State disturbance and pointer shift in protective quantum measurements

We investigate the disturbance of the state of a quantum system in a protective measurement for finite measurement times and different choices of the time-dependent system-apparatus coupling function. The ability to minimize this state disturbance is essential to protective measurement. We show that for a coupling strength that remains constant during the measurement interaction of duration $T$, the state disturbance scales as $T^{-2}$, while a simple smoothing of the coupling function significantly improves the scaling behavior to $T^{-6}$. We also prove that the shift of the apparatus pointer in the course of a protective measurement is independent of the particular time dependence of the coupling function, suggesting that the guiding principle for choosing the coupling function should be the minimization of the state disturbance. Our results illuminate the dynamics of protective measurement under realistic circumstances and may aid in the experimental realization of such measurements.

Figure 1

Normalized time-dependent system-apparatus coupling functions $g\left(t\right)$. The horizontal axis is in units of the measurement time $T$ and the vertical axis is in units of $1/T$. (a) Constant system-apparatus coupling $g\left(t\right)$ as defined in Eq. (5). (b) Constant system-apparatus coupling with a linear turn-on and turnoff, each of duration $\Delta T$, shown here for $\Delta T/T=0.2$. (c) Triangular system-apparatus coupling, corresponding to a linear turn-on and turnoff, each of duration $\Delta T=T/2$. (d) System-apparatus coupling following a raised-cosine function as defined in Eq. (38).

Figure 2

Comparison of the squared Fourier transform $|\tilde{g}\left({\omega }_{mn}T\right){|}^2$ for the following system-apparatus coupling functions $g\left(t\right)$ shown in Fig. 1: constant coupling $g\left(t\right)=1/T$ given by Eq. (5) (solid line), triangular $g\left(t\right)$ given by Eq. (32) with $\Delta T=T/2$ (dashed line), and raised-cosine function given by Eq. (38) (dotted line). The dimensionless quantity $|\tilde{g}\left({\omega }_{mn}T\right){|}^2$ describes the dependence of the transition probability ${\mathcal{P}}_m^{\left(1\right)}\left(T\right)$ on the dimensionless parameter ${\omega }_{mn}T$, which quantifies the ratio of the measurement time $T$ to the time scale ${\omega }_{mn}^{-1}$ associated with the transition $|n\rangle \to |m\rangle$. (a) Plot of $|\tilde{g}\left({\omega }_{mn}T\right){|}^2$ for $T\gtrsim {\omega }_{mn}^{-1}$. (b) Decay of the envelope of $|\tilde{g}\left({\omega }_{mn}T\right){|}^2$ for $T\gg {\omega }_{mn}^{-1}$.