Quantum measurement of a double quantum dot coupled to two kinds of environment

We theoretically study the quantum measurement of a double quantum dot coupled to environments by Bloch equations with an additional vector. Using a quantum point contact as a detector, we accurately calculate the current, the Fano factor, and the waiting time, respectively, to characterize the dynamical properties in two kinds of environments. In the dissipative case, the asymmetrical Fano factor is enhanced with the increase in deocoherence rates and suppressed as the growth of relaxation rates, and the super-Poissonian noise is mainly due to the effects of a dynamical channel blockade and quantum coherence. In the pure dephasing case, the symmetrical Fano factor is magnified, which can be attributed to the quantum Zeno effect. Moreover, we show that the distribution of the average waiting time exhibits good agreement with the variation tendency of the current. Our paper provides an effective method in handling quantum measurement.

Figure 1

Schematic for the DQD monitored by a QPC detector. The position of the electron in the left or right dot generates two different states of the detector (1) and (2). ${\mu }_{1,2}$'s denote the chemical potential in the left and right reservoirs.

Figure 2

The current and Fano factor versus the level displacement $\varepsilon$ in the dissipative environment. (a) The current and (b) the Fano factor for different relaxation rates: ${\mathrm{\Gamma }}_r=0.1\mathrm{\Delta }$ (the black solid curve), ${\mathrm{\Gamma }}_r=0.2\mathrm{\Delta }$ (the black dashed curve), and ${\mathrm{\Gamma }}_r=0.3\mathrm{\Delta }$ (the black dotted curve). (c) The current and (d) the Fano factor for different decoherence rates: ${\mathrm{\Gamma }}_d=0.1\mathrm{\Delta }$ (the red curve), ${\mathrm{\Gamma }}_d=0.2\mathrm{\Delta }$ (the blue curve), and ${\mathrm{\Gamma }}_d=0.3\mathrm{\Delta }$ (the green curve).

Figure 3

The average waiting time $\langle \tau \rangle$ calculated from the Bloch equations for (a) different relaxation rates: ${\mathrm{\Gamma }}_r=0.1\mathrm{\Delta }$ (the black solid curve), ${\mathrm{\Gamma }}_r=0.2\mathrm{\Delta }$ (the black dashed curve), and ${\mathrm{\Gamma }}_r=0.3\mathrm{\Delta }$ (the black dotted curve) and (b) different decoherence rates: ${\mathrm{\Gamma }}_d=0.1\mathrm{\Delta }$ (the red curve), ${\mathrm{\Gamma }}_d=0.2\mathrm{\Delta }$ (the blue curve), and ${\mathrm{\Gamma }}_d=0.3\mathrm{\Delta }$ (the green curve).

Figure 4

The current and Fano factor versus the level displacement $\varepsilon$ in the pure dephasing environment. (a) The current and (b) the Fano factor for different dephasing rates: ${\mathrm{\Gamma }}_{\varphi }=0.1\mathrm{\Delta }$ (the black solid curve), ${\mathrm{\Gamma }}_{\varphi }=0.2\mathrm{\Delta }$ (the black dashed curve), and ${\mathrm{\Gamma }}_{\varphi }=0.3\mathrm{\Delta }$ (the black dotted curve). (c) The current and (d) the Fano factor for different decoherence rates: ${\mathrm{\Gamma }}_d=0.1\mathrm{\Delta }$ (the red curve), ${\mathrm{\Gamma }}_d=0.2\mathrm{\Delta }$ (the blue curve), and ${\mathrm{\Gamma }}_d=0.3\mathrm{\Delta }$ (the green curve).

Figure 5

The average waiting time $\langle \tau \rangle$ calculated from the Bloch equations for (a) different dephasing rates: ${\mathrm{\Gamma }}_{\varphi }=0.1\mathrm{\Delta }$ (the black solid curve), ${\mathrm{\Gamma }}_{\varphi }=0.2\mathrm{\Delta }$ (the black dashed curve), and ${\mathrm{\Gamma }}_{\varphi }=0.3\mathrm{\Delta }$ (the black dotted curve) and (b) different decoherence rates: ${\mathrm{\Gamma }}_d=0.1\mathrm{\Delta }$ (the red curve), ${\mathrm{\Gamma }}_d=0.2\mathrm{\Delta }$ (the blue curve), and ${\mathrm{\Gamma }}_d=0.3\mathrm{\Delta }$ (the green curve).