Continuous Quantum Measurement: Inelastic Tunneling and Lack of Current Oscillations

We study the dynamics of a charge qubit, consisting of a single electron in a double well potential coupled to a point-contact (PC) electrometer, using the quantum trajectories formalism. Contrary to previous predictions, we show formally that, in the sub-Zeno limit, coherent oscillations in the detector output are suppressed, and the dynamics is dominated by inelastic processes in the PC. Furthermore, these reduce the detector efficiency and induce relaxation even when the source-drain bias is zero. This is of practical significance since it means the detector will act as a source of decoherence. Finally, we show that the sub-Zeno dynamics is divided into two regimes: low and high bias in which the PC current power spectra show markedly different behavior.

Figure 1

Schematic of the qubit and PC showing lead energy bands. Electrons tunneling from the source ($S$) to the drain ($D$) may do so elastically or inelastically, depicted by arrows. Different transitions induce different jumps, $P_i$, on the qubit.

Figure 2

Equilibrium ground state occupation probability for a qubit near a PC (solid line), and for a thermalized qubit, ${\rho }_{gg}^{\mathrm{therm}}(\infty )$, at a temperature $T=eV/2$ (dashed line).

Figure 3

Current power spectra, $\tilde{S}(\omega )=[(2\varphi {\nu }^2{\Delta }^2)/({ϵ}^2\delta I_{\mathrm{loc}}^2)][S(\omega )-S(\infty )]$. For a given $\nu$ the plot is valid in the region $eV/\varphi \ll 1/{\nu }^2$. Note the sharp change in the power spectrum at $\varphi =eV$.