Long-Lived Quasistationary Coherences in a $V$-type System Driven by Incoherent Light

We present a theoretical study of noise-induced quantum coherences in a model three-level $V$-type system interacting with incoherent radiation, an important prototype for a wide range of physical systems ranging from trapped ions to biomolecules and quantum dots. By solving the quantum optical equations of motion, we obtain analytic expressions for the noise-induced coherences and show that they exhibit an oscillating behavior in the limit of large excited level spacing $\mathrm{\Delta }$ ($\mathrm{\Delta }/\gamma \gg 1$, where $\gamma$ is the radiative decay width). Most remarkably, we find that in the opposite limit of small level spacing $\mathrm{\Delta }/\gamma \ll 1$, appropriate for large molecules, (a) the coherences can survive for an extremely long time $\tau =(2/\gamma )(\mathrm{\Delta }/\gamma {)}^{-2}$ before eventually decaying to zero, and (b) coherences at short times can be substantial. We further show that the long-lived coherences can survive environmental relaxation and decoherence, suggesting implications to the design of quantum heat engines and to incoherent light excitation of biological systems.

Figure 1

(a) The energy level structure of the $V$-type system. The shaded area represents the effects of environment-induced relaxation and decoherence on the excited states $a$ and $b$ with $\mathrm{\Delta }={\epsilon }_a-{\epsilon }_b$. The real part of the two-photon coherence ${\rho }_{ab}^R(t)$ (b) and the ratio $\mathcal{C}=|{\rho }_{ab}|/({\rho }_{aa}+{\rho }_{bb})$ (inset). Full lines—exact result (numerical calculations), dashed lines—analytical result obtained in this work in the $\mathrm{\Delta }/\gamma \gg 1$ limit ($\gamma /2\pi =1\mathrm{GHz}$, $\mathrm{\Delta }/\gamma =24$). The incoherent pumping rate $r/\gamma =\overline{n}=0.063$, where $\overline{n}=1/[e^{\hslash {\omega }_{ac}/k_{B}T}-1]$ is the thermal occupation number ($k_{B}T=0.5\mathrm{eV}$ for sunlight), and ${\omega }_{ac}=1.41\mathrm{eV}$. Results for $p=1/2$ are also shown [green (light grey) line].

Figure 2

Excited-state population and coherence dynamics in the $\mathrm{\Delta }/\gamma \ll 1$ limit ($\mathrm{\Delta }/\gamma =0.024$, $\gamma /2\pi =1\mathrm{GHz}$, $r/\gamma =0.1\overline{n}$). Exact results (full line), analytic results (dashed line), results for $p=1/2$ [green (light grey) line].

Figure 3

Effects of environmental relaxation and decoherence on $V$-type system dynamics: (a) The $\mathcal{C}$ ratio and the real part of the coherence (inset) as a function of time for $\mathrm{\Delta }/\gamma =24$ and ${\gamma }_d/\gamma =0$, 5, 50. The decoherence rates ${\gamma }_d$ are shown next to each curve. (b) The real part of the coherence for $\mathrm{\Delta }/\gamma =0.024$ and ${\gamma }_d/\gamma =0$, 5, 50. The imaginary part of the coherence is negligible compared to the real part.