Nanoscale Atomic Density Microscopy

Quantum simulations with ultracold atoms typically create atomic wave functions with structures at optical length scales, where direct imaging suffers from the diffraction limit. In analogy to advances in optical microscopy for biological applications, we use a nonlinear atomic response to surpass the diffraction limit. Exploiting quantum interference, we demonstrate imaging with superresolution of $\lambda /50$ and excellent temporal resolution of 500 ns. We characterize our microscope’s performance by measuring the ensemble-averaged probability density of atoms within the unit cells of an optical lattice and observe the dynamics of atoms excited into motion. This approach can be readily applied to image any atomic or molecular system, as long as it hosts a three-level system.

Popular Summary

The wavelike nature of light limits the resolution of conventional imaging devices such as microscopes. To get around this limitation, researchers have developed a variety of techniques known as superresolution imaging to visualize such things as a single atom on a surface or the dynamics of individual molecules within a living cell. Here, we bring superresolution imaging to an ensemble of ultracold atoms, which allows us to “see” individual atoms with an unprecedented resolution of about 11 nm—50 times the resolution of conventional lenses—and observe changes in the atomic wave functions.

Our target is a 1D chain of ytterbium atoms trapped in an optical lattice. Two lasers work to change the spin state of the atoms via a common excited state, which generates interference between the excitation pathways. By varying the strength of each of the two lasers, we can change the spin mixture from all up to all down, for example. We then count the atoms by shining another laser on them. Since only one spin state absorbs the laser light, we can effectively choose which atoms to count. This lets us map the atomic wave function with a subwavelength spatial resolution and a temporal resolution of about 500 ns.

By bringing superresolution imaging to cold atomic systems, we add a new technique to the atomic physics toolbox. This will enable new direct probes of the atomic wave function in a variety of many-body systems.

Figure 1

Principle of our nanoscale atomic density microscope. (a) Configuration of the control field ${\mathrm{\Omega }}_c(x)$ and probe field ${\mathrm{\Omega }}_p$ coupling a $\mathrm{\Lambda }$ system composed of $|g_1⟩$, $|g_2⟩$, and $|e⟩$. Population in $|g_2⟩$ is measured via a cycling transition connecting the imaging state $|i⟩$. (b) Wave function density $|\psi (x){|}^2$ in $|g_1⟩$ in the lattice of interest $V(x)$. (c) The spin-state composition is transferred to $|g_2⟩$ near the nodes of ${\mathrm{\Omega }}_c(x^{\prime }-x)$ with probability given by $f(x^{\prime }-x)$ (narrow red peaks) and $|g_1⟩$ elsewhere. The width of $f(x^{\prime }-x)$ is determined by the relative strength of the two light fields $\epsilon ={\mathrm{\Omega }}_p/{\mathrm{\Omega }}_c$ [see Eq. (2)]. (d) $f(x^{\prime }-x)$ maps $|\psi (x){|}^2$ onto the population in $|g_2⟩$, $n(x)$, which can be selectively measured via state-dependent imaging. By stepping through different positions $x$ and measuring $n(x)$, we can reconstruct $|\psi (x){|}^2$.

Figure 2

Measurements of the ground-state wave function within the unit cell of an optical lattice with different shapes. (a) The orange points show $n(x)$ for atoms in a sinusoidal lattice measured with $\epsilon =0.05$. The green points represent $n(x)$ in a Kronig-Penney lattice measured with $\epsilon =0.1$. Number fluctuations between realizations result in number uncertainties of 5%. The solid lines are calculations using measured Rabi frequencies to determine the lattice depth normalized to the same atom number. Inset: Schematic of different lattice potentials and corresponding $|\psi (x){|}^2$. (b) FWHM $w$ of $n(x)$ in a sinusoidal lattice as a function of the lattice depth. Black points show experimental data with $\epsilon =0.05$, and the blue line is a calculation including the 800-ns measurement time. The error bars are 1 standard deviation from the Gaussian fits.

Figure 3

Wave-function dynamics within the unit cell of an optical lattice. We excite the (a) sloshing motion and (b) breathing motion of ${|\psi (x)|}^2$ appears to have a wrong scale in a $140E_R$-deep sinusoidal lattice by suddenly changing either the position or the depth of the lattice potential. $n(x)$ is plotted at different hold times [1 to $14\mu \mathrm{s}$ in steps of $1\mu \mathrm{s}$ for (a) and 1.5 to $9.5\mu \mathrm{s}$ in steps of $1\mu \mathrm{s}$ for (b)] after the sudden change. The points are experimental data with $\epsilon =0.05$, and the blue curves represent calculations of $n(x)$ based on the independently measured lattice parameters. Typical number uncertainties are 5% due to fluctuations from shot to shot.

Figure 4

Spatial resolution of the microscope. We create a narrow wave function $|\psi (x){|}^2$ (FWHM 26 nm) by exciting the breathing motion of atoms in a deep sinusoidal lattice and measure $n(x)$ at the focus point [see Fig. 3] as a function of $\epsilon$. The measured $w$ from a Gaussian fit with a vertical offset to the $n(x)$ (see upper panel for typical wave-function measurements) is plotted against $\epsilon$ as the gray open circles, with the error bars showing 1 standard deviation from the fitting. These data are then deconvolved with the calculated wave function $|\psi (x){|}^2$ to find the intrinsic resolution $\sigma$ and plotted as the black closed circles. The error bars are dominated by the systematic uncertainties in the width of $|\psi (x){|}^2$. The blue curve is the calculated width of $f(x)$ at different $\epsilon$.

Figure 5

Level structure of the ${}^1{S}_0$, ${}^3{P}_1$, and ${}^1{P}_1$ manifolds of ${}^{171}\mathrm{Yb}$: $\mathrm{\Delta }$ is the single-photon detuning and ${\mathrm{\Delta }}_{\mathrm{HFS}}\sim 6\mathrm{GHz}$ is the ${}^3{P}_1$ hyperfine splitting.

Figure 6

Experimental sequences. (a) Probing the ground-state wave function of a sinusoidal lattice. (b) Probing the ground-state wave function of a KP lattice. (c) Probing the dynamics after a sudden change in the lattice position. (d) Probing the dynamics after a sudden change in lattice depth. The drawings are not to scale.

Figure 7

The optimal amplitude waveform for ${\mathrm{\Omega }}_{c2}(t)$ for ${\mathrm{\Omega }}_{c1}=250\mathrm{\Gamma }$, ${\mathrm{\Omega }}_p=25\mathrm{\Gamma }$.

Figure 8

(a) The temperature of the atoms after ten complete STIRAP pulses for different values of $r$ and ${\mathrm{\Omega }}_p$ and (b) different values of ${\mathrm{\Omega }}_{\mathrm{rms}}$ and $T_r$

Figure 9

(a) Optical pulse sequence used to optimize the temporal overlap between the ${\mathrm{\Omega }}_{ci}$ and ${\mathrm{\Omega }}_p$ light fields. (b) The population of the atoms in the $|g_2⟩$ state as a function of $t_{\text{shift}}$ with ${\mathrm{\Omega }}_{c1}=250\mathrm{\Gamma }$ and ${\mathrm{\Omega }}_p=50\mathrm{\Gamma }$.