Monitoring quantum transport: Backaction and measurement correlations

We investigate a tunnel contact coupled to a double quantum dot (DQD) and employed as a charge monitor for the latter. We consider both the classical limit and the quantum regime. In the classical case, we derive measurement correlations from conditional probabilities, yielding quantitative statements about the parameter regime in which the detection scheme works well. Moreover, we demonstrate that not only the DQD occupation but also the corresponding current may strongly correlate with the detector current. The quantum-mechanical solution, obtained with a Bloch-Redfield master equation, shows that the backaction of the measurement tends to localize the DQD electrons, and thus significantly reduces the DQD current. Moreover, it provides the effective parameters of the classical treatment. It turns out that already the classical description is adequate for most operating regimes.

Figure 1

Quantum point contact in the tunnel regime acting as a charge monitor for an undetuned but biased DQD. Electrons on the latter increase the tunnel barrier and thus reduce the detector current.

Figure 2

Classical frequency-dependent correlation coefficients between the detector current and (a) the occupation of the left dot, (b) the occupation of the right dot, and (c) the symmetrized current through the DQD for various interdot rates ${\Gamma }_{12}$. The detector is characterized by a bare rate ${\gamma }_0={10}^8\Gamma$ and the sensitivities ${\tilde{s}}_1=0.2$ and ${\tilde{s}}_2=0$, i.e., it couples to only the left dot. The inset in panel (a) shows the frequency-dependent Fano factor of the detector current for ${\Gamma }_{12}=0.01\Gamma$, where the dashed lines mark the crossover region between the plateaus. The horizontal line in panel (c) marks the upper limit $1/2$ discussed in the text.

Figure 3

Classical correlation coefficients depicted in Fig. 2 as a function of the frequency $\omega$ and the interdot rate ${\Gamma }_{12}$ while all other parameters are the same. The dashed lines at $\omega =\Gamma$ and at $\omega ={\omega }_{\max }$ mark the crossover between the different regimes discussed in the text.

Figure 4

Classical correlation coefficients between the detector and (a) the occupation of the left dot, (b) the current through the left DQD barrier, and (c) the Ramo-Shockley current as a function of the detector sensitivities ${\tilde{s}}_1$ and ${\tilde{s}}_2$. The frequency $\omega =50\Gamma$ corresponds to the middle of the second plateau at which time-resolved measurement is possible. The interdot rate is ${\Gamma }_{12}=0.1\Gamma$, while all other parameters are as in Fig. 2.

Figure 5

Quantum-mechanical version of the correlation coefficients between the detector current and (a) the occupation of the left dot, (b) the current through the left DQD barrier, and (c) the symmetrized current through the DQD for various values of the sensitivity $s_2$ while $s_1=0.2$ is fixed. The interdot tunnel coupling is $T_{12}=10\Gamma$, while all other parameters are as in Fig. 2. The thin red lines in panel (a) show the corresponding classical correlation coefficients for ${\Gamma }_{12}$ determined by the quantum-to-classical mapping in Eq. (27).

Figure 6

Quantum-mechanical version of the correlation coefficients shown in Fig. 4 as a function of the detector sensitivities $s_1$ and $s_2$. The frequency is $\omega =50\Gamma$, the tunnel rate is $T_{12}=10\Gamma$, while all other parameters are as in Fig. 5. Notice that in the regime depicted, the classical and the quantum-mechanical detector sensitivities relate as $s_{\ell }\approx {\tilde{s}}_{\ell }/2$.