Phase space tomography of cold-atom dynamics in a weakly corrugated potential

We demonstrate tomographic reconstruction of the phase space distribution of atoms oscillating in a harmonic trap with weak potential corrugation caused by nanoscale imperfections in an atom chip. We find that deformations in these distributions are highly sensitive to anharmonic components of the potential. They are explained in terms of angular velocity dispersion of isoenergetic phase space trajectories. We show that the method is applicable for probing classical and quantum dynamics of cold atoms, and we note its importance for future technological applications.

Figure 1

(top) The atom chip (gold surface) acts as a mirror for the magneto-optical trap and for imaging the atomic cloud. Three wires are used to generate the potentials used in this experiment. (bottom) The trapping fields are harmonic potentials (smooth curves) generated by the “trapping” wire (Cu) whose center is $\approx 1.2$ mm from the atomic cloud; the harmonic potentials may be displaced along the $x$ axis (solid vs dashed curves) by currents through a pair of gold U-shaped “displacement” wires (U) located $320\mu \mathrm{m}$ from the atoms. The weak corrugation potential (wavy line along the $x$ axis), which is highly exaggerated compared to the harmonic potentials, is imposed by a small current through the atom chip “corrugation” wire (Z) located $20\mu \mathrm{m}$ from the atoms. External coils creating bias fields along the $x$ and $y$ axes are not shown.

Figure 2

(a)–(c) Reconstructed phase space distributions with a 5-mA current in the corrugation wire, an $85-\mu \mathrm{m}$ oscillation amplitude, and oscillation times $t_1$ as shown. The distribution in (d) is obtained with the corrugation wire turned off, while that shown in (e) is obtained by translating the equilibrium position of the cloud by $73\mu \mathrm{m}$ using the displacement wires. (f) A typical in situ image and its corresponding one-dimensional density distribution, selected from the data set used for reconstructing the $t_1=500\mathrm{ms}$ phase space distribution shown in (c).

Figure 3

The potential corrugation at a distance of $20\mu \mathrm{m}$ from the corrugation wire (Z in Fig. 1). The potential is shown for a corrugation-wire current of $40\mathrm{mA}$ as reconstructed from a series of measurements, like those in the inset, that overlap along the $x$ axis. The right-hand scale shows the potential as it would correspond to the corrugation-wire current of $5\mathrm{mA}$ used in the oscillation experiments. The inset shows a typical image of the atomic cloud optical density (OD) distribution for one particular current in the displacement wires.

Figure 4

Frequency shift $\delta \omega$ (solid curve) as a function of the longitudinal energy of an atom in a harmonic trap centered at $x=0$ in the presence of the corrugation potential shown in Fig. 3 for a current of $5\mathrm{mA}$. The energy is in units of $E_{\mathrm{shift}}$ (see text). Energy distribution (dashed curve) for an atomic ensemble prepared in the initial trap centered at $x_{\mathrm{shift}}$. The ensemble contains atoms whose energies span oscillation frequencies $\delta \omega /{\omega }_0\approx -0.015$ to $\delta \omega /{\omega }_0\approx +0.009$ (red dots). The evolution that follows leads to the simulated phase space distributions shown in Fig. 5.

Figure 5

Phase space distributions obtained by simulation and experiment for oscillation times $t_1=500\mathrm{ms}$. (a) The simulation was based on preparing an ensemble of atoms with a thermal distribution in the initial trap and then adding a proper phase space angle according to $\delta \omega \left(E\right)$ in Fig. 4. (b) Newton's equations of motion were solved numerically for each atom, and collisions were included. We also repeated the tomography algorithm and took into account the finite resolution of the optical system. The similarity between (a) and (b) shows that for a collision rate of about $2{\mathrm{s}}^{-1}$ there is no significant smearing of the phase space distribution, even after $500\mathrm{ms}$. (c) The experimental results [repeated from Fig. 2] are explained to a high degree by the theory [(a) and (b)]. (d) An alternative maximum-likelihood tomography algorithm [42] gives similar results.

Figure 6

Phase space distribution for a trapped noninteracting BEC split into a superposition of two momenta. (a) Wigner function reconstructed by keeping the BEC in the harmonic potential for times $t_j=(j-1)\pi /\omega ,j=1,\cdots ,60$ and then allowing free expansion for a flight time of $t_f=30\mathrm{ms}$. (b) Wigner function calculated directly from the wave function [Eq. (15)]. The green background color represents values close to zero, while red is positive and blue is negative.

Figure 7

Effect of a small quartic anharmonicity on the phase space evolution of a trapped single-particle quantum system. (a)–(d) Evolution of the phase space density for different times in the trap. (e) Uncertainties in position (blue line) and momentum (black line, expressed as $p/m\omega$) as a function of time; note that the minimum position uncertainty occurring near $t=130\mathrm{ms}$ is $\approx 30\%$ below the minimum-uncertainty initial state.

Figure 8

(a) Scanning-electron-microscope images showing one edge of the corrugation wire (Z in Fig. 1). The lower scan is a copy of the upper scan, displaced one period to the right. The vertical white lines are drawn through the peaks of the upper scan and show that the second period is $\approx 10\%$ longer than the other periods due to a fabrication error occurring every $160\mu \mathrm{m}$ along the wire. An additional imperfection, almost imperceptible in the image, is also indicated as a transverse shift of $\approx 60$ nm. (b) Close-up view showing that random edge imperfections are $\lesssim 20$ nm. The wire is $8\mu \mathrm{m}$ wide and $0.5\mu \mathrm{m}$ thick. (c) Corrugation potential amplitudes as a function of distance from the corrugation wire, as calculated for the $5-\mu \mathrm{m}$ edge modulation (solid black curve) and for the $160-\mu \mathrm{m}$-period fabrication errors exhibited in (a): $0.5-\mu \mathrm{m}$ longitudinal stretch (blue dotted curve) and 60-nm transverse shift (red dashed curve). The deepest corrugations observed in the experimental potential (Fig. 3) are quantitatively reproduced by the latter curve at a distance of $20\mu \mathrm{m}$.