Quantum synchronization as a local signature of super- and subradiance

We study the relationship between the collective phenomena of super- and subradiance and spontaneous synchronization of quantum systems. To this aim we revisit the case of two detuned qubits interacting through a pure dissipative bosonic environment, which contains the minimal ingredients for our analysis. By using the Liouville formalism, we are able to find analytically the ultimate connection between these phenomena. We find that dynamical synchronization is due to the presence of longstanding coherence between the ground state of the system and the subradiant state. We finally show that, under pure dissipation, the emergence of spontaneous synchronization and of subradiant emission occur on the same time scale. This reciprocity is broken in the presence of dephasing noise.

Figure 1

Schematic representation of two independent channels connecting the excited state $|E\rangle =:|ee\rangle$ (energy equal to ${\omega }_0$) to the ground state $|G\rangle =:|gg\rangle$ (energy equal to $-{\omega }_0$) [see Eq. (A14)]. The channel passing through the state $|S_L\rangle =|S_R{\rangle }^{*}$ [whose oscillations are associated with $V_i/2$; see Eq. (A11)] is characterized by a decay rate towards the ground state equal to ${\gamma }_0+V_r$ (${\gamma }_0=\frac{{\gamma }_1+{\gamma }_2}{2}$). The channel passing through the state $|A_L\rangle =|A_R{\rangle }^{*}$ (whose oscillations are associated with $-V_i/2$) is characterized by a decay rate towards the ground state equal to ${\gamma }_0-V_r$. For simplicity, the channels connecting $|E\rangle$ and $|G\rangle$ through the coherences between $|S_L\rangle$ and $|A_L\rangle$ are not shown.

Figure 2

Plot of ${\kappa }_S$ as a function of $\delta$ and $s_{12}$ in units of ${\gamma }_0$. The values of the other parameters are ${\gamma }_1={\gamma }_2={\gamma }_0$ and ${\gamma }_{12}=0.8{\gamma }_0$.

Figure 3

Time evolution of ${\langle {\sigma }_1^x\rangle }_t$ (dashed line) and ${\langle {\sigma }_2^x\rangle }_t$ (solid line) for the initial state $\frac{1}{2}\left(\right|e\rangle +|g\rangle )\left(\right|e\rangle +|g\rangle )$. The values of the parameters are ${\gamma }_1=1.1{\gamma }_0$, ${\gamma }_2=0.9{\gamma }_0$, ${\gamma }_{12}=0.95{\gamma }_0$, $s_{12}=0.6{\gamma }_0$, $\delta ={\gamma }_0$, and ${\omega }_0=10{\gamma }_0$. In this case ${\kappa }_S^{-1}{\gamma }_0\approx 1.2$, while the frequency of synchronization is ${\nu }_S=9.26{\gamma }_0$. In the inset we plot the evolution of the synchronization measure $C_{{\langle {\sigma }_1^x\rangle }_t,{\langle {\sigma }_2^x\rangle }_t}\left(\mathrm{\Delta }t\right)$ for $\mathrm{\Delta }t=2/{\gamma }_0)$ (dashed curve) and time delayed one (solid line).

Figure 4

Time evolution of ${\langle {\sigma }_1^x\rangle }_t$ (dashed line) and ${\langle {\sigma }_2^x\rangle }_t$ (solid line). Time is scaled with ${\gamma }_0$. The initial state is (a) $\frac{1}{\sqrt{2}}\left(\right|G\rangle +|A_R\rangle )$ and (b) $\frac{1}{\sqrt{2}}\left(\right|G\rangle +|S_R\rangle )$. The parameters have the same values as in Fig. 3.

Figure 5

Time evolution of $I\left(t\right)/2{\gamma }_0$ (logarithmic scale) for several initial conditions (see the legend; the curves have different styles and thicknesses). Time is scaled with ${\gamma }_0$. The parameters have the same values as in Fig. 3. Here $|S\rangle$ and $|A\rangle$ are the symmetric and antisymmetric states, respectively, and $|S_{\delta }\rangle$ and $|A_{\delta }\rangle$ are the one-excitation states diagonalizing the dressed Hamiltonian $H_S+H_{LS}$. The IA case is the one of independent atoms having individual decay rates equal to ${\gamma }_0$, starting from an arbitrary one-excitation state.

Figure 6

Comparison between the synchronization measure $C_{{\langle {\sigma }_1^x\rangle }_{t^{*}},{\langle {\sigma }_2^x\rangle }_{(t^{*}+\delta t)}}\left(\mathrm{\Delta }t\right)$ ($\mathrm{\Delta }t=2/{\gamma }_0$) and the ratio $R_I\left(t^{*}\right)$, both computed at $t^{*}=5/{\gamma }_0$ for the initial state $\frac{1}{2}\left(\right|e\rangle +|g\rangle )\left(\right|e\rangle +|g\rangle )$. The values of the parameters are ${\gamma }_1={\gamma }_2={\gamma }_0$, $s_{12}=0.6{\gamma }_0$, $\delta ={\gamma }_0$, and ${\omega }_0=10{\gamma }_0$. For ${\gamma }_{12}={\gamma }_0$, the synchronization parameter is such that ${\kappa }_S^{-1}{\gamma }_0\approx 1.2$, while for ${\gamma }_{12}=0.3{\gamma }_0$, ${\kappa }_S^{-1}{\gamma }_0\approx 4.3$.