Simultaneous readout of two charge qubits

We consider a system of two solid state charge qubits, coupled to a single readout device, consisting of a single-electron transistor (SET). The conductance of each tunnel junction is influenced by its neighboring qubit, and thus the current through the transistor is determined by the qubits’ state. The full counting statistics of the electrons passing the transistor is calculated, and we discuss qubit dephasing, as well as the quantum efficiency of the readout. The current measurement is then compared to readout using real-time detection of the SET island’s charge state. For the latter method we show that the quantum efficiency is always unity. Comparing the two methods a simple geometrical interpretation of the quantum efficiency of the current measurement appears. Finally, we note that full quantum efficiency in some cases can be achieved measuring the average charge of the SET island, in addition to the average current.

Figure 1

A Coulomb blockade island in a single-electron transistor geometry. The conductance of each tunnel junction is influenced by a nearby charge qubit. (See text.)

Figure 2

The four nonzero tunneling rates for electrons, on and off the SET island. The applied source-drain voltage $eV$ creates a difference in chemical potentials between the left and right lead $eV={\mu }_L-{\mu }_R$.

Figure 3

An illustration of the capacitative coupling between a double quantum dot qubit and low transparency point contact. (a) When the electron is located in the dot closest to the PC, the barrier is higher, leading to a lower tunneling rate ${\Gamma }^{-}=\Gamma -\delta \Gamma$. (b) For the qubit in the other state, the barrier is lower and the tunneling rate higher ${\Gamma }^{+}=\Gamma +\delta \Gamma$.

Figure 4

(Color online) (a) The average current through the SET for the different initial qubit product states, with no qubit interdot coupling $(\Delta =0)$, and $\delta {\Gamma }_L=\delta {\Gamma }_R=0.2\Gamma$. (b) The corresponding probability distributions $P(m,t)$ for the number of electrons $m$ having passed the SET at time $t=60∕\Gamma$.

Figure 5

(Color online) A schematic representation of the different two-dimensional rate distributions corresponding to the four different qubit product states $A–D$. A current measurement is equivalent to projecting the distributions on the axis $I_a$ in the case of negligible Coulomb energy, and on $I_b$ in the case of large Coulomb energy.