Effects of a quantum measurement on the electric conductivity: Application to graphene

We generalize the standard linear-response (Kubo) theory to obtain the conductivity of a system that is subject to a quantum measurement of the current. Our approach can be used to specifically elucidate how back-action inherent to quantum measurements affects electronic transport. To illustrate the utility of our general formalism, we calculate the frequency-dependent conductivity of graphene and discuss the effect of measurement-induced decoherence on its value in the dc limit. We are able to resolve an ambiguity related to the parametric dependence of the minimal conductivity.

Figure 1

Frequency dependence of the conductivity of clean single-layer graphene when the current is unsharply quantum measured. The chemical potential is fixed at $\mu /k_{B}T=1$, and the value for measurement-induced decoherence assumed for each curve is indicated. The dashed line is at $\pi /8$. As the coupling to the measurement device is decreased, the curves more closely resemble the result found for clean graphene.

Figure 2

ac conductivity of single-layer graphene, for fixed $\hslash \Gamma /k_{B}T=1$ and different values of chemical potential $\mu$ (measured from the Dirac point).

Figure 3

(a) dc conductivity plotted as a function of chemical potential for different values of $\Gamma$. The conductivity is linear in the chemical potential for large values of $\mu /k_{B}T$. (b) dc conductivity shown as a function of $\hslash \Gamma /k_{B}T$ for different values of the chemical potential. For $\mu >k_{B}T$ and $\hslash \Gamma <\mu$, the conductivity is inversely proportional to the effective rate of decoherence, in analogy with the behavior expected from inelastic impurity scattering. In contrast, for $\hslash \Gamma >\mu$ there is direct proportionality. This behavior was exhibited as long as $\mu \gtrsim k_{B}T$, whereas curves for $\mu \le k_{B}T$ practically coincide.