Non classical correlations of two interacting qubits coupled to independent reservoirs R. Migliore CNR-INFM, Research Unit CNISM of Palermo Dipartimento di Scienze Fisiche ed Astronomiche Università di Palermo, Italy M. Scala, M.A. Jivulescu, M. Guccione, L.L. Sánchez-Soto, A. Messina dsfa CNR-INFM

  The system and its Hamiltonian   Two coupled qubits interacting with two independent bosonic baths.   Counter-rotating terms are present in the interaction Hamiltonian describing the qubit-qubit coupling.   Microscopic derivation of the Markovian master equation in the weak damping limit   The system dynamics   Stationary state at general temperatures   Behavior of the entanglement at zero temperature presenting the phenomena of sudden death and sudden birth as well as the presence of stationary entanglement for long times.   Effect of nonzero temperature on the entanglement dynamics.   Conclusive remarks OUTLINE M. Scala et al., J. Phys A: Math. and Theor. 41, (2008); M. Scala et al., in preparation. R. Migliore et al., Phys. Stat. Sol. B 246, 1013 (2009).

THE SYSTEM AND ITS HAMILTONIAN We consider two spin-1/2 like interacting systems (qubits) coupled with their own (uncorrelated) bosonic environments Reservoir 2 T2T2 22 Reservoir 1 T1T1 11 Counter-rotating terms included in the interaction play a central role in the dynamics of the entanglement between the two systems. (1)

The four-level energy spectrum of the bipartite system  II II II By exploiting the fact that the Hamiltonian H S, in the uncoupled basis {|00>; |11>; |10>; |01>}, is block diagonal, it is straightforward to find its energy spectrum and the relative eigenstates & Here The allowed transitions here sketched are characterized by the Bohr frequencies

Microscopic derivation of the Markovian master equation in the weak damping limit   From this Hamiltonian model, performing both the Born-Markov and the rotating wave approximation, we find that the evolution of the two two-state systems is described by the equation: and the Kubo-Martin-Schwinger relation holds. Assuming that the two reservoirs are independent and that both are in a thermal state, with temperatures T 1 and T 2 respectively, one has: with i.e. when All the jump processes involve transitions between dressed states of the open system under study, described by the following operators relative to the coupling of the first (second) qubit with its own reservoir: for the transitions b  a and d  c for the transitions c  a and d  b (2)

Rearranging the ME, we obtain a system of differential coupled equations, describing the time evolution of the populations of the dressed states |a>, | b>, | c> and | d> namely and of the corresponding coherences:

The decay rates ( i  I,II ) and the cross terms are given by: When the temperatures of the two reservoirs T 1 and T 2 are both zero, the rates and vanish. Physically this means that there is no possibility to create excitations in the bipartite system due to the interaction with the reservoirs. The excitation rates and the cross terms are obtained by substituting, with the corresponding quantities

Analytical solution I: the stationary state We prove the existence of the following stationary solution, by imposing and the normalization condition there exists a stationary entanglement traceable back to the presence of counter-rotating terms in the interaction Hamiltonian describing the coupling between the two two-state systems. when T 1 = T 2 = 0 K   aa,ST = 1 We note, that the analysis of this Hamiltonian model allows to bring to the light the fact that, when T 1  T 2, it is wrong to apply the principle of detailed balance in order to derive the stationary solution of the master equation of the system. This is due to the fact that the excitation rates are not related to the corresponding rates from the usual Boltzmann factor. We note, that the analysis of this Hamiltonian model allows to bring to the light the fact that, when T 1  T 2, it is wrong to apply the principle of detailed balance in order to derive the stationary solution of the master equation of the system. This is due to the fact that the excitation rates are not related to the corresponding rates from the usual Boltzmann factor. Initial state: |11> Linear entropy S L (  ) =1-Tr[  2 ]

Analytical solution II: dynamics of the entanglement at zero temperature Exploiting the knowledge of the master equation solutions (when T 1 =T 2 =0 K) we determine the amount of entanglement between the two-state systems and the entanglement dynamics, by analyzing the concurrence, a function introduced by Wootters and defined as : eigenvalues of the matrix no entanglement maximal entanglement

RESULTS I: INITIAL STATE |01> concurrence  1 = 2 =10 flat spectrum ==0.1 T 1 =T 2 = 0 K Damped Rabi-oscillations + stationary entanglement due to the presence of counter-rotating terms in the  x (1)  x (2) interaction Hamiltonian which are responsible for the presence of the component |11> in the ground state |a> t

RESULTS II: INITIAL STATE |11> Birth and death of the entanglement  1 = 2 =10 ; flat spectrum: ==0.1 ; T 1 =T 2 = 0 K Sudden death and birth of entanglement, phenomena which are well known in the recent literature on dissipative two-qubits dynamics. Note that also in this case the stationary entanglement is due to the structure of the qubit-qubit interaction Hamiltonian and not to the presence of a common environment. concurrence t

EFFECT OF NONZERO TEMPERATURES I By means of Laplace transforms, it is possible to obtain the solution of the master equation at generic T 1 and T 2 initial state |01> Effect on the oscillations initial state |11> Effect on the short-time dynamics T 1 =T 2 =10 mK T 1 =T 2 =20 mK T 1 =T 2 =30 mK The oscillatory phenomena, indicating coherence, are robust enough with respect to the temperature increasing concurrence t (sec)  1 = 2 =10GHz; ohmic spectrum:  1 = 2 =0.1 T 1 =T 2 =10 mK T 1 =T 2 =20 mK T 1 =T 2 =30 mK

Initial state |11>: Effect on the long-time dynamics concurrence T 1 =T 2 =5 mK T 1 =T 2 =10 mK T 1 =T 2 =15 mK The stationary entanglement due to non-resonant interaction is less robust with respect to temperature: anyway it is experimentally detectable. t (sec) EFFECT OF NONZERO TEMPERATURES II The larger the coupling constant λ the larger the amount of stationary entanglement.

SUMMING UP  Derivation of the master equation for two coupled qubits interacting with two independent reservoirs.  Stationary solution: in general, for T 1  T 2, the detailed balance principle is not satisfied.  Dynamics at zero temperature: Damped Rabi oscillations and stationary entanglement due to the counter-rotating terms in the qubit-qubit interaction Hamiltonian.  Initial condition |11>: phenomena of sudden death and birth of entanglement.  Nonzero values for the reservoirs temperatures may destroy stationary entanglement, which is anyway visible for a reasonable temperature range. WORK IN PROGRESS…  Decoherence in superconducting systems:  Structured environments  Non-markovian dynamics WORK IN PROGRESS…  Decoherence in superconducting systems:  Structured environments  Non-markovian dynamics M. Scala et al., J. Phys A: Math. and Theor. 41, (2008); M. Scala et al., in preparation. R. Migliore et al., Phys. Stat. Sol. B 246, 1013 (2009).