Probing the spectral density of a dissipative qubit via quantum synchronization

The interaction of a quantum system, which is not accessible by direct measurement, with an external probe can be exploited to infer specific features of the system itself. We introduce a probing scheme based on the emergence of spontaneous quantum synchronization between an out-of-equilibrium qubit, in contact with an external environment, and a probe qubit. Tuning the frequency of the probe leads to a transition between synchronization in phase and antiphase. The sharp transition between these two regimes is locally accessible by monitoring the probe dynamics alone and allows one to reconstruct the shape of the spectral density of the environment.

Figure 1

Schematics of the model: a system $q$, interacting with its own thermal environment, is connected to an external probe $p$.

Figure 2

Dynamical synchronization of $\langle {\sigma }_q^x\rangle$ (red solid line) and $\langle {\sigma }_p^x\rangle$ (black dashed line) for ${\omega }_p=1.2{\omega }_q$ and $\lambda =0.2{\omega }_q$. The bath spectral density is Ohmic: $J\left(\omega \right)=2{\gamma }_0\omega {\omega }_c^2/({\omega }_c^2+{\omega }^2)$ with cutoff frequency ${\omega }_c=20{\omega }_q$ and ${\gamma }_0={10}^{-2}{\omega }_q$ and $T=0$. The initial state is $\left|\psi \right(0)\rangle =(|0\rangle +|1\rangle \left)\right(|0\rangle +|1\rangle )/2$. The inset shows the synchronization measure ${\mathcal{C}}_{{\sigma }_p^x,{\sigma }_q^x}\left(t\right)$ for ${\omega }_p/{\omega }_q=1.2$ (black solid line), ${\omega }_p/{\omega }_q=0.8$ (red dashed line), and ${\omega }_p/{\omega }_q=1$ (gray dotted line) as examples of SS in phase and antiphase and the absence of SS, respectively.

Figure 3

(a) Synchronization frequency as a function of ${\omega }_p/{\omega }_q$ for $s=2$ and (b) absolute value of ${\mathcal{C}}_{{\sigma }_p^x,{\sigma }_q^x}(t,\tau )$ as a function of ${\omega }_p/{\omega }_q$ and $s$ for $\lambda =0.2{\omega }_q$ and spectral density $J\left(\omega \right)=2{\gamma }_0{\omega }^s{\omega }_c^2/({\omega }_c^2+{\omega }^{2s})$. The white line in (b) is the analytical solution of Eq. (13).

Figure 4

Absolute value of the Fourier transform $F\left(\omega \right)$ of $\langle {\sigma }_p^x\left(t\right)\rangle$. The three panels refer to the three synchronization lines of Fig. 7: (a) ${\omega }_p/{\omega }_q=0.8$, (b) ${\omega }_p/{\omega }_q=1$, and (c) ${\omega }_p/{\omega }_q=1.2$. As in Fig. 7, the bath is Ohmic, and $\lambda =0.2{\omega }_q$. The blue dotted lines are taken calculating $F\left(\omega \right)$ during the time window $T_1=\{0,110{\omega }_q\}$, the red dashed lines concern the time window $T_2=\{100,210{\omega }_q\}$, and, finally, the black solid lines refer to $T_3=\{200,310{\omega }_q\}$.

Figure 5

Absolute value of the Fourier transform $F\left(\omega \right)$ of $\langle {\sigma }_p^x\left(t\right)\rangle$ in the initial transient (black solid line) and after SS has occurred (red dashed line). In the last case the peak linewidth can be determined. In the case simulated here the system-bath coupling is stronger than in the previous cases (${\gamma }_0=2.5\times {10}^{-2}{\omega }_q$).

Figure 6

Dynamical synchronization as a function of the temperature. The probe frequency is set to ${\omega }_p/{\omega }_q=0.8$ (see Fig. 2) for the initial state and the bath. The red dashed line is the one obtained at $T=0$. It is compared with the case where $T=1$ (black line) and with $T=10$ (gray solid line). As we can notice, in principle, the finite-temperature regime does not necessarily prevent the system from becoming synchronized. However, if $T$ is too high, the system reaches the effective stationary state before synchronization can take place. To give a rough estimation, considering a spin-spin coupling around $\simeq 100$ MHz (see, for instance, Ref. [29]), $T=1$ (in our units) would correspond to a few dozen $\mu \text{K}$.

Figure 7

Synchronization diagram as a function of the ratio ${\omega }_p/{\omega }_q$ and of the coupling $\lambda$. The synchronization has been measured at $t=350{\omega }_q^{-1}$ assuming an Ohmic bath with a frequency cutoff ${\omega }_c=20{\omega }_q$.

Figure 8

Comparison between $\mathcal{C}$ (thick lines) and ${10}^2\langle {\sigma }_{+}^{\left(q\right)}\left(t\right){\sigma }_{-}^{\left(p\right)}\left(t\right)\rangle$ (thin lines). The environment is assumed to have the power-law spectral density $J\left(\omega \right)\sim {\omega }^s$ away from the frequency cutoff. The colors correspond to different values of $s$: $s=0.5$ (black lines), $s=1$ [blue (gray) dashed lines], $s=1.5$ [red (dark gray) solid lines], and $s=2$ [green (light gray) solid lines]. The qubit-probe coupling is set to $\lambda =0.2{\omega }_q$, and all the data are taken at $t=300{\omega }_q$.