Entanglement generation and protection by detuning modulation

We introduce a protocol for steady-state entanglement generation and protection based on detuning modulation in the dissipative interaction between a two-qubit system and a bosonic mode. The protocol is a global-addressing scheme which only requires control over the system as a whole. We describe a postselection procedure to project the register state onto a subspace of maximally entangled states. We also outline how our proposal can be implemented in a circuit-quantum electrodynamics setup.

Figure 1

Two qubits interact with the same cavity mode (in contact with a thermal reservoir, decay rate $\kappa$) acting as a common bus. An inhomogeneous global potential, setting a time-dependent detuning is also shown.

Figure 2

(Color online) Fidelity of the two-qubit state $\varrho$ for the full-resonant case with the symmetric state ${\mid s⟩}_{12}$ (solid line) and the antisymmetric one ${\mid a⟩}_{12}$ (identical to zero) against the dimensionless interaction time $\tau =g^{2}t∕\kappa$.

Figure 3

(Color online) $\mathcal{C}\left(\rho \right)$ against $\tau$ for the initial state ${\mid \uparrow ⟩}_{12}$ and $\Delta \left(\tau \right)=10\Theta (\tau -{\tau }_0)$. We have considered ${\tau }_0=1.5,2.5,3.5$ (dashed, solid, and dotted line, respectively). The inset shows the behavior of the detuning functions.

Figure 4

(Color online) Fidelities ${\mathcal{F}}_{s,a}$ for a detuning modulation strategy based on a Heaviside function with ${\tau }_0=2.5$. The solid line is for ${\mathcal{F}}_s$, the dotted one for ${\mathcal{F}}_a$. At $\tau ={\tau }_0$, an antisymmetric state component is developed. For large $\tau$, both the antisymmetric and symmetric state fidelities are stabilized.

Figure 5

(Color online) Comparison between the concurrence obtained by using a Heaviside function $10\Theta (\tau -2.5)$ (dashed line) and the smooth detuning function $10{(1+e^{6(2.5-\tau )})}^{-1∕2}$ (solid line). The inset shows the time behavior of the detuning functions.

Figure 6

(Color online) Concurrence (red curve, left vertical scale) superimposed on the detuning function ${\Delta }_{sw}\left(\tau \right)$ (black curve, right scale) against $\tau$ for the qubits initially prepared in the superradiant state ${\mid \uparrow ⟩}_{12}$. In this plot we have considered $N=16$. No entanglement is ever present between the qubits.

Figure 7

(Color online) Concurrence against $\tau$ for a modulated detuning with $g=0.3\kappa$, $\gamma ={10}^{-3}\kappa$, and $\overline{n}=0.06$. The time axis has been extended in order to show that a concurrence close to 0.2 is present for $\tau$ up to 100. The slow entanglement decay is due to the presence of a nonzero $\gamma$, while the smaller steady-state value is largely due to the thermal nature of the bath $(\overline{n}\ne 0)$.

Figure 8

(Color online) Comparison between the entanglement in the postselected state with and without the detuning-modulation protocol (solid and dashed curves, respectively). A Heaviside modulation is considered in the solid line case.