1.2.5. Ruling Out Multi-Order Interference in Quantum Mechanics[4]

This article ruled out third- and higher-order interference and providing a bound on the accuracy of Born's rule.

The probability density to find a particle at position $\vec{r}$ and at time $t$ is given by $$P(\vec{r}, t)=\Psi^* (\vec{r}, t)\Psi(\vec{r}, t)=|\Psi(\vec{r}, t)|^2$$ A double-slit diffraction experiment is a direct consequence of this rule; the probability to detect a particle at r after passing through an aperture with two slits, A and B, is given by $$\begin{align} P_{AB}(\vec{r})&=|\Psi_A(\vec{r})+\Psi_B(\vec{r})|^2\\ &=|\Psi_A(\vec{r})|^2+|\Psi_B(\vec{r})|^2+\Psi_A^*(\vec{r})\Psi_B(\vec{r})+\Psi_A(\vec{r})\Psi_B^*(\vec{r})\\ &=P_A+P_B+I_{AB} \end{align}$$ The corresponding (second-order) interference term can be defined as $$I_{AB}:=P_{AB}-P_A-P_B$$ We define the third- order interference term $I_{ABC}$ for a three-path configuration (mutually exclusive) as the deviation of $P_{ABC}$ from the sum of the individual probabilities and the second-order interference terms: $$I_{ABC}:=P_{ABC}-P_{AB}-P_{BC}-P_{AC}+P_A+P_B+P_C$$

The nonzero interference term $I_{AB}$ is expected in all wave theories, including quantum mechanics. The next higher-order (i.e., three-path) interference term $I_{ABC}$ will be zero in all wave theories, with a square-law relation between the field energy (or probability density) and field amplitude, which is the case in quantum mechanics with Born's rule. Moreover, if there is no interference at a certain level in the hierarchy, the higher-order terms must vanish as well.

We expect the three-path interference term $I_{ABC}$ to be zero, with the advantage of being independent of many experimental parameters, thus enabling a more precise null test for Born's rule. $$\varepsilon =p_{ABC} − p_{AB} − p_{AC} − p_{BC}+p_A + p_B + p_C − p_0$$

$$\begin{align} \delta&=|I_{AB}|+|I_{BC}|+|I_{AC}|\\ &=|p_{AB}-p_A-p_B+p_0|+|p_{BC}-p_B-p_C+p_0|+|p_{AC}-p_A-p_C+p_0| \end{align}$$

To measure $\kappa\equiv\frac{\varepsilon}{\delta}$ in various optical power regimes, we used different types of photon sources.

With a null experiment, a very careful analysis of random and systematic errors must be undertaken, as our bound on the amount of three-path interference will be directly related to the level of experimental uncertainty.

We are able to bound the magnitude of the third-order interference term to less than $10^{-2}$ of the regular expected second-order interference, at several detector positions. Thus, our experiment is able to rule out the existence of third-order in- terference terms (and, in effect, any higher-order interference terms) up to this bound.