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Jaynes-Cummings model of a pair of two-level atoms with dipole interaction

N. A. Enaki, V. I. Koroli, Sergiu Bazgan

Published Jan 1, 2015• 0 citations
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Abstract

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].

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TWO-PHOTON QUANTUM-STATISTICAL PROPERTIES OF THE SINGLE-MODE CAVITY FIELD INTERACTING WITH A PAIR OF COLD ATOMS

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Photon spectra and statistics in generalized Jaynes-Cummings model with photon losses and atom motion

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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.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Mar 3, 2006•0 citations
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].
We study interaction between a pair of indistinguishable two-level atoms and the single-mode cavity field. It is supposed that the pair of two-level atoms is laser cooled and trapped into the ground vibrational state, in which vibrational quantum number ⟨nv⟩=0\langle n_v\rangle =0⟨nv​⟩=0 . Two Jaynes–Cummings models are investigated. One is the Jaynes–Cummings model with intensity-dependent coupling and the another is the two-photon Jaynes–Cummings model of a pair of indistinguishable two-level atoms. It should be noted that in the present model, at initial moment t=0t=0t=0 , in intensity-dependent Jaynes–Cummings model the cavity field is prepared in Holstein–Primakoff SU(1,1) CS, while in two-photon Jaynes–Cummings model it is prepared in the squeezed vacuum state. Moreover, at initial moment t=0t=0t=0 , pair of atoms is supposed to be in the first excited state ∣e1⟩\vert e_1\rangle ∣e1​⟩ in both models. By using exact analytical solutions for state-vectors of the coupled atom-field systems amplitude-squared squeezing of the quantized cavity field is examined as a function of the ∣ξ∣\vert \xi \vert ∣ξ∣ parameter. In this situation, in both models higher-order squeezing has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the second-order squeezing.
The interaction between the pair of cold two-level atoms and the single-mode cavity field is investigated. The two-level atoms in the pair are supposed to be indistinguishable. This problem generalizes the two-photon Jaynes-Cummings model of a single two-level atom interacting with the squeezed vacuum. The model of the pair of indistinguishable two-level atoms is equivalent to the problem of the equidistant three-level radiator with equal dipole moment matrix transition elements between the adjacent energy levels. Supposing that at the initial moment the field is in the squeezed vacuum state we obtain the exact analytical solution for the atom-field state-vector. By using this solution the quantum-statistical and squeezing properties of the radiation field are investigated. The obtained results are compared with those for the single two-level atom system. We observe that in the model of the pair of cold two-level atoms the exact periodicity of the squeezing revivals is violated by the analogy with the single two-level atom one.
An exact solution is found for the collective model of two identical two-level atoms that resonantly interact with a two-mode quantum electromagnetic field in an ideal cavity via two-photon nondegenerate transitions. In the case under study, at the initial moment, both field modes are in the coherent state and atoms are in the excited state. The time dependences of the atomic probabilities, the mean number of photons in the modes, and the statistics and squeezing of the photon modes are studied based on the exact solution.
An exact analytical approach to study the time evolution of the interaction between a system of N two-level atoms with a single-mode cavity field is proposed. Supposing that at the initial moment the system of radiators is prepared in a superposition of states, the wave functions of the atom–field interaction for three and four atoms are presented exactly. In the case of an odd number of atoms in the cavity, the quantum Rabi frequencies have non-zero values; otherwise one zero value of the quantum Rabi frequency is characteristic of the even number of atoms. It was shown that with an increase of the odd number of atoms the non-zero quantum Rabi frequencies in the system increase too. It was established that a single atom has one quantum Rabi frequency, three atoms have two frequencies and five atoms have three. For the even number of atoms, a zero value of eigenstates of radiators in the interaction with a single cavity mode is observed. In this case the number of collective levels in the system is odd and the trapping states of atoms in the cavity are possible. From analytical solutions it follows that it is possible to realize the micro-maser regimes for three atoms choosing the flying time for which the radiators pass totally in the ground state. The dependence of the flying time on the number of atoms is established as well as the cavity state for which the atomic ensemble comes back to its initial state. The behavior of quantum fluctuations of photon numbers and atomic inversion during collective absorption and radiation is exactly analytically and numerically studied. The exact quantum solutions for these qubits, formed from two-level atoms, can be used in quantum processing of information encapsuled in the initial states of the atomic ensemble and the cavity field.
Photon statistics in some fully quantized models of the interaction of a two-level atom with a single-mode radiation field have been studied using the operator equations of motion. Expressions for the photon number distribution and the mean photon number are presented for various initial conditions. It is found that the mean photon number may show decays and revivals of coherence similar to those of the atomic inversion in the coherent-state Jaynes-Cummings model. Application of these models to the study of multiphoton laser, absorption, and emission processes is also discussed.
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.