QLunch: Frederik Nathan
Speaker: Frederik Nathan, NBI
Title: Universal Lindblad Equation for open quantum systems
Abstract: Most, if not all, quantum systems in the real world are open: coupled to their surrounding environment in some way. Accounting for this fact in the theoretical description of quantum systems is a nontrivial problem subject of ongoing work. Here, Lindblad, or GKSL, equations play a crucial role, by providing the most general CPTP formulation of open quantum system dynamics in the Markovian regime where the system-bath coupling is weak enough (relative to the timescale of environmental fluctuations) that the density matrix effectively evolves through a time-local equation of motion. Moreover, Lindblad equations offer valuable intuition and efficient solutions via stochastic wavefunction evolution.
Historically, obtaining a Lindblad equation for a problem required two approximations: (1) the Born-Markov approximation, which assumes the system-environment coupling weak relative to the correlation time of the environmental fluctuations, and (2) the secular approximation, which further assumes the effective system-environment coupling weak relative to all Bohr frequencies (level splittings) of the system. While the Born-Markov approximation is broadly relevant, the secular approximation is very restrictive and precludes a wide range of important problems that have dense spectra, including many-body or multi-component systems, systems with nonlinear potentials, and driven quantum systems.
In this talk, I show that the secular approximation is not necessary to obtain an accurate Lindblad description of an open quantum systems. To this end, I present a universal Lindblad equation (ULE), that only makes use of the Born-Markov approximation. The ULE involves a single jump operator with a quasi-local structure for extended systems, whose support is controlled by the product of the Lieb Robinson velocity of the system and the correlation time of the bath. I provide rigorous bound on the approximation induced error of the ULE, which are controlled by a dimensionless Markovianity parameter measuring the ratio of bath correlation time and the characteristic timescale of the system-environment coupling. I use this to establish a bound for the deviation of the ULE steady state from the exact steady-state of a given problem, and show that the deviation scales to zero with the Markovianity parameter. I discuss past and future potential applications of the ULE in quantum information processing, solid state physics, and beyond.