Steady-state entanglement of harmonic oscillators via dissipation in a single superconducting artificial atom

The correlated emission lasing (CEL) is experimentally demonstrated in harmonic oscillators coupled via a single three-level artificial atom [Phys. Rev. Lett. 115, 223603 (2015)] in which two-mode entanglement only exists in a certain time period when the harmonic oscillators are resonant with the atomic transitions. Here we examine this system and show that it is possible to obtain the steady-state entanglement when the two harmonic oscillators are resonant with Rabi sidebands. Applying dressed atomic states and Bogoliubov-mode transformation, we obtain the analytical results of the variance sum of a pair of Einstein-Podolsky-Rosen (EPR)-like operators. The stable entanglement originates from the dissipation process of the Bogoliubov modes because the atomic system can act as a reservoir in dressed state representation. We also show that the entanglement is robust against the dephasing rates of the superconducing atom, which is expected to have important applications in quantum information processing.

Figure 1

(a) The illustration of a simplified circuit model in which the artificial atom couples to a transmission line resonator (TLR) via capacitance $C_c$. Two harmonic oscillators with frequencies ${\nu }_1$ and ${\nu }_2$ are effectively coupled with the atom via capacitances $C_1$ and $C_2$, respectively. (b) The simplified level structure of the system. The two oscillators couple to the two transitions $|3\rangle \to |2\rangle$ and $|2\rangle \to |1\rangle$ with effective coupling constants $g_1$ and $g_2$, respectively. A classical microwave field is applied to the transition $|3\rangle \to |1\rangle$. $\mathrm{\Omega }$ represents the Rabi frequency of the pumping field and ${\mathrm{\Delta }}_{jk}(j,k=1,2,3,j>k)$ are corresponding detunings.

Figure 2

The dressed state population evolution as a function of normalized detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ for two cases: (a) ${\gamma }_{31}={\gamma }_{21}={\gamma }_{33}={\gamma }_{22}=\gamma$ and (b) ${\gamma }_{31}=2\gamma ;{\gamma }_{21}=0.2\gamma ;{\gamma }_{33}={\gamma }_{22}=\gamma$.

Figure 3

The variance sum $V_1$ and $V_2$ of two Bogoliubov modes versus the normalized detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ for (a) $\mathrm{\Delta }<0$; (b) $\mathrm{\Delta }>0$ by choosing $\kappa =0.1\gamma$ (dotted line), $0.05\gamma$ (dashed line), $0.01\gamma$ (solid line), $g_1=g_2=\gamma$. The other parameters in (a), (b) are the same as those in Figs. 2 and 2, respectively.

Figure 4

The dephasing effect of the levels $|3\rangle$ and $|2\rangle$ on the steady-state variance sum $V_{1,2}$ for (a) $\mathrm{\Delta }<0$; (b) $\mathrm{\Delta }>0$ by choosing ${\gamma }_{33}={\gamma }_{22}=0.1\gamma$ (solid line), $\gamma$ (dashed line), $g_1=g_2=\gamma$. Other parameters in (a), (b) are the same as those in Figs. 2 and 2, respectively.

Figure 5

The dressed state population evolution as a function of the normalized detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ by choosing different dephasing rates ${\gamma }_{22}$ and ${\gamma }_{33}$. In case (i), the dephasing rates are chosen as ${\gamma }_{33}={\gamma }_{22}=0.1\gamma$ and in (ii) ${\gamma }_{33}={\gamma }_{22}=\gamma$. The other parameters are the same as those in Fig. 2.

Figure 6

The variance sum $V_{1,2}$ as a function of the normalized detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ by choosing the experimental parameters as ${\gamma }_{31}=2\pi \times 8 \mathrm{MHz},{\gamma }_{21}=2\pi \times 1.5\mathrm{MHz}, {\gamma }_{32}=2\pi \times 3$ MHz, ${\gamma }_{22}=2\pi \times 6\mathrm{MHz}, {\gamma }_{33}=2\pi \times 4\mathrm{MHz}, \kappa =2\pi \times 0.63\mathrm{MHz}, g_1=2\pi \times 90$ MHz, $g_2=2\pi \times 78\mathrm{MHz}$.

Figure 7

The variance sums $V_{1,2}$ of the Bogoliubov modes $b_{1,2}$ and the variance sums $Q_{1,2}$ of the original modes $a_{1,2}$ for (a) $\mathrm{\Delta }<0$ and (b) $\mathrm{\Delta }>0$ as a function of the normalized detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ by choosing $\kappa =0.1\gamma$. Other parameters are the same as those in Fig. 3

Figure 8

The output correlation spectra for (a) $Q_1^{\mathrm{out}}\left(\omega \right)$ and (b) $Q_2^{\mathrm{out}}\left(\omega \right)$ as a function of transformation frequency $\omega$. The normalized detuning in (a) is chosen as $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}=-1$ and in (b) $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}=1$. The cavity decay rates are chosen as $\kappa =0.1\gamma$ (solid line), $\kappa =0.05\gamma$ (dashed line), and $\kappa =0.01\gamma$ (dotted line). Other parameters are the same as those in Fig. 2.