Decoherence effects on weak value measurements in double quantum dots

We study the effect of decoherence on a weak value measurement in a paradigm system consisting of a double quantum dot continuously measured by a quantum point contact. Fluctuations of the parameters controlling the dot state induce decoherence. We find that, for measurements longer than the decoherence time, weak values are always reduced within the range of the eigenvalues of the measured observable. For measurements at shorter time scales, the measured weak value strongly depends on the interplay between the decoherence dynamics of the system and the detector backaction. In particular, depending on the postselected state and the strength of the decoherence, a more frequent classical readout of the detector might lead to an enhancement of weak values.

Figure 1

(a) Scheme of a double quantum dot with a nearby QPC measuring its charge states, labeled as $|L\rangle$,$|R\rangle$. (b),(c) Scheme of the detection mechanism. The conductance of the detector is sensitive to the qubit's occupancy: The tunneling amplitude in the QPC is $\Omega +\delta \Omega$ when the right dot is occupied (b) and $\Omega -\delta \Omega$ when the left dot is occupied (c).

Figure 2

(a) Qubit plus detector interaction with a classical pointer reading out the number $m_k$ of electrons having penetrated to the detector's right reservoir at certain discrete times $t=t_k$. Illustration of the continuous readout (b) and coherent (c) measurement regimes, respectively.

Figure 3

Schematic evolution of $\mathbf{v}\left(s\right)$ and $\mathbf{n}\left(s\right)$ on the Bloch sphere for (a) $\mathbf{m}\ne \mathbf{k}$ and (b) $\mathbf{m}=\mathbf{k}$. (c) Weak value as a function of $\tau$ for different $\gamma$; (d) WV for different $\gamma$ at ${\tau }_1=0.10\mathrm{ns}$ and ${\tau }_2=1.31\mathrm{ns}$, marked in panel (c). In all plots the parameters are chosen as ${\epsilon }_0=20\mu \mathrm{eV}$, ${\Delta }_0=3\mu \mathrm{eV}$, $\mathbf{k}\approx -0.89{\mathbf{e}}_x+0.45{\mathbf{e}}_z$, $\mathbf{v}\left(0\right)=(0.5,-0.33,0.80)$, and $\mathbf{n}=(-0.20,0.75,0.63)$.

Figure 4

Weak value as a function of $\tau$ for (a) different $\mathbf{k}$ ($\theta =0$ corresponds to $\mathbf{k}\parallel \mathbf{m}$) and (b) different $\mathbf{n}$ ($\varphi =0$ corresponds to $\mathbf{n}\parallel \mathbf{m}$). (c) Same as in (a) in the presence of multiple decoherence sources, where $\gamma =1.0\mu \mathrm{eV}$, $\mathbf{k}=\frac{1}{\sqrt{2}}{\mathbf{e}}_x+\frac{1}{\sqrt{2}}{\mathbf{e}}_z$, ${\mathbf{k}}_1=\frac{1}{\sqrt{2}}{\mathbf{e}}_x$, ${\mathbf{k}}_2=\frac{1}{\sqrt{2}}{\mathbf{e}}_z$, ${\gamma }_1=\frac{1}{\sqrt{2}}\mu \mathrm{eV}$ ${\gamma }_2=\frac{1}{\sqrt{2}}\mu \mathrm{eV}$, ${\overline{\mathbf{k}}}_1=\mathbf{m}$, ${\overline{\mathbf{k}}}_2={\mathbf{m}}_{\perp }$, ${\overline{\gamma }}_1=(\mathbf{k}·\mathbf{m})\mu \mathrm{eV}$, ${\overline{\gamma }}_2=(\mathbf{k}·{\mathbf{m}}_{\perp })\mu \mathrm{eV}$; all the other parameters are as in Fig. 3.

Figure 5

Weak value for continuous detection for (a) different $\gamma$ with $D=0.270\mathrm{eV}$ and (b) different $D$ with $\gamma =0.1\mu \mathrm{eV}$; other parameters as in Fig. 3. In all plots $\delta \Omega /\Omega =0.001$.

Figure 6

Weak value for continuous detection and coherent readout for different $\gamma$ and pre- and postselection states $\mathbf{n}$, $\mathbf{v}$; $\gamma =1.0\mu \mathrm{eV}$ in (a),(c) and $\gamma =0.1\mu \mathrm{eV}$ in (b),(d); $\mathbf{v}\left(0\right)=(0.5,-0.33,0.80)$, $\mathbf{n}=(-0.20,0.75,0.63)$ in (a),(b) and $\mathbf{v}\left(0\right)=(-0.5,0.33,-0.80)$, $\mathbf{n}=(0.50,0.20,0.84)$ in (c),(d). In all plots $D=0.270\mathrm{eV}$, $\delta \Omega /\Omega =0.001$ and the other parameters as in Fig. 3.

Figure 7

Validity perturbation. (a) Left-hand side of Eq. (33) and (b) the corresponding WV for $\gamma =1.0\mu \mathrm{eV}$, $D=0.270\mathrm{eV}$, $\delta \Omega /\Omega =0.001$; all other parameters are as in Fig. 3.