Preparing and Probing Atomic Number States with an Atom Interferometer

We describe the controlled loading and measurement of number-squeezed states and Poisson states of atoms in individual sites of a double well optical lattice. These states are input to an atom interferometer that is realized by symmetrically splitting individual lattice sites into double wells, allowing atoms in individual sites to evolve independently. The two paths then interfere, creating a matter-wave double-slit diffraction pattern. The time evolution of the double-slit diffraction pattern is used to measure the number statistics of the input state. The flexibility of our double well lattice provides a means to detect the presence of empty lattice sites, an important and so far unmeasured factor in determining the purity of a Mott state.

Figure 1

(a) Action of a double well beam splitter. Different colors indicate the lack of phase coherence between double well sites. (b) Optical analog of our atom interferometer with an example diffraction pattern resulting from a single input mode.

Figure 2

(a) Dumping the population initially in $|L⟩$ into highly excited, possibly unbound bands. (b) The resulting TOF image: the population initially in $|R⟩$ appears in the first BZ of the $\lambda$-lattice; the population initially in $|L⟩$ appears in higher BZs as a ring around the first BZ. (c) Calibration of the BS in Fig. 1. After applying the BS at a given value of $\delta x/\lambda$, the population in the $|L⟩$ (red) or $|R⟩$ (blue) sites are measured by dumping the other site and measuring the fraction of atoms remaining in the first BZ. The value of $\delta x/\lambda$ where the two curves cross corresponds to a $50/50$ BS.

Figure 3

(a) A double-slit interference pattern (inset) and integrated density profile after 13 ms TOF. (b) Example $y$-integrated diffraction pattern as a function of time for tilts $V=4.90(3)\mathrm{kHz}$ (left) and $V=-4.90(3)\mathrm{kHz}$ (right). (c) $C(t)$ for $N=1$ (black ○), $N=2$ (green $\times$), and $⟨N⟩=1$ (blue ▪). Solid lines are a fit to Eq. (2). Note: The plotted $C(t)$ for $N=2$ was scaled by 2. (d) $\varphi (t)$ for the cases shown in (c). The $\varphi (t)$ points which were derived from a signal with $2\times C(t)<0.1$ are shaded in light green. $\varphi (t=0)=0$ for all three cases, but each has been shifted for visual clarity.

Figure 4

We use Eq. (2) to fit $C(t)$ for the fast and slow load data. (a) Extracted $f_1$ for varied slow load times. The uncertainties are from the fit to Eq. (2). (b) Using Eq. (2) to fit the fast load $C(t)$ we extract values for $f_1$, $f_2$, $f_3$, and $f_4$. From those coefficients we determine $⟨N⟩$, the peak number density of atoms in $\lambda$-lattice sites, as a function of $N_{\mathrm{BEC}}$. The line and the gray shaded area are the peak number density scalings predicted by the TF approximation (with no adjustable parameters) and our estimated uncertainty in the calibration of $N_{\mathrm{BEC}}$ (factor of $\pm 1.5$). The uncertainties on the individual points are from a combination of the uncertainties in $f_i$ and from the fit determining $⟨N⟩$.