Time-delay interferometry is put forward to improve the signal-to-noise ratio of space-borne gravitational wave detectors by canceling the large laser phase noise with different combinations of measured data. Based on the Michelson data combination, the sensitivity function of the detector can be obtained by averaging the all-sky wave source positions. At present, there are two main methods to encode gravitational wave signal into detector. One is to adapt gravitational wave polarization angle depending on the arm orientation in the gravitational wave frame, and the other is to divide the gravitational wave signal into plus and cross polarizations in the detector frame. Although there are some attempts using the first method to provide the analytical expression of sensitivity function, only a semianalytical one could be obtained. Here, starting with the second method, we demonstrate the equivalence of both methods. First time to obtain the full analytical expression of sensitivity function, which provides a fast and accurate mean to evaluate and compare the performance of different space-borne detectors, such as LISA and TianQin.

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The atom interferometry is a promising tool for the detection of the gravitational wave (GW) in the mid-frequency band ranging from 0.01 Hz to 10 Hz. Particularly, the multi-arm atom interference detectors can help to extract better data with the removal of fake detection. The existing models for GW detection with atom interferometry, however, are mainly focused on the one-dimensional situation, and cannot be directly applied to the multi-arm frame of the GW detection. In this work, we develop a universal theoretical model for GW detectors based on atom interferometry, in which the eikonal equation under general relativity is adopted to calculate the perturbation of the GW on light propagation, and the dependence of the response function on the azimuth is also discussed. We further analyze the dominant noise sources, which limits the detection capabilities of detector. This work provides a complete model for GW detection with atom interferometry, with the ability to analyze the performance of the multi-arm atom interference detectors.

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