Optimal pulse propagation in an inhomogeneously gas-filled hollow-core fiber

We study optical pulse propagation through a hollow-core fiber filled with a radially inhomogeneous cloud of cold atoms. A copropagating control field establishes electromagnetically induced transparency. In analogy to a graded index fiber, the pulse experiences microlensing and the transmission spectrum becomes distorted. Based on a two-layer model of the complex index of refraction, we can analytically understand the cause of the aberration, which is corroborated by numerical simulations for a radial Gaussian-shaped function. With these insights, we show that the spectral distortions can be rectified by choosing an optimal detuning from one-photon resonance.

Figure 1

Atomic level scheme in Raman configuration with transition frequencies ${\omega }_{31}={\omega }_3-{\omega }_1$ and ${\omega }_{32}={\omega }_3-{\omega }_2$ interacting with the control pulse ${\mathrm{\Omega }}_c$ and probe pulse ${\mathrm{\Omega }}_p$. The one- and two-photon detunings are denoted by $\mathrm{\Delta }$ and $\delta$, respectively. Spontaneous decay rates ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ couple the excited state $|3\rangle$ to the ground states $|1\rangle$ and $|2\rangle$. Residual perturbations are accounted for by a ground-state decoherence rate ${\mathrm{\Gamma }}_g$. Experimentally, the presence of level $|4\rangle$ can be relevant, but will not be considered at present.

Figure 2

Propagation of a probe pulse ${\mathcal{E}}_p$ (red) through a HCF of length $2l+L$ with core radius $a_0$. The fiber is filled with an inhomogeneous cold gas of density $n_a(r,z)$. Along the propagation direction, the fiber is divided into the entrance and exit zones $①$ and $③$, where the atomic density rises slowly to its homogeneous value $n_a\left(r\right)$ in $②$. The polarizability is controlled by the copropagating immutable control beam ${\mathcal{E}}_c$ (blue). The inhomogeneous, frequency dependent susceptibility will alter the mode shape and phase of the probe pulse.

Figure 3

Real part ${\chi }^{\prime }\left(\delta \right)$ (solid red) and imaginary part ${\chi }^{\prime \prime }\left(\delta \right)$ (dashed blue) of the complex susceptibility versus two-photon detuning $\delta$ for (a) $({\mathrm{\Omega }}_c,\mathrm{\Delta },{\mathrm{\Gamma }}_g)=(0,0,0)\mathrm{\Gamma }$, (b) $({\mathrm{\Omega }}_c,\mathrm{\Delta },{\mathrm{\Gamma }}_g)=(1,0,0)\mathrm{\Gamma }$, (c) $({\mathrm{\Omega }}_c,\mathrm{\Delta },{\mathrm{\Gamma }}_g)=(1,1,0)\mathrm{\Gamma }$, with ${\chi }_0^{\left(1\right)}=0.1$.

Figure 4

Real-part of the two-layer potential $\text{Re}\left[w\right(r,\delta \left)\right]$ (solid black) versus radius $r$ for different two-photon detunings: (a) $\delta =-0.2\mathrm{\Gamma }$, (b) $\delta =-0.1\mathrm{\Gamma }$, (c) $\delta =0\mathrm{\Gamma }$, (d) $\delta =0.2\mathrm{\Gamma }$ with other parameters $w_0=12.5$, ${\mathrm{\Omega }}_c=\mathrm{\Gamma }$, $\mathrm{\Delta }=0$, $r_1=0.301$. The corresponding radial intensities $|u^{\left(1\right)}{|}^2$ are shown for a Gaussian (blue, solid) and a two-layer potential (red, dashed) in arbitrary units. The radially weighted intensity $r^{\prime }{\left|u^{\left(1\right)}\right|}^2$ (dashed $•$) are for the two-layer potential in arbitrary units. The yellow area is proportional to the potential energy of Eq. (30).

Figure 5

(a)–(c) Complex dispersion relation ${\varepsilon }^{(1,2)}\left(\delta \right)$ of the two-layer model versus two-photon detuning $\delta$: $\text{Re}\left[{\varepsilon }^{\left(1\right)}\right]$ ($•$, solid), $\text{Re}\left[{\varepsilon }^{\left(2\right)}\right]$ ($▾$, dashed), $\text{Im}\left[{\varepsilon }^{\left(1\right)}\right]$ ($◼$, solid), $\text{Im}\left[{\varepsilon }^{\left(2\right)}\right]$ ($▸$, dashed). (d)–(f) Corresponding intensities of eigenmodes $|u^{\left(1\right)}{(r;\delta =0.2\mathrm{\Gamma })|}^2$ (black) and $|u^{\left(2\right)}{(r;\delta =0.2\mathrm{\Gamma })|}^2$ (black, dashed) versus radius $r$ for different atomic distribution widths $r_1=1$ (a,d), 0.7 (b,e), 0.35 (c,f). The overlap between $|u^{\left(i\right)}{|}^2$ and the atomic medium is highlighted in gray shades. Note the different sign in the curvatures of $\text{Re}\left[{\varepsilon }^{\left(m\right)}\right]$ for $r_1=0.35$, indicating a different sign of the microlensing effect. The other parameters are $({\mathrm{\Omega }}_c,\mathrm{\Delta },{\mathrm{\Gamma }}_g)=(1,0,0)\mathrm{\Gamma }$ and $w_0=12.5$.

Figure 6

(a) Complex energies versus two-photon detuning $\delta$: $\text{Re}\left[{\varepsilon }^{\left(1\right)}\right]$ for two-layer (solid $•$) and Gaussian (dashed $⧫$) potential as well as $\text{Im}{\left[\varepsilon \right]}^{\left(1\right)}$ for both potential types (solid $◼$, dashed $▾$). For comparison, we superimposed the input amplitude ${\mathcal{E}}^{\left(\mathrm{in}\right)}\left(\delta \right)$ (black, solid) in arbitrary units with $w_0=12.5$, ${\mathrm{\Omega }}_c=\mathrm{\Gamma }$, $r_1=0.301$, $\mathrm{\Delta }=0$ and other parameters as in Figs. 4 and 5. (b) Transmitted power $P_p^{\left(\text{out}\right)}\left(t\right)$ in the two-layer potential versus time $t$ for different one-photon detunings: $\mathrm{\Delta }/\mathrm{\Gamma }=-1\left(•\right),-0.5(◂),0\left(⧫\right),0.5\left(◼\right)$ with $r_1=0.301$, $w_0=25.0$, ${\mathrm{\Omega }}_c=\mathrm{\Gamma }$, $d_{opt}=100$, ${\tau }_p=150\mathrm{ns}$.

Figure 7

(a) Refraction strength functions ${\mathcal{C}}^{(1,2,3)}\left(r_1\right)$ vs width of gas filled core $r_1$. ${\mathcal{C}}^{\left(1\right)}$ shows the local minimum $C^{\left(1\right)}(r_1=0.301)=-0.0183$ ($◂$). ${\mathcal{C}}^{\left(2\right)}$ exhibits two zero crossings, indicating a complete mitigation of lensing effects ($▸$), while ${\mathcal{C}}^{\left(m\right)}$ for higher modes is an increasingly oscillating function with diminishing amplitude ($m=3$, $•$). (b) Contour plot of ${\mathrm{\Delta }}_{\text{opt}}^{\left(1\right)}(v,n_0)$ for a specific experimental system using rubidium atoms (${}^{87}\mathrm{Rb}{D}_1$ line, $S_{F{F}^{\prime }}=1/6$) and $C^{\left(1\right)}(r_1=0.301)$.

Figure 8

Intensity transmission $T\left(\delta \right)$ vs detuning $\delta$ normalized to the EIT window width [see Eq. (51)] for different potential depths $w_0=5$ (a), 25 (b), 50 (c) with $r_1=0.301$ and different optical densities $d_{opt}=5\left(•\right),25(◂),250\left(▸\right)$. The one-photon detuning is $\mathrm{\Delta }=0$.