Noninvasive Quantum Measurement of Arbitrary Operator Order by Engineered Non-Markovian Detectors

The development of solid-state quantum technologies requires the understanding of quantum measurements in interacting, nonisolated quantum systems. In general, a permanent coupling of detectors to a quantum system leads to memory effects that have to be taken into account in interpreting the measurement results. We analyze a generic setup of two detectors coupled to a quantum system and derive a compact formula in the weak-measurement limit that interpolates between an instantaneous (text-book type) and almost continuous—detector dynamics-dependent—measurement. A quantum memory effect that we term “system-mediated detector-detector interaction” is crucial to observe noncommuting observables simultaneously. Finally, we propose a mesoscopic double-dot detector setup in which the memory effect is tunable and that can be used to explore the transition to non-Markovian quantum measurements experimentally.

Figure 1

Non-Markovian cross-correlation measurement. Two detectors $a$ and $b$ are coupled to a system for a finite time and read out at times $t_a>t_b$. Detector $b$ collects information directly on the system and on detector $a$ (which is transmitted through the system). The presence of detector $b$ after its readout (light blue) also contributes to the information which detector $a$ collects on detector $b$ at a later time $t_a$.

Figure 2

(a) Couplings and correlations between the different variables. The intersystem couplings ${\eta }_{\alpha }$ connect the detector variables ${\widehat{D}}_{\alpha }$ to the system variables $\widehat{A}$ and $\widehat{B}$ at equal times. Within the system and within the detectors, information is transmitted nonlocally in time via the response functions $\chi$. (b) Schematic representation of the second-order processes which contribute to the cross-correlation measurement. The straight arrows indicate the direction and causality of the time flow.

Figure 3

Ratio of the weights ${\xi }_a$ and ${\xi }_s$ of the antisymmetric and symmetric contributions to Eq. (9) as a function of the detectors’ frequency ${\mathrm{\Omega }}^{\prime }$ and damping $\lambda$ in units of the system frequency $\mathrm{\Omega }$ at zero detector temperature.

Figure 4

Sketch of the setup. (a) Two double-dot detectors are capacitively coupled to a system, and the occupation of one of the dots is recorded via a bypassing current. (b) The state of each detector can be set by additional contacts [not shown in (a)], which are biased to adjust the occupation difference of the high and low energy eigenstate $n^{+}$ and $n^{-}$. (c) Weights ${\xi }_s$ and ${\xi }_a$ as a function of the occupation difference of the energy eigenstates $\mathrm{\Delta }{n}_{\alpha }=n_{\alpha }^{-}-n_{\alpha }^{+}$, ${\epsilon }_{\alpha }=0$, no damping.

Figure 5

Markovian coupling: Prefactors of $S_{AB}^0$ and ${\chi }_{AB}^0$ as a function of the angle $\varphi$ for $⟨{\widehat{\sigma }}_z⟩=1$ and $⟨{\widehat{\sigma }}_x⟩=⟨{\widehat{\sigma }}_y⟩=0$.