Quantum speed limits in open systems: non-Markovian dynamics without rotating-wave approximation

Abstract

We derive an easily computable quantum speed limit (QSL) time bound for open systems whose initial states can be chosen as either pure or mixed states. Moreover, this QSL time is applicable to either Markovian or non-Markovian dynamics. By using of a hierarchy equation method, we numerically study the QSL time bound in a qubit system interacting with a single broadened cavity mode without rotating-wave, Born and Markovian approximation. By comparing with rotating-wave approximation (RWA) results, we show that the counter-rotating terms are helpful to increase evolution speed. The problem of non-Markovianity is also considered. We find that for non-RWA cases, increasing system-bath coupling can not always enhance the non-Markovianity, which is qualitatively different from the results with RWA. When considering the relation between QSL and non-Markovianity, we find that for small broadening widths of the cavity mode, non-Markovianity can increase the evolution speed in either RWA or non-RWA cases, while, for larger broadening widths, it is not true for non-RWA cases.

Figure 1. Quantum speed limit time versus different parameter γ (in units of ω₀).

The initial state is pure state. Different QSL time definitions are shown. are derived from the operator norm, trace norm and Hilbert-Schmidt norm in Ref. . The actual driving time τ = 30.

Figure 2

(a) Quantum speed limit time versus different parameter γ (in units of ω₀). The broadening-width parameter λ = 0.03 (in units of ω₀). The initial state is chosen as a mixed state in Eq. (19) with mixed parameter p = 0.8. (b) Measure of non-Markovianity M versus different parameter γ. The cases of non-RWA (black solid line with dots) and RWA (blue solid line with circles) are plotted. The actual driving time τ = 30.

Figure 3

(a) Quantum speed limit time versus different parameter γ (in units of ω₀). The broadening-width parameter λ = 0.1 (in units of ω₀). (b) Measure of non-Markovianity versus different parameter γ. The cases of non-RWA (black solid line with dots) and RWA (blue solid line with circles) are plotted. The initial state and other parameters are chosen as Fig. 2.

Figure 4

(a) Quantum speed limit time versus different parameter γ (in units of ω₀). The broadening-width parameter λ = 0.6 (in units of ω₀). (b) Measure of non-Markovianity versus different parameter γ. The cases of non-RWA (black solid line with dots) and RWA (blue solid line with circles) are plotted. The initial state and other parameters are chosen as Fig. 2.

Figure 5

(a) For the non-RWA case, we plot quantum speed limit time versus different parameter γ (in units of ω₀). The broadening-width parameter λ = 0.6 (in units of ω₀). Different initial states are studied by choosing different parameters p = 1, 0.7, 0.4, 0.1. For comparison, we plot the RWA case in (b).