Bath-induced coherence and the secular approximation

Finding efficient descriptions of how an environment affects a collection of discrete quantum systems would lead to new insights into many areas of modern physics. Markovian, or time-local, methods work well for individual systems, but for groups a question arises: Does system-bath or intersystem coupling dominate the dissipative dynamics? The answer has profound consequences for the long-time quantum correlations within the system. We consider two bosonic modes coupled to a bath. By comparing an exact solution against different Markovian master equations, we find that a smooth crossover of the equations of motion between dominant intersystem and system-bath coupling exists—but it requires a nonsecular master equation. We predict singular behavior of the dynamics and show that the ultimate failure of nonsecular equations of motion is essentially a failure of the Markov approximation. Our findings support the use of time-local theories throughout the crossover between system-bath-dominated and intersystem-coupling-dominated dynamics.

Figure 1

Cartoon of the system we consider. (a) Two bosonic modes of frequencies ${\omega }_a$ and ${\omega }_b$ coupled collectively to a single bath. As illustrated, we take a superohmic bath with an exponential cutoff when an explicit form is required. (b) The equivalent problem of two coupled modes with a bath coupling to only one of the modes.

Figure 2

(a) Evolution of coherence $F_{ab}$ with time (vertical) and detuning (horizontal), found from the exact solution. We use a superohmic density of states with an exponential cutoff, $J\left(\omega \right)=J_0{e}^{z-z\omega /{\omega }_0}{(\omega /{\omega }_0)}^z$. This form is written such that its peak value is at $\omega ={\omega }_0$, and $J\left({\omega }_0\right)=J_0$. We choose $z=3$, and we measure all energies and times in units such that ${\omega }_a=1$. In these units ${\omega }_0=0.9,J_0=0.001$, and the thermal occupation is controlled by $k_{B}T=0.52$. The horizontal axis is the detuning, found by varying ${\omega }_b$ for a fixed ${\omega }_a$. (b) Vertical slice at ${\omega }_b=0.9$ corresponding to $\mathrm{\Delta }=0.05$, showing a comparison between the exact and the Bloch-Redfield theories. (c) Horizontal slice at $t=200$. In the secularized theory the coherence $F_{ab}=0$ for all times.

Figure 3

Comparison of steady-state values of $F_{ij}$ between the exact solution (solid line), BR master equation (dashed line), and SpBR master equation (dotted line), plotted for a higher bath density of states $J_0=0.02$, with all other parameters as for Fig. 2.