Entanglement of a qubit coupled to a resonator in the adiabatic regime

We discuss the ground state entanglement of a bi-partite system, composed by a qubit strongly interacting with an oscillator mode, as a function of the coupling strength, the transition frequency and the level asymmetry of the qubit. This is done in the adiabatic regime in which the time evolution of the qubit is much faster than the oscillator one. Within the adiabatic approximation, we obtain a complete characterization of the ground state properties of the system and of its entanglement content.

Figure 1

The lower adiabatic potential for $D=10$ and $\alpha =2$. The dashed line refers to the symmetric, $W=0$, case (dashed line), while the solid line refers to $W=1$. The case of frozen qubit $(W=D=0)$ would have given a pair of independent parabolas instead of the adiabatic potentials $U_{l,u}$ of Eq. (10).

Figure 2

Normalized ground state wave function for the oscillator in the lower adiabatic potential, for $D=10$ and $\alpha =2$ and with $W=0$ (dashed line) and $W=0.1$ (solid line).

Figure 3

The dependence of the ground state expectation value $b_x=⟨{\sigma }_x⟩$ as a function of the parameter $\alpha$, for $D=10$.

Figure 4

The $z$ component of the Bloch vector as a function of $\alpha$, for $D=10$.

Figure 5

The tangle $\tau$ as a function of $\alpha$ for $D=10$. Different curves, corresponding to different values of $W$, indicate the entanglement progressively decreases with increasing the asymmetry.

Figure 6

The tangle $\tau$ as a function of $\alpha$ for $W=0.1$ and different values of $D$.

Figure 7

The tangle as a function of $\alpha$ in the symmetric case $W=0$ for different values of the qubit tunneling amplitude $D$. One can appreciate that the result of Eq. (32) is indeed reached asymptotically.