Realization of a deeply subwavelength adiabatic optical lattice

We propose and describe our realization of a deeply subwavelength optical lattice for ultracold neutral atoms using $N$ resonantly Raman-coupled internal degrees of freedom. Although counterpropagating lasers with wavelength $\lambda$ provided two-photon Raman coupling, the resultant lattice period was $\lambda /2N$, an $N$-fold reduction as compared to the conventional $\lambda /2$ lattice period. We experimentally demonstrated this lattice built from the three $F=1$ Zeeman states of a ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate, and generated a lattice with a $\lambda /6=132\mathrm{nm}$ period from $\lambda =790\mathrm{nm}$ lasers. Lastly, we show that adding an additional rf-coupling field converts this lattice into a superlattice with $N$ wells uniformly spaced within the original $\lambda /2$ unit cell.

Figure 1

Experimental and conceptional schematic for $N=3$. (a) Experimental geometry and level diagram. A BEC is illuminated by a pair of laser beams that complete two-photon Raman transitions between $N$ internal states. (b) Computed adiabatic potentials illustrating the spatial subdivision of this lattice. The dashed curves were computed for $\hslash \mathit{\Omega }=(1.0,1.0,1.0)\times E_R$ and $\mathit{\delta }=\mathbf{0}$; the solid curves were computed for $\hslash \mathit{\Omega }=(1.0,1.25,0.75)\times E_R$ and $\mathit{\delta }=\mathbf{0}$. The ternary diagram to the right shows the color scheme used throughout this paper to indicate the fractional composition of states. (c) Internal state–momentum coupling diagrams showing nodes describing states (labeled by internal state and momentum) connected by links denoting the laser-induced momentum kicks on each link (left) transferred to a single link (right) after the state-dependent gauge transformation.

Figure 2

Enlarged BZ. The top axis shows the frequency shift $\delta \nu$ used to effect a moving lattice that populated the desired $\hslash q$ state. Top: Computed band structure. Each solid curve is shaded in accordance with the population in the $|-1\rangle , |0\rangle , |+1\rangle$ using the ternary diagram in Fig. 1. The dashed curves depict the bare free particle dispersion absent Raman coupling. Bottom: Internal state composition of the lowest band. Different regions of the BZ were explored by starting in the three $|m\rangle$ states. The solid curves plot the outcome of a full simulation of our experimental protocol. We determine the Raman coupling strengths to be $\hslash \mathit{\Omega }=(0.54\left(1\right),0.29\left(1\right),0.61\left(1\right))\times E_R$ and detunings $\hslash |{\delta }_m|\le 0.07\left(1\right)E_R$.

Figure 3

(i) Subwavelength lattice and (ii) superlattice. (a) Momentum-coupling graphs as in Fig. 1, with red links from Raman coupling and black links from rf coupling. Panels (b)–(e) share the same parameters with $\hslash \mathit{\Omega }=(0.87,0.98,0.82)\times E_R$ and $\hslash \mathit{\delta }=(0.08,0.08,0.15)\times E_R$, with columns (i) and (ii) depicting $\hslash {\mathit{\Omega }}_{\mathrm{rf}}=\mathit{0}$ and $\hslash {\mathit{\Omega }}_{\mathrm{rf}}=(0,0.7,0)\times E_R$, respectively. (b) Computed adiabatic potentials with rf phase ${\varphi }_0=\pi /4$. (c) Computed band structure for the experimental parameters [shaded per Fig. 1]. (d) Measured probability in $|-1\rangle$ as a function of time and crystal momentum, for a system initially in $|-1\rangle$. Dark and light color tones indicate high and low probability, respectively. (e) PSDs obtained from time traces such as in (c). Each PSD is derived from the time evolution of all three internal states, averaged over five time series. For the Raman-only data (i) we started in the $|m=-1\rangle$ state only, while for the Raman and rf data in (ii) we increased the signal-to-noise ratio by combining data starting in all three internal states. The horizontal structures present in both data sets around $2.5E_R$ result from laboratory technical noise.

Figure 4

Stroboscopic evolution starting in $|m=0\rangle$. During each pulse the Raman lasers were applied for $50\mu \mathrm{s}$ and then removed for $16.8\mu \mathrm{s}$, giving a total pulse duration close to $h/\left(4E_R\right)$. Top: Absorption images showing the initial $k=0$ state and diffracted orders for 2 and 13 pulses, respectively. The symbols plot the population $|m=0\rangle , |+1\rangle$, and $|-1\rangle$, colored blue, orange, and gray, respectively. Bottom: The solid curves depict our full lattice model using calibrated couplings $\hslash \mathit{\Omega }=(0.58\left(1\right),0.57\left(1\right),0.58\left(1\right))\times E_R$ and detunings $\hslash \mathit{\delta }=(0,-0.03\left(1\right),0)\times E_R$. The detuning of the initial state ${\delta }_0$ was the only fit parameter. The dashed curve plots the prediction of the simple lattice model with a depth $\hslash ({\mathrm{\Omega }}_0+{\mathrm{\Omega }}_{-1})=1.15\left(2\right)E_R$ and the same $-0.03E_R$ energy shift of the initial state.