Time-local Heisenberg-Langevin equations and the driven qubit

Abstract

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 (...

Open quantum systems with delayed coherent feedback

“…Appendix A. Derivation of Eq. (26) We briefly discuss here the reason why Eq. (26) is true whenever Φ (m) is divisible, for all m. We can define an auxiliary map Φ t that satisfies…”

Revisiting Quantum Feedback Control: Disentangling the Feedback‐Induced Phase from the Corresponding Amplitude

“…The first setup we consider is a two-level system (2LS) in front of a mirror. [39,47,52,58,63,69,70] The 2LS is pumped with a pulsed laser field Ω(t) which controls the emission statistics via its pulse area A. Looking at the emission without feedback, we observe that for a pulse with A = the 2LS is inverted and acts as a single-photon source.…”

“…As a consequence, the non-Markovianity of the dynamics needs to be taken into account. [37][38][39][40][41][42][43][44][45][46] Various setups to control quantum few-level systems via timedelayed feedback have been studied theoretically and it has been shown that it is possible to control characteristic quantities such as the photon-photon correlation and the concurrence which functions as a measure of entanglement. [47][48][49][50][51][52][53][54][55][56][57][58][59] In these systems, in general, the control parameters that can be used to evoke the desired behavior are the delay time and the characteristic frequency which, depending on the considered setup, can be, for example, the frequency of an involved optical transition or the frequency of a cavity mode.…”

“…[58] If the previously studied systems are extended in a way that allows its application, our scheme potentially enables and simplifies the control of the emerging phenomena which are instanced to inspire new applications in the field of coherent quantum feedback. [39,53,[60][61][62][63] Subsequently, in Section 4 we derive our new control scheme in detail and compare it to the scheme in which only the delay time is used as a control parameter. We demonstrate that the application of a microwave pump field opens up new possibilities in potential experimental realizations and allows a fast and efficient stabilization of the excitation as well as population trapping.…”

Stabilizing quantum coherence against pure dephasing in the presence of time-delayed coherent feedback at finite temperature

“…An example of such a structured reservoir that we consider is a coherent, feedback-type, reservoir-LB interaction, which can be used to steer the dynamics of the TLE with the goal of coherence preservation. Moreover, even though there are numerical methods that in principle are capable of characterizing the effect of quantum coherent feedback at a finite temperature [65,66], to our knowledge, this is the first work to report on the exact influence of this effect on the amount of recovered coherence.…”