Enabling quantum non-Markovian dynamics by injection of classical colored noise

The non-Markovian nature of quantum systems recently turned to be a key subject for investigations on open quantum system dynamics. Many studies, from its theoretical grounding to its usefulness as a resource for quantum information processing and experimental demonstrations, have been reported in the literature. Typically, in these studies, a structured reservoir is required to make non-Markovian dynamics emerge. Here, we investigate the dynamics of a qubit interacting with a bosonic bath and under the injection of a classical stochastic colored noise. A canonical Lindblad-like master equation for the system is derived by using the stochastic wave function formalism. Then, the non-Markovianity of the evolution is witnessed by using the measure of Andersson, Cresser, Hall, and Li. We evaluate the measure for three different noises and study the interplay between environment and noise pump necessary to generate quantum non-Markovianity, as well as the energy balance of the system. Finally, we discuss the possibility to experimentally implement the proposed model.

Figure 1

(a) $\eta \left(t\right)$, defined in Eq. (52), and (b) ${\gamma }_1\left(t\right)$, defined in Eq. (55), as a function of time using the exponential pump for three different temperatures: $T=0.10{\omega }_0$ (short-dashed red line), $T=0.33{\omega }_0$ (long-dashed blue line), which characterizes the limit to find RHP Markovian dynamics in the system and $T=0.50{\omega }_0$ (dot-dashed yellow line). Note that the condition to achieve RHP non-Markovianity is ${\eta }_{\min }<0$. For all the curves used, $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.91, \gamma =1$.

Figure 2

Average energy of the system, in units of ${\omega }_0$, with ($E_P$, blue solid line) and without ($E_{NP}$, red dashed line) exponential pump. Inset shows average energy difference, defined as $\mathrm{\Delta }E=E_P-E_{NP}$. (a) RHP Markovian regime, with $T=0.33{\omega }_0$, and (b) RHP non-Markovian regime, with $T=0.10{\omega }_0$. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.91, \gamma =1$. We can observe that the energy differences in the two regimes are equivalent; thus, the energy cost to generate RHP non-Markovianity is low.

Figure 3

(a) Decay rate ${\gamma }_1\left(t\right)$, Eq. (55) as a function of time in RHP non-Markovian regime using the exponential pump for two different temperatures, $T=0.1{\omega }_0$ (short-dashed red line) and $T=0.3{\omega }_0$ (long-dashed blue line) (b) ACHL measure $f\left(t\right)$, Eq. (10), as a function of time in the same regime, for the temperatures $T=0.1{\omega }_0$ (red solid line with circles) and $T=0.3{\omega }_0$ (blue solid line with triangles). The value obtained for the measure were ${\mathcal{N}}_{\mathrm{ACHL}}=0.0546$ for $T=0.1{\omega }_0$ and ${\mathcal{N}}_{\mathrm{ACHL}}=0.0046$ for $T=0.3{\omega }_0$. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.91, \gamma =1$.

Figure 4

Average energy of the system with ($E_P$, blue solid line) and without ($E_{NP}$, red dashed line) squared exponential pump. Inset shows the average energy difference, defined as $\mathrm{\Delta }E=E_P-E_{NP}$. (a) RHP Markovian regime, with $T=0.33{\omega }_0$, and (b) RHP non-Markovian regime, with $T=0.10{\omega }_0$. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.47, \gamma =1$. We can observe again that the energy differences in the two regimes are equivalent.

Figure 5

(a) Decay rate ${\gamma }_1\left(t\right)$, Eq. (55), as a function of time in RHP non-Markovian regime using the square exponential pump for two different temperatures: $T=0.1{\omega }_0$ (short-dashed red line) and $T=0.3{\omega }_0$ (long-dashed blue line). (b) ACHL measure $f\left(t\right)$, Eq. (10), as a function of time in the same regime, for the temperatures $T=0.1{\omega }_0$ (red solid line with circles) and $T=0.3{\omega }_0$ (blue solid line with triangles). The value obtained for the measure were ${\mathcal{N}}_{\mathrm{ACHL}}=0.0585$ for $T=0.1{\omega }_0$ and ${\mathcal{N}}_{\mathrm{ACHL}}=0.0040$ for $T=0.3{\omega }_0$. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.47, \gamma =1$.

Figure 6

Average energy of the system with ($E_P$, blue solid line) and without ($E_{NP}$, red dashed line) power-law pump. Inset shows the average energy difference, $\mathrm{\Delta }E=E_P-E_{NP}$, in (a) RHP Markovian regime, with $T=0.33{\omega }_0$, and (b) RHP non-Markovian regime, with $T=0.10{\omega }_0$. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=10, \mathrm{\Omega }=1.35, \gamma =1$. We can observe also in this case that the energy differences in the two regimes are equivalent.

Figure 7

(a) Decay rate ${\gamma }_1\left(t\right)$, Eq. (55), as a function of time in RHP non-Markovian regime using the power-law pump for two different temperatures: $T=0.1{\omega }_0$ (short-dashed red line) and $T=0.3{\omega }_0$ (long-dashed blue line) (b) ACHL measure $f\left(t\right)$, Eq. (10), as a function of time in the same regime, for the temperatures $T=0.1{\omega }_0$ (red solid line with circles) and $T=0.3{\omega }_0$ (blue solid line with triangles). The value obtained for the measure were ${\mathcal{N}}_{\mathrm{ACHL}}=0.0532$ for $T=0.1{\omega }_0$ and ${\mathcal{N}}_{\mathrm{ACHL}}=0.0058$ for $T=0.3{\omega }_0$. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=10, \mathrm{\Omega }=1.35, \gamma =1$.

Figure 8

(a) Decay rate ${\gamma }_1\left(t\right)$, Eq. (55), as a function of time in RHP non-Markovian regime for different noises. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.91$ (OU, green dashed line); $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.47$ (SE, red solid line); $\mathrm{\Delta }\omega =2, {\tau }_c=10, \mathrm{\Omega }=1.35$ (PL, dot-dashed blue line). (b) Average energy for different noises. The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.91$ (OU, short-dashed green line); $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.47$ (SE, red solid line); $\mathrm{\Delta }\omega =2, {\tau }_c=10, \mathrm{\Omega }=1.35$ (PL, dot-dashed blue line); average energy of the system without pump (long-dashed black line). Inset shows a comparison of average energies on a better scale. All the curves used have $\gamma =1, T=0.1{\omega }_0$.

Figure 9

ACHL measure $f\left(t\right)$ for different noises. (a) The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.91$ (OU, red dashed line); $\mathrm{\Delta }\omega =2, {\tau }_c=4.25, \mathrm{\Omega }=0.47$ (SE, green solid line); $\mathrm{\Delta }\omega =2, {\tau }_c=10, \mathrm{\Omega }=1.35$ (PL, dot-dashed blue line). (b) The parameters used were $\mathrm{\Delta }\omega =2, {\tau }_c=10, \mathrm{\Omega }=1.35$. All curves used $\gamma =1, T=0.1{\omega }_0$.