Direct measurement of the energy spectrum of a quantum dot qubit

Abstract

The mapping between gate voltages applied to a double quantum dot, and the parameters of a Hubbard-like Hamiltonian, is of utmost importance for understanding and operating spin qubits. State-of-the-art techniques for measuring Hamiltonian parameters (e.g., detuning axis pulsed spectroscopy, DAPS) provide details about energy levels; however, tunnel coupling estimates typically reveal only a small portion of the full Hamiltonian. Here, we demonstrate a Hamiltonian-agnostic technique for measuring the double dot energy spectrum over a wide energy range, at every value of the detuning, called delta-axis spectroscopy (DAXS). We apply the DAXS method to obtain the energy spectrum of a Si/SiGe double quantum dot and use this data to extract the diagonal and off-diagonal couplings of a 15-level Hubbard-like Hamiltonian, demonstrating very good agreement with the experimental measurements.

Figures (9)

• Figure 1: Overview of measurement regime and spectroscopy (a) A false-colored scanning electron micrograph of a device lithographically identical to the one used in this experiment. Gates are labeled P(B)[A] for Plunger(Barrier)[Accumulation]. (b) Stability diagram of the (1,3)-(0,4) charge configuration plotted as a function of $\varepsilon$ and $\delta$. The inset shows the same charge regime as a function of the two plunger gate voltages. (c) Energy dispersion corresponding to the (1,3)-(0,4) charge regime. (d) Average of four DAXS measurements of the double dot utilizing both reservoirs. (e) PGS data for both $P_\text{L}$ (top) and $P_\text{R}$ (bottom) with corresponding double-dot schematics. The voltage pulsing, $V_\text{P}$, is represented with solid lines at the bottom of the pulse and dashed lines at the top, in units of peak to peak voltage. Green and purple arrows depicting first visible splittings are shown here and also in (d). (f,g) Double-dot diagrams depicting schematic wavefunctions, following Ref. burkardSemiconductorSpinQubits2023, on both the left (f) and right (g) sides of the excited state anticrossing shown in (c).
• Figure 2: DAXS fitting and and extracted couplings. (a) DAXS data overlaid with fits of eigenvalues from the 15x15 Hamiltonian, shown in Ref. SI describing the levels and couplings. (b) Plot of extracted couplings as a function of the center barrier gate voltage, $B_\text{C}$. (c) Plot of extracted couplings over 5 different trials, showing scan-to-scan variability of a single tuning where $B_\text{C}=660$$mV$. One standard deviation is displayed as shaded regions for each coupling and the average is a dashed line.
• Figure 3: Identifying relevant dot states. (a) Overlaid DAXS data with colored arrows pointing to states corresponding to the right quantum dot. (b) Plotting $\delta$ vs. $A_\text{R}$ with pulsed gate voltages applied. Vertical lines with colored arrows correspond to dot states while diagonal lines correspond to resonances in the leads. Compensation is applied to the $P_\text{R}$ as well as $B_\text{R}$ voltages to keep the lines of dot states vertical and hold tunnel rates relatively constant, respectively. (c) Schematic depicting a picture of the Fermi level of the leads and the resonances due to the quasi-1D behavior mottonenProbeControlReservoir2010. Red and blue correspond to how lead states may look at different lead voltages. (d) A combined magnetospectroscopy and PGS scan used to examine the behavior of excited singlet and triplet states in a magnetic field.
• Figure S1: Steps of the DAXS Fitting Procedure. (a) Example hand-drawn curve used to produce initial guess values of $\delta$ for the Lorentzian fits of each column of data in the image. Similar curves are drawn over each of the other visible eigenstates that are fitted to. (b) Calculated peak locations from the Lorentzian fitting procedure. Each color corresponds to a unique eigenstate. (c) Calculated eigenstates (solid lines) from fitting the Hamiltonian matrix to the Lorentzian data (colored points) from (b).
• Figure S2: Plot of Hamiltonian eigenvalues. Eigenvalues from the fit shown in Fig. \ref{['figS1']}, with the range extended. State curves that are not included in the fitting process are shown as partially transparent, and states where the electron is loaded purely into the left/right dot are denoted with dashed black lines.