Timing in quantum measurements of position and momentum

The prototype for a simultaneous measurement of two conjugate variables was originally introduced by Arthurs and Kelly in 1965. It relies on coupling the quantum particle to be probed to two additional systems, which serve as measurement pointers. In this contribution we investigate an extended scheme to measure position and momentum of a massive particle. By considering an explictly time-dependent coupling we can quantify the timing of the measurement. We investigate how the noise resulting from such a measurement process depends on the interaction strength and the size of the pointers. In particular, we focus on the question of which measurement timing minimizes the corresponding uncertainty product.

Figure 1

Pointer-pointer coupling coefficient ${\Gamma }_{\xi }$, Eq. (29), using the interaction coefficients Eqs. (33) and (34) with parameters $t_0=0.5$ and $w=5\times {10}^{-2}$. Note that the time $t\to \infty$, for which ${\Gamma }_{\xi }(t)$ is depicted here, lies well outside the interaction region. For a delay time $t_{\mathrm{d}}>0$ the interaction with the $\xi$-pointer precedes the $\pi$-pointer interaction, which results in smaller values of ${\Gamma }_{\xi }$ and a diminishing influence of the momentum pointer on the position pointer. For $t_{\mathrm{d}}<0$ the opposite is true.

Figure 2

Time dependence of the uncertainty product $\Delta \mathcal{X}\Delta \mathcal{P}$ (thick line) for (a) weak interaction ${\xi }_0={\pi }_0=\mu /10$, (b) balanced interaction ${\xi }_0={\pi }_0=\mu$, and (c) strong interaction ${\xi }_0={\pi }_0=10\mu$. Here both interaction pulses, Eqs. (33) and (34), were assumed to be equally strong and the offset $t_0=0.5$ as well as the width $w=5\times {10}^{-2}$ were chosen such that the interaction at $t=0$ is negligible. The timing was optimized for the time delay $t_{\mathrm{d}}$ in such a way that it minimizes the uncertainty product. For comparison, we also show the optimally delayed interaction pulses $\xi (t)$ (solid red line) and $\pi (t)$ (dashed blue line). Note that for weak as well as for strong interactions relative to the mass ratio $\mu$, Eq. (13), the optimal time delay is approximately zero, while in (b) one notices a significantly earlier momentum coupling. Note further that the order of magnitude of the uncertainty product for a weak interaction (a) differs considerably from the one in (b) and (c). The latter two almost touch the Arthurs and Kelly result, Eq. (42). Moreover, we realize that the minimum of the uncertainty product (cross lines) always lies within the interaction window. This minimum indicates the readout time at which we have to read the pointers. In (c) we even have to read the pointers before the interaction has reached its maximum strength.

Figure 3

Optimal time delay $t_{\mathrm{d}}^{(\mathrm{opt})}$ between the two interaction pulses $\xi (t)$ and $\pi (t)$, Eqs. (33) and (34), as a function of equal interaction strengths ${\xi }_0={\pi }_0$. All other parameters were chosen as in Fig. 2. Note that for weak as well as for strong interactions relative to the mass ratio $\mu$, Eq. (13), the optimal time delay tends to zero, while it drops significantly around ${\xi }_0\approx \mu$. The inset shows this region in more detail. Note further that $t_{\mathrm{d}}^{(\mathrm{opt})}$ is always negative, suggesting that an early momentum interaction is generally preferable.

Figure 4

Minimum of the uncertainty product $\Delta \mathcal{X}\Delta \mathcal{P}$, Eqs. (38) and (39), as a function of the interaction strength ${\xi }_0={\pi }_0$. Note that in general $t_{\mathrm{d}}^{(\mathrm{opt})}$ as well as the optimal readout time differ for different interaction strengths (compare Fig. 2). Only for ${\xi }_0\ll \mu$ is the uncertainty product considerably larger than the Arthurs and Kelly limit, Eq. (42). For stronger interactions the uncertainty product converges to this limit. From the inset we note a clear dip around ${\xi }_0\approx \mu$. Here, the corresponding time delay is significantly less than zero (compare Fig. 3), i.e., a nonsimultaneous measurement is at hand here, as discussed in the main text. For comparison, we also show $\Delta \mathcal{X}\Delta \mathcal{P}$ for the corresponding simultaneous measurement, that is, for $t_{\mathrm{d}}=0$ (dashed line). Clearly in this case the simultaneous measurement is less optimal.