Nonlinearity-induced entanglement stability in a qubit-oscillator system

We consider a system composed of a qubit interacting with a quartic (undriven) nonlinear oscillator (NLO) through a conditional displacement Hamiltonian. We show that even a modest nonlinearity can enhance and stabilize the quantum entanglement dynamically generated between the qubit and the NLO. In contrast to the linear case, in which the entanglement is known to oscillate periodically between zero and its maximal value, the nonlinearity suppresses the dynamical decay of the entanglement once it is established. While the entanglement generation is due to the conditional displacements, as noted in several works before, the suppression of its decay is related to the presence of squeezing and other complex processes induced by two- and four-phonon interactions. Finally, we solve the respective Markovian master equation, showing that the previous features are preserved also when the system is open.

Figure 1

Time dependence of the entanglement negativity for different values of $k$ in the absence of nonlinearities in the NLO potential. Starting from the separated state given by Eq. (11) ($\alpha =2$) the system becomes entangled ($0<t<2\pi$), reaching a maximum at $t=\pi$. Finally, at $t=2\pi$ the system returns to its original state, thus the negativity is zero. In this and all the figures, $t$ is a scaled time, corresponding to the actual time multiplied by ${\omega }_o$.

Figure 2

Dynamics of the reduced density operator for the qubit state in the Bloch Sphere (top view) with $k=0.5,\alpha =2,\delta =0$. We can see that in the absence of nonlinearities the qubit dynamics remains periodically for the whole evolution. Here, the leftmost (lowermost) point of the $x$ axis ($y$ axis) represents the state $|+\rangle \left(\frac{1}{\sqrt{2}}\left(\right|\uparrow \rangle \right)+i|\downarrow \rangle )$.

Figure 3

The figure shows the Wigner function $W(x,y)$ of the reduced density operator for the NLO associated with the Eq. (12). The single peak of the initial coherent states separates into two components, each associated with a different qubit eigenstate. Specifically, the solid line arrow (dashed line arrow) indicates the component associated with $|\downarrow \rangle (|\uparrow \rangle )$. The Wigner function is defined as $W(x,y)=\frac{1}{\pi \hslash }{\int }_{-\infty }^{\infty }\langle x+x^{\prime }|{\widehat{\rho }}_{\text{osc}}|x-x^{\prime }\rangle e^{-2iy{x}^{\prime }/\hslash }d{x}^{\prime }$ and the axis are given accordingly.

Figure 4

(a) negativity as function of time $t$ for $k=1/100$ and $\delta =1/1000$ $(\alpha =2)$. We compare the entanglement using an analytical expression (solid line) [Eq. (25)] and the numerical one (dashed line) using Eq. (22). The dotted line is the evolution in the absence of nonlinearity. As we can see the inclusion of the nonlinear term increases the entanglement reaching a time-plateau or stabilization zone. (b) We compare the analytical expression to the numerical solution for the same set of parameters for larger times.

Figure 5

(a) The top figure shows the Wigner function $\left[W\right(x,y\left)\right]$ for the oscillator state for $t=0,\pi /2,\pi ,3\pi /2,2\pi$. The state considered here is the one in Eq. (25). As we can see, in the first cycle, due to the small values of $k$ and $\delta$ both components of the qubit remains superposed during all times, showing squeezing in the quadratures $\{x,y\}$. (b) The bottom figure shows the state at $t=50\pi$, as we see the state becomes complex evidencing negatives values during the dynamics.

Figure 6

The main plot shows the normalized uncertainty relation $\Delta {\widehat{x}}_r\Delta {\widehat{y}}_r/\Delta {\widehat{x}}_0\Delta {\widehat{y}}_0$ for the first cycle in the weak-coupling regime. The sudden increasing in time shows the short period in which the squeezing remains valid. The subplot shows the individual normalized variance, the quadrature $x_r$ ($y_r$) becomes linearly increasing (decreasing).

Figure 7

In the main plot we show the numerical results for the negativity for $k=0.5,\alpha =2$, and varying $\delta$. In contrast to $\{k,\delta \}\ll 1$, here we have achieved a higher entanglement as well as a faster stabilization zone. In the subplot we compare the entanglement generated for $k=0.5,\alpha =2,\delta =1/100$ using an approximated Hamiltonian, the solid line is for a full Hamiltonian without approximation. The dashed line considers only number states in the quartic potential, and finally the dotted line considers up to four-phonon transitions in the quartic potential.

Figure 8

Here we provide a pictorial explanation for the entanglement enhancement for $k=0.5,\alpha =2,\delta =1/100$ at different times $t=2\pi ,4\pi ,6\pi ,10\pi ,15\pi$. In panels (a) and (b), we plot the Wigner function for the oscillator state for each spin component $W_{\uparrow ,\downarrow }(x,y)$. In panel (c), we show the product between $W_{\uparrow }(x,y)$ and $W_{\downarrow }(x,y)$. The number $w_p$ in the corner corresponds to the integration of the product over all the $xy$ phase space [Eq. (30)]. The small overlap between $W_{\uparrow }(x,y)$ and $W_{\downarrow }(x,y)$ then shows that the states corresponding to the latter are quasi-orthogonal, thus allowing for the establishment of maximal entanglement.

Figure 9

We illustrate the reduced density qubit operator in the Bloch sphere (top view) for two cycles $0\le t\le 4\pi$. The qubit shows a strong precession in the dynamics.

Figure 10

Negativity for the open quantum system for different values of the dissipation ratio $\gamma$ and different values of $\delta$. Here $k=0.5,\alpha =2$.