Double-dot charge stability diagram

The honeycomb map that defines a semiconductor spin-qubit device: charge-sensor signal against the two plunger-gate voltages, with each cell a fixed electron occupation (N_L, N_R) of the double dot.

For every gate setting the dot settles into whichever integer occupation minimizes the electrostatic energy,

E = (1/2) Ec_L N_L^2 + (1/2) Ec_R N_R^2 + Ecm N_L N_R

• N_L (a_LL V_L + a_LR V_R) - N_R (a_RL V_L + a_RR V_R),

so the field is piecewise constant with sharp boundaries where the ground state switches. The mutual charging energy Ecm is what splits the crossings into the characteristic honeycomb vertices rather than a simple square grid.

This one is a stress test for a mesh renderer rather than for a color scale: the data is flat plateaus separated by one-cell discontinuities, so any smoothing or interpolation would blur exactly the boundaries the measurement is about. pcolormesh maps one cell to one cell, and the plateau edges stay sharp.

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import numpy as np
import polars as pl
import plotpress

EC_L, EC_R, EC_M = 1.0, 1.0, 0.28      # charging energies (arbitrary units)
A_LL, A_LR = 1.0, 0.30                 # gate-to-dot lever arms
A_RL, A_RR = 0.30, 1.0
N_MAX = 6                              # occupations considered per dot

v_left = np.linspace(0.0, 5.0, 340)
v_right = np.linspace(0.0, 5.0, 340)
VL, VR = np.meshgrid(v_left, v_right)

# Energy of every candidate (N_L, N_R), then keep the lowest at each gate point.
occupations = [(nl, nr) for nl in range(N_MAX + 1) for nr in range(N_MAX + 1)]
energies = np.stack([
    0.5 * EC_L * nl ** 2 + 0.5 * EC_R * nr ** 2 + EC_M * nl * nr
    - nl * (A_LL * VL + A_LR * VR) - nr * (A_RL * VL + A_RR * VR)
    for nl, nr in occupations
])
ground = np.argmin(energies, axis=0)

n_left = np.array([o[0] for o in occupations])[ground]
n_right = np.array([o[1] for o in occupations])[ground]

# A nearby charge sensor couples more strongly to the left dot than the right.
sensor = -(0.62 * n_left + 0.38 * n_right)

# One row per swept (V_L, V_R) gate point -- sorted before the reshape below
# so the pivot back to a grid is correct regardless of row order.
sweep = pl.DataFrame({
    "v_left": VL.ravel(),
    "v_right": VR.ravel(),
    "sensor": sensor.ravel(),
}).sort(["v_right", "v_left"])

v_left_axis = sweep["v_left"].unique().sort().to_numpy()
v_right_axis = sweep["v_right"].unique().sort().to_numpy()
sensor = sweep["sensor"].to_numpy().reshape(v_right_axis.size, v_left_axis.size)

fig, ax = plotpress.subplots(figsize=(6.8, 5.8))
mesh = ax.pcolormesh(v_left_axis, v_right_axis, sensor, cmap="cividis")
bar = fig.colorbar(mesh, ax=ax)
bar.set_title("sensor\n(a.u.)")
ax.set_aspect("equal")
ax.set_xlabel("left plunger V_L (a.u.)")
ax.set_ylabel("right plunger V_R (a.u.)")
ax.set_title("Charge stability honeycomb of a double quantum dot")
fig.tight_layout()