Direct Observation of a Sub-Poissonian Number Distribution of Atoms in an Optical Lattice

We report single-site resolution in a lattice with tunneling between sites, allowing for an in situ study of stochastic losses. The ratio of the loss rate to the tunneling rate is seen to determine the number fluctuations, and the overall profile of the lattice. Sub-Poissonian number fluctuations are observed. Deriving the lattice beams from a microlens array results in perfect relative stability between beams.

Figure 1

The experimental system. (a) The microlens array. (b) The array of 2000 foci generated by the microlens array. These foci are similar to delta functions. (c) Lens. (d) The Fourier plane where the desired lattice beams are selected. (e) Aspheric lens used for absorption imaging, as well as producing the lattice. (f) The lattice sites. (h) The vertical confinement beam. (i) Aspheric lens used for absorption imaging, as well as focusing the vertical confinement beam. (j) An image of the available lattice beams in (d). The four beams circled in blue are used in this experiment. The fifth vertically propagating beam circled in green is combined with the other four to provide vertical confinement for diagnostic purposes. The dashed circle indicates the radius of (e). Only beams within this line are usable. (k) Large thermal cloud in the 5-beam lattice, imaged through (e). (l) The horizontal planes of the 5-beam lattice, as viewed through (i). Gravity is indicated by “g”. The green ellipses in (k) and (l) indicate the region of the lattice used for the experiment, which employs 4 lattice beams and the vertical confinement beam. (k) and (l) verify the uniformity of the lattice beams.

Figure 2

In situ images of the lattice, with single-site resolution. (a) The condensate in the vertical confinement beam only ($V_0=0$). The image is in a horizontal plane. (c) The partially divided lattice with a depth $V_0/h=0.7\mathrm{kHz}$ and chemical potential $\mu /h=2\mathrm{kHz}$. (e) The lattice with a depth $V_0/h=7\mathrm{kHz}$ and $\mu /h=3\mathrm{kHz}$. The wave function is evanescent between sites. (b), (d), and (f) are as in (a), (c), and (e), but imaged in a vertical plane.

Figure 3

Profiles of the lattice in situ, and time-of-flight images. The vertical axes in (a)–(f) are in units of atoms per lattice site. (a) Profiles of the lattice in the $z$ direction, summed over the pixels composing the central column of lattice sites. The upper, middle, and lower traces correspond to Figs. 2a, 2c, 2e. (c) Smoothed $z$ profiles. The individual lattice sites have been smoothed away to better show the overall profile. The upper, middle, and lower profiles correspond to $V_0/h=0.7\mathrm{kHz}$, 1.0 kHz, and 7 kHz, respectively. Several profiles for each depth are shown. Below (above) the horizontal lines, the wave function of each lattice site can be considered to be 1D (3D) [29]. The dash-dotted, dashed, and dotted lines correspond to the lower, middle, and upper profiles. (e) Smoothed $z$ profiles showing the decay of the lattice after a 10 ms ramp-up to $V_0/h=7\mathrm{kHz}$. One profile is shown with and without smoothing. Times between 5 and 1000 ms after the ramp-up are shown. The blue curves show the theoretical ground-state profiles. The dash-dotted line shows the transition from 1D to 3D lattice sites. The inset shows the population of the central lattice site from each of the profiles. The solid curve of the inset shows a fit taking one and 3-body losses into account. The 3-body loss rate is an adjustable parameter in the fit. (b), (d), and (f) $x$ profiles corresponding to the $z$-profiles of (a), (c), and (e). (g) The lattice after 5 msec time-of-flight, imaged in a horizontal plane. (h) 5 msec time-of-flight from the vertical confinement beam only ($V_0=0$). The diameter is larger than $2\pi /a$, causing the overlap of the orders in (g). (i) and (j) Same as (g) and (h) but with 3 msec time-of-flight, viewed in a vertical plane. (i) and (j) have opposite aspect ratios because the in situ aspect ratio reverses when the lattice is turned on.

Figure 4

In situ observation of sub-Poissonian number fluctuations in individual lattice sites. (a) The solid curve is a histogram of the fluctuations for 6 images like Fig. 2e. The dashed curve indicates the Poissonian distribution. The vertical axis indicates the total number of sites in 6 images with a given value of $\delta N_i$. (b) The solid curve is a histogram of the fluctuations for 6 images like Fig. 2c, with less losses than in (a). The dashed curve indicates the Poissonian distribution. (c) The difference between the measured histogram and the Poissonian distribution. The solid and dashed curves corresponds to (a) and (b). (d) The optical lattice of Fig. 2e, showing $\delta N_i$, which is the population in each site, relative to the mean population for each site. The color scale indicates the number of atoms.