We present an elementary derivation and generalisation of a recently reported method of simulating feedback in open quantum systems. We use our generalised method to simulate systems with multiple delays, as well as cascaded systems with delayed backscatter. In addition, we derive a generalisation of the quantum regression formula that applies to systems with delayed feedback, and show how to use the formula to compute two-time correlation functions of the system as well as output field properties. Finally, we show that delayed coherent feedback can be simulated as a quantum teleportation protocol that requires only Markovian resources, pre-shared entanglement, and time travel. The requirement for time travel can be avoided by using a probabilistic protocol.
We present an algorithm to simulate genuine, measurement-conditioned quantum trajectories for a class of non-Markovian systems, using a collision model for the environment. We derive two versions of the algorithm, the first corresponding to photodetection and the second to homodyne detection with a finite local oscillator amplitude. We use the algorithm to simulate trajectories for a system with delayed coherent feedback, as well as a system with a continuous memory.
The time-local master equation for a driven boson system interacting with a boson environment is derived by way of a time-local Heisenberg-Langevin equation. Extension to the driven qubit fails-except for weak excitation-due to the lost linearity of the system-environment interaction. We show that a reported time-local master equation for the driven qubit is incorrect. As a corollary to our demonstration, we also uncover odd asymptotic behavior in the "repackaged" time-local dynamics of a system driven to a far-from-equilibrium steady state: the density operator becomes steady while time-dependent coefficients oscillate (with periodic singularities) forever. Issues of formal exactness aside, the current move from given quantum systems-an atom or radiation field in a cavity-to engineered systems, drives a more pragmatic interest in the theory of non-Markovian open quantum systems. While generalizations of input-output theory to the non-Markovian regime have been considered [5][6][7], more commonly the Schrödinger picture is adopted, where, after the work of Hu et al. Treatments in the[8], time-local master equations are derived [9][10][11][12][13][14][15][16]. Of particular interest in this paper are the time-local master equation for a driven boson system in interaction with a boson environment (first derived in [11]) and the equation for spontaneous emission from a two-state system, or qubit [9][10][11][12]. They invite a conflation: a time-local master equation for the driven qubit.The driven qubit is an important example, considering its role in quantum information science and the numerous physical realizations. The question of a time-local master equation is an old one. It is raised in Sec. IVA of Ref. [11], where, after first noting obstacles to its derivation, the author mentions an equation reported in a preprint [18], following with: "Such a relatively simple result seems to be inconsistent with the conclusion reached above about the difficulty of dealing with the two level atom problem, and is hence an issue that requires further consideration." The noted equation is absent from the published version of [18] (Ref. [10]), which can be seen to endorse the call for "further consideration."The "relatively simple result" substitutes qubit raising and lowering operators for the creation and annihilation operators in the time-local master equation for the driven boson system. Having appeared first in [18], it reappears in a recent work of Shen et al. [17], along with a derivation from Feynman-Vernon influence functional theory which invokes a coherent state representation in Grassmannian variables for the qubit state. We show in this paper that the derived time-local master equation is incorrect; it is not influence functional theory and coherent-state path integrals that yield a successful derivation, but linearity, which for the driven qubit is lost. While in spontaneous emission [9][10][11][12] linearity is effectively retained-due to the one-quantum truncation-for the driven qubit, multiphoton scattering (...
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