Abstract
We derive frequency correlation and exit probability expressions for photons generated via spontaneous parametric downconversion (SPDC) in nonlinear waveguides that exhibit linear scattering loss. Such loss is included within a general Hamiltonian formalism by connecting waveguide modes to reservoir modes with a phenomenological coupling Hamiltonian, the parameters of which are later related to the usual loss coefficients. In the limit of a low probability of SPDC pair production, the presence of loss requires that we write the usual lossless generated pair state as a reduced density operator, and we find that this density operator is naturally composed of two photon, one photon, and zero photon contributions. The biphoton probability density, or joint spectral intensity (JSI), associated with the two-photon contribution is determined not only by a phase matching term, but also by a loss matching term. The relative sizes of the loss coefficients within this term lead to three qualitatively different regimes of SPDC JSIs. If either the pump or generated photon loss is much higher than the other, the side lobes of the phase matching squared sinc function are washed out. On the other hand, if pump and generated photon loss are appropriately balanced, the lossy JSI is identical to the lossless JSI. Finally, if the generated photon loss is frequency dependent, the shape of the JSI can be altered more severely, potentially leading to generated photons that are less frequency correlated though also produced less efficiently when compared to photons generated in low-loss waveguides.
Highlights
INTRODUCTIONWith an eye toward future calculations, we see four key advantages to employing this “backward Heisenberg picture” approach, so-named because it evolves operators backward in time to ensure that their associated Schrodinger picture states correctly evolve forward in time
We limit ourselves here to consideration of photon pair generation via spontaneous parametric downconversion (SPDC), as this allows us to consider pump losses separately from generated photon losses; we expect many of the results presented here to carry over to photon pair generation via spontaneous four-wave mixing, a topic we intend to explore in detail in future work
We show that the common expression for the biphoton probability density, or joint spectral intensity (JSI), in which it is composed of just a pump pulse spectrum term and a phase matching term, should contain a loss matching term that can strongly modify its shape
Summary
With an eye toward future calculations, we see four key advantages to employing this “backward Heisenberg picture” approach, so-named because it evolves operators backward in time to ensure that their associated Schrodinger picture states correctly evolve forward in time It works within a wavevector-time framework, rather than frequency-time or position-time, enabling extensions beyond effectively one-dimensional devices to two-dimensional and three-dimensional structures.
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