Phase readout of a charge qubit capacitively coupled to an open double quantum dot

We study the dynamics of a charge qubit that is capacitively coupled to an open double quantum dot. Depending on the qubit state, the transport through the open quantum dot may be resonant or off-resonant, such that the qubit affects the current through the open double dot. We relate the initial qubit state to the magnitude of an emerging transient current peak. The relation between these quantities enables the readout of not only the charge but also the phase of the qubit with sufficient resolution.

Figure 1

(Color online) (a) Sketch of the qubit-meter setup consisting of an open and a closed double quantum dot. The closed double dot ($D_1$ and $D_2$) is occupied with one electron and forms a charge qubit. An electron in dot $D_1$ effectively shifts the onsite energy of dot $D_3$, such that dots $D_3$ and $D_4$ are tuned into resonance. Thus the current through the open double dot ($D_3$ and $D_4$) is sensitive to the state of the qubit. (b) Current as a function of the meter energy bias $\epsilon ={\epsilon }_4-{\epsilon }_3$ without coupling to the qubit, $U=0$, for interdot tunneling ${\Delta }_{\text{m}}=0.2\text{meV}$ and wire-lead coupling $\Gamma =0.2\text{meV}$. The marked values correspond to the energy shift induced by a qubit electron localized in dot $D_1$ and $D_2$, respectively, for the interaction strength $U=1\text{meV}$.

Figure 2

(Color online) Charge readout for ${\Gamma }_{\text{L}}={\Gamma }_{\text{R}}=0.2\text{meV}$, ${\Delta }_{\text{q}}={\Delta }_{\text{m}}=0.2\text{meV}$, and $U=\epsilon =1\text{meV}$. (a) Transient dynamics of the qubit population $⟨{\sigma }_z⟩=⟨n_1-n_2⟩$ and (b) corresponding current for various initial localizations. (c) Height of the current peak as a function of the initial occupation $\text{cos}\left(\theta /2\right)$ for different relative phases $\varphi$. (d) Charge-readout visibility ${\nu }_{\theta }$. The dashed-dotted line marks the “optimal” visibility, for which backaction is ignored; see Fig. 1b.

Figure 3

(Color online) Phase readout of a qubit with energy splitting ${\Delta }_{\text{q}}=0.2\text{meV}$ coupled to a meter with the same parameters as in Fig. 2. The qubit states are at initial time equally populated, i.e., $\theta =\pi /2$. (a) Transient qubit dynamics and (b) current for various initial phases $\varphi$ and qubit-meter interaction $U=\epsilon =1\text{meV}$ and ${\Delta }_{\text{q}}=0.2\text{meV}$. (c) Height of the current peak as a function of the initial phase. (d) Readout visibility as a function of the qubit-meter interaction strength $U$.

Figure 4

(Color online) Phase-readout visibility for a meter with ${\Gamma }_{\text{L}}={\Gamma }_{\text{R}}=0.2\text{meV}$ and tunnel matrix element ${\Delta }_{\text{m}}=0.2\text{meV}$. Comparison of symmetric coupling $\left(U=\epsilon \right)$ and coupling between the left dots $\left(U^{\prime }=0\right)$ for qubit energy splitting (a) ${\Delta }_{\text{q}}=3{\Delta }_{\text{m}}$ and (b) ${\Delta }_{\text{q}}=2{\Delta }_{\text{m}}$. (c) Visibility as a function of the interaction strength $U$ for symmetric coupling $U^{\prime }=U$, bias $\epsilon =1.5\text{meV}$ and various qubit splittings.