Problem II (30 marks). Let us recall the double-slit experiment setting with a device measuring which slit the atom goes through (please refer to lecture notes). Without the measurement device, the spatial wavefunction of the atom passing the two slits is assumed to be Zallur)+(4L)]. In the presence of an cavity aiming to detect which-pathway information, the wavefunction of the total system (particle of interest plus the measuring device) would be tallur)le) + | UL)\g)], where le) and (g) represent the excited state or the ground state of the additional cavity as the measuring device. (a) By calculating the reduced density matrix for the particle undergoing the double-slit experiment, show that no double slit interference should be expected. (b) Using this understanding and some simple equations if needed, qualitatively describe how system-environment interaction would in general reduce the visibility of double- slit interference patterns. (c) If we now measure the cavity's state in terms of the basis of talle)+9)] and only keep a subset of cases where the cavity is found to be in the state of talle) + \9)]. Under these assumptions show that conditional upon this subset of data, we do expect double- slit interference pattern to reappear again. This is roughly the basis of the physics underlying the so-called "quantum erasure". Problem II (30 marks). Let us recall the double-slit experiment setting with a device measuring which slit the atom goes through (please refer to lecture notes). Without the measurement device, the spatial wavefunction of the atom passing the two slits is assumed to be Zallur)+(4L)]. In the presence of an cavity aiming to detect which-pathway information, the wavefunction of the total system (particle of interest plus the measuring device) would be tallur)le) + | UL)\g)], where le) and (g) represent the excited state or the ground state of the additional cavity as the measuring device. (a) By calculating the reduced density matrix for the particle undergoing the double-slit experiment, show that no double slit interference should be expected. (b) Using this understanding and some simple equations if needed, qualitatively describe how system-environment interaction would in general reduce the visibility of double- slit interference patterns. (c) If we now measure the cavity's state in terms of the basis of talle)+9)] and only keep a subset of cases where the cavity is found to be in the state of talle) + \9)]. Under these assumptions show that conditional upon this subset of data, we do expect double- slit interference pattern to reappear again. This is roughly the basis of the physics underlying the so-called "quantum erasure