A lossless beamsplitter has certain (complex-valued) probability amplitudes for sending an incoming photon into one of two possible directions. We use elementary laws of classical and quantum optics to obtain general relations among the magnitudes and phases of these probability amplitudes. Proceeding to examine a pair of (nearly) single-mode wavepackets in the number-states n1 and n2 that simultaneously arrive at the splitter's input ports, we find the distribution of photon-number states at the output ports using an argument inspired by Feynman's scattering analysis of indistinguishable Bose particles. The result thus obtained coincides with that of the standard quantum-optical treatment of beamsplitters via annihilation and creation operators â and â. A simple application of the Feynman method provides a form of justification for the Bose enhancement implicit in the well-known formulas ân=nn1 and ân=n+1n+1.

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For there to be an arrival time and an exit time, the incoming electromagnetic fields must be in the form of multimode wavepackets with large cross-sectional areas and long durations to properly approximate single-mode photon-number states—hence our use of the qualifier “nearly” when referencing single-mode wavepackets. When discussing two packets that arrive simultaneously at the input ports 1 and 2 of a beamsplitter, we envision identical packets whose leading edges arrive simultaneously at the entrance ports. In a thought experiment, one may imagine packets as large as they need to be for the approximation to reach the desired level of accuracy. In practice, optical wavepackets with cross-sectional areas of about 1cm2 and durations of a few nanoseconds will probably suffice.
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The symmetry is not needed for our initial discussion, but will be invoked later, when we extend the argument to the case of incoming wavepackets with n1 and n2 photons simultaneously arriving at ports 1 and 2. Strictly speaking, even in the latter case, the symmetry of the splitter is merely convenient but not necessary for the validity of the argument.
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Assigning a probability to stimulated emission when none of the incident photons participate in the process might seem questionable. Recall, however, that the quantum rule â0=1 allows for the stimulation of the excited atom by vacuum fluctuations.
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