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Direct observation of number squeezing in an optical lattice

We present an in-situ study of an optical lattice with tunneling and single lattice site resolution. This system provides an important step for realizing a quantum computer. The real-space images show the fluctuations of the atom number in each site. The sub-Poissonian distribution results from the approach to the Mott insulator state, combined with the dynamics of density-dependent losses, which result from the high densities of optical lattice experiments. These losses are clear from the shape of the lattice profile. Furthermore, we find that the lattice is not in the ground state despite the momentum distribution which shows the reciprocal lattice. These effects may well be relevant for other optical lattice experiments, past and future. The lattice beams are derived from a microlens array, resulting in lattice beams which are perfectly stable relative to one another.

momentum distribution which shows the reciprocal lattice. These effects may well be relevant for other optical lattice experiments, past and future. The lattice beams are derived from a microlens array, resulting in lattice beams which are perfectly stable relative to one another.
In a ground-state optical lattice with a negligible on-site repulsive interaction energy U relative to the hopping energy J, the wavefunction of each atom spreads across the entire lattice, giving the same phase to each lattice site. The uncertainty in the phase for each lattice site φ σ is much less than unity. Due to the small interaction However, optical lattices often exhibit significant losses, at higher rates than expected 8,15,18,19 . Such losses can change N σ (see Appendix). If the loss rate 1 In Ref. 20 it was demonstrated that electron microscopy has the potential for studying an optical lattice with magnificent resolution, although averaging over many instances of the experiment is currently required. Here, we employ a type of absorption imaging of light to study a 2D optical lattice, an array of lattice sites lying in a horizontal plane. Each lattice site has a slightly elongated cigar shape. For the highest lattice depths reported here, the vertical radius of each site is 2.3 times the horizontal radius. Our lattice can be considered as a hybrid between an array of focused dipole traps 21 and an optical standing wave 1,15,22 . The lattice is a greatly reduced image of an array of foci created by a microlens array, as shown in Fig. 1.
However, this image is filtered in the Fourier plane, allowing for a variety of possible lattices. Fig. 1l is an image of the Fourier plane, showing the Fourier transform of the array of foci created by the microlens array. Each of the discrete points in the Fourier plane represents an available lattice beam, which will propagate in a unique direction at the location of the atoms. For our experiment, we use the 4 points circled in blue, giving a lattice period of a = 2.0 microns. All of the lattice beams of Fig. 1l have the same frequency. This does not create a stability problem due to slight relative motions of the beams, because the lattice is an image of the microlens array, an object whose shape does not change. In order to verify the uniformity of the lattice beams, we add a 5 th beam, circled in green in Fig. 1l. This vertically propagating beam has two effects on the lattice. Firstly, it supports the lattice against gravity by dividing it into planes, and secondly, it increases the lattice period by a factor of 2 . Fig. 1m,n show a large thermal cloud which has been loaded into the 5-beam lattice, and imaged from two directions. Fig. 1m shows that the lattice is uniform over the region indicated by the green ellipse, which is the region of the 4-beam lattice occupied by the condensate in the rest of this work.
In the 4-beam lattice, an additional laser beam with an elliptical cross section confines the condensate in the horizontal plane, as shown in Fig. 2a,b. We refer to this beam as the vertical confinement beam (Fig. 1k). There is no magnetic trapping potential. The vertical confinement beam has a waist (1/e 2 radius) of 3. The reciprocal lattice period a / 2π is clearly visible in the images. However, such a momentum space image is not sufficient to prove that the phase is uniform 16,27 .
These time-of-flight images have a somewhat different appearance than the images in previous works 4 . This is a result of the fact that the momentum spacing between orders, indicated by a / 2π in Fig. 3g, is smaller than the momentum distribution of the cloud released from the vertical confinement beam in the absence of the lattice, seen in Fig. 3h. Thus, the various orders expand and overlap, forming the pattern of Fig. 3g. Imaging the lattice in time-of-flight from the horizontal direction in Fig. 3i, a reversal is seen in the overall aspect ratio of the momentum distribution when the lattice is turned on, relative to Fig. 3j. This occurs since turning on the lattice reverses the in-situ aspect ratio by converting the horizontal pancakeshaped trap into slightly elongated vertical ellipsoids. However, the figure shows that most of the lattice sits on a curve of constant µ . The phase of each lattice site can therefore evolve at almost the same rate, maintaining an extended wavefunction with almost constant phase across the lattice. We thus see that even a lattice which is not in the ground state can have a uniform phase.
In conclusion, we have imaged an optical lattice in-situ, with single-site resolution. We find that losses affect the statistics of the number fluctuations between lattice sites, which can limit the usefulness of the Mott transition to produce a uniform distribution of atoms in a lattice. On the other hand, such losses can be useful for preparing exotic quantum phases 16 . Such losses could possibly be avoided however, by using one or two atoms per site 3,8 . We also find that an almost uniform phase across the lattice does not necessarily imply that the lattice is in the ground state. The in-situ imaging of the current system could be exploited in the study of non-linear effects such as 2D gap solitons 28 . By rotating the microlens array 22 , quantum phase transitions of vortex states could also be explored [29][30][31][32] . Single-site addressability could readily be added to the system. By combining such a system with single-atom sensitivity, a quantum computer could be created 13 .
We thank A. Auerbach, E. Polturak, , R. Pugatch, E. Altman, and Y. Kafri for helpful discussions. This work was supported by the Israel Science Foundation.   At the equilibrium value of N δ , the rates (2) and (3) will be equal. Note that equating (2) and (3) is not self-consistent, since the former was derived with constant (S6) For 3-body losses, (6) gives N N N 53 . 0 18 / 5 = ≈ δ . A numerical simulation of the stochastic decay process gives a value which is larger by roughly 15%.