The thermodynamics of a quantum system interacting with an environment that can be assimilated to a harmonic oscillator bath has been extensively investigated theoretically. In recent experiments, the system under study however does not interact directly with the bath, but though a cavity or a transmission line. The influence on the system from the bath is therefore seen through an intermediate system, which modifies the characteristics of this influence. Here we first show that this problem is elegantly solved by a transform, which we call the Vernon transform, mapping influence action kernels on influence action kernels. We also show that the Vernon transform takes a particularly simple form in the Fourier domain, though it then must be interpreted with some care. Second, leveraging results in quantum thermodynamics we show how the Vernon transform can also be used to compute the generating function of energy changes in the environment. We work out the example of a system interacting with two baths of the Caldeira-Leggett type, each of them seen through a cavity.
Higher-order quantum thermodynamics: equilibrium and causal structure
Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University（赫瑞-瓦特大学工程与物理科学学院光子与量子科学研究所）；School of Physics, Trinity College Dublin（都柏林三一学院物理学院）；Fakultät für Mathematik, Universität Wien（维也纳大学数学学院）；Department of Physics, University of Helsinki（赫尔辛基大学物理系）；HUN-REN Wigner Research Centre for Physics（匈牙利科学院维格纳物理研究中心）；Algorithmiq Ltd（Algorithmiq有限公司）；Mathematical Institute, Slovak Academy of Sciences（斯洛伐克科学院数学研究所）；Center for Quantum Engineering, Research, and Education, TCG CREST（TCG CREST量子工程、研究与教育中心）；QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院QICI量子信息与计算倡议）；Quantum Group, Department of Computer Science, University of Oxford（牛津大学计算机科学系量子组）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
Comparison between time-independent and time-dependent quantum systems in the context of energy, Heisenberg uncertainty, average energy, force, average force and thermodynamic quantities
