermiticity, convergence to the initial state for small times, and convergence to the ground state for large times. We compute some matrix elements of the density operator for the case of weak damping and find that their analytic structure becomes much simpler. Finally, it is shown that the exact atomic density matrix converges to the state of maximum von Neumann entropy if the time, the square of the initial electromagnetic energy density, and the inverse of the cavity-damping parameter tend to infinity equally fast. The initial condition for the atom can be chosen freely, whereas the field may start from either a coherent or a photon-number state.
We have assumed that atoms are placed in given positions of the standing wave of cavity. Based on modern optical achievements it is possible to localize a small number of emitters (atoms, ions, molecules and even nuclei) in various geometrical systems like optical cavities, quantum dots, optical lattices and dielectric or disordered media. It is supposed that at initial moment t = 0 pair of atoms is prepared in superposition of ground state |g⟩, first excited |e1⟩, second |e2⟩ and third excited |e3⟩ states and quantized cavity field in coherent state |ξ⟩. By using exact analytical solution for statevector of the coupled atom-field system found with the help of Schrodinger equation quantum-statistical properties of the quantized cavity field are examined as a function of the |ξ| parameter. Exact analytical solutions for the atomic population inversion ⟨Sz⟩, mean photon number ⟨n⟩ and their fluctuations σ are obtained by using this solution. In this situation quantum statistics of cavity field has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the micromaser model [2] and two-photon JCM of a threelevel atom [3].