Superposition Principle & Higher Order Interference (SP-HOI): Concepts
Superposition principle and the double slit interference
According to Richard Feynman, the Young's double slit experiment with single particles captures the essence of mysteries associated with Quantum Mechanics. The probability of finding the particle at the detector plane in a double slit experiment is usually given by \( |\psi_A + \psi_B |^2\), where \( \psi_A \) is the probability amplitude for the case when the particle allowed to go through only slit A and \(\psi_B\) is probability associated with the particle to go through slit B. To describe the intereference pattern i.e. the intensity when both slit A and slit B are open as \( |\psi_A + \psi_B |^2\), we have used two assumptions namely Superposition Principle and the Born rule. The naive superposition principle states that wavefunction when both slits are open is given by the superposition of the wavefunction for the individual slits i.e. \(\psi_{AB} = \psi_A + \psi_B \). Born rule states that, given the wavefunction is \( \psi_{AB}\), the probability of finding the particle is given by \( |\psi_{AB}|^2\).
Fundamental tests of quantum mechanics must test the validity of application of superposition principle and the Born rule. Born rule cannot be directly tested since we cannot measure the wavefunction directly. To test superposition principle directly as well, we need wavefunctions. All we can directly measure in an experiment without these two assumptions is the initensity. Thus, we need Born rule and Superposition principle together, so that we can have an equation with all the terms being intensities. We know that the difference between the intensity when two slits are open and sum of the initensity when individual slit is open is non zero. \( \kappa_2 = I_{AB}-|_{A}-I_{B} = 2\sqrt{I_A I_B} \cos(\phi) \). The disadvantage is that \(\kappa_2\) is non zero and depends on phase. Thus without accurate modelling fundamental test of superposition principle and Born rule cannot be carried out in double slit interference experiment.
Triple-slit interference
For the case of triple slit interference, however, we find that the sum and difference of intensity for various combination of slits gives us \( \kappa_3 = 0\) \( \kappa_3 = I_{ABC} - 2 I_{AB}-2 I_{BC}- 2 I_{CA} + I_{A} + I_{B}+I_{C} = 0\).
The advantage of obtaining a zero is that the above equation can be tested empirically without the need for any modelling. If in a triple slit experiment, we obtain the intensities for various slit combinations, then we can directly compute \( \kappa_3 \). If it turns out to be non-zero then we can comment that the application of the naive superposition principle together with the Born rule is not valid.
Boundary Conditions in Interference Experiments
Strictly speaking the superposition principle applies to solution of linear differential equations. If \( \psi_A\) is the solution to a linear differential equation and \(\psi_B\) is another, then any linear superposition \(\alpha \psi_A + \beta \psi_B\) is also a solution. However, linear differential equations are well posed if appropriate boundary conditions are specified. Now, the boundary conditions for the differential equation ( Maxwell's equation / Schrodinger's equation) when two slits are open is not the same for the case when only one slit is open. Since the boundary condition changes for the different combination of slits we do not expect the solution to the differential equation for two slits to be linear combination of solutions for individual slits.
We can show the same solving the Maxwell Equation/Schrodinger equation for double slits and and individual slits. Due to boundary conditions being different the naive application of superposition principle to interference experiment is wrong. Thus we expect \( \kappa_3 \) to be non-zero as well. But since we have been using superposition principle it must be giving us a well approximated answer i.e. the deviation from the actual solution must be small. One can compute the deviation using numerically solving differential equations for various interferometric scenarious but they are computationally very expensive.
Non-classical paths in Feynman path integral formalism
We can also use the Feynman path integral formalism for computing the \( \kappa_3 \) analytically (or with less computational resources if needed). Usually, the probability amplitude when two slits are open is computed by summing over all the paths from the source to the detector that crosses the slit once. If we restrict ourselves to these terms path integral formalism predicts \(\kappa_3\) to be zero. However, one can consider higher terms where the Feynman path hugs more than one slit. Such Feynman paths along with all higher order paths that crosses the slit plane multiple times together are called 'Non-classical' Feynman paths.
The contribution of such paths bring a change in the net amplitude and hence make \(\kappa_3\) non-zero.
Experiments
Path integral formalism with controntibution from non-classical paths show that the maximum deviation from superposition principle goes as \( \approx 0.03 \frac{\lambda^{3/2}}{w \sqrt{d}}\), where \(d\) and \(w\) are inter-slit distance and slit width respectively and \(\lambda \) is the wavelength. Thus unless we make very tiny slits, for optical wavelengths \(\kappa_3\) would be very small to be measured above errors. However, if we perform the experiment in microwaves we expect the deviation to be higher than the noise floor. Note that the applicability of superposition principle is as much of concern to quantum mechanics as it is for classical wave mechanics.
More about the experiment performed in this video