In the quantum eraser double slit experiment, does the photon (or wavefunction) pass through one slit or both slits when different polarizers are placed over the slits?
According to quantum physics, when certain different polarizers are placed over the slits in the double-slit experiment (for instance, one vertical and one horizontal polarizer, or one circular clockwise and one circular counter-clockwise), thus "marking" each photon with which-way information, the photon indeed passes through only one slit, resulting in no interference. If you cover up one of the slits, you'll observe the very same absence of interference.
Quantum mechanics is a theory that can only predict probability distributions. It cannot predict trajectories. It is ruled by differential equations which have as solutions the wavefunctions, and the complex conjugate square of the wavefunction gives the probability of a specific, photon, electron, to be at (x,y,z,t) given the boundary conditions of the problem.
The importance of the boundary conditions is demonstrated clearly in this experiment.
In the experiment you refer to, both the boundary conditions and the initial states are changed during the experiment and it is not surprising that the interference disappears when the boundary condition constrains the slit through which a photon passed. The fact that by further manipulation an interference pattern appears, is again due to the wavefunction's probabilistic waving.
Change of boundary conditions changes the wavefunctions, the boundary conditions are not only "photon scatter from two slits" , but "photon scatter from two slits and all paraphernalia like polarizers upstream and downstream". If one really wanted to solve for the specific quantum mechanical problem all these should be taken into account when picking the wavefunction that describes the probability of the photon scatter.
Single photon double slit experiments for classrooms can be seen here.
It is the photon wavefunction that has a probability for the photon to pass through a slit, and has a sinusoidal dependence that gives interference patterns in the accumulation. The photon, when detected is detected as a point particle interaction with a screen or a ccd or a photomultiplier. One has never seen experimentally the signal of an elementary particle spread in space. Thus it is not the photon that is waving, but its probability of manifesting at (x,y,z,t).