Quantized conductance through a dissipative atomic point contact

Signatures of quantum transport are expected to quickly vanish as dissipation is introduced in a system. This dissipation can take several forms, including that of particle loss, which has the consequence that the total probability current is not conserved. Here, we study the effect of such losses at a quantum point contact (QPC) for ultracold atoms. Experimentally, dissipation is provided by a near-resonant optical tweezer the power and detuning of which control the loss rates for the different internal atomic states as well as their effective Zeeman shifts. We theoretically model this situation by including losses in the Landauer-Büttiker formalism over a wide range of dissipative rates. We find good agreement between our measurements and our model, both featuring robust conductance plateaus. Finally, we are able to map out the atomic density by varying the position of the near-resonant tweezer inside the QPC, realizing a dissipative scanning gate microscope for cold atoms.

Figure 1

Experimental realization of a Landauer-Büttiker setup with losses. (a) ${}^6\mathrm{Li}$ atoms in the lowest ($|\downarrow \rangle$, orange) and third-lowest ($|\uparrow \rangle$, blue) hyperfine states are allowed to flow between two reservoirs connected by an optically defined QPC (gray). A near-resonant optical tweezer (red) with waist $w_s=2.0\left(1\right)\mu \mathrm{m}$ introduces different loss rates ${\mathrm{\Gamma }}_{\uparrow }$ and ${\mathrm{\Gamma }}_{\downarrow }$ as well as a spin-dependent potential $V_s$ inside the QPC. A far-detuned gate beam (dashed circle) locally increases the chemical potential ${\mu }_{\text{res}}$ imposed by the reservoirs by $V_g$. (b) In a Landauer-Büttiker picture and in the absence of a temperature bias, net transport can be attributed to the excess of particles in the left reservoir not compensated by the right reservoir (orange) due to a chemical potential difference ${\mu }_L-{\mu }_R>0$. Losses on the other hand involve the energy integration of the reservoir Fermi-Dirac distributions $f_L\left(E\right)$ and $f_R\left(E\right)$, including states below the Fermi level that do not contribute to a net current (brown). Conductance and loss are furthermore weighted by the energy-dependent transmission $\mathcal{T}\left(E\right)$ and loss $\mathcal{L}\left(E\right)$ coefficients of the mesoscopic channel.

Figure 2

Potentials and scattering rates at a magnetic field of $B=568\mathrm{G}$ for the $|\downarrow \rangle$ and $|\uparrow \rangle$ states for different frequencies $\nu$ of the light where the mean potential $(V_{\downarrow }+V_{\uparrow })/2$ vanishes. The lower frequency axes are offset by the $D_2$ transition frequency at zero magnetic field ${\nu }_{D_2}$ [33]. The intensities in each pair of graphs are chosen to give a constant amplitude of the spin potential $|V_{\downarrow }-V_{\uparrow }|=k_B\times 0.6\mu \mathrm{K}$. Panels (a) and (b) correspond to ${\sigma }^{-}$ polarization and an intensity of $3\mathrm{W}/{\mathrm{m}}^2$. The top frequency axis has been offset by the mean of the resonance frequencies of state $|\downarrow \rangle$ and $|\uparrow \rangle$ for the main $D_2{\sigma }^{-}$ transition: $\overline{\delta }=\nu -({\nu }_{\downarrow }+{\nu }_{\uparrow })/2$. Panels (c) and (d) correspond to ${\sigma }^{+}$ polarization and an intensity of $136\mathrm{W}/{\mathrm{m}}^2$ close to a narrow $D_2$ resonance appearing in the Paschen-Back regime. Panels (e) and (f) correspond to ${\sigma }^{+}$ polarization and an intensity of $1.2\times {10}^4\mathrm{W}/{\mathrm{m}}^2$ between the $D_1$ and $D_2$ lines.

Figure 3

Measured and simulated evolution of the relative atom number imbalance and of the atom number. (a) Experimental relative imbalance and (b) normalized atom number for spin $|\downarrow \rangle$ (orange, closed symbols) and spin $|\uparrow \rangle$ (blue, open symbols), obtained for a mean chemical potential $V_g+{\mu }_{\text{res}}=k_B\times 0.61\left(2\right)\mu \mathrm{K}$, with a near-resonant beam power $P_s=20\left(6\right)\mathrm{pW}$ corresponding to an intensity $I_s=0.13\left(4\right)I_{\text{sat}}$, where $I_{\text{sat}}$ is the $D_2$-line saturation intensity and at a scattering length $a=0\left(7\right)a_0$. The initial atom numbers for each spin state are $N_{0,\downarrow }=117\times {10}^3$ and $N_{0,\uparrow }=110\times {10}^3$. Here and in the following, error bars correspond to the standard error of the mean of three measurements. (c) Simulation results for the relative atom number difference $\mathrm{\Delta }N/N$ and (d) the normalized atom number with spin-dependent potential $V_s=k_B\times 0.25\mu \mathrm{K}$, gate potential $V_g=k_B\times 0.38\mu \mathrm{K}$, temperature $T=60\mathrm{nK}$, and initial conditions $N_{0,\downarrow /\uparrow }=115\times {10}^3$ and $\mathrm{\Delta }{N}_0/N_0=0.41$. Losses occur mostly for $|\downarrow \rangle$ atoms.

Figure 4

Conductance plateaus are preserved at a lossy QPC. (a) Breakdown of the simulated numbers of lost and transmitted atoms as a function of their energy using the parameters of Fig. 3 integrated over a transport time of 4 s. This highlights one reason for the robustness of transport observables. The total number of lost (transmitted) atoms (first bar of the chart) can be decomposed into two parts, shown in the filled (hatched) region: (i) those which do not contribute to transport [Eqs. (18) and (25), brown region in Fig. 1 and second bar of the chart] and (ii) those which could contribute to transport [Eqs. (17) and (24), orange region in Fig. 1 and third bar of the chart]; see text for details. The relative fraction of lost atoms in each category, indicated on top of each bar, shows that the losses are less important for the atoms participating in transport (due to their higher velocities) than for the other atoms, hence conductance is weakly affected by the losses. (b) Quasi-1D potentials along the transport direction $y$ for each spin state $V_{\sigma }\left(y\right)$. The chemical potential of Fig. 3 and panel (a) is indicated as dashed lines. (c) Conductance $G$ of each spin state at scattering length $a=91\left(7\right)a_0$ vs local chemical potential $V_g+{\mu }_{\mathrm{res}}$ with a near-resonant beam intensity of $I_s=0.13\left(4\right)I_{\text{sat}}$. Fits by a Landauer model are shown as solid curves and indicate a spin-dependent potential of $V_s=k_B\times 0.25\left(2\right)\mu \mathrm{K}$.

Figure 5

Validity of the Landauer model for large dissipation. Conductance $G$ at a scattering length $a=91\left(7\right)a_0$ for spin $|\downarrow \rangle$ (orange, closed symbols) and spin $|\uparrow \rangle$ (blue, open symbols) with near-resonant light (a) on resonance with $|\uparrow \rangle$ at detuning $\overline{\delta }=-81.3\mathrm{MHz}$, (b) at detuning $\overline{\delta }=-40.6\mathrm{MHz}$, (c) at detuning $\overline{\delta }=40.6\mathrm{MHz}$, and (d) on resonance with $|\downarrow \rangle$ at detuning $\overline{\delta }=81.3\mathrm{MHz}$. Solid curves show a Landauer prediction using the fit parameters of Fig. 4 extended to different detunings. The different detunings at which the conductance is measured are indicated by dashed gray lines in Figs. 2 and 2.

Figure 6

Dissipative scanning gate microscopy. Integrated atom loss in state $|\downarrow \rangle$ after a time $\mathrm{\Delta }t=4\mathrm{s}$ as a function of the near-resonant beam position $(x_s,y_s)$ and corresponding two-dimensional density $n_{\text{2D}}$, showing a high-resolution map of the QPC. The exact value of $n_{\text{2D}}$ should be obtained by deconvolving this map with the spatial profile of the near-resonant beam, which is beyond the scope of this paper. The near-resonant beam has a detuning of $\overline{\delta }=40\mathrm{MHz}$, a narrower waist of $w_s=1.02\left(5\right)\mu \mathrm{m}$, and a power $P_s=4\left(1\right)\times {10}^1\mathrm{pW}$, corresponding to a peak intensity of $I_s=24\left(8\right)\mathrm{W}/{\mathrm{m}}^2$ and therefore a large photon absorption rate $\mathrm{\Gamma }=9\left(3\right)\times {10}^4{\mathrm{s}}^{-1}$ for state $|\downarrow \rangle$.