Master equation for high-precision spectroscopy

The progress in high-precision spectroscopy requires one to verify the accuracy of theoretical models, such as the master equation describing spontaneous emission of atoms. For this purpose, we apply the coarse-graining method to derive a master equation of an atom interacting with the modes of the electromagnetic field. This master equation naturally includes terms due to quantum interference in the decay channels and fulfills the requirements of the Lindblad theorem without the need for phenomenological assumptions. We then consider the spectroscopy of the $2S-4P$ line of atomic hydrogen and show that these interference terms, typically neglected, significantly contribute to the photon count signal. These results can be important in understanding spectroscopic measurements performed in recent experiments for testing the validity of quantum electrodynamics.

Figure 1

Time-integration regions used to evaluate Eqs. (8) and (9). The blue rectangle is the region with both ${\tau }_1$ and ${\tau }_2$ integrated over $[t,t+{\tau }_c]$. (a) Integration region leading to Eq. (14) after extending the integration over ${\tau }_1$ to $(-\infty ,\infty )$, corresponding to the limit ${\tau }_c/{\tau }_R\to \infty$ while shrinking the integration interval over ${\tau }_2$, corresponding to ${\tau }_c/{\tau }_A\to 0$ (see the arrows). (b) New integration region: The yellow box indicates the region with ${\tau }_1^{\prime }={\tau }_1-{\tau }_2\in [-{\tau }_c,{\tau }_c]$ and ${\tau }_2^{\prime }={\tau }_1+{\tau }_2-2t\in [0,2{\tau }_c]$, which preserves the symmetries of the cross-damping coefficients, thus warranting the Lindblad form. The blue and yellow integration regions are equivalent, being the correlation function in the integrand different from zero only within the red small stripe of width $\sim 2{\tau }_R\ll {\tau }_c$.

Figure 2

(a) Relevant states for the spectroscopy of the $2S-4P$ transition in hydrogen. The hyperfine structure is displayed in the standard notation where states with different $F\left(M_F\right)$ quantum numbers are vertically (horizontally) displaced. The atom is assumed to be homogeneously broadened, initially prepared in the $|2{}^{2}S_{1/2},F=0,M_F=0\rangle$ state and probed by a laser that is detuned by $\delta$ from the transition $|2{}^{2}S_{1/2},F=0,M_F=0\rangle \to |4{}^{2}P_{1/2},F=1,M_F=0\rangle$. The laser is linearly polarized along $z$ and propagates along $x$. The decay channels to the $1S,2S,3S$, and $3D$ manifolds are taken into account, and the steady-state signal is obtained after a time $t=500{\gamma }_{\mathrm{tot}}^{-1}$ with ${\gamma }_{\mathrm{tot}}$ as the total linewidth. The laser Rabi frequencies are $\sim {10}^{-3}{\gamma }_{\mathrm{tot}}$. (b) Processes that lead to a vanishing line pulling when detecting over the $4\pi$ angle with no polarization filter.

Figure 3

Lower panels: line pulling in the $2S-4P$ hyperfine transition in hydrogen driven by a nearly resonant laser which is linearly polarized along $z$ and propagates along the $x$ axis. The different curves correspond to four different detection regions (a)–(d). The curves (a)–(c) are plotted as a function of the detection solid angle $\mathrm{\Omega }$, whereas (d) is plotted as a function of the azimuthal detection angle $\theta$. The line pulling of the $4{}^{2}P_{1/2}\left(4{}^2{}_{3/2}\right)$ resonance is displayed by the solid blue (dotted red) line. Upper panels: the detection regions are (a) a conic region around the $y$ axis, (b) a double cone with symmetry axis $z$, and (c) the corresponding inverted double cone (analogous to the one of Ref. [30]). For these cases $\theta$ denotes the opening angle, whereas in (d) photons are detected in a stripe of width $\delta \theta$ at an azimuthal angle $\theta$.

Figure 4

Line pulling for the detection scheme of Fig. 2 and $\mathrm{\Omega }=0$ in units of the maximum line pulling ${\mathrm{\Delta }}_L^{\max }$ [indicated by the arrow in Fig. 2] and as a function of the coarse-graining time ${\tau }_c$. The line pulling remains constant in the interval $[{10}^{-13},{10}^{-11}]\mathrm{s}$ and decreases when ${\tau }_c$ approaches the inverse linewidth of the transition.