Frequency-Filtered Photon Correlations

Abstract

In quantum optics, the standard approach for measuring and calculating frequency-filtered photon correlations is to filter the source field of interest with a Lorentzian-type filter, e.g., a tunable single-mode cavity or detector atom. However, given the inverse relation between a filter’s bandwidth and temporal response, there is trade-off between the frequency isolation and temporal response of the filter. A broad bandwidth results in a faster temporal response with more accurately measured photon correlations, yet the slow decaying tails of a Lorentzian distribution can allow for non-target frequency photons to pass through the filter. Conversely, a narrow filter bandwidth results in more effective frequency isolation, yet a slow temporal response, potentially changing the nature of the emitted photon correlations. The aim of this work is to develop a theoretical filtering technique that is simple to implement and offers an effective method of calculating frequency-filtered photon correlations. We model our filter as a multi-mode array filter, which consists of an array of tunable single-mode cavities that are equally spaced in frequency. By introducing a mode-dependent phase modulation, we produce a near rectangular frequency response, allowing us to increase the filter bandwidth – and thus the temporal response – without sacrificing frequency isolation. To ensure the filter has no effect on the evolution of the source system, we couple the source system using a cascaded quantum open systems approach. The complete lack of back-action of the filter onto the source system allows us to derive a closed set of operator moment equations for source and filter system operators. This provides an extremely effective and computationally efficient way to calculate frequency-filtered first- and second-order correlation functions. By coupling the target field into two multi-mode array filters, we can set the resonance of the two filters to two different transitions, and thus calculate frequency-filtered cross-correlation functions. We demonstrate this novel filtering method by applying it to two different driven quantum systems: a resonantly driven two-level atom and a three-level ladder-type atom driven at two-photon resonance. We present results of frequency-filtered power spectrum to demonstrate the improved frequency isolation of the multi-mode array filter over the single-mode filter. We then present results for the single-mode and multi-mode array filtered second-order auto- and cross-correlation functions. These are compared against expressions derived in the secular approximation. The improved frequency isolation of the multi-mode array filter allows us to investigate new areas of frequency-filtered photon correlations, such as two-photon leapfrog processes, and the effect of vanishing bandwidth on filtered auto-correlation functions.