Monitored quantum transport through a disordered one-dimensional conductor

We formulate a quantum master equation for the many-particle density matrix of electrons propagating through a single-mode conductor, combining elastic scattering by disorder with time-resolved projective measurements that monitor the outcome of scattering events. The full counting statistics of transmitted electrons has a binomial distribution function, whose mean ${\cal T}$ and variance ${\cal T}(1-{\cal T})$ determine the conductance and shot noise power, respectively. Monitoring suppresses the phase coherence responsible for one-dimensional localization: The decay with conductor length $L$ of the typical transmission probability crosses over at $L\simeq \ell_\phi$ from the exponential $e^{-L/\xi}$ (with localization length $\xi$) to the Ohmic $1/L$ decay. Numerical solution of the master equation gives, for weak monitoring, a logarithmic dependence $\ell_\phi\simeq \xi\ln(v_{\rm F}\tau_\phi/\xi)$ of the coherence length $\ell_\phi$ on the mean time $\tau_\phi$ between measurements.