Uncertainty in the context of multislit interferometry

A pair of uncertainty relations relevant for quantum states of multislit interferometry is derived, based on the mutually commuting “modular” position and momentum operators and their complementary counterparts, originally introduced by Aharonov and co-workers. We provide a precise argument as to why these relations are superior to the standard Heisenberg uncertainty relation at expressing the complementarity between spatial localization and the appearance of fringes. We further support the argument with explicit computations involving wave functions specifically tailored to the interference setup. Conceptually developing the idea of Aharonov and co-workers, we show how the modular momentum should reflect the given experimental setup, yielding a refined observable that accurately captures the fine structure of the interference pattern.

Figure 1

Intensity profiles associated with the double-slit state ${\psi }_2$ are depicted, in (a) position space and (b) momentum space. Note in particular that ${\widehat{\psi }}_2\left(k\right)$ vanishes at $k=(j+1/2)K$, with integer $j$.

Figure 2

In (a) $Q_T$ is depicted in position space; it is defined indirectly through relations (8) and (5). Panel (b) illustrates $P_{\mathrm{mod}}$ in momentum space, as defined in (9).

Figure 3

The slit number $m$ versus the standard deviation of $P_{\mathrm{mod}}$ in state ${\psi }_m$ is depicted (crosses on dotted line) for $T=5$, and also $P_{\mathrm{mod}}\left(m\right)$ (dots on dashed line), which is discussed in Sec. 6c.

Figure 4

The interference patterns associated with ${\psi }_2$, ${\psi }_4$, and ${\psi }_8$ are depicted; their recursive relationship is illustrated. Panel (a) shows $|{\widehat{\psi }}_2\left(k\right){|}^2$, the first interference state and the sinc envelope (dotted line). Panel (b) shows $|{\widehat{\psi }}_4\left(k\right){|}^2$ (solid line); $|{\widehat{\psi }}_2\left(k\right){|}^2$ (dotted) serves as the envelope to $|{\widehat{\psi }}_4\left(k\right){|}^2$. Panel (c) shows $|{\widehat{\psi }}_8\left(k\right){|}^2$ (solid line); $|{\widehat{\psi }}_4\left(k\right){|}^2$ (dotted) serves as the envelope to $|{\widehat{\psi }}_8\left(k\right){|}^2$. Note that the depicted states are not normalized relative to each other so as to better demonstrate the shape and recursive relationship of the depicted states.

Figure 5

Values of fringe width associated with uniformly illuminated apertures versus the number of illuminated slits $m$, for $T=5$. The values assigned by $\Delta (P_{\mathrm{mod}},{\psi }_m)$ are represented using crosses, whereas using dots we indicate the values of $\Delta (P_{\mathrm{mod}}\left(m\right),{\psi }_m)$. Compare Fig. 3.

Figure 6

Convergence of the uncertainty product (41) for uniformly illuminated apertures of even slit number.

Figure 7

The two different ways of arranging the slits of an aperture in pairs are depicted for an aperture of eight slits. In (a), the usual way of pairing up slits across the origin is shown; this results in a sum. The numbers indicate the distance between the two members of the pair; they are in units of “slit separations.” In (b) an alternative way of pairing the slits is shown, which underlies the product form. The numbers indicate the distance of the successive pairings in units of “slit separations,” down to the unit of a single pair of slits.