In-situ Imaging of a Single-Atom Wave Packet in Continuous Space

Abstract

The wave nature of matter remains one of the most striking aspects of quantum mechanics. Since its inception, a wealth of experiments has demonstrated the interference, diffraction or scattering of massive particles. More recently, experiments with ever increasing control and resolution have allowed imaging the wavefunction of individual atoms. Here, we use quantum gas microscopy to image the in-situ spatial distribution of deterministically prepared single-atom wave packets as they expand in a plane. We achieve this by controllably projecting the expanding wavefunction onto the sites of a deep optical lattice and subsequently performing single-atom imaging. The protocol established here for imaging extended wave packets via quantum gas microscopy is readily applicable to the wavefunction of interacting many-body systems in continuous space, promising a direct access to their microscopic properties, including spatial correlation functions up to high order and large distances.

Figures (11)

• Figure 1: Preparation and in-situ imaging of single-atom wave packets.(a) Measurement scheme: Individual atoms are prepared close to the harmonic oscillator ground state of individual sites of a triangular optical lattice created by a self-interfering laser beam with wave vectors $\mathbf{k_1,k_2}$ and $\mathbf{k_3}$. Wave packets initially trapped in the lattice wells, characterized by a Gaussian probability density distribution $|\psi_0(\mathbf{r})|^2$, are released in a plane, allowing them to expand for a given time. For imaging after expansion, the lattice is quickly ramped up again, projecting the wave packet, and Raman sideband cooling is applied to pin the atom on a single site. Resulting atomic positions are recorded through site-resolved fluorescence imaging. From many repetitions of identically prepared wave packets we create histograms of the projected positions with a discretization given by the lattice structure, resulting in a measured probability distribution $|\psi(\mathbf{r},t)|^2$. (b) Experimental single-atom resolved image. The top-right panel shows a subregion containing an individual atom. The bottom-right panel displays an enlarged region of the image over which the reconstructed triangular lattice structure with a spacing of $a_L = 709$ nm is shown as white dots. (c) Experimental configuration of the oblate optical dipole trap confining the atoms to a two-dimensional plane, the Raman beams (R$_1$, R$_2$ and RP) used for cooling and imaging, and the microscope objective. (d) Top view of the experimental configuration, showing the geometry of the optical lattice beam.
• Figure 2: Experimental sequence and image analysis.(a) Raman sideband cooling scheme for $^6$Li atoms. The two Raman beams ($R_1$ and $R_2$) are blue-detuned by $\Delta = h\cdot 3$ GHz with respect to the first electronically excited state $|e\rangle$ and connect the two hyperfine levels $|g_1\rangle$ and $|g_2\rangle$ in the ground state manifold, split by $E_\mathrm{HF} = h\cdot 228.2$ MHz. A resonant repumper (RP) beam connects $|g_1\rangle$ and $|e\rangle$. In the presence of a deep optical lattice, RSC brings the atoms to lower harmonic oscillator eigenstates, indicated by eigenvalue $n$, while producing fluorescence photons for single-particle imaging through the microscope objective, serving as both a preparation and detection method. (b) Experimental sequence for the preparation, expansion and pinning of the single-atom wave packets. After the wave packets have been localized in a deep optical lattice and their initial positions recorded in a first image (1), RSC is turned off and the lattice depth is adiabatically ramped down to a variable value $U_0$ to adjust the width of their initial momentum distribution. The lattice is then suddenly switched off and the wave packets expand for a time $t$ after which we take a second image (2) to record the new atom positions. (c) Two single atom images taken in a single experimental realization (left and right panels). The center panel schematically shows the most likely assignment of how the atoms moved between the two images. (d) Relative likelihood of the 4000 most likely assignments based on a combined likelihood computation, for $\omega = 2\pi \times 600(30)\,\mathrm{kHz}$ ($U_0 = 0.38\,U_\mathrm{max}$) and an expansion time of $10\,\mu$s. The image insets show the assignments ranked 1$^\mathrm{st}$, 50$^\mathrm{th}$ and 2000$^\mathrm{th}$. Initial (white dots) and assigned final (red dots) positions are connected by arrows. The top-right inset shows the relative likelihood in a linear scale.
• Figure 3: Expansion dynamics of a single-particle Gaussian wave packet.(a) Probability density distribution as a function of expansion time $t$ for wave packets prepared at different lattice depths $U_0$ before release. Hexagons represent the lattice sites of the triangular pinning lattice, with the central site indicating the position of each atom in the first single-atom image. The histograms are obtained after 100 measurements, each with around $20$ to $50$ identically prepared single-atom wave packets. (b) Cuts of the two-dimensional histograms at $U_0 = 0.23\,U_\mathrm{max}$ for $t=3\,\mu$s, $5\,\mu$s, $8\,\mu$s, and $12.5\,\mu$s. (c) Data points show the width of the Gaussian wave packet ($\sigma$) extracted from the probability density distribution as a function of $t$ for each preparation lattice depth. Error bars indicate the standard error of the maximum likelihood estimate and are typically smaller than the symbol size. Dashed lines represent linear fits to the experimental data. (d) Solid points show the average harmonic oscillator eigenvalues $\langle n \rangle =[0.42(3),0.47(3),0.49(3),0.46(3),0.45(3)]$ for each preparation, corresponding to the lattice depth $U_0/U_\mathrm{max} = 0.072(5), 0.136(5), 0.228(7), 0.376(10)$ and $0.61(2)$. The open points are the values extracted from the linear fits in (b) using Eq. (\ref{['eq:expand']}) before the correction due the non-instantaneous release is taken into account (see text). The horizontal red dashed line together with the light red band indicate the average of all five values $\langle n \rangle=0.46(3)$.
• Figure 4: Continuum pinning dynamics. Effect of the lattice ramp time on the probability to be correctly pinned on the closest lattice site when projecting from continuous space ($p_\mathrm{ideal}$, yellow data points), and the width of the measured Gaussian probability density in units of lattice spacing ($\sigma/a_\mathrm{L}$, red data points) for $U_0/U_\mathrm{max} = 0.23$ and an expansion time of $4\,\mu$s. The vertical grey band displays the ramp time interval where $p_\mathrm{ideal}$ is maximal and the vertical dashed line shows the ramp time used for the data presented in Fig. \ref{['fig:mosaic']}. The yellow dotted line indicates $p_\mathrm{ideal}=1$, representing perfect pinning from continuum. The inset shows a semi-logarithmic plot of $p_\mathrm{ideal}$, and $\sigma$ normalized to the initial width, with the horizontal red band indicating a $\pm 5\%$ interval. Dashed lines represent guides to the eye for the respective data sets. Error bars give the standard error of the maximum likelihood estimates.
• Figure S1: Simulated time-evolution of the motional degrees of freedom during RSC. Atoms start in a thermal superposition of harmonic oscillator states with $\left\langle n_x \right\rangle = \left\langle n_y \right\rangle = 2$. (a) Evolution of the average quantum number $\left\langle n_{x,y}\right\rangle$ and temperature over time. Temperature is obtained by fitting the $(n_x,n_y)$ populations to a Boltzmann distribution. (b) Evolution of the fraction of atoms with $n_x=0$ (grey line), and the fraction of atoms in the absolute ground state $\left|{n_x=0,n_y=0}\right\rangle$ (black line).