Adiabatic elimination is a standard tool in quantum optics, which produces an effective Hamiltonian for a relevant subspace of states, incorporating effects of its coupling to states with much higher unperturbed energy. It shares with techniques from other fields the emphasis on the existence of widely separated scales. Given this fact, the question arises whether it is feasible to improve on the adiabatic approximation, similarly to some of those other approaches. A number of authors have addressed the issue from the quantum optics/atomic physics perspective, and have run into the issue of non-hermiticity of the effective Hamiltonian improved beyond the adiabatic approximation, which poses conceptual and practical problems. Here, we first briefly survey methods present in the physics literature. Next, we rewrite the problems addressed by the adiabatic elimination technique to make apparent the fact that they are singular perturbation problems from the point of view of dynamical systems. We apply the invariant manifold method for singular perturbation problems to this case, and show that this method produces the equation named after Bloch in nuclear physics. Given the wide separation of scales, it becomes intuitive that the Bloch equation admits iterative/perturbative solutions. We show, using a fixed point theorem, that indeed the iteration converges to a perturbative solution that produces in turn an exact Hamiltonian for the relevant subspace. We propose thus several sequences of effective Hamiltonians, starting with the adiabatic elimination and improving on it. We show the origin of the non-hermiticity, and that it is inessential given the isospectrality of the effective non-hermitian operator and a corresponding effective hermitian operator, which we build. We propose an application of the introduced techniques to periodic Hamiltonians.
Explicit formulas for adiabatic elimination with fast unitary dynamics
Laboratoire de Physique de l’Ecole normale supérieure（巴黎高等师范学院物理实验室）；ENS-PSL（巴黎文理研究大学高等师范学院）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Centre Automatique et Systèmes (CAS)（自动与系统中心）；Mines Paris, Université PSL（巴黎矿业学院，巴黎文理研究大学）；Sorbonne Université（索邦大学）；Université Paris Cité（巴黎西岱大学）；Dept. of Electronics and Information Science, UGent（根特大学电子与信息科学系）
Realizing leakage elimination operator-based adiabatic speedup on a superconducting quantum processor
College of Physics and Optoelectronic Engineering, Ocean University of China（中国海洋大学物理与光电工程学院）；China Mobile (Suzhou) Software Technology Co., Ltd（中国移动（苏州）软件技术有限公司）；School of Sciences, Qingdao University of Technology（青岛理工大学理学院）
Adiabatic elimination and Wigner function approach in microscopic derivation of Open Quantum Brownian Motion
University of KwaZulu-Natal（夸祖鲁-纳塔尔大学）；North-West University（西北大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；Stellenbosch University（斯泰伦博斯大学）
Collective atomic correlations in absorptive optical bistability without adiabatic elimination: exemplifying nonclassicality from a linearized treatment of fluctuations
我们确定了吸收光学双稳中集体原子自由度的非相干谱、压缩特性和二阶关联函数。这是通过在正P表示中的线性化Fokker-Planck方程来实现的，由[H. J. Carmichael，Phys.Rev.A33，3262（1986）]，其不采用绝热消除。我们重点关注弱和强腔内激发的机制，解决好腔和坏腔限制以及集体强耦合的限制。利用与系综耦合的辅助谐振腔对腔内场的绝热消除，通过形成集体发射通道来探测原子关联。我们比较了相应的表达式的前向散射光的实验结果，讨论了稳态半经典双稳态曲线的下分支和上分支之间的关键差异。我们的分析是在远离转折点的稳定状态下进行的，在那里可以自洽地获得解析表达式，表明与经典行为有明显的偏离。
Effective Hamiltonians in Cavity and Waveguide QED from Transition-Operator Diagrammatic Perturbation Theory
Telecom Paris, Institut Polytechnique de Paris（巴黎电信学院，巴黎理工学院）；Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS - Université de Bourgogne Franche-Comté（勃艮第卡诺跨学科实验室，CNRS联合研究单位6303，勃艮第弗朗什-孔泰大学）；Technical University of Munich, TUM School of Natural Sciences, Physics Department（慕尼黑工业大学，自然科学学院，物理系）；Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften（瓦尔特·迈斯纳研究所，巴伐利亚科学院）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Centre for Nanosciences and Nanotechnology, Université Paris-Saclay（纳米科学与工程中心，巴黎萨克雷大学）
