Robust spatial coherence $5\mu \mathrm{m}$ from a room-temperature atom chip

We study spatial coherence near a classical environment by loading a Bose-Einstein condensate into a magnetic lattice potential and observing diffraction. Even very close to a surface ($5\mu \mathrm{m}$), and even when the surface is at room temperature, spatial coherence persists for a relatively long time ($\ge 500\mathrm{ms}$). In addition, the observed spatial coherence extends over several lattice sites, a significantly greater distance than the atom-surface separation. This opens the door for atomic circuits, and may help elucidate the interplay between spatial dephasing, interatomic interactions, and external noise.

Figure 1

Experimental configuration. (a) An artist's view of the trapped cloud of atoms a few $\mu \mathrm{m}$ from the surface of an atom chip. The atoms are trapped below the surface to allow their spatial distribution to evolve after release from the trap without being adsorbed onto the chip. (b) An optical microscope image of the current-carrying “snake wire” (500-nm-thick Au) that creates a magnetic lattice potential along the $x$ axis. The main “trapping wire” is not shown. (c) The modulated potential along the $x$ axis is depicted together with the harmonic potential produced by the trapping wire, giving rise to the atom density profile sketched in (a). The trapping wire is used to adjust the distance $d$ of the trap from the snake wire, with progressively stronger potential modulation controlled by reducing $d$ from $9.0\mu \mathrm{m}$ (no modulation) to $4.0\mu \mathrm{m}$ (greatest modulation shown). The modulation amplitude may also be fine-tuned by adjusting the snake-wire current. Not shown is the radial confinement potential that prevents the atoms from hitting the surface. The imaging beam propagates along the $y$ axis but we do not provide in situ images of the modulated atom density because of insufficient imaging resolution ($\sim 7\mu \mathrm{m}$).

Figure 2

Experimental drifts and the effect of jitter. (a) Horizontal ($x$) position of the center of the atomic cloud, measured in situ during the first 3–$5\mathrm{h}$ of two experimental runs performed several weeks apart (black and red squares). The blue lines depict linear fits to a fast movement during an initial $\approx 1$-h warm-up period, followed by a slower drift for the remainder of these experimental runs. (b) Loss of contrast that can be caused by residual “jitter” (i.e., shot-to-shot variability in the in situ horizontal position) and inaccuracies in our post-correction procedures. The blue curves are calculated by applying random displacements along $x$ to a short-term average of 30 consecutive experimental cycles, with standard deviations of $2.0\mu \mathrm{m}$ (dashed) and $3.6\mu \mathrm{m}$ (solid). The extent of smearing due to assumed jitter is consistent with data obtained by averaging over 200 experimental cycles acquired over several days (red curve).

Figure 3

Experimental signal. Diffraction pattern observed after a trap holding time of $t=100\mathrm{ms}$ for 30 consecutive experimental cycles (no post-selection or post-correction is used). (a) Average of images acquired by absorption imaging. (b) A cut through the center of (a). The pronounced contrast of $\approx 0.6$ is highly unlikely to result from any random processes, thus demonstrating spatial coherence. The origin of the observed asymmetry is discussed in Sec. 5c. Only a narrow vertical integration band is required for obtaining a relatively high signal-to-noise ratio, but a wider band would not significantly reduce the observed contrast since the fringes are straight and parallel to the vertical axis $z$.

Figure 4

Robust spatial coherence. (a) Repeating the cut of Fig. 3 for trap holding times of $t=30–500$ ms, averaged over all post-selected experimental cycles for each holding time. Here we include a 100-$\mu \mathrm{m}$-wide wide band along the vertical ($z$) axis in order to improve the signal-to-noise ratio. We find that the first-order peaks are “locked” at about $\pm 15\mu \mathrm{m}$, independent of $t$, for the $>1000$ images collected in this figure. The progressively declining OD for increasing $t$ indicates an atom lifetime of $500\pm 50\mathrm{ms}$, consistent with measured spin-flip rates for this experiment, which are mostly due to technical noise [39]. (b) Data points [same color code as in (a)] show the optical density difference (“OD diff”) between the diffraction side-peak maxima and minima, averaged over all the experimental images obtained for each holding time. Error bars are extracted from a bootstrapping procedure [40]. The black curve simulates how the same OD difference, after averaging a given number of images, would drop towards zero for a random distribution of phases amongst the potential wells. The shaded band around this curve shows a $1\sigma$ standard deviation for the average that would be caused by such random phases. The data lie 2-5 $\sigma$ outside this band.

Figure 5

Persistent spatial coherence. (a) Average contrast of the two first-order fringes for all images containing $1100<N<1900$ atoms. Error bars are extracted from a bootstrapping procedure [40] that yields the experimental statistical errors for each holding time. The shaded band shows systematic errors corresponding to several different ranges of $N$ (see text). (b) Average contrast of the two fringes, showing a systematic dependence on $N$, hence requiring the restricted range used in (a). See text for possible sources of this dependence. The dashed line is an exponential fit to guide the eye; error bars are omitted for clarity. Color code and symbol types as in Fig. 4.

Figure 6

Coherence length. Comparison of the observed contrast (dashed line) to two simple numerical models for the contrast expected when averaging 30 images. The BEC is assumed to be coherent over a given number of lattice sites $n$ and random for the remainder (blue points), or it is assumed that the phase correlation drops exponentially over a $1/e$ distance of $n$ sites (yellow points). In both cases, the observed contrast corresponds to a spatial coherence extending over $\approx 4$ sites, i.e., $15\mu \mathrm{m}$.

Figure 7

Simulated evolution during the holding time. (a) Initial ground-state BEC with $N=1500$ atoms in the nearly harmonic (unmodulated) potential. (b)–(e) Re-distribution of the atomic density as a function of the holding time $t$ in the magnetic lattice, shown as atomic column densities in the $x-z$ plane. (f) Atomic density per unit volume (red) and unit length (black) are shown as an average over $t=5,10,\cdots ,30\mathrm{ms}$. In this simulation, the chemical potential slightly exceeds the barrier height (Sec. 6). The density between the barriers becomes smaller with horizontal positions further from the center due to the rising longitudinal harmonic potential.

Figure 8

Simulated evolution during the launch. The cloud is ejected from the magnetic lattice by increasing the current in the snake wire from $5.5$ to $18\mathrm{mA}$ in $100\mu \mathrm{s}$ (Sec. 2a). (a) Trajectory of the center of mass of the cloud during the launch. The cloud arrives at the outer turning point (created by the trapping wire which is still on) and starts to accelerate back towards the atom chip before all currents are turned off (see text). Dots along the trajectory depict times at which the atomic density is shown in (b). Full release occurs after $2.3\mathrm{ms}$. (b) The near-field diffraction pattern is seen to start forming already during the launching stage, while the atomic cloud starts to expand rapidly along the radial direction. Our optical resolution is currently unable to resolve detailed features of the expanding cloud during this period (see Fig. 9 for the corresponding far-field simulation results.)

Figure 9

Simulated diffraction pattern after release. (a) The shape of the atomic cloud after $12\mathrm{ms}$ of free-fall in the $x-z$ plane, and (b) a cut through the center of the cloud along the horizontal direction $x$. The finite resolution of an imaging system similar to ours is included (Appendix pp1). The observed asymmetry between the first-order peaks can arise from slight asymmetries in the magnetic potential near the snake wire due to the well-known rotation of the cloud by the $\mathsf{Z}$-shaped trapping wire (Sec. 5c). Since this simulation was done with the full magnetic potential, the latter rotation is introduced automatically.

Figure 10

Focusing effect. (a) Single-shot image of a BEC, launched and released under the same conditions used throughout this work, and acquired after $12\mathrm{ms}$ of free fall (time-of-flight). The BEC is initially held in a trap $9\mu \mathrm{m}$ from the surface, where there is no potential modulation, and hence no fringes are seen in the image. (b) A cut through the center of (a) with a measured Thomas-Fermi full width of $18\mu \mathrm{m}$, compared to a width of about $35\mu \mathrm{m}$ calculated with no focusing. The thermal tails are seen to be unfocused.

Figure 11

A simple model for the formation of an asymmetric diffraction pattern. (a) A symmetric diffraction pattern with two first-order peaks is formed by a Fourier transform of a wave function with a periodic amplitude and a constant phase. (b) A noisy diffraction pattern is obtained by applying a random phase between the sites. (c) and (d) Asymmetry between the two first-order peaks is obtained when a linear phase difference is applied between the sites (while keeping a constant phase within each site). Here (c) and (d) are obtained from a phase difference of $0.5\mathrm{rad}$ and $1\mathrm{rad}$ between adjacent sites, respectively (see text for possible sources of phase differences between sites). Note that, in addition to asymmetry, the fringe patterns in (c) and (d) also shift horizontally relative to (a).

Figure 12

Spatial coherence for low chemical potential. Diffraction pattern observed after a trap holding time of $t=500\mathrm{ms}$ for an average of all 29 experimental cycles having $N=400\pm 130$ atoms, corresponding to the smallest number of atoms in Fig. 5. Note that the chemical potential is most likely below the central potential barrier, as depicted in the inset (dashed line, see text).

Figure 13

Atom-surface distance measurements: imaging the atomic cloud in the trap near the chip. (a) This simulated image shows the cloud and its reflection from the chip surface; a vertical cut through the center of the cloud is shown in (b). (c) and (d) The corresponding cloud and vertical cut from experimental measurements after a holding time of $t=30\mathrm{ms}$ in the modulated potential. The detection efficiency for the reflected cloud image is assumed to be reduced by 30% relative to the direct image. The peak positions of the simulation are chosen to match those of the experimental measurements and show a systematic difference between the measured atom-surface distance and the actual distance of about $0.8\mu \mathrm{m}$ (see text).

Figure 14

Contrast vs time for varying atom numbers $N$. Here we use an alternative analysis to that of Fig. 5 by analyzing data samples that are chosen to conform to a certain mean $N$. Each sample consists of exactly 30 experimental cycles that are closest to the means $\langle N\rangle$ shown. While the dependence of contrast on $N$ [as presented in Fig. 5] is evident, it is also clear that the qualitative behavior of the data remains constant. The chosen mean values could not be taken outside the presented range since there would not be enough images for some of the holding times.