Retrieving qubit information despite decoherence

The time evolution of a qubit, consisting of two single-level quantum dots, is studied in the presence of telegraph noise. The dots are connected by two tunneling paths with an Aharonov-Bohm flux enclosed between them. Under special symmetry conditions, which can be achieved by tuning gate voltages, there develops partial decoherence: at long times, the off-diagonal element of the reduced density matrix (in the basis of the two dot states) approaches a nonzero value, generating a circulating current around the loop. The flux dependence of this current contains full information on the initial quantum state of the qubit, even at infinite time. Small deviations from this symmetry yield a very slow exponential decay toward the fully decoherent limit. However, the amplitudes of this decay also contain the full information on the initial qubit state, measurable either via the current or via the occupations of the qubit dots.

Figure 1

The two-dot qubit with tunneling channels $u$ and $d$.

Figure 2

(Color online) The averages of $x,y$ and $z$ [Eq. (49)] for $\varphi =-.3\pi ,J_d=.5,{\zeta }_d=.3,q_J=q_{\zeta }=.5,{\zeta }_{\Delta }=0,p_{+}=p_{-}=1/2$. The initial qubit state [Eq. (1)] is given by $\alpha =\gamma =.25\pi$. All energies and inverse times are in units of $\lambda$. The full (dashed) thick lines correspond to $\Delta =0$ (0.2). The full (dashed) thin lines represent the exact (approximate) asymptotic behavior. The derivation of the dashed lines, for $\Delta \ne 0$, is described in Sec. 5.

Figure 3

(Color online) The flux dependence of the asymptotic average current (in units of $J_0$) in the symmetric case, for $q_J=q_{\zeta }=1/2$ and $\alpha =.25\pi$. Increasing dashes correspond to $\gamma =0,\pi /4$ and $\pi /2$.

Figure 4

(Color online) The flux dependence of the amplitude $z_0$ for the same parameters as for the dashed lines in Fig. 2 but with $\gamma$ as in Fig. 3.