Synchronized coherent charge oscillations in coupled double quantum dots

We study coherent charge oscillations in double quantum dots tunnel coupled to metallic leads. If two such systems are coupled by Coulomb interaction, there are, in total, six (instead of only two) oscillation modes of the entangled system with interaction-dependent oscillation frequencies. By tuning the bias voltage, one can engineer decoherence such that only one of the six modes, in which the charge oscillations in both double quantum dots become synchronized in antiphase, is singled out. We suggest using waiting-time distributions and the $g^{\left(2\right)}$-correlation function to detect the common frequency and the phase locking.

Figure 1

Model. The top DQD (blue) and the bottom DQD (green) are both tunnel coupled to left and right metallic leads with tunnel-coupling strengths ${\mathrm{\Gamma }}_{\text{L}}$ and ${\mathrm{\Gamma }}_{\text{R}}\ll {\mathrm{\Gamma }}_{\text{L}}$, respectively. The applied bias is ${\mu }_{\text{L}}-{\mu }_{\text{R}}=eV$. Once an electron tunnels into a DQD, it oscillates coherently between the left and right quantum dots before it tunnels out again. The parameters of the top and bottom DQDs are identical, except for the tunneling amplitude ${\mathrm{\Delta }}_{\text{T}}<{\mathrm{\Delta }}_{\text{B}}$. In addition, the DQDs are coupled via the Coulomb interaction $U$.

Figure 2

Synchronization. (a) and (b) Waiting-time distributions $w_{\text{T}}\left(\tau \right)$ (blue) and $w_{\text{B}}\left(\tau \right)$ (green) for the top and bottom DQDs, respectively. In (a), the DQDs are decoupled ($U=0\mathrm{\Gamma }$), so that the detected coherent oscillations are independent of each other with distinct, but different, frequencies. In (b), the Coulomb interaction is $U=6\mathrm{\Gamma }$, and the waiting-time distributions are almost identical, $w_{\text{T}}\left(\tau \right)\approx w_{\text{B}}\left(\tau \right)$; that is, the coherent charge oscillations are synchronized. The respective insets show the Fourier-transformed waiting-time distributions ${\widehat{w}}_{\text{T}}\left(\omega \right)$ (blue) and ${\widehat{w}}_{\text{B}}\left(\omega \right)$ (green), which have clear peaks at the individual frequencies ${\omega }_{\text{T}}$ and ${\omega }_{\text{B}}$ in (a) and at the common frequency ${\omega }_{\text{S}}$ in (b). The frequencies are indicated with dashed lines. The parameters are $eV=5\mathrm{\Gamma }$, $k_{\text{B}}T=0.2\mathrm{\Gamma }$, $\varepsilon =0.5\mathrm{\Gamma }$, ${\mathrm{\Delta }}_{\text{T}}=2\mathrm{\Gamma },{\mathrm{\Delta }}_{\text{B}}=3\mathrm{\Gamma }$, $W=25\mathrm{\Gamma }$, ${\mathrm{\Gamma }}_{\text{L}}=0.95\mathrm{\Gamma }$, and ${\mathrm{\Gamma }}_{\text{R}}=0.05\mathrm{\Gamma }$.

Figure 3

Frequency and phase locking. (a) and (b) Fourier-transformed waiting-time distributions ${\widehat{w}}_{\text{T}}\left(\omega \right)$ and ${\widehat{w}}_{\text{B}}\left(\omega \right)$ as a function of both frequency $\omega$ and Coulomb interaction $U$. In (a), the peak (blue) indicating the coherent charge oscillations of the top DQD gets gradually shifted to smaller frequencies as the Coulomb interaction $U$ increases. In (b), the peak (green) indicating the coherent charge oscillations in the bottom DQD first shifts to higher frequencies before it diminishes. As the interaction increases, a new peak appears with the same frequency as the top DQD. (c) $g_{\text{BT}}^{\left(2\right)}\left(\tau \right)$-correlation function as a function of both time $\tau$ and Coulomb interaction $U$, indicating temporal correlations between a tunneling-in event in the top DQD and a tunneling-out event in the bottom DQD. The maxima at $\tau =2\pi n/{\omega }_{\text{S}}$ with $n=0,1,2,...$ (depicted in red) suggest a phase relation of $\pi$ between the oscillations. In (a)–(c), the insets show cross sections for $U=0\mathrm{\Gamma }$ and $U=6\mathrm{\Gamma }$. The remaining parameters are the same as in Fig. 2.

Figure 4

(a) All six oscillation frequencies (solid lines) are shown. The common synchronization frequency ${\omega }_{\text{S}}$ is depicted in red, and the remaining frequencies are depicted in gray. The blue and green dashed lines indicate the individual oscillation frequencies ${\omega }_{\text{T}}$ and ${\omega }_{\text{B}}$, respectively, and the red dashed line shows the leading term of the synchronization frequency ${\omega }_{\text{S}}\approx {\mathrm{\Delta }}_{\text{T}}{\mathrm{\Delta }}_{\text{B}}/U$ for strong interactions $U$. (b) Decoherence rates ${\gamma }_{\psi {\psi }^{\prime }}$ of the coherences $|\psi \rangle \langle \psi \prime |$ with $|\psi \rangle ,|{\psi }^{\prime }\rangle \in \{|\text{LL}\rangle |\text{LR}\rangle ,|\text{RL}\rangle ,|\text{RR}\rangle \}$ as a function of the applied bias $eV$. For $eV\lesssim 2U$, the coherences involving $|\text{LL}\rangle$ (blue, green, and orange) decohere fast compared to the rest. Only for high bias voltages $eV\gtrsim 2U$ do all six coherences decohere slowly. (c) Probabilities ${\left({\rho }_{\text{st}}\right)}_{\psi }$ to find the system in the state $|\psi \rangle$. For $eV\lesssim 2U$, the system is most likely in either $|\text{LR}\rangle$ or $|\text{RL}\rangle$, while for $eV\gtrsim 2U$, the system is most likely in $|\text{LL}\rangle$. The dotted vertical lines at $eV=5\mathrm{\Gamma }$ in (b) and (c) indicate the bias voltage used in Figs. 2 and 3 to obtain the synchronized coherent charge oscillations. (d) We use $eV=20\mathrm{\Gamma }$ and find that the waiting-time distributions $w_{\text{T/B}}\left(\tau \right)$ show beats and all six frequencies are visible in the Fourier transform ${\widehat{w}}_{\text{T/B}}\left(\omega \right)$ (see the inset). The remaining parameters are the same as in Fig. 2.