Narrow dip inside a natural linewidth absorption profile in a system of two atoms

Absorption spectrum of a system of two closely spaced identical atoms displays, at certain preparation, a dip that can be much narrower than the natural linewidth. This preparation includes (i) application of a strong magnetic field at an angle $\alpha$, that is very close to the magic angle ${\alpha }_0=arccos(1/\sqrt{3})\approx {54.7}^{\circ }$, with respect to the direction from one atom to another, and (ii) in-plane illumination by a laser light in the form of a nonresonant standing wave polarized at the same angle $\alpha$. Both qualitative and quantitative arguments for the narrow dip effect are presented.

Figure 1

Absorption profiles (1) at ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=\mathrm{\Gamma }$ and certain values of splitting $\mathrm{\Delta }={\omega }_{20}-{\omega }_{10}$ of levels $|1\rangle$ and $|2\rangle$. Parameter $\overline{\omega }$ (2) is equal in this case to the middle frequency $\frac{1}{2}({\omega }_{10}+{\omega }_{20})$.

Figure 2

Arrangement of the external fields creating conditions for the effect named in the title. The magnetic field applied at $z$ direction $[\alpha \approx {\alpha }_0=arccos(1/\sqrt{3})\approx 54.7^{\circ }]$ provides minimization of the dipole-dipole interaction of two identical atoms A. The gradient of electric component $\mathcal{E}cosk(x-x_0)cos\omega t$ of the laser field induces a gap between transition frequencies of these atoms. Quantitatively, the magnitude of the magnetic field and the amplitude of the laser field are to provide inequalities given by Eq. (12) below.

Figure 3

Energies of eigenstates of a system of two atoms as functions of the magnetic field $H$ applied at the magic angle (see Fig. 2). (a) A relatively narrow scale of $H$ variation is presented. (b),(c) A large scale is shown by solid curves for the dependencies of splittings (9) and (10), respectively, on the magnetic field. Also, the fractions of corresponding admixtures defined as ${\mathcal{P}}_{\mathrm{admix}}^{\left(M\right)}={\sum }_{M^{\prime }\ne M}{\left|〈{\mathcal{Q}}_{\mathrm{s},\mathrm{a}}^{\left(M^{\prime }\right)}|{\mathcal{R}}_{\mathrm{s},\mathrm{a}}^{\left(M\right)}〉\right|}^2$ are shown by the dashed curves.

Figure 4

Illustration of the shifts of states (4) by a field different in positions of two atoms. Quantities $E_{\mathrm{S}}$ and ${\mathrm{\Delta }}_{\mathrm{S}}$ relate to the shifts of $|e\rangle \to |g\rangle$ transition frequencies. As for the signs of their values, a case is shown where $\delta <0$, so $E_{\mathrm{S}}>0$ and ${\mathrm{\Delta }}_{\mathrm{S}}>0$.

Figure 5

Dependencies of all relevant quantities on the difference of Stark shifts for two states $|1\rangle$ and $|2\rangle$ with $M=0$. (a) The rates of spontaneous decay starting from ${\mathrm{\Gamma }}_1\ll \mathrm{\Gamma }$ [37] and ${\mathrm{\Gamma }}_2\approx 2\mathrm{\Gamma }$ for, respectively, antisymmetric (subradiant) and symmetric (superradiant) entangled states approach to $\mathrm{\Gamma }$ as the value of ${\mathrm{\Delta }}_{\mathrm{S}}$ grows. (b) Naturally, the splitting between states $|1\rangle$ and $|2\rangle$ and the resulting width of the dip in the absorption profile grow with ${\mathrm{\Delta }}_{\mathrm{S}}$.

Figure 6

Diagram of states for the problem under treatment in Sec. 4. Interactions of two discrete states $|1\rangle$ and $|2\rangle$ with two continua are described by functions $F_{1,2}\left(x\right)$ and $f_{1,2}\left(y\right)$. For their general definition, see Eq. (20). Their definition for our system of two atoms in Secs. 2 and 3 is given by Eqs. (15), (24), (25), (A5), and (A9).

Figure 7

The dashed curve shows the spectrum $\mathcal{A}\left(\omega \right)$ [see Eq. (1) with ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=\mathrm{\Gamma }$]. The distance $\mathrm{\Delta }\approx 0.46\mathrm{\Gamma }$ between the two states is chosen so that the FWHM of $\mathcal{A}\left(\omega \right)$ would be equal to $\mathrm{\Gamma }/10$. The solid curve takes into account a small correction of Sec. 4e1 [Eq. (40)] introduced by $\gamma \approx 0.0055$ ($kr=0.25$ in our numerical example), but ignore corrections of Sec. 4e2.

Figure 8

The solid curve shows the spectrum $\mathcal{A}\left(\omega \right)$ with inclusion of the quantity $\mathcal{I}$ (44). All other parameters are the same as in Fig. 7. The frequency scale is compressed twice, as compared to that in Fig. 7, and is the same as in Fig. 1.

Figure 9

Examples of spectra $\mathcal{A}\left(\omega \right)$ calculated using Eqs. (30, 31, 32) and (39). Equality ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2$ and all other conditions of Sec. 3 are kept. The parameter $\sigma /\mathrm{\Gamma }$ [see Eq. (3)] that gives an estimate for the dip's relative width is equal to $0.1$ (curve 1, same as in Fig. 7), $0.05$ (curve 2), $0.02$ (curve 3), and $0.01$ (curve 4). (The frequency scale is extended twice, as compared to that in Fig. 7, and fourfold, as compared to that in Fig. 8.)

Figure 10

Examples of spectra $\mathcal{A}\left(\omega \right)$ calculated using Eqs. (30, 31, 32) and (39) for different values of the parameter $kr$ at the fixed value of the ratio $\sigma /\mathrm{\Gamma }=0.1$. Curve 1: $kr=0.4$; 2: $kr=0.6$; 3: $kr=0.8$; 4: $kr=1.0$.

Figure 11

Examples of spectra $\mathcal{A}\left(\omega \right)$ calculated using Eqs. (30, 31, 32) and (39) for different values of the ratio $\sigma /\mathrm{\Gamma }$ at the fixed value of the parameter $kr=2.2$. Curve $1-\sigma =0.2$; $2-\sigma =0.3$; $3-\sigma =0.4$; $4-\sigma =0.5$.