Abstract
We develop a new computational tool and framework for characterizing the scattering of photons by energy-nonconserving Hamiltonians into unidirectional (chiral) waveguides, for example, with coherent pulsed excitation. The temporal waveguide modes are a natural basis for characterizing scattering in quantum optics, and afford a powerful technique based on a coarse discretization of time. This overcomes limitations imposed by singularities in the waveguide-system coupling. Moreover, the integrated discretized equations can be faithfully converted to a continuous-time result by taking the appropriate limit. This approach provides a complete solution to the scattered photon field in the waveguide, and can also be used to track system-waveguide entanglement during evolution. We further develop a direct connection between quantum measurement theory and evolution of the scattered field, demonstrating the correspondence between quantum trajectories and the scattered photon state. Our method is most applicable when the number of photons scattered is known to be small, i.e. for a single-photon or photon-pair source. We illustrate two examples: analytical solutions for short laser pulses scattering off a two-level system and numerically exact solutions for short laser pulses scattering off a spontaneous parametric downconversion (SPDC) or spontaneous four-wave mixing (SFWM) source. Finally, we note that our technique can easily be extended to systems with multiple ground states and generalized scattering problems with both finite photon number input and coherent state drive, potentially enhancing the understanding of, e.g., light-matter entanglement and photon phase gates.
Highlights
A central object of study in quantum optics is a finite-dimensional quantum system coupled to a bath with an infinite number of degrees of freedom
Quantum optical methods initially focused on the dynamics of the system, i.e. by tracing out the state of the waveguide, often using assumptions that are violated in practice
The canonical Lindblad master equation governing these reduced dynamics [2] was originally derived under the assumption that the state of the bath and the system factorizes at all times
Summary
A central object of study in quantum optics is a finite-dimensional quantum system (e.g. an atom, quantum dot, superconducting circuit) coupled to a bath with an infinite number of degrees of freedom. We illustrate our technique first for a single waveguide coupled to a low-dimensional system (Fig. 1a) and extend the formalism to handle multiple waveguides (Fig. 1b) This technique, to our knowledge, is the first to provide a general method of obtaining the scattered state vector in the case where coherent laser pulses are incident on a system and source the energy of the scattered photons. Our interest in this problem is to understand the dynamics of few-photon sources as potential state generators for quantum communication or computation applications [49] To this end, we detail how our technique based on coarse-graining of time [45, 46, 48, 50] can be used to show how a coherently driven two-level system acts as a single- or two-photon source and spontaneous parametric downconversion or four-wave mixing act as photon pair sources. IV we discuss two prototypical examples of single- and two-photon sources, based on two-level systems and spontaneous parametric downconversion or four-wave mixing
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