The Statistics Properties of Orthogonal Coherent State Interacting with Two-level Atom

Jingqiu Chen (Corresponding author) Institute of Optoelectronics Science and Engineering Huazhong University of Science and Technology Wuhan 430074, China College of Physics Science and Technology Shenzhen University Guangdong 518060, China E-mail: chenjq@szu.edu.cn Qiao Gu International Institute of Biophysics Kaiserslautern100004, Germany Wenda Peng Institute of Optoelectronics, Shenzhen University Guangdong 518060, China Abstract In this paper, the statistics properties of the statistics properties of orthogonal coherent state Interacting with two-level atom have been discussed, the statistics properties of both atom and field have been discussed. A set of conclusion have been gotten that atom inversion exit the collapse,calm,revival and chaotic behavior. The filed exit several unclassical effect: Sub-poisson photon distribution, Antibanching effect and 2 ) ( x ∆ squeezing effect.


Introduction
Jayness-Cummings model(JCM)( F.W. CUMMINGS, 1965) has been studied many times because the relatively realistic way that it represents the quantum physics of a resonant interact ion .It is the simplest fully quantized model of quantum optics, quantum electronics, and resonance physics .From this model one hopes to know more properties of single mode filed interaction with a two level atom.For example, the coherent-state Jaynes-cummings model has been studied by J.H.Eberly, et al (J.H.Eberly, N.B.Narozhny And J.J. Sanchez-Mondragon, 1980).aset of equations characterizing the Jaynes-Cummings model ,which can be used to describe the dynamic and statistical aspect of the system ,have presented by Qiao.Gu(QIAO QU, 2003).
In general, when people study the statistics properties of JCM ,both atom and field should be discussed.
(1)The atomic inversion is given by expectation value of atomic inversion operator z σ .z σ which its scale between -1 and +1 represents the degree of excitation of a two-level system.
Based on the definition the quadrature operators x and p a a x The fluctuations of the radiation field are represented by (QIAO GU, J. ZHANG, 1989).
, the field has squeezed effect,which it is unclassical effect.
The photon statistical properties of the filed if characterized by the Mandel factor can be written by Sub-poisson is a unclassical effect.
The Second-order Coherence degree is defined below: Antibanching -effect is uncalssical effect.
the properity of the coherent state and the squeezing state filed has been discussed (QIAO QU, 2003).The Orthogonal Coherent state (PENG SHI AN, GUO GUANG CAN, 1990) interacting two-level atom has never been heart to report.we will discuss its statistics properties .
The arragement of this paper is as follows.We first introduced general solution of the Jaynes-Cummings Model (QIAO QU, 2003), then we discussed the statistical properties of the orthogonal coherent state interacting with two-level atom

General Solution of the Jaynes-Cummings Model
A two-level atom description is valid in Fig. 1. if the two atomic levels involved are resonant or nearly resnant with driving field ,while all other levels are highly detuned.
The two-level atom is characterized by the ground state b and an excited state a .
The Hamiltonian system of Jaynes-Cummings model is where a and + a are annihilation and creation operators of the field mode , z σ is the atomic inversion operator and ± σ are the atomic raising and lowering operators .ω and 0 ω are the frequencies of the field mode and of the two level atom, repectively, g is a coupling constant ,and h is Planck's constant divied by π 2 . where We can solve the eigenequation of H: where E and φ represent the eigenvalue and the corresponding eigenstate.We assume that φ be written as a Eq.(2.5) display a set of the linear and homogeneous equations on A and B,and the necessary and sufficient condition for occurring of the non-zero solution is that the coefficient determinant is equal to zero, its results is the eigenvalues where being the detuning.substituting (2.6)into(2.5)andusing(2.4),weobtain the coefficients (2.7) where n θ is defined by (2.9) We have then the eigenstates Which are usually called "dresses states".we observe that There is another states that is not included,namely.no photon in the field with atom at the downer state b g , 0 = φ (2.12)It is the lowest energy and is called ground state.The corresponding eigenvalue g E is calculated from the eigenequation Each of the dressed states ± n φ is orthogonal to the ground state g φ ,and all of them are complete so that 1 ) ( We now consider an arbitrary field-atom system described by a time dependent state vector Assume that the system is initially in state ) 0 ( Φ , which can be expended by using the completeness relation (2.14)

Orthogonal Coherent State Interact with Two-Level Atom
For convenient our work, we briefly review the orthogonal coherent state .theorthogonal coherent state is one of the eigenstates of the operator of square of annihilation 2 a (PENG SHI AN, GUO GUANG CAN, 1990).
Because coherent state α and the α − are not orthogonal .So Peng Shi An et al (WILLIAM H.LOUISELLl, 1982) by use the Gram-Schmidt method, gained a new state By representation of number state we can get that: We now consider a system with initial state that atom is in the upper level a .and the field is orthogonal coherent state describing by (3.2).so the initial state can be written as: is the probability amplitude of finding n photons in thermal equilibrium state. After Substitute (3.15) into (3.16),using (2.2).
We get the result (QIAO QU, 2003): As the same process, we can get: 20) Eg. (2.20) displays a general solution [5]of the Jaynes_Cummings model in terms of the dressed states Figure 1. a two-level atom model