Coherent Electron Optics with Ballistically Coupled Quantum Point Contacts

The realization of integrated quantum circuits requires precise on-chip control of charge carriers. Aiming at the coherent coupling of distant nanostructures at zero magnetic field, here we study the ballistic electron transport through two quantum point contacts (QPCs) in series in a three terminal configuration. We enhance the coupling between the QPCs by electrostatic focusing using a field effect lens. To study the emission and collection properties of QPCs in detail we combine the electrostatic focusing with magnetic deflection. Comparing our measurements with quantum mechanical and classical calculations we discuss generic features of the quantum circuit and demonstrate how the coherent and ballistic dynamics depend on the details of the QPC confinement potentials.

Figure 1

(a) Atomic force microscope image of the sample; Ti/Au gates on GaAs surface (dark). Gate voltages $V_1,V_2,V_{\mathrm{L}}$ are used to define below in the 2DES ${\mathrm{QPC}}_{1,2}$ and a lens. Source-drain voltage $V$ is applied across ${\mathrm{QPC}}_1$; current $I$ is measured through ${\mathrm{QPC}}_2$; the region in between is grounded via 4 Ohmic contacts (squares). The horizontal dashed line is the principal axis of the lens with aperture angle $\alpha \simeq 55°$. (b) Individual linear response pinch-off curves $G(V_{1,2})$ of ${\mathrm{QPC}}_{1,2}$ ($V_{2,1}=V_{\mathrm{L}}=0$), corrected for lead resistance $R_{\mathrm{lead}}$. (c) Energy spacings between subsequent subbands.

Figure 2

(a) Magnetic deflection with two QPCs in series: measured detector current $I_{N,N}$ versus perpendicular magnetic field $B$ for both QPCs tuned to the $N$th conductance plateau with $N=1,\dots ,7$. Two data sets (gray, blue) correspond to opposite current directions; vertical shifts $I_{\mathrm{off}}(N)$ for clarity. Measured in (b) versus calculated in (c) transmission differences $\mathrm{\Delta }{T}_{N,M=7}$. Model curves in (c) for perfect symmetry and zero lens potential (red dashed) and with corrections of the QPC positions and accounting for the piezoelectric dip of the lens potential (solid blue lines). Maxima and minima of $\mathrm{\Delta }{T}_{N,M=7}$ are marked in panel (b) with red (black) triangles. Dashed gray lines [identical in (b) and (c)] connect the $n$th maxima for odd (even) $N$ and the $n$th minima for even (odd) $N$.

Figure 3

(a) Measured serial transmission through both QPCs, $T(V_L,B)$ for $N=M=7$ and ${\mathrm{QPC}}_1$ as emitter. (b) $T(V_L)$ for various magnetic fields. For $B-B_0=0$ (blue) a pronounced maximum indicates focusing. Red dashed line: $T(V_L)/10$ without QPCs ($V_1=V_2=0$). (c) Calculated $T(V_L,B)$ as described in the main text. The dashed lines in (a) and (c) are identical. (d) Calculated current density emitted by ${\mathrm{QPC}}_1$ modeled as a hard wall potential for $N=7$ at $B=0$ and $V_L=0$ and neglecting the electrostatic potential dip at the lens waist. Solid lines: approximate extension of the lens potential for $V_L\simeq -0.64\mathrm{V}$.