The effects of vacuum fluctuations on teleportation of quantum Fisher information

The teleported quantum Fisher information of the phase parameter of atomic state is studied in consideration of vacuum fluctuations. Our results show that the teleported information is determined by the wavelength of the atoms as well as the distance of teleportation. When the wavelength of the atoms is much smaller than the teleportation distance, the teleported information decays with time and the decay rates are determined by the spontaneous emission rate of the atoms. However, when the wavelength of the atoms is much larger than the teleportation distance, the teleported information remains unchanged with time. The information of the phase parameter of atomic state has been absolutely transmitted.


The procedure of teleportation
We consider two atoms were initially prepared in state | 〉 = | 〉 | 〉 − | 〉 | 〉 A ( 1 0 0 1 ) 1 2 1 2 1 2 . Here |1〉 1,2 and |0〉 1,2 are the excited and ground state of the atoms. The atom one was sent to position P(x 0 , y 0 , z 0 ) and the atom two was sent to position Q(x 0 , y 0 , z 0 + L). So the trajectories of the static two atoms can be described as  Here we note (t i , x i , y i , z i ), (i = 1, 2) are the time-space coordinates of the two atoms. At first, let us treat the atoms as closed quantum systems. So the Hamiltonian of the two atoms can be written as . Here we use QFI to measure the information transmission of factor φ from position P to position Q. For a two-level system, the state of the system can be expressed in the Bloch sphere representation as where ω ω ω ω = ( , , ) 3 is the Bloch vector and σ σ σ σ = ( , , ) 3 denotes the Pauli matrices. As a result, the QFI of factor X can be expressed in a simple form 24 So the QFI of factor φ for each of the four states at position Q can be calculated and the QFI for each state has the same result The result is also same as the QFI of the third atomic state |ψ〉 before the teleportation. The QFI of factor φ has been absolutely transmitted from position P to position Q.
However, in reality, every quantum system is open quantum system and the vacuum fluctuations are the unavoidable environment. So the pair of entangled atoms at position P and position Q are also affected by the vacuum fluctuations. As a result, for time τ, the two atoms will not remain in state | 〉 = | 〉 | 〉 − | 〉 | 〉 A ( 1 0 0 1 ) For simplicity, we consider the two-atom system interacting with a bath of fluctuating scalar fields in the Minkowski vacuum. The total Hamiltonian of such a system can then be written as

A F I
Here H A is the Hamiltonian of the two atoms as we has discussed before. H F denotes the Hamiltonian of the scalar fields and the interaction Hamiltonian H I is taken in analogy to the electric dipole interaction in the weak coupling limit as 25 with μ denoting the coupling constant and Φ denotes the strength of scalar field. We use ρ ρ = ⊗ (0) (0) 0 0 tot to denote the initial state, which means the atoms and the environment are uncorrelated at the beginning as supposed. Here |0〉 is the Minkowski vacuum state of the scalar fields, and is the initial state of the two-atom system. In the weak-coupling limit, the reduced dynamic of the two-atom system takes the Kossakowski-Lindblad form [26][27][28] ( ) denote the field correlation functions and P denotes the principal value. Then αβ C ij ( ) can be written explicitly as Allowing for the trajectories (1), the correlation functions can be written as where  denotes a small constant with → + 0  . The Fourier transforms of the above correlation functions are Then the coefficients of the dissipator in the master equation can be obtained directly as ( 1 0 0 1 ), , then a set of equations being used to describe the time evolution of the populations of some matrix elements, which are decoupled from other matrix elements, can be obtained as 29,30 As a result, the state of initially entangled two-atom system at time τ becomes ρ τ ρ τ and the total state of the three atoms before teleportation becomes the product state of |ψ〉 〈 ψ| and ρ τ ρ τ . Now let us perform the Bell measurement at position P. Then, the teleported state becomes one of the four states The results show that causing by the interaction between vacuum fluctuations and the pair of atoms, the teleported QFI of parameter φ decreases with time and the decay rates depend on When the wavelength of the atoms is much smaller than the distance between position P and Q (ΩL → ∞ ), ∆ = Γ A 4 0 . So the decay rates of the QFI depend on the spontaneous emission rate of the atoms. However, when the wavelength of the atoms is much larger than the distance between position P and Q (ΩL → 0), Δ A → 0. The atoms interact with the field modes in a collective and coherent way and the teleported QFI of parameter φ remains sin 2 θ, which means the QFI of factor φ has been absolutely transmitted from position P to position Q in the existence of vacuum fluctuations.
For θ = π/2, the teleported QFI for each measurement has the same result F φ = e −8ΔAτ . We find that the teleported QFI decreases exponentially and the decay rate becomes 8Δ A. We plot Δ A (in the unit of Γ 4 0 ) as the function of ΩL. We find that with the increase of atomic distance in comparing with the wavelength of the atom, the decay rate increases oscillatorily.

Conclusion
In conclusion, we have studied the teleported QFI of phase parameter of atomic state by using two initially entangled static two-level atoms, which coupled to a bath of fluctuating vacuum scalar field in two different places. Our results show that for given time, the teleported QFI is determined by the wavelength of the atoms as well as the distance of teleportation. When the wavelength of the atoms is much smaller than the teleportation distance, the teleported information decays with time and the decay rates are determined by the spontaneous emission rate of the atoms. However, when the wavelength of the atoms is much larger than the teleportation distance, the QFI of the phase parameter of atomic state remains sin 2 θ and the information of the phase parameter of atomic state has been absolutely transmitted.