Markovian evolution of strongly coupled harmonic oscillators

We investigate how to model Markovian evolution of coupled harmonic oscillators, each of them interacting with a local environment. When the coupling between the oscillators is weak, dissipation may be modeled using local Lindblad terms for each of the oscillators in the master equation, as is commonly done. When the coupling between oscillators is strong, this model may become invalid. We derive a master equation for two coupled harmonic oscillators that are subject to individual heat baths modeled by a collection of harmonic oscillators and show that this master equation in general contains nonlocal Lindblad terms. We compare the resulting time evolution with that obtained for dissipation through local Lindblad terms for each individual oscillator and show that the evolution is different in the two cases. In particular, the two descriptions give different predictions for the steady state and for the entanglement between strongly coupled oscillators. This shows that when describing strongly coupled harmonic oscillators, one must take great care in how dissipation is modeled and that a description using local Lindblad terms may fail. This may be particularly relevant when attempting to generate entangled states of strongly coupled quantum systems.

Figure 1

Average number of excitation quanta $n=\langle {\widehat{a}}^{†}\left(t\right)\widehat{a}\left(t\right)\rangle =\langle {\widehat{b}}^{†}\left(t\right)\widehat{b}\left(t\right)\rangle$ for each oscillator, each of them interacting with an environment in an identical way, calculated using the master equations (7) (red solid line) and (51) (green thick solid line), plotted as a function of time. Each oscillator is initially in a vacuum state and ${\Gamma }_a={\Gamma }_b=\omega$/100. (a) $\kappa =\lambda =\omega /20$ and (b) $\lambda =\omega /3$ and $\kappa =0$. Time is in units of $1/\omega$.

Figure 2

Logarithmic negativity plotted as a function of time, calculated using numerical solutions of the master equations (7) (red solid line) and (51) (green thick solid line). Each oscillator is initially in a vacuum state and ${\Gamma }_a={\Gamma }_b=\omega$/100. (a) $\kappa =\lambda =\omega /20$ and (b) $\lambda =\omega /3$ and $\kappa =0$. Time is in units of $1/\omega$.

Figure 3

Time dependence of the quantum fidelity between the two one-mode states of each oscillator computed from the numerical solutions of Eqs. (7) and (51) for ${\Gamma }_a={\Gamma }_b=\omega /100$, when (a) $\kappa =\omega /20$ and (b) $\lambda =\omega /20$. Each oscillator is initially in the vacuum state and time is in units of $1/\omega$.

Figure 4

Steady-state value of (a) the logarithmic negativity calculated using the local Lindblad-type dissipation approach (red solid line) and the bath-induced dissipation approach (green thick solid line) and (b) the quantum fidelity between the two one-mode states of each oscillator obtained using the approaches of Secs. 3 and 4, plotted as a function of $\overline{n}$. Other physical parameters are chosen such that ${\Gamma }_a={\Gamma }_b=\omega /100$, $\lambda =\omega /3$, and $\kappa =0$.