Photonic spectrum of bichromatic optical lattices

We study the photonic spectrum of a one-dimensional optical lattice possessing a double primitive cell, when the atoms are well localized at the lattice minima. While a one-dimensional lattice with a simple Wigner-Seitz cell always possesses a photonic band gap at the atomic resonance, in this configuration the photonic transmission spectrum may exhibit no, double, or multiple photonic band gaps depending on the ratio between the interparticle distance $\varrho$ inside the cell and the cell size $a$. The transmission spectra of a weak incident probe are evaluated when the atoms are trapped in free space and inside an optical resonator for realistic experimental parameters.

Figure 1

A possible optical realization of the double-period 1D lattice, considered in this work, can be obtained by using a monochromatic standing wave with wavelength $\lambda$, to which two laser beams are superposed, such that they are rotated by angles of $60°$ and $120°$ with respect to the axis of the lattice. Here, $d_1=\varrho$, $d_1+d_2=\lambda$, and the size of the Wigner-Seitz cell is $a=\lambda$.

Figure 2

(Color online) (a) Polariton dispersion for a 1D bichromatic lattice for different values of $\varrho$ obtained by summing over 40 BZs, when the atoms are trapped in a hollow fiber, whose fundamental mode is Gaussian with waist $w=5\mu \text{m}$ [21]. The curves are evaluated for ${\omega }_1={\omega }_2={\omega }_0-10\gamma$, considering the $D2$ line of ${}^{85}\text{R}\text{b}$ atoms, with $\lambda =780\text{nm}$ and $\gamma =2\pi \times 6\text{MHz}$. The solid black line is plotted for comparison, and corresponds to the case in which atoms and field are not coupled $\left({\mathcal{G}}_j=0\right)$. (b) Photonic band gap (numbered from lower to higher frequency) as a function of $\varrho /a$, compared with the analytical prediction obtained from Eq. (22).

Figure 3

(Color online) Same as in Fig. 2 but for ${\omega }_1={\omega }_2={\omega }_0-530\gamma$.

Figure 4

(Color online) Same as in Fig. 2 but for ${\omega }_1={\omega }_0-10\gamma$ and ${\omega }_2={\omega }_0+530\gamma$.

Figure 5

(Color online) Same as in Fig. 2 but for ${\omega }_1={\omega }_0-530\gamma$ and ${\omega }_2={\omega }_0+530\gamma$. Note that gaps 1 and 3 overlap.

Figure 6

(Color online) The absolute squares of the transmission coefficient (left panel) and of the reflection coefficient (right panel) as functions of $\left({\omega }_p-\omega \right)/\gamma$ are plotted for a lattice of ${}^{85}\text{R}\text{b}$ atoms with areal density $n_s=5.7\times {10}^{-2}\mu {\text{m}}^{-2}$, trapped by laser beams detuned to the blue of resonance by $10\gamma$ and for ${10}^6$ atomic planes.

Figure 7

(Color online) Same as Fig. 6, but $\varrho =0.2\lambda$.

Figure 8

(Color online) Same as Fig. 6, but for $\varrho =0.24\lambda$.

Figure 9

(Color online) The absolute square of the transmission coefficient as a function of $\left({\omega }_p-{\omega }_0\right)$ (in units of $\gamma$) is plotted for a lattice of ${}^{85}\text{R}\text{b}$ atoms with areal density $n_s=5.7\times {10}^{-2}\mu {\text{m}}^{-2}$, trapped by laser beams detuned to the blue of resonance by $10\gamma$, for 200 planes, in the presence of a cavity of length $L=85\text{mm}$ and finesse $\mathcal{F}=170000$, and for $\phi =\pi /2$. Note that for $\varrho =0$ the atoms are trapped at the nodes of the resonator and hence do not interact with the cavity field.

Figure 10

(Color online) Same as Fig. 9, but for $\phi =0$.