Steady-state Fano coherences in a V-type system driven by polarized incoherent light

We explore the properties of steady-state noise-induced (Fano) coherences generated in a three-level V-system continuously pumped by polarized incoherent light in the absence of coherent driving. By solving the nonsecular Bloch-Redfield quantum master equation, we obtain the ratio of the stationary coherences to excited-state populations, $\mathcal{C}=(1+\frac{{\mathrm{\Delta }}^2}{\gamma (r+\gamma )}{)}^{-1}$, which quantifies the impact of steady-state coherences on excited-state dynamical observables of the V-system. The ratio is maximized when the excited-state splitting $\mathrm{\Delta }$ is small compared to either the spontaneous decay rate $\gamma$ or the incoherent pumping rate $r$. We demonstrate that while the detrimental effects of a strongly decohering environment generally suppress the coherence-to-population ratio by the factor $\simeq {\gamma }_d/\gamma$, an intriguing regime exists where the $\mathcal{C}$ ratio displays a maximum as a function of the dephasing rate ${\gamma }_d$. We attribute the surprising dephasing-induced enhancement of stationary Fano coherences to the environmental suppression of destructive interference of individual incoherent excitations generated at different times. We clarify the physical basis for the steady-state Fano coherence, whose imaginary part is identified with the nonequilibrium flux across a pair of qubits coupled to two independent thermal baths or, equivalently, the spontaneous emission flux from the right qubit to the right bath, unraveling a direct connection between the seemingly unrelated phenomena of incoherent driving of multilevel quantum systems and nonequilibrium quantum transport in qubit networks. We further establish the equivalence between the two-qubit system and a V-system, each of whose transitions is driven simultaneously by both baths. The real part of the steady-state Fano coherence is found to be proportional to the deviation of excited-state populations from their values in thermodynamic equilibrium, making it possible to observe signatures of steady-state Fano coherences in excited-state populations. Finally, we put forward an experimental proposal for observing steady-state Fano coherences by detecting the total fluorescence signal emitted by Calcium atoms excited by polarized versus isotropic incoherent light. Our analysis paves the way toward further theoretical and experimental studies of nonequilibrium coherent steady states in thermally driven atomic and molecular systems and for the exploration of their potential role in quantum thermodynamics and in biological processes.

Figure 1

(a) Energy level diagram of the V-system composed of a single ground state $g$ and two nearly degenerate excited levels $a$ and $b$. In our proposed experimental realization of the V-system, the ground state corresponds to the ${}^1\mathrm{S}_0$ level of atomic Ca and the excited states corresponds to the ${}^1\mathrm{S}_1$ levels of atomic Ca with the total electron orbital angular momentum $J=1$ and its projection $m_J=\pm 1$ on the magnetic field axis. The excited-state energy splitting $\mathrm{\Delta }$ is continuously tunable with an external magnetic field. The rates of spontaneous decay (wavy arrows) are given by ${\gamma }_i$ ($i=a,b$), and those of incoherent pumping (straight arrows) by $r_i={\gamma }_i\overline{n}$. (b) An alternative representation of the V-system driven by polarized incoherent light as a pair of coherently coupled qubits interacting with two independent thermal baths at different temperatures. (c) An equivalent representation of the two-qubit model as a V-system, in which the transitions $g\leftrightarrow a$ and $g\leftrightarrow b$ are driven simultaneously by the left and right baths (see text for details).

Figure 2

Time evolution of excited-state populations and Fano coherences in the V-system weakly [(a) and (b)] and strongly [(c) and (d)] driven by $x$-polarized incoherent light. (a) Overdamped regime with $\overline{n}={10}^{-3}$ and $\mathrm{\Delta }/\gamma ={10}^{-1}$. (b) Underdamped regime with $\overline{n}={10}^{-3}$ and $\mathrm{\Delta }/\gamma =10$; (c) $\overline{n}={10}^3$ and $\mathrm{\Delta }/\gamma ={10}^2$, and (d) $\overline{n}={10}^2$ and $\mathrm{\Delta }/\gamma =2\times {10}^2$. The steady-state values of Fano coherences are shown by horizontal dashed lines. Note that the excited-level populations shown in (a) and (b) grow monotonically, whereas the real and imaginary Fano coherences in (b) and (d) display underdamped coherent oscillations.

Figure 3

Schematic depiction of the unitary time evolution and decay of a single incoherent excitation (a) and of an ensemble of incoherent excitations (b). The incoherent excitations in (b) are assumed undamped for simplicity.

Figure 4

(a) Steady-state Fano coherence plotted as a function of the average number of thermal photons $\overline{n}$ at fixed values of the excited-state splitting $\mathrm{\Delta }/\gamma$. (b) Same as in panel (a) plotted vs $\mathrm{\Delta }/\gamma$ at fixed values of $\overline{n}$ corresponding to the weak, intermediate, and strong pumping regimes. (c) Two-dimensional contour plot of steady-state Fano coherence as a function of $\mathrm{\Delta }/\gamma$ and $\overline{n}$.

Figure 5

Contour plots of steady-state Fano coherence as a function of the excited-state splitting $\mathrm{\Delta }/\gamma$ and of the decoherence rate ${\gamma }_d/\gamma$ under weak (a) and strong (b) $x$-polarized incoherent pumping. The values of the average photon number used are $\overline{n}=0.01$ (a) and $\overline{n}=100$ (c). (c) shows the steady-state Fano coherence as a function the decoherence rate and $\overline{n}$ at a fixed $\mathrm{\Delta }/\gamma =10$.

Figure 6

Steady-state coherence-to-population ratio $\mathcal{C}$ plotted as a function of dimensionless decoherence rate ${\gamma }_d/\gamma$ in the weak-pumping limit [(a) and (c)] and in the strong-pumping limit [(b) and (d)]. The values of the $\mathrm{\Delta }/\gamma$ are indicated in each panel.

Figure 7

(a) The ratio of fluorescence intensities calculated for Ca atoms driven by $x$-polarized vs isotropic incoherent light in different regimes: (a) $\overline{n}={10}^{-2}$ and $\mathrm{\Delta }/\gamma ={10}^{-1}$; (b) $\overline{n}={10}^{-2}$ and $\mathrm{\Delta }/\gamma =10$; (c) $\overline{n}={10}^3$ and $\mathrm{\Delta }/\gamma =10$; (d) $\overline{n}=10$ and $\mathrm{\Delta }/\gamma ={10}^2$. The spontaneous decay lifetime used in the calculations is ${\tau }_s=1/\gamma ={10}^{-9}\mathrm{s}$.