Asymmetric coupling between two quantum emitters

We study a prototypical model of two coupled two-level systems, where the competition between coherent and dissipative coupling gives rise to a rich phenomenology. In particular, we analyze the case of asymmetric coupling, as well as the limiting case of chiral (or one-way) coupling. We investigate various quantum optical properties of the system, including its steady-state populations, power spectrum, and second-order correlation functions, and outline the characteristic features which emerge in each quantity as one sweeps through the nontrivial landscape of effective complex couplings. Most importantly, we reveal instances of population trapping, unexpected spectral features, and strong emission correlations.

Figure 1

(a) Sketch of the energy ladder of two coupled 2LSs, which is necessarily restricted to three rungs $N=\{0,1,2\}$, in the weak-coupling regime (left) and the strong-coupling regime (right) [cf. Eq. (10)]. On the right the four red arrows label the two transitions between the $N=2$ and $N=1$ rungs (called $A$ and $B$) and the two transitions between the $N=1$ and the $N=0$ rungs (called $C$ and $D$). (b) Cartoon of the system under investigation: a pair of 2LSs (labeled 1 and 2) with coherent coupling (red arrows), incoherent pumping (orange arrows), dissipative coupling (light green arrows), and self-damping decay (dark green arrows) [cf. Eq. (11)].

Figure 2

Sketches of the system of two coupled two-level systems, denoted by 1 and 2, in different coupling regimes [cf. Eqs. (14)]. We denote coherent (dissipative) coupling by red (green) arrows and the common bath by a green rectangle. We consider (a) coherent [cf. Eq. (14a)], (b) dissipative [cf. Eq. (14b)], (c) chiral I and II (top and bottom, respectively) [cf. Eqs. (14c) and (14d)], and (d) asymmetric coupling.

Figure 3

Mean populations in the coherent coupling regime, as a function of the pumping rate $P_1$, in units of the damping rate ${\gamma }_0$ [cf. Eqs. (16)]. (a) Case of symmetric pumping ($P_2=P_1$), which is independent of the coherent coupling strength $g$. (b) Asymmetric pumping case ($P_2=0$), with $g={\gamma }_0$. The labeling of the mean population of the state $|i,j\rangle$ is displayed in the legend in (a) and states with $N=\{0,1,2\}$ excitations are shown with increasingly thick lines.

Figure 4

Second-order cross correlator in the coherent coupling regime at zero delay $g_{12,\mathrm{co}}^{\left(2\right)}\left(0\right)$, as a function of the pumping rate $P_1$, in units of the decay rate ${\gamma }_0$ [cf. Eq. (19)]. We show results with asymmetric pumping ($P_2=0$) for increasingly strong coherent coupling strengths $g$ (increasingly thick colored lines).

Figure 5

Spectrum of 2LS-1 in the coherent coupling regime $S_1^{\mathrm{co}}\left(\omega \right)$, in units of the inverse damping rate ${\gamma }_0^{-1}$, for increasingly strong coherent coupling strengths $g$ (increasingly thick colored lines) [cf. Eq. (26)].

Figure 6

Mean populations in the dissipative coupling regime as a function of the pumping rate $P_1$, in units of the damping rate ${\gamma }_0$ [cf. Eqs. (27)]: (a) dissipative coupling strength $\gamma =3{\gamma }_0/4$ and (b) maximal coupling $\gamma ={\gamma }_0$. The labeling of the mean population of the state $|i,j\rangle$ is displayed in the legend in (a) and states with $N=\{0,1,2\}$ excitations are shown with increasingly thick lines. In the figure, we consider asymmetric pumping, with $P_2=0$.

Figure 7

Second-order cross correlator in the dissipative coupling regime at zero delay $g_{12,\mathrm{ds}}^{\left(2\right)}\left(0\right)$ as a function of the pumping rate $P_1$, in units of the decay rate ${\gamma }_0$ [cf. Eq. (28)]. Increasingly strong dissipative coupling strengths $\gamma$ are denoted by increasingly thick colored lines. We show results with (a) symmetric pumping ($P_2=P_1$) and (b) asymmetric pumping ($P_2=0$).

Figure 8

Spectrum of 2LS-1 in the dissipative coupling regime $S_1^{\mathrm{ds}}\left(\omega \right)$, in units of the inverse damping rate ${\gamma }_0^{-1}$, for increasingly strong dissipative coupling strengths $\gamma$ (increasingly thick colored lines) [cf. Eq. (31)].

Figure 9

Mean populations in the chiral-1 coupling regime as a function of the pumping rate $P_1$, in units of the damping rate ${\gamma }_0$ [cf. Eqs. (33)]. We show results with (a) symmetric pumping ($P_2=P_1$) and (b) asymmetric pumping ($P_2=0$) for the dissipative coupling strength $\gamma ={\gamma }_0/2$. The labeling of the mean population of the state $|i,j\rangle$ is displayed in the legend in (a) and states with $N=\{0,1,2\}$ excitations are shown with increasingly thick lines.

Figure 10

Second-order cross correlator in the chiral-1 coupling regime $g_{12,\mathrm{ch},\mathrm{I}}^{\left(2\right)}\left(0\right)$ at zero delay as a function of the pumping rate $P_1$, in units of the decay rate ${\gamma }_0$ [cf. Eq. (34)]. Solid lines show symmetric pumping ($P_2=P_1$) for increasingly strong dissipative coupling strengths $\gamma$ (increasingly thick colored lines). The dashed line shows an asymmetric pumping case ($P_2=0$), which is independent of $\gamma$.

Figure 11

Spectrum of 2LS-1 in the chiral I coupling regime $S_1^{\mathrm{ch},\mathrm{I}}\left(\omega \right)$, in units of the inverse damping rate ${\gamma }_0^{-1}$ [cf. Eq. (37)].

Figure 12

Mean population $n_1$ of the first 2LS as a function of the relative phase $\theta -\varphi$ for increasingly strong coherent coupling strengths $g$ (increasingly thick solid lines) [cf. Eq. (38a)]: (a) case of symmetric pumping rates ($P_2=P_1$) and (b) asymmetric pumping case ($P_2=0$). The dashed gray line shows the population of an isolated 2LS $n_{\mathrm{iso}}$ [cf. Eq. (15)]. In the figure, $P_1={\gamma }_0/10$ and the dissipative coupling strength $\gamma ={\gamma }_0/2$, so the blue line fulfills the chiral magnitude condition of $g/\gamma =1/2$.

Figure 13

Second-order cross correlator in the asymmetric coupling regime at zero delay $g_{12}^{\left(2\right)}\left(0\right)$ as a function of the relative phase $\theta -\varphi$. We show results with (a) symmetric pumping rates ($P_2=P_1$) and (b) asymmetric pumping rates ($P_2=0$) for increasingly strong coherent coupling strengths $g$ (increasingly thick colored lines) [cf. Eq. (40)]. In the figure, $P_1={\gamma }_0/10$ and the dissipative coupling strength $\gamma ={\gamma }_0/2$, so the blue line fulfills the chiral magnitude condition of $g/\gamma =1/2$.

Figure 14

Relative (a) frequencies ${\omega }_p-{\omega }_0$ and (b) broadenings ${\gamma }_p$, both in units of the damping rate ${\gamma }_0$, as a function of the relative phases $\theta -\varphi$, with $g/\gamma =1/2$ [cf. Eqs. (41)]. The thin pink lines show the dissipative (coherent) coupling strength $\gamma ={\gamma }_0/2$ ($g=3{\gamma }_0/8$). Thick orange lines show $\gamma ={\gamma }_0$ ($g={\gamma }_0/2$). Solid (dashed) lines show the index $p=C$ ($D$) [cf. Eq. (22)]. (c) and (d) Spectrum of 2LS-1 in the asymmetric coupling regime $S_1\left(\omega \right)$, in units of the inverse damping rate ${\gamma }_0^{-1}$ [cf. Eq. (21)]. We show results for $\theta -\varphi =\{0,\pi /4,\pi /2\}$ (increasingly thin lines) and (c) $\gamma =3{\gamma }_0/4$ and $g=3{\gamma }_0/8$, corresponding to the pink lines in (a) and (b), and (d) $\gamma ={\gamma }_0$ and $g={\gamma }_0/2$, corresponding to orange lines in (a) and (b).

Figure 15

(a) Mean population $n_{\mathrm{iso}}$ of an isolated 2LS as a function of the pumping rate $P_1$, in units of the damping rate ${\gamma }_0$ [cf. Eq. (15)], and (b) spectrum $S_{\mathrm{iso}}\left(\omega \right)$ of an isolated 2LS, in units of the inverse damping rate ${\gamma }_0^{-1}$, as a function of the frequency $\omega$ (which is measured from the 2LS resonance frequency ${\omega }_0$) [cf. Eq. (23)].

Figure 16

Mean populations in the dissipative coupling regime as a function of the pumping rate $P_1$, in units of the damping rate ${\gamma }_0$ [cf. Eqs. (27)], for (a) dissipative coupling strength $\gamma =3{\gamma }_0/4$ and (b) maximal coupling $\gamma ={\gamma }_0$. The labeling of the mean population of the state $|i,j\rangle$ is displayed in the legend in (a) and states with $N=\{0,1,2\}$ excitations are shown with increasingly thick lines. In the figure, we consider symmetric pumping ($P_2=P_1$).