Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator

Using method of quantum trajectories we study the behavior of two identical or different superconducting qubits coupled to a quantum dissipative driven resonator. Above a critical coupling strength the qubit rotations become synchronized with the driving field phase and their evolution becomes entangled even if two qubits may significantly differ from one another. Such entangled qubits can radiate entangled photons that opens new opportunities for entangled wireless communication in a microwave range.

Figure 1

(Color online) Spectral density $S\left(\nu \right)$ of two driven qubits as a function of system parameters for identical ${\Omega }_1/{\omega }_0={\Omega }_2/{\omega }_0=1.2$ (top row) and different ${\Omega }_1/{\omega }_0=1.1$, ${\Omega }_2/{\omega }_0=1.2$ (bottom row) qubits: (a) and (b) $n_p=15$, $\omega /{\omega }_0=1$; (c) and (d) $g=0.04$, $\omega /{\omega }_0=1$; and (e) and (f) $n_p=15$, $g=0.04$. Here and below $\lambda /{\omega }_0=0.02$.

Figure 2

(Color online) The average quantities of number of photons in the resonator $⟨n⟩$ (top), total spin polarization $S_z$ (middle), and the concurrence of two qubits $C$ (bottom) vs the driving frequency $\omega$. Left: identical qubits with ${\Omega }_1/{\omega }_0={\Omega }_2/{\omega }_0=1.2$, right: different qubits with ${\Omega }_1/{\omega }_0=1.1$, ${\Omega }_2/{\omega }_0=1.2$. Other parameters are $\lambda /{\omega }_0=0.02$, $n_p=15$, and $g=0.04$. Symbols mark the values of the total spin $S_z$: $S_z>0.2$ (red/gray $+$), $\left|S_z\right|\le 0.2$ (black dots), and $S_z<-0.2$ (blue/black $\times$). Dashed curves on top panels show the resonance dependence $n_{S_z}\left(\omega \right)$ (see text) with the resonance shift $\Delta {\omega }_{S_z}=1.35S_{z}g{\omega }_0\sqrt{\left(1-{⟨S_z⟩}^2\right)/\left(n_{S_z}+1\right)}$.

Figure 3

(Color online) Time evolution of system quantities: average photon number $⟨n⟩$ (top); components of total spin polarization $S_{x,y,z}$ (middle) where $x,y,z$ components correspond to red, green, and blue colors [curves from top to bottom at $\omega t/2\pi =1.5\times {10}^4$ (left) and ${10}^4$ (right)]; concurrence $C$ (bottom). Left: two identical qubits with ${\Omega }_{1,2}/{\omega }_0=1.2$, right: two different qubits with ${\Omega }_1/{\omega }_0=1.1$ and 1.2. Data are shown at stroboscopic moments of time with driving phase $\phi =\omega t\left(\text{mod}2\pi \right)=0$, Here $\omega /{\omega }_0=1$, other parameters are as in Fig. 2.

Figure 4

(Color online) Rows from top to bottom: average photon number $⟨n⟩$ vs $S_z$; concurrence $C$ vs $S_z$; and resonator phase ${\varphi }_o$ and qubits phases ${\varphi }_{1,2}$ vs the driving force phase $\phi$. Columns from left to right: (i) two identical qubits with ${\Omega }_{1,2}/{\omega }_0=1.2$, $g=0.04$, (ii)–(iii) two different qubits, with ${\Omega }_1/{\omega }_0=1.1$, ${\Omega }_2/{\omega }_0=1.2$, $g=0.04$ and 0.001, all tree columns correspond to driving frequency $\omega /{\omega }_0=1.01$. The last column (iv) shows data for two different qubits with ${\Omega }_1/{\omega }_0=1.1$, ${\Omega }_2/{\omega }_0=1.2$, and $g=0.04$ but the driving frequency $\omega /{\omega }_0=1$ [cf. (ii)]. Colors of points are explained in the text. Other parameters of simulations are: $\lambda /{\omega }_0=0.02$, $n_p=15$.