Scanning Gate Microscope for Cold Atomic Gases

We present a scanning probe microscopy technique for spatially resolving transport in cold atomic gases, in close analogy with scanning gate microscopy in semiconductor physics. The conductance of a quantum point contact connected to two atomic reservoirs is measured in the presence of a tightly focused laser beam acting as a local perturbation that can be precisely positioned in space. By scanning its position and recording the subsequent variations of conductance, we retrieve a high-resolution map of transport through a quantum point contact. We demonstrate a spatial resolution comparable to the extent of the transverse wave function of the atoms inside the channel and a position sensitivity below 10 nm. Our measurements agree well with an analytical model and ab initio numerical simulations, allowing us to identify a regime in transport where tunneling dominates over thermal effects. Our technique opens new perspectives for the high-resolution observation and manipulation of cold atomic gases.

Figure 1

An atomic scanning gate microscope. (a) Close-up view of the channel region: mesoscopic reservoirs, 2D regions, and the QPC. The scanning gate is realized by a tightly focused, repulsive laser beam that is scanned across the structure. (b) The gate (green circle) locally modifies the potential landscape shown for the parameters used in the simulation, Fig. 2. The green square indicates the region mapped in Fig. 2. Engineering of the QPC and the scanning gate. The QPC is formed at the intersection of two repulsive laser beams (indicated in green) having a nodal line at the center. A ${\mathrm{TEM}}_{01}$-like laser mode propagating along the $-x$ axis forms the 2D regions. The beam passing through a lithographic mask is shrunk onto the 2D region and creates the QPC. The scanning gate is holographically shaped and moved (blue beam) by a DMD and projected onto the atoms with a microscope objective. The red beam creates an attractive potential to locally shift the chemical potential.

Figure 2

Scanning gate map of a single-mode QPC. Measured (a) and calculated (b) conductance $G$ normalized to the background value $G_0$ as a function of the position of the scanning gate imposed onto the QPC. Because of the weak attraction among the particles, the measured background of $1.3/h$ is larger than the universal conductance of $1/h$ [38]. The scanning gate has a strength of $620(1)\mathrm{nK}{k}_B$. The reservoir acting as the source is located on the left and the sink on the right side of the map. The simulation considers a single free parameter, the chemical potential in the reservoirs. The temperature and the chemical potential bias are 58(7) nK and $99(4)\mathrm{nK}{k}_B$, respectively, calibrated independently [23]. (c) Longitudinal cuts through the scanning gate maps, along the dashed lines in (a) and (b). The measured conductances are represented by the points and the simulation by the line.

Figure 3

Transverse scans through the QPC’s center in the single-mode regime. (a) Transverse ground state probability distribution with FWHM $l$ of $0.5(2)\mu \mathrm{m}$ in the absence of the tip. (b) Conductance normalized to the measured background value $G_0$ of $1.1/h$ as a function of the tip position $x$ and strength $V_0$. The dotted lines indicate representative cuts shown in (d). (c) Prediction by the analytical model including the chemical potential as the only free parameter, while the temperature and chemical potential bias are independently calibrated to 58(5) nK and $102(3)\mathrm{nK}{k}_B$, respectively. The fitted chemical potential $\mu$ is indicated in the plot. Solid lines mark realizations with the same ratio $\hslash \mathrm{\Omega }(x)/2\pi k_{B}T$. (d) Transverse scans for different tip strengths $V_0$. The solid curves indicate best fits with the analytical model. (e) Fermi-Dirac distribution $f_{L/R}(E)$ of each reservoir centered around the chemical potential ${\mu }_{L/R}$. They are smeared over the energy scale $4k_{B}T$ (dark shaded area), defined by the tangential line (dashed line) at the inflection point. The transmission $\mathcal{T}(E)$ through a parabolic barrier with height $V(x)$ and the associated energy scale $2\hslash \mathrm{\Omega }(x)/\pi$ (light shaded area) is shown in the tunneling-dominated regime, for a ratio $\hslash \mathrm{\Omega }(x)/2\pi k_{B}T$ of 1.4 (regime discussed in Ref. [23]).

Figure 4

Resolution and sensitivity of the scanning gate technique. (a) FWHM of the transverse scans through the QPC’s center, shown in Fig. 3, as a function of the tip strength. Beside the experimental points, the dashed and solid lines represent the predictions of the analytical model for zero and a finite temperature of 58(5) nK. For a comparison, the FWHM of the obstacle in the transverse direction (dashed dotted line) and of the transverse probability distribution (horizontal dotted line), depicted in Fig. 3, are indicated. (b) Derivative of the measured conductance with respect to the position of the tip as a function of its position $x$ and its strength $V_0$. (c) Representative cuts along the dashed lines in (b). Conductance measurements are maximally sensitive to the position, for the strongest gate, where the derivative gets extremal.