Triplet-assisted leakage during singlet-triplet qubit readout with a quantum point contact

Abstract: Quantum point contact readout theory for singlet-triplet qubits in a lateral double quantum dot is extended by including tunneling of triplet configurations into a higher-energy level of the neighboring dot. This additional channel creates energetically allowed leakage pathways that modify the branch-dependent charge and current-noise signatures, even when the Pauli blockade remains effective within the ground-state manifold. The model contains two single-particle levels in each dot. The resulting singlet and triplet block structure is derived together with a Lindblad master equation. Quantum-jump simulations are then used to resolve the dynamics of individual readout events. A complementary Liouvillian steady-state analysis identifies the regime in which tunneling to the excited level qualitatively changes the readout signatures, with the crossover determined by the level spacing.

• The paper extends QPC readout theory by adding excited orbitals to a GaAs double quantum dot and uses Lindblad dynamics, quantum-jump trajectories, and Fano spectra to track leakage pathways.
• When the excited-level spacing falls below the QPC bias, triplet-assisted transitions create a low-current shelving state and a sharp zero-frequency noise peak, causing triplet noise to exceed singlet noise.
• The results show that keeping the QPC bias below the excited-state threshold preserves conventional Pauli spin-blockade contrast, while near-threshold operation can reveal hidden leakage through current-noise spectroscopy.

Overview

The paper extends the standard theory of quantum point contact (QPC) readout of singlet-triplet ($S$–$T$) qubits in lateral double quantum dots (DQDs) by including a second single-particle level in each dot (2605.02785). Prior master-equation treatments of $S$–$T$ readout via Pauli spin blockade (PSB) — notably those of Barrett and Stace and subsequent phonon-assisted extensions — assume a single active orbital per dot. The author shows that when an excited orbital lies within or near the QPC bias window, triplet configurations can tunnel into doubly occupied states via the excited level, creating leakage pathways that qualitatively alter the branch-dependent charge and current-noise signatures even though PSB remains effective within the ground-state manifold.

Model

The system is a lateral GaAs DQD capacitively coupled to a QPC placed next to the right dot, so that QPC conductance depends on right-dot occupation. Each dot hosts a ground ($g$) and first excited ($e$) orbital separated by energy $\epsilon$, with spin-conserving interdot tunneling $t_g = t_e = t_c = t$, local Hubbard repulsion $U$, and neglected interdot Coulomb coupling. The two-electron Hilbert space decomposes into singlet and triplet blocks: three singlets per orbital manifold plus four mixed-manifold singlets (ten states total), and one triplet per manifold plus four mixed-manifold triplets (six states). The Hamiltonian is diagonalized within each block, yielding eigenstates $\{s_1,\dots,s_{10}\}$ and $T$0, with an effective exchange $T$1 appearing in the mixed-manifold triplet sector.

The QPC is modeled as two noninteracting reservoirs whose tunneling amplitude is modulated by right-dot occupation, following the standard Barrett–Stace scheme. In the DQD eigenbasis, this produces pure-dephasing (diagonal) and inelastic (off-diagonal) backaction channels.

Methods

The reduced dynamics is governed by a Born–Markov–secular Lindblad equation with transition-resolved jump operators $T$2, where the ramp function $T$3 encodes the energetically allowed tunneling window set by the QPC bias $T$4. Downward transitions are always open; upward transitions require $T$5.

Two complementary analyses are performed. First, quantum-jump (Monte Carlo wave-function) trajectories resolve individual readout events, post-selected onto singlet-like or triplet-like steady states from a coherent initial superposition $T$6. Second, the steady-state Fano spectrum $T$7 is computed exactly via Liouvillian spectral decomposition combined with the quantum regression theorem. A modal pathway diagnostic ranks transition pairs by their excitation of slow Liouvillian modes weighted by current visibility, supplemented by effective population-transfer rates and inter-state current contrasts; reduced connected pathway models built from the ranked pairs validate the interpretation numerically.

Results

Unless stated otherwise, parameters are $T$8 meV, $T$9 meV, $S$0 meV, with level spacings $S$1 meV (above bias) and $S$2 meV (below bias).

For $S$3, conventional PSB phenomenology is recovered: the triplet Fano spectrum is flat and Poissonian, $S$4, while the singlet retains a broad low-frequency enhancement from current-contrasting intra-manifold shelf cycles ($S$5 and $S$6). Notably, the triplet slow modes are actually longer-lived than the singlet ones at this spacing, but have negligible current visibility — the flat spectrum reflects invisibility, not absence of slow dynamics.

For $S$7, the qualitative reversal occurs. The triplet develops a sharp, large zero-frequency noise peak driven by a shelving mechanism: rare transitions through the state $S$8 (mainly $S$9 and $T$0) connect a low-current shelved state to the conducting $T$1 branch, producing telegraphic switching concentrated near $T$2. The singlet remains super-Poissonian but with excess noise spread over several comparable shelf cycles rather than a single dominant switch. The two branches therefore encode distinct dynamical mechanisms in spectral shape, not merely different noise magnitudes.

The zero-frequency Fano factor versus $T$3 exhibits a clear crossover: as $T$4 decreases below the bias window, triplet noise rises sharply until it overtakes the singlet, reversing the usual PSB intuition that the blocked branch is quieter. Reduced pathway models reproduce both branches' full-model results closely, confirming that minimal shelving networks capture the physics. Quantum-jump estimates agree with the Liouvillian spectra over the resolved window, with residual mismatch confined to the narrowest structure at $T$5, where correlation times exceed trajectory lengths.

Limitations and open questions

Several assumptions bound the applicability of these conclusions. The model neglects interdot Coulomb repulsion, treats spin as passive (no magnetic-field gradients, exchange splitting of the qubit, or hyperfine/phonon relaxation channels), assumes identical dots, sets all tunneling amplitudes equal, and uses energy-independent QPC densities of states. The secular approximation requires transition energies well separated from QPC-induced rates, which holds for the parameters studied but constrains extrapolation to smaller $T$6 or stronger coupling. The quantum regression theorem underlying the analytic spectra inherits the Markovian assumptions of the master equation. The paper does not quantify how the leakage-induced noise degrades readout fidelity or assignment times in an actual measurement protocol, nor does it treat valley states relevant to silicon implementations where excited-level effects are experimentally documented. Whether the shelving-mediated noise can be exploited as a quantitative spectroscopic probe of excited-level admixture, rather than merely a diagnostic signature, remains open.

Conclusion

Including excited orbitals in the DQD model does not merely renormalize QPC readout response but can invert the branch-noise hierarchy central to $T$7–$T$8 qubit measurement: once excited-level-assisted transitions enter the detector bias window, triplet shelving generates strong low-frequency super-Poissonian noise that exceeds the singlet's. The level spacing relative to the QPC bias emerges as the key design parameter — keeping the bias below the excited-state threshold preserves conventional PSB contrast, while operating near it enables current-noise spectroscopy of hidden leakage channels as a complement to average-current measurements.