Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 6 / c

Let $S$ denote immediate transmission through the BS1 subsystem between Bob's lower-right port and Alice's upper-left port; let $L$ denote transmission after one visit to each internal mirror. From the same beam splitter rules as in b, the corresponding amplitudes in either direction areup to a common fixed two-mirror propagation phase. The minus sign in $L$ comes from the $1^{\prime }\to 1$ or reverse $1\to 1^{\prime }$ scattering. The two paths selected by D0's time gate are $L$ on the outward journey followed by $S$ on the return, and $S$ outward followed by $L$ on the return. They have the same total delay and fixed propagation phase, so their probability amplitudes interfere.

For the first history, Alice applies $e^{i{\varphi }_A}$ to the late component, while Bob's return-time modulation is not encountered in the return delay arm. For the second, Alice's early component is unmodulated and Bob applies $e^{i{\varphi }_B}$ in the return delay arm. Bob must time his modulation to the return passage, leaving the outward reference preparation unchanged; this is implicit in the stated encoding. Each enabled modulator is traversed twice, so its $\pi /2$ single-pass phase becomes the stipulated $\pi$.

Include the amplitude $c$ to enter BS1 through BS2, and the amplitude of modulus $c$ to reach D0 through BS2 on return. Both histories therefore contain six 50:50 beam splitter factors. Ignoring one common unobservable global phase,Hence the unconditional probability per emitted photon, with no renormalization after discarding paths, isFor the allowed values $0,\pi$, this is $1/16$ when ${\varphi }_A={\varphi }_B$ and zero otherwise. The ignored trajectories exit other ports or other time bins; excluding them from D0's acceptance does not increase the success probability. Equal mirror reflection phases, channel phases and Alice's mirror phase multiply both accepted histories equally and cancel from the interference condition. A residual relative phase would need calibration.