Conveyor-Mode Shuttling in Quantum and Granular Systems

• Conveyor-mode shuttling is a transport regime where a moving confinement potential carries electrons or particles, setting it apart from bucket-brigade or gravity-driven methods.
• In quantum-dot devices, engineered sinusoidal and digitally synthesized waveforms create traveling potentials that enable high-speed, spin-coherent electron shuttling.
• In granular and Brownian systems, the moving boundary or collective cluster dynamics regulate particle flow, demonstrating versatile applications beyond quantum transport.

Conveyor-mode shuttling denotes a transport regime in which particles are carried by a moving confinement potential or a moving boundary, rather than transferred between static sites or discharged by gravity alone. In Si/SiGe quantum-dot devices, the term refers to translating a quantum-dot minimum so that a single electron or spin qubit remains in essentially one moving dot while the potential travels along a gate-defined channel (Seidler et al., 2021, Struck et al., 2023). In other settings, the same term has been used for granular discharge controlled by a conveyor belt placed directly below a silo outlet and for collective particle shuttling by a condensed cluster in an asymmetric narrow channel (Fan et al., 2023, Pototsky et al., 2010).

1. Definition and transport regimes

In the quantum-dot literature, conveyor-mode shuttling means creating a moving confinement potential so that “the potential minimum itself is translated smoothly along the device, and the electron’s wavefunction follows adiabatically” (Volmer et al., 2 Mar 2026). The electron is therefore transported by a moving quantum dot, not by a sequence of discrete tunnel events between fixed dots. This definition is also formulated as moving “a single electrostatic potential minimum itself” through a linear array of gates, with the electron spin carried along by the translating dot center $x_{\rm QD}(t)$ (Nagai et al., 28 Feb 2025).

This mode is routinely contrasted with bucket-brigade transport. In bucket-brigade schemes, the electron is adiabatically transferred between static dots by pulsing local tunnel couplings and detunings; in conveyor mode, a travelling electrostatic wave carries the electron in one moving minimum (Struck et al., 2023). It is also distinct from surface-acoustic-wave transport: the Si/SiGe implementations discussed here are electrostatic and do not rely on a moving piezoelectric potential (Nagai et al., 28 Feb 2025).

A second, mechanically grounded usage appears in granular matter. There, conveyor-mode discharge means that a silo outflow is “controlled not by gravity alone but by a moving boundary – a conveyor belt directly under the outlet” (Fan et al., 2023). A related but older usage appears in stochastic transport, where a condensed cluster in a ratchet channel acts as a shuttle that moves between docking stations under a low-frequency ac drive (Pototsky et al., 2010).

Taken together, these usages indicate a common operational idea: transport is set by the motion of the carrier landscape itself, whether that landscape is an electrostatic minimum, a moving extraction boundary, or a collectively moving cluster.

2. Quantum-dot implementations and waveform synthesis

A canonical Si/SiGe implementation employs a one-dimensional electron channel with a periodic gate pattern wired into four phase-shifted control sets. In one representative device, 17 “clavier” finger gates are grouped into four interleaved sets $S1$–$S4$, and the moving potential is generated by

$$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$

so that the quarter-period phase shift creates a travelling wave along the channel (Volmer et al., 2 Mar 2026). In the proof-of-principle QuBus device, every fourth gate shares the same drive and the electrostatic wavelength is $\lambda=280\,\mathrm{nm}$, giving the conveyor speed $v_{\mathrm S}=f\lambda$ (Seidler et al., 2021).

This four-phase control structure is the defining scalability feature of the platform: the number of high-level shuttling signals does not grow with shuttle length. The same logic underlies later experiments on spin-coherent transport, where a travelling wave on $S1$–$S4$ moved a separated electron of an EPR pair through a 420–560 nm section of a quantum bus (Struck et al., 2023).

Several control refinements generalize the basic sinusoidal drive. One route is digital synthesis at cryogenic temperature. A digital-controlled conveyor-belt method replaces per-gate analog sinusoids by a small number of DC voltage levels, a cryogenic switch matrix, and local RC low-pass filters, producing near-sinusoidal conveyor waveforms while keeping the room-temperature wiring count essentially independent of qubit number (Nagai et al., 28 Feb 2025). Another route modifies not only the average speed but the detailed velocity profile. In a smooth-velocity protocol, the carrier waveform is frequency-modulated so that the dot velocity follows a Tukey-window profile,

$$V_n(t)=V_0 \cos\Bigl[\omega_0 t+\phi_n+\omega_0 \int^t s(\tau)\,d\tau\Bigr]+B_n,$$

with $s(t)=(v(t)-v_0)/v_0$, thereby shaping the shuttling spectrum without changing the underlying conveyor architecture (Nagai et al., 1 Jun 2026).

At the architectural level, conveyor mode is no longer restricted to straight 1D rails. A T-junction device links two independently driven shuttle lanes without adding extra control lines beyond the four channels per conveyor belt, and a 2D “clavette” grid generalizes the same travelling-pocket principle to omnidirectional motion (Beer et al., 7 Jan 2026, Németh et al., 2024).

3. Valley physics, non-adiabaticity, and coherence

In Si/SiGe, the principal microscopic complication is the valley degree of freedom. In the local valley eigenbasis, the moving-dot Hamiltonian acquires both a longitudinal splitting and a transverse non-adiabatic term,

$S1$0

with

$S1$1

Accordingly, the shuttling velocity profile directly drives valley excitation through the term $S1$2 (Nagai et al., 1 Jun 2026).

This valley dynamics matters because the spin splitting depends on valley state. In one formulation, the full Hamiltonian includes

$S1$3

so that population in different valley states causes different spin precession frequencies and hence spin–valley entanglement (Nagai et al., 1 Jun 2026). A related long-distance treatment writes $S1$4, with the resulting $S1$5-type spin–valley coupling as the dominant spin-decoherence channel during shuttling (David et al., 2024).

The resulting error mechanism is not confined to isolated hot spots. In alloy-disorder-dominated Si/SiGe, the intervalley coupling $S1$6 is a spatially varying complex field; the corresponding valley splitting $S1$7 inevitably develops minima along long trajectories (Németh et al., 2024). For constant-speed shuttling over $S1$8, improved disorder modeling and realistic noise ranges lead to typical errors of $S1$9, even though nuclear-spin dephasing is strongly reduced by motional narrowing at meter-per-second speeds (David et al., 2024).

A useful perturbative limit expresses the valley-excitation probability through the spectral weight of the shuttling velocity,

$S4$0

which makes explicit why abrupt start–stop profiles are problematic: high-frequency sidelobes of $S4$1 couple directly to the valley splitting frequency (Nagai et al., 1 Jun 2026). This is the analytical basis for later velocity-shaping schemes.

4. Control optimization, disorder avoidance, and material dependence

A central development after the first conveyor demonstrations is the shift from constant-speed transport to explicitly optimized trajectories. One route uses a very low-dimensional ansatz,

$S4$2

with as few as four Fourier components. For $S4$3 shuttling, this reduces the median infidelity by orders of magnitude relative to constant speed; at $S4$4, $S4$5 harmonics yield a median infidelity of about $S4$6 in the modeled ensemble (David et al., 2024). A complementary analytical strategy maps velocity-shape design onto window-function design and shows that Tukey or Hann profiles strongly suppress the high-frequency sidelobes responsible for valley excitation, significantly reducing average spin infidelity in the moderate-to-low disorder regime $S4$7 (Nagai et al., 1 Jun 2026).

A second strategy is geometric rather than spectral: avoid the problematic valley minima altogether. In a multichannel shuttler, electrons can tunnel between parallel conveyor channels to achieve transverse displacements of order 100 nm, but transfer fidelity is limited by valley-phase mismatch and disorder-sensitive interchannel tunnelling (Németh et al., 2024). A fully omnidirectional 2D shuttler is more favorable. For a $S4$8 unit cell of “clavette” gates, electrostatic and master-equation simulations identify an operating window roughly at $S4$9 and $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$0 where $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$1 and pocket leakage over $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$2 can be kept below $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$3 (Németh et al., 2024).

Full-device simulations across materials reinforce that conveyor mode is not automatic. In a physics-informed optimization study spanning FD-SOI, SiMOS, and Si/SiGe, successful conveyor operation is associated with a single smoothly moving dot, nearly constant velocity, and nearly constant ground-state energy; FD-SOI without J-gates instead exhibits a tunnelling event where the ground-state gap collapses, indicating failure of true conveyor mode (Sokolov et al., 8 Oct 2025). In realistic SiMOS, low clavier-gate voltages cause “conveyor-belt shuttling to collapse to the bucket-brigade mode,” because the additional oxide screening weakens every second gate layer; raising the confinement restores conveyor-belt operation (Turner et al., 3 Dec 2025).

Device imperfections further stratify the viable regime. In a QuBus with sparse singly charged defects, the dominant modeled spin error channel is defect-induced orbital excitation combined with orbital-dependent $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$4-factors and phonon-mediated relaxation; for weak confinement and a worst-case defect location the expected dephasing can reach about $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$5, whereas strong confinement suppresses it to $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$6 (Ciroth et al., 3 Dec 2025). In SiMOS, negative oxide defects are comparatively benign, whereas positive defects at the Si/SiO$$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$7 interface can capture passing electrons or induce strong orbital excitation, especially at low conveyor gate bias (Turner et al., 3 Dec 2025). Digital waveform synthesis addresses a different bottleneck: by generating near-sinusoidal conveyor waveforms from a few DC levels at cryogenic temperature, it preserves valley-shuttling fidelity while greatly reducing room-temperature wiring overhead and allowing scaling analyses up to $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$8 qubits (Nagai et al., 28 Feb 2025).

5. Experimental milestones, metrology, and routing

The first proof-of-principle demonstration in Si/SiGe showed conveyor-mode single-electron shuttling in a 420 nm channel with a single-electron shuttling fidelity of $$V_{\mathrm{S},i}(t) = A_i \cos\!\left(2\pi f t - (i-1)\frac{\pi}{2}\right)+B_i,\quad i=1,\dots,4,$$9, including reversal of direction, using only four sinusoidal control signals independent of channel length (Seidler et al., 2021). That experiment also used the smooth SET response during slow shuttling to detect the continuity of the electron motion and to map potential disorder along the bus.

Spin-coherent operation followed. In a later experiment, a conveyor-mode bus separated and rejoined an EPR spin pair while boosting the shuttle velocity by a factor of 10000 relative to the earlier conveyor demonstration. The measured dephasing-induced spin-shuttle infidelity was estimated as $\lambda=280\,\mathrm{nm}$0 for a nominal total shuttle distance of 560 nm, and spin entanglement remained detectable after several loops corresponding to an accumulated distance of $\lambda=280\,\mathrm{nm}$1 (Struck et al., 2023). A characteristic feature was motional narrowing: the ensemble dephasing time increased with shuttle distance because the moving electron averaged over spatial variations of the effective Zeeman splitting.

Conveyor-mode transport has also become a metrology tool for valley physics. One method maps local valley splitting in a $\lambda=280\,\mathrm{nm}$2 area by detecting magnetic-field-dependent anticrossings of ground and excited valley states, using entangled spin pairs as probes; it achieves sub-$\lambda=280\,\mathrm{nm}$3eV energy accuracy and nanometer lateral resolution (Volmer et al., 2023). A related method maps local $\lambda=280\,\mathrm{nm}$4-factor differences with precision better than $\lambda=280\,\mathrm{nm}$5 over a $\lambda=280\,\mathrm{nm}$6 region, observes two $\lambda=280\,\mathrm{nm}$7-factors per site associated with the two valley states, and reconstructs the complex intervalley coupling $\lambda=280\,\mathrm{nm}$8 along shuttle trajectories (Volmer et al., 2 Mar 2026). These measurements reframe conveyor mode as a scanning probe of spin–valley disorder, not merely a transport primitive.

Routing has progressed from linear buses to network elements. A T-junction device linking two independently driven shuttle lanes realizes inter-lane charge transfer without any extra fast lines beyond the four channels per conveyor belt. The measured inter-lane transfer fidelity is

$\lambda=280\,\mathrm{nm}$9

at an instantaneous electron velocity of $v_{\mathrm S}=f\lambda$0 (Beer et al., 7 Jan 2026). The same device controls the filling of 54 quantum dots by a small set of atomic pulses and swaps electron patterns, a charge-level precursor of a native spin-qubit SWAP operation (Beer et al., 7 Jan 2026). This progression—from single-lane proof-of-principle to spin-coherent shuttling, disorder metrology, and T-junction routing—defines the present experimental core of conveyor-mode shuttling.

6. Broader usage beyond quantum devices

In granular matter, conveyor-mode shuttling denotes a mechanically controlled discharge regime rather than a moving quantum dot. In a quasi-2D silo filled with elliptical cylinders, a conveyor belt placed directly below the outlet acts as a moving boundary that sets throughput. Relative to free discharge, the belt reduces the flow rate, enlarges stagnant zones, strongly increases relative velocity fluctuations at low belt speed, and suppresses orientational order at the orifice (Fan et al., 2023). The orientational order is quantified by

$v_{\mathrm S}=f\lambda$1

and one of the notable observations is that slow belt motion does not produce a smooth, uniformly slower flow: it produces intermittent bursts together with a central dip in $v_{\mathrm S}=f\lambda$2, particularly for large orifices (Fan et al., 2023).

A different non-quantum usage appears in Brownian transport through asymmetric narrow channels. There, attracting particles in single file condense into a compact cluster whose center of mass moves collectively through a static ratchet potential under a quasistatic square-wave drive (Pototsky et al., 2010). In the strong-attraction limit, the cluster behaves as a rigid rod with center-of-mass equation

$v_{\mathrm S}=f\lambda$3

so that the effective pinning force depends sensitively on the cluster length $v_{\mathrm S}=f\lambda$4 (Pototsky et al., 2010). Because the rectified current changes sign as $v_{\mathrm S}=f\lambda$5 varies, adding or removing one particle can reverse the direction of net motion; this enables a shuttle cycle between source and sink docking stations in which the condensed cluster loads cargo on one side and unloads it on the other (Pototsky et al., 2010).

These broader uses suggest that conveyor-mode shuttling is not tied to a single material platform or microscopic mechanism. The shared structure is transport governed by a moving carrier landscape—electrostatic, mechanical, or collective—whose dynamics determine throughput, fluctuations, and state preservation.