The capacity of transmitting atomic qubit with light

The quantum information transfer between a single photon and a two-level atom is considered as a part of a quantum channel. The channel is a degradable channel even when there are decays of the atomic excited state and the single photon state, as far as the total excitation of the combined initial state does not exceed 1. The single letter formula for quantum capacity is obtained. Recent experiment has realized the coherent transfer of coherent state of light to and from the hyperfine states of an atom trapped within the modes of high finesse optical cavity[1]. This may provide the basic nodes of quantum networks. In the proposal for the implementation of quantum networks[2], each node is a quantum system that stores and locally processes quantum information in quantum bits, atomic internal states with long coherent times serve as these ’stationary’ qubits, exchange of information between the nodes of the network is realized by the transmission of photons (’flying’ qubits) over optical fiber. In the the experimental setting, a ’Λ’ type three-level atom is used. A three-level system can be reduced to a twolevel system adiabatically for far detuning system [2]. Thus a problem of basic interests is the interface of light and two-level atom. We will treat the quantum information transfer process as a quantum channel (or a part of quantum channel) for the first time. The whole process involves transfer of atomic state to field, the optical transmission and transfer quantum state back to atom. The field is restricted to single photon state and in each step of transfer the part that receives quantum state is prepared as ground state or vacuum field state. Field decay and atomic excited state decay are included in our model. The state transfer: The qubit is a two-level atom with ground state |↓〉 and excited state |↑〉. It interacts with a single-mode near-resonant cavity field prepared in vacuum state |0〉. The dynamics of the system is given by the Jaynes-Cumming interaction Hamiltonian [3] (h̄ = 1) Ĥ = ν(a†a+ 1 2 ) + ω 2 σz + g(a σ− + aσ+), (1) where ν is the frequency of the field, ω is the atomic transition frequency between the two levels, the coupling constant between the atom and field is g, a and a† are the annihilation and creation operators of the field, σ− and σ+ are the atomic state flip operators defined as σ− |↑〉 = |↓〉 , σ− |↓〉 = 0, σ+ |↓〉 = |↑〉 , σ+ |↑〉 = 0, σz = σ+σ− − σ−σ+ and σz |↑〉 = |↑〉 , σz |↓〉 = − |↓〉. The evolution operator of the system is Û = exp(−iĤt) = e−iν {cos(Ω̂t) (2) − i Ω̂ sin(Ω̂t)[ ∆ 2 σz + g(a σ− + aσ+)]}, (3) where ξ̂ = a†a + (1 + σz)/2 is the operator of the total number of excitation of the system. ∆ = ω − ν is the detuning, and Ω̂ = √ g2ξ̂ + ∆ 2 4 . For initial states |↓ 0〉 or |↑ 0〉 , the evolved states are Û |↓ 0〉 = e |↓ 0〉 , Û |↑ 0〉 = e−iνt{[cos (Ωt) − i sin(Ωt) ∆ 2Ω ] |↑ 0〉−i sin(Ωt) g Ω |↓ 1〉}, respectively, where Ω = √ g2 + ∆ 2 4 . For a general atomic density matrix ρA = (1− p) |↓〉 〈↓| +r |↓〉 〈↑| +r∗ |↑〉 〈↓|+ p |↑〉 〈↑|, the evolved system state is ÛρA⊗|0〉 〈0| Û †. At some proper time t, the atomic state is traced out, leaving the optical field state ρB = TrA(ÛρA ⊗ |0〉 〈0| Û †). In the photon number basis |0〉 , |1〉, it reads ρB = [ 1− p |h1(t)|], rh1(t) rh1(t), p |h1(t)| ]