A theoretical particle-number conserving quantum field theory based on the concept of imaginary time is presented and applied to the scenario of a coherent atomic laser field at ultra-cold temperatures. The proposed theoretical model describes the analytical derivation of the frequency comb spectrum for an atomic laser realized from modeling a coherent atomic beam of condensate and non-condensate quantum field components released from a trapped Bose-Einstein condensate at a given repetition phase and frequency. The condensate part of the atomic vapor is assumed to be subjected to thermal noise induced by the temperature of the surrounding thermal atomic cloud. This new quantum approach uses time periodicity and an orthogonal decomposition in a complex-valued quantum field representation to derive and model the quantum field's forward- and backward-propagating components as a standing wave field in the same unique time and temperature domain without singularities at finite temperatures. The complex-valued atom laser field, the resulting frequency comb, and the repetition frequency distribution with the varying shape of envelopes are numerically monitored within a quantitative Monte-Carlo sampling method, as a function of temperature and trap frequency of the external confinement.

翻译：提出了一种基于虚时概念的粒子数守恒量子场论理论方法，并将其应用于超低温相干原子激光场场景。该理论模型描述了由从囚禁玻色-爱因斯坦凝聚体中释放的凝聚态与非凝聚态量子场分量组成的相干原子束建模所实现的原子激光频率梳光谱的解析推导。假设原子蒸汽的凝聚部分受到周围热原子云温度引起的热噪声影响。这一新的量子方法利用时间周期性和复值量子场表示下的正交分解，将量子场的前向与后向传播分量推导并建模为同一独特时空域中无奇点的驻波场。通过定量蒙特卡洛采样方法，数值监测了复值原子激光场、所得频率梳及包络形状变化的重复频率分布随外部囚禁势的温度和阱频的变化。