Enhancement of non-Markovianity due to environment-induced indirect interaction

Abstract

Non-Markovian effects are often significant when the system-environment coupling is not weak. Indeed, we find that the non-Markovianity is negligible for a single two-level system undergoing pure dephasing via a weak interaction with a harmonic-oscillator environment. In this paper, we show that, within the framework of pure dephasing, the non-Markovianity displayed by a two-level system can, in fact, be far more pronounced. To demonstrate that this is indeed the case, we consider a pure dephasing model where a collection of two-level systems interacts with a common environment. We obtain analytically the dynamics of the collection of the two-level systems, and then take a partial trace over all the two-level systems except one. This remaining single two-level system exhibits markedly non-Markovian dynamics, even when the system-environment coupling is weak. This is due to the indirect interaction between the two-level systems, induced by their interaction with the common environment. In fact, this indirect interaction can not only increase the non-Markovianity by orders of magnitude, but also qualitatively change the characteristics of the non-Markovian behavior. For instance, for a single two-level system undergoing pure dephasing, the dynamics are Markovian for both Ohmic and sub-Ohmic environments. This is markedly not the case when we consider multiple two-level systems. These findings provide insights into controlling decoherence in multi-qubit quantum systems and have implications for quantum technologies where non-Markovianity can be a resource rather than a limitation.

Figures (17)

• Figure 1: (a) Plot of the BLP measure of non-Markovianity, $\mathcal{N}_{\text{BLP}}$, as a function of the Ohmicity parameter $s$ for a single TLS (that is, $N = 1$). We are working in dimensionless units with $\hbar = 1$ throughout, and here we have set $G = 1$ and $\omega_c = 3$. Throughout this paper, unless stated otherwise, we are in the zero temperature regime. To calculate the non-Markovianity measure $\mathcal{N}_{\text{BLP}}$, we need to integrate over time; here we have set the upper limit of this integral to be $T = 20$. (b) Same as (a), but using two TLSs and then tracing out one of these. The dynamics are now influenced by an indirect interaction.
• Figure 2: Same as Fig. \ref{['BLP vs s']}, except that we are now looking at the non-Markovianity measure $\mathcal{N}_{\text{Entropy}}$. Once again, (a) shows the non-Markovianity measure for a single TLS, while (b) includes the effect of the indirect interaction.
• Figure 3: Time evolution of the indirect-interaction induced factor $\Delta(t)$ for $s=0.1$ (red dashed), $s=1.5$ (blue dashed-dotted), and $s=3$ (black solid), with $G=1.0$ and $\omega_c=3$. For $s = 0.1$, $\Delta(t)$ does not become a linear function at long times, but it does become linear for $s = 1.5$ and $s = 3$.
• Figure 4: Time evolution of the decoherence function $\Gamma(t)$ with $s=0.1$ (red dashed), $s=1.5$ (blue dashed-dotted), and $s=3$ (solid black), for coupling strength $G=1$ and cutoff frequency $\omega_c=3$. For $s = 0.1$, $\Gamma(t)$ increases without bound, while $\Gamma(t)$ approaches a constant value for $s = 1.5$ and $s = 3$ at longer times. The inset shows the behavior of $\Gamma(t)$ for $s = 3$ at smaller times to emphasize that $\Gamma(t)$ can decrease over some time interval for $s = 3$. It is precisely this decrease that leads to non-Markovianity for a single TLS without any indirect interaction. As $s$ increases further, the behavior of $\Gamma(t)$ remains qualitatively the same as for $s = 3$.
• Figure 5: Non-Markovianity measure $\mathcal{N}_{\text{BLP}}$ as a function of the total evolution time $T$. In (a), we have only a single TLS, so there is no indirect interaction. In (b), we have taken $N = 2$, so indirect interaction is present. As before, we are working in dimensionless units with $\hbar = 1$. We have $G=1$ and $\omega_c=3$. The solid black curve is for $s=3$, the dotted-magenta curve is for $s=2$, the dashed-dotted blue curve is for $s=1$, and the red-dashed curve is for $s=0.5$. In the inset, the same dashed-dotted blue curve is for $s=1$, and the red-dashed curve is for $s=0.5$. The non-Markovianity for the Ohmic environment saturates very slowly since the decoherence factor $\Gamma(t)$ grows relatively slowly; this saturation is clearer if we use a larger value of $G$.