“Stückelberg Interferometry” with Ultracold Molecules

We report on the realization of a time-domain “Stückelberg interferometer”, which is based on the internal-state structure of ultracold Feshbach molecules. Two subsequent passages through a weak avoided crossing between two different orbital angular momentum states in combination with a variable hold time lead to high-contrast population oscillations. This allows for a precise determination of the energy difference between the two molecular states. We demonstrate a high degree of control over the interferometer dynamics. The interferometric scheme provides new possibilities for precision measurements with ultracold molecules.

Figure 1

(a) Molecular energy structure below the dissociation threshold showing all molecular states up $\ell =8$. The relevant states for the present experiment (solid lines) are labeled $|g⟩$, $|g^{\prime }⟩$, and $|l⟩$. Molecules in state $|g⟩$ or $|l⟩$ are detected upon dissociation as shown in (b) and (c). The crossing used for the interferometer is the one between $|g^{\prime }⟩$ and $|l⟩$ near 11.4 G. Initially, ultracold molecules are generated in state $|g⟩$ on the Feshbach resonance at 19.8 G.

Figure 2

(a) Scheme of the Stückelberg interferometer. By ramping the magnetic field over the avoided crossing at $B_c$ at a rate near the critical ramp rate $R_c$ the population in the initial molecular state is coherently split. $\Delta E$ is the binding energy difference at the given hold field $B_0$. After the hold time $\tau$ a reverse ramp coherently recombines the two populations. The populations in the two “output ports” are then determined as a function of acquired phase difference $\varphi \propto \Delta E\times \tau$. (b) Corresponding magnetic-field ramp.

Figure 3

Series of dissociation patterns showing about one oscillation period with $\Delta E/h=155\mathrm{kHz}$ at a hold field of 11.19 G. From one picture to the next the hold time $\tau$ is increased by $1\mu \mathrm{s}$. The first and the last of the absorption images mainly show dissociation of $l$-wave molecules, whereas the central image shows predominant dissociation of $g$-wave molecules.

Figure 4

Interferometer fringes for magnetic hold fields $B_0$ below the crossing of $|g^{\prime }⟩$ and $|l⟩$. The $g$-wave molecular fraction is plotted as a function of the hold time $\tau$. Sinusoidal fits give the oscillation frequency as indicated.

Figure 5

Interferometrically measured binding energy difference $\Delta E$ in the region of the crossing between states $|g^{\prime }⟩$ and $|l⟩$ as a function of magnetic field. Solid circles: Standard ramp sequence of the interferometer. Open circles: Inverted scheme for field values above the crossing. The one-sigma statistical error from the sinusoidal fit is less than the size of the symbols. The solid curve is a hyperbolic fit to the experimental data. Inset: Oscillation at 26.6(3) kHz for a hold field $B_0=11.34\mathrm{G}$ right on the crossing.