Entanglement for two qubits interacting with a thermal field Mikhail Mastyugin The XXII International Workshop High Energy Physics and Quantum Field Theory June 24 – July 1, 2015 Samara, Russia

The magnetic flux through the hole superconductor takes discrete values Josephson tunneling contact - two superconductors S 1 and S 2, which separated by a thin dielectric layer Josephson predicted 2 effects: The dependence of the superconducting current through the tunneling barrier of the phase difference at the contact I c – The critical current Feedback voltage at the contact with the derivative of the phase difference time Alternating current oscillates at a frequency Concepts of superconducting qubits

Josephson coupling energy Josephson inductance The charge (Coulomb) energy one-contact flux qubit and his basic state The potential energy U (φ) when β 1 - two minimums Main characteristics of Josephson junctions: Flux qubits:

three-contact flux qubit the potential energy U (φ) in the case of low inductance β << 1 and f = 0, then the potential has two minimums at the points: Standing in the local minima correspond to the two currents Superconducting qubit stream with Josephson junctions :

Two superconducting qubits interact with a superconducting electric "resonator" (LC-circuit) The scheme transitions in a three-level artificial atom Δ-type and effective two-level atom with degenerate two- photon transition. A qubit interact with a superconducting electric "resonator" (LC-circuit), the second qubit is outside the cavity

The Hamiltonian interaction The evolution operator The reduced density matrix Initially, the resonator field is in single-mode thermal field The atoms are in the form of coherent states Influence of atomic coherence and dipole-dipole interaction on the entanglement of two qubits induced by thermal noise

The matrix elements of the evolution operator criterion Perez Horodetskih or "negative"

The initial coherent atomic not entangled states Fig. 1. Time-dependent parameter entanglement for different initial coherent atomic states:, (dashed line) and (solid line). The average number of photons in the mode. The constant dipole-dipole interaction of atoms.

Fig.2. The time dependence of the parameter for the entanglement of incoherent and coherent the initial states of atoms. The first case corresponds to the dashed, and the second - a solid line. The average number of photons in the mode. The constant dipole-dipole interaction of atoms. Phase states of atoms in both cases are the same.

Figure 3. The time dependence of the parameter entanglement for coherent initial states of atoms and different values of the constant dipole-dipole interaction: α = 0 (dashed line) and α = 0,1 (solid line). The average number of photons in the mode Figure 4. The time dependence of the parameter entanglement for coherent initial states of atoms and different values of the relative phase of the atomic states (solid line) and (dashed line). The average number of photons in the mode The constant dipole-dipole interaction of atoms α = 0,1.

The initial coherent atomic entangled states Fig.5. Time-dependent parameter entanglement for initial entangled states (solid line), (dashed line), (dotted). The average number of photons in the mode, constant dipole-dipole interaction α=0,1 and Fig.6. Time-dependent parameter entanglement for initial entangled states (solid line), (dashed line), (dotted). The average number of photons in the mode, constant dipole-dipole interaction α=0,1 and

Hamiltonian starting the atomic density matrix Initially, the resonator field is in single-mode thermal field Atoms are in coherent states eigenfunctions Entanglemen of two supercondacted qubets one of wich is a traped in a cavity

Here Where energy eigenvalues Where

reduced atomic density matrix criterion Perez Horodetskih or "negative"

Fig. 1. The time dependence of the parameter ɛ (t) with and. The initial atomic states: (a), (b) and (c)

Fig. 2. The time dependence of the parameter ɛ (t) for and α = 1. The initial atomic state. Fig. 3. The time dependence of the parameter ɛ (t) for and α = 1. The initial atomic state.