Dispersion relations for stationary light in one-dimensional atomic ensembles

We investigate the dispersion relations for light coupled to one-dimensional ensembles of atoms with different level schemes. The unifying feature of all the considered setups is that the forward and backward propagating quantum fields are coupled by the applied classical drives such that the group velocity can vanish in an effect known as “stationary light.” We derive the dispersion relations for all the considered schemes, highlighting the important differences between them. Furthermore, we show that additional control of stationary light can be obtained by treating atoms as discrete scatterers and placing them at well-defined positions. For the latter purpose, a multimode transfer matrix theory for light is developed.

Figure 1

Level diagrams of the three schemes that we consider. The blue circles on state $|a\rangle$ indicate that the atoms are assumed to be initialized in this state. The arrows with small wiggly lines originating on the excited states $|b_{\pm }\rangle$ and $|b\rangle$ indicate spontaneous emission with a decay rate ${\mathrm{\Gamma }}^{\prime }$. The arrows between different states indicate either quantum fields (wiggly lines), or classical drives (double straight lines). The small horizontal arrows on each of these coupling arrows indicate the propagation direction. All the transitions are assumed to couple equally to both the right-moving and left-moving fields, but the arrows pointing only in a single direction on the classical drives for $\mathrm{dual}-\mathsf{V}$ and dual-color schemes instead indicate that the externally applied drives propagate in the shown direction. The excited level $|b\rangle$ for the dual-color scheme is shifted vertically in order to be able to clearly show all the different detunings.

Figure 2

Log-log plot of the dispersion relations for the different setups. The upper solid black curve is for EIT [see Eq. (37)]. The lower solid red curve is the quadratic dispersion relation for the $\mathrm{dual}-\mathsf{V}$ setup (or the secular approximation for the $\mathrm{\Lambda }$-type scheme) given by Eq. (31). The dashed green curves are the dispersion relations for the truncations of Eqs. (41) with increasing number of the Fourier components of ${\sigma }_{ab}$ and ${\sigma }_{ac}$. The dispersion relations for small $\text{Re}\left[q\right]/n_0$ alternate between linear and quadratic depending on the truncation. The solid blue curve is the analytical limit of these dispersion relations given by Eq. (56). The lower dashed cyan curve is for the dual-color scheme with ${\mathrm{\Delta }}_{\text{d}}/\mathrm{\Gamma }=1$. It overlaps the quadratic $\mathrm{dual}-\mathsf{V}$ curve, so that the difference is not visible. The common parameters for all the curves are ${\mathrm{\Gamma }}_{\text{1D}}/\mathrm{\Gamma }=0.1, {\mathrm{\Delta }}_{\text{c}}/\mathrm{\Gamma }=-90$, and ${\mathrm{\Omega }}_0/\mathrm{\Gamma }=1$. The curves are obtained by using a real $\delta$, calculating complex $q$, and then plotting $\delta /\mathrm{\Gamma }$ as a function of $\text{Re}\left[q\right]/n_0$. The alternative approach: using real $q$, calculating complex $\delta$, and then plotting $\text{Re}\left[\delta \right]/\mathrm{\Gamma }$ as a function of $q/n_0$ will produce results that are indistinguishable for this parameter regime (big ${\mathrm{\Delta }}_{\text{c}}/\mathrm{\Gamma }$ and ${\mathrm{\Delta }}_{\text{d}}/\left|{\delta }_{\text{S}}\right|$). For all the dispersion relations we pick the branches such that $\text{Re}\left[q\right]/n_0>0$.

Figure 3

Log-log plot of the dispersion relations calculated analytically with the continuum model and numerically with the discrete model for randomly placed atoms. The solid black (upper), red (lower), and blue (in between) curves are as in Fig. 2 and are shown for reference. The middle dashed magenta curve is for $\mathrm{\Lambda }$-type scheme computed numerically with the discrete model. It overlaps with the solid blue curve (the same dispersion relation computed analytically), so that the difference is not visible. The lower dashed green curve is the quadratic dispersion relation for the $\mathrm{dual}-\mathsf{V}$ scheme found numerically with the discrete model. The solid cyan and dashed yellow curves that are almost vertical for small $\delta /\mathrm{\Gamma }$ show the linear dispersion relation for the $\mathrm{dual}-\mathsf{V}$ scheme. The solid cyan curve is the analytical result given by Eq. (27), while the dashed yellow curve is computed numerically with the discrete model. Both the numerical curves for the two dispersion relations for the $\mathrm{dual}-\mathsf{V}$ scheme (linear and quadratic) overlap with the respective analytical solutions, so that the difference is not visible. The common parameters are ${\mathrm{\Gamma }}_{\text{1D}}/\mathrm{\Gamma }=0.1, {\mathrm{\Delta }}_{\text{c}}/\mathrm{\Gamma }=-90, {\mathrm{\Omega }}_0/\mathrm{\Gamma }=1, k_0/n_0=\pi /2$, and $N_{\text{u}}={10}^4$ [i.e., $L_{\text{u}}=({10}^4/2)\pi /k_0$].

Figure 4

Placement of atoms for periodic ensembles. At the top, the standing wave of the classical drive is plotted. In the table below, the crosses indicate the positions of the atoms in the standing wave of the classical drive for different values of the number of atoms per unit cell $N_{\text{u}}$. The thick crosses are the atoms in the chosen unit cell, and the thin crosses are the other atoms in the ensemble. The particular choice of the unit cell (gray) is such that the atoms with the nonzero classical drive are taken first (when propagating from the left), and the last atom is placed on the node of the classical drive (at $k_{0}z=\pi /2$) which effectively makes it a two-level atom.

Figure 5

Log-log plot of the dispersion relations for $\mathrm{\Lambda }$-type atoms with different placement of the atoms within the unit cell. The solid black (upper), red (lower), and blue (in between) curves are as in Fig. 2 and are shown for reference. The dashed green curves are for ensembles with regularly placed atoms (see Fig. 4) for the period lengths $N_{\text{u}}=2,4,8,16,32$. The dash-dotted magenta curves are for the same setups, but with a shifted standing wave of the classical drive: $\mathrm{\Omega }\left(z\right)={\mathrm{\Omega }}_{0}cos(k_{0}z+\phi )$ with $\phi =\pi /\left(2N_{\text{u}}\right)$. The common parameters are ${\mathrm{\Gamma }}_{\text{1D}}/\mathrm{\Gamma }=0.1, {\mathrm{\Delta }}_{\text{c}}/\mathrm{\Gamma }=-90, {\mathrm{\Omega }}_0/\mathrm{\Gamma }=1$. The density of the atoms $n_0$ is related to the spacing between the atoms $d$ by $n_0=1/d$. The distance $d$ depends on the desired period length and is given by $d=\pi /\left(N_{\text{u}}{k}_0\right)$ (plus any integer multiple of $\pi /k_0$).

Figure 6

Plot of transmittance ${\left|t\right|}^2$ and reflectance ${\left|r\right|}^2$ of ensembles with $N=4\times {10}^4$ atoms. The dotted magenta and solid blue curves are respectively the transmittance and reflectance of an ensemble with regularly placed $\mathrm{\Lambda }$-type atoms and $N_{\text{u}}=2$ (with the placement shown in Fig. 4). The dash-dotted cyan and dashed green curves are respectively the transmittance and reflectance of an ensemble with randomly placed $\mathrm{\Lambda }$-type atoms and is averaged over 100 ensemble realizations. The other parameters are ${\mathrm{\Gamma }}_{\text{1D}}/\mathrm{\Gamma }=0.1, {\mathrm{\Delta }}_{\text{c}}/\mathrm{\Gamma }=-90, {\mathrm{\Omega }}_0/\mathrm{\Gamma }=1, k_0/n_0=\pi /2$. The interval around $\delta =0$ is shown in more detail in Fig. 7.

Figure 7

(a) Same as Fig. 6, but zoomed in around $\delta =0$. Additionally, the reflectance for an ensemble with randomly placed $\mathrm{dual}-\mathsf{V}$ atoms is plotted (dashed red), and it completely overlaps the reflectance for the regularly placed $\mathrm{\Lambda }$-type scheme. The $\mathrm{dual}-\mathsf{V}$ scheme has the same parameters except that ${\mathrm{\Omega }}_0$ is multiplied by $\sqrt{2}$ to make the dispersion relation equal to the one of the regularly placed $\mathrm{\Lambda }$-type scheme. (b) Dispersion relations for $\mathrm{\Lambda }$-type scheme: regularly placed (solid blue) and randomly placed (dashed green). The dispersion relation was calculated numerically with the transfer matrix formalism for the regularly placed ensemble, and using Eq. (56) for the randomly placed ensemble. The two horizontal dotted lines at $\mathrm{Re}\left[q\right]/n_0=\pi /N$ and $\mathrm{Re}\left[q\right]/n_0=2\pi /N$ give the condition for the first and the second high transmission resonance. At each intersection (for $\delta >0$) of these horizontal lines with the dispersion relation curves, vertical dotted lines are drawn, which can be seen to coincide with the high transmission resonances in (a).