Quantum Wave-Packet Expansion and Refocusing
A Reversible Pathway for Controlled Electron Transport
John Swygert
November 9, 2026
DOI: To be assigned
Abstract
Quantum uncertainty is usually presented as a restriction on simultaneous localization, yet the associated spreading and refocusing of coherent wave packets can also be treated as an engineering resource. This paper proposes a testable electron-transport protocol in which a localized electronic state is prepared, coherently broadened to access a selected pathway, guided toward a target, and refocused for readout or collection. The proposal is not a reversal of the uncertainty principle, nor does it imply that spatial expansion follows automatically from shortening a pulse. It instead asks whether controlled phase evolution and field shaping can convert temporary delocalization into useful spatial selectivity, transfer probability, or timing performance. A minimal model, measurable figures of merit, experimental comparisons, failure modes, and falsification criteria are specified.
The present extension uses reported measurements from Maier et al. (2026) as quantitative anchors and proposes additional experiments. Numerical bandwidth and Gaussian-propagation examples below are explicitly illustrative; they are not claimed experimental observations.
1. Research Question and Scope
Can a coherent electron wave packet be deliberately expanded, steered, and refocused in a repeatable sequence such that transport to a chosen collection region is improved relative to a matched control? “Reversible” here means approximately reversible coherent evolution under suitable control, not thermodynamic reversibility, perfect recovery, or reversal of elapsed time. The proposal concerns electron wave functions and their measured distributions, not literal expansion or contraction of spacetime.
The motivating physical context is ultrafast laser-driven electron motion and tunnelling near nanoscale metallic structures. A reported attosecond-scale response in such systems makes temporal control relevant, but this manuscript does not treat any single recent experiment as evidence that the proposed full cycle has already been demonstrated.
2. Physical Basis
For a quantum state with position and momentum standard deviations σx and σp, σxσp ≥ ħ/2. Strong initial localization commonly entails a broad momentum distribution. Under free evolution, different momentum components acquire different phases; the resulting position distribution may broaden. An external potential or suitably timed field can alter those phases and, in favorable cases, produce a later focus. This is analogous in principle to dispersion management and wave-packet lensing, although the electron implementation faces distinct Coulomb, decoherence, and material constraints.
Temporal pulse width and spectral bandwidth are likewise linked for a pulse, but the energy–time relation is not simply a second canonical position–momentum commutator. Crucially, shortening an electron emission pulse does not universally force greater spatial width: spatial spreading depends on the prepared state, dispersion relation, geometry, and subsequent evolution. The engineering opportunity lies in shaping these variables, not in assuming an automatic one-to-one conversion of time precision into distance.
2.1 Quantitative Anchoring to Maier et al. (2026)
The experimental values and geometrical context in this section are taken from Maier et al. (2026), Nature Photonics, doi:10.1038/s41566-026-01932-0; the following Gaussian calculations are independent illustrative benchmarks, not fits to those measurements.
Maier et al. report isolated tunnelling transients shorter than 1 fs using phase-controlled near-infrared fields at an atomically sharp tip–Ag(100) junction. Their measured current decay length along the tip–surface separation is approximately 8.7 Å at 171 pJ pulse energy, versus 5.8 Å at 93 pJ, 4.3 Å at 55 pJ, and 3.8 Å at 36 pJ. At 93 pJ, imaging a single Cu adatom establishes lateral confinement on a scale of approximately 6 Å. These values are experimentally reported spatial-response measures, not a direct measurement of a freely propagating Gaussian wave packet’s standard deviation.
For an illustrative transform-limited Gaussian temporal amplitude, ΔE·Δt ≥ ħ/2 when Δt is the standard deviation of the corresponding time distribution. Taking Δt = 0.5 fs as an illustrative pulse-width convention gives ΔE ≥ 0.658 eV; at 0.25 fs it gives ΔE ≥ 1.316 eV. These are bandwidth scale estimates, not measured electron energy spreads in Maier et al., and the pulse-width convention and nonstationary tunnelling dynamics must be specified before comparison. An attosecond response delay of about 500 as is not interchangeable with the pulse duration or an uncertainty Δt.
A quantitative device prediction requires a driven, open-system junction calculation fitted to the measured optical pulse envelope, tip–surface barrier parameters, tunnelling decay lengths, and phase-dependent currents. Free-particle dispersion alone cannot establish useful expansion or refocusing in this apparatus.
A separate free-particle benchmark is σx(t) = σx(0)·sqrt[1 + (ħt/(2mσx(0)^2))^2] for an initially minimum-uncertainty, unchirped Gaussian. With σx(0) = 3 Å and t = 0.5 fs, this model predicts σx(t) ≈ 3.15 Å, not a large free-flight expansion. It is only a sanity check: the experimental junction is a driven, barrier-coupled, multiband metallic environment, and its reported decay lengths reflect excitation and tunnelling matrix elements as well as wave-function extent. A credible device prediction requires fitting a driven open-system junction model to the actual pulse, field, and barrier parameters.
The engineering objective is therefore not to assume an enormous spatial expansion from a short timing window. It is to tune the measured field-dependent extension and phase-sensitive transfer into a useful, target-selective routing response.
3. Proposed Five-Stage Protocol
Stage A — Prepare
Prepare a reproducible, spatially localized electronic state or short coherent electron pulse. Characterize initial position distribution, momentum/energy spectrum, phase coherence, and timing jitter.
Stage B — Expand
Allow a designed interval of coherent evolution, optionally assisted by a shaped electromagnetic field, to broaden the spatial probability distribution or couple amplitude into otherwise weakly accessed transport modes.
Stage C — Select a Pathway
Use a controlled potential landscape, aperture, barrier, or near-field structure to favor propagation toward a specified target while suppressing competing channels. This step must be distinguished from mere broadening: increased access alone is not directed transport.
Stage D — Refocus
Apply a second phase-shaping interaction or electron-optical lens to concentrate probability at a collection region and prescribed arrival-time window. A focus is not necessarily the original state; its fidelity must be measured.
Stage E — Collect and Reset
Measure transferred charge, spatial concentration, timing, energy deposited, and loss. Reset the apparatus and repeat to establish cycle-to-cycle reproducibility, stability, and any cumulative heating or damage.
4. Minimal Model
4.1 Minimal Open-System Junction Model and Simulation Plan
Use a finite-dimensional tight-binding representation with source state |s⟩, junction or bridge states |j⟩, and two collection channels |L⟩ and |R⟩. A minimal driven Hamiltonian is H(t) = Σi εi(t)|i⟩⟨i| + Σ⟨i,j⟩ [Jij(t)|i⟩⟨j| + h.c.], where optical phase, envelope, and gate/bias control enter εi(t) and/or Jij(t). The Expand stage changes the accessible-state distribution; Select modifies relative couplings or site energies; Refocus uses a delayed, phase-controlled interaction. Model the reservoirs explicitly or with calibrated injection and extraction operators.
Evolve the reduced density matrix using dρ/dt = −(i/ħ)[H(t),ρ] + Σk γk(LkρLk† − ½{Lk†Lk,ρ}). Illustrative channels are site dephasing Lφ,j = |j⟩⟨j|, inelastic relaxation Lr,ij = |i⟩⟨j|, and irreversible collection Lc,L/R = |sinkL/R⟩⟨L/R|. The sink states keep track of collected probability; their rates must be fitted or bounded by data rather than chosen to produce a desired effect.
Fit the model first to baseline current-versus-gap decay, field-phase dependence, pulse response, and thermal sensitivity. Then predict target and off-target collected charge for all five stages and each matched control, with the same injected population and total drive energy. Vary dephasing, extraction rates, target asymmetry, pulse phase/delay, and field strength; report uncertainty bands and parameter identifiability. A coherent advantage should vanish or weaken as experimentally constrained dephasing increases, while classical focusing and heating-only baselines are separately evaluated.
The phenomenological tight-binding model does not directly identify the observed 8.7 Å and 3.8 Å current decay lengths with wave-packet widths. Those lengths constrain the effective barrier transmission versus gap and must be reproduced by an appropriate junction-specific tunnelling calculation before quantitative device claims are made.
Let ψ(x,t) denote a normalized electron wave function and H(t) = p²/(2m) + V(x,t) a simplified effective Hamiltonian. The controlled evolution over one cycle is ψf = U5U4U3U2U1ψ0, with Uk the unitary evolution for each coherent stage. Real devices require an open-system description including emission statistics, scattering, dissipation, and detection. Define a target region R and final arrival window T. The target probability is Ptarget = ∫R |ψ(x,tf)|² dx (or its time-resolved detection analogue).
The operational goal is not to reduce all uncertainties simultaneously. It is to maximize Ptarget, or a task-specific figure of merit, under fixed input energy, device geometry, elapsed time, and allowed loss. An approximate reversibility check compares the prepared and reconstructed states using fidelity F = |⟨ψ0|ψreconstructed⟩|² when state reconstruction is feasible. High fidelity and high useful transport probability are separate objectives and may conflict.
5. Experimental Design
A plausible platform is a phase-coherent electron source coupled to a nanostructured barrier or electron-optical region, with ultrafast field control and spatially resolved collection. Begin with a baseline pulse and static transport geometry. Compare it against: (i) expansion only; (ii) focusing only; (iii) expansion followed by focusing without directional selection; and (iv) the complete expansion–selection–refocusing sequence. Match initial charge, energy budget, total transport time, and collection geometry as closely as practicable.
Sweep the delay, amplitude, and phase of the second control field; target position; barrier width; and environmental scattering conditions. Record full output distributions, not only the most favorable collected signal. A delayed peak at the intended collector without corresponding improvement in total transfer efficiency might indicate redistribution rather than genuine transport gain.
5.1 Direct Comparison with the Reported Tip–Surface Experiment
Baseline apparatus: retain a low-temperature, ultrahigh-vacuum scanning tunnelling microscope with an atomically sharp metallic tip, Ag(100) surface, phase-stable near-infrared pulses, attosecond-adjustable pulse delay, and carrier-envelope-phase (CEP) modulation. The original experiment measured CEP-sensitive tunnelling current and distance decay, and mapped a Cu adatom. It did not implement a distinct downstream refocusing stage or multiple selectable collection terminals.
Required thermal control: repeat the full protocol and all matched controls at multiple repetition rates and duty cycles while matching average optical power, optical fluence where appropriate, and measured tip/substrate temperature. Track thermally driven drift and thermoelectric currents separately from carrier-envelope-phase-sensitive transport.
An initial proof-of-principle can compare two patterned collection regions or two locally addressable surface targets under identical tip height, bias, optical fluence, spectral bandwidth, and thermal conditions. A second, independently controlled phase or potential interaction is required to operationally distinguish “refocus” from simply changing the first tunnelling barrier. Spatially resolved readout must demonstrate where charge is delivered, not only that total tunnelling current changes.
Four prespecified controls: (1) static-field routing with no timed expansion/refocusing pulse; (2) a focusing-only second interaction without the expansion stage; (3) expansion and selection with the second interaction disabled or phase-randomized; and (4) a matched-energy, matched-spectrum two-pulse sequence with the pathway selector removed. Randomize the control order and match average power, charge injection, junction geometry, and temperature.
Success requires a joint outcome: improved target selectivity AND a nontrivial absolute target yield under matched injected charge, optical energy, and measurement bandwidth. A high selectivity ratio achieved solely by extinguishing off-target current, while target charge also collapses, is not a transport improvement. Report both quantities with uncertainty intervals and prespecified thresholds.
A strong signal is a target-specific, reproducible CEP- and delay-dependent enhancement of absolute collected charge, accompanied by a complementary off-target suppression. Move or exchange the selected target: the enhancement must follow it. Quantify phase contrast, target-to-off-target selectivity, absolute target yield per injected electron, heat load, and uncertainty intervals. Distinguish a genuine multi-stage interference effect from photon-assisted barrier lowering, thermal drift, and ordinary electron-optical focusing.
6. Metrics and Predicted Signatures
Primary metrics: target collection probability, off-target probability, spatial full width at half maximum, arrival-time distribution, transmitted charge per unit input energy, coherence visibility, and cycle-to-cycle variance. A useful composite selectivity ratio is S = Ptarget/(Poff-target + ε), with ε a declared regularization constant. Any claim of advantage should report both absolute collected charge and normalized efficiency; selective postselection can otherwise create a misleading appearance of success.
A positive result would show a reproducible, phase- and delay-dependent refocusing maximum at the selected collector, outperforming matched controls on a prespecified metric. Moving the intended target should move the refocused distribution predictably. Loss of the effect under controlled dephasing would strengthen the interpretation that coherent wave-packet manipulation, rather than an ordinary classical field-driven trajectory, is responsible.
7. Constraints, Failure Modes, and Alternatives
Potential obstacles include rapid decoherence in metallic environments, electron–electron interactions, inelastic scattering, imperfect phase control, limited collection efficiency, space-charge effects, field-induced emission artifacts, and heat generation. Apparent benefits might instead arise from conventional electrostatic focusing, energy filtering, detector gating, or altered emission probability. These alternatives must be measured explicitly. A protocol that works once but fails after repeated cycles does not establish a reusable transport architecture.
No fundamental conservation law is bypassed. Expanding and refocusing probability distributions cannot create charge or energy, guarantee tunnelling, or enable superluminal signalling. A complete energy and entropy accounting is required before claiming efficiency gains.
7.1 Coherence, Irreversibility, and Open-System Conditions
Coherent expansion and refocusing require phase information to survive over the combined Expand–Select–Refocus interval. Define τcoh operationally from the decay of an interference or CEP-sensitive contrast observable under matched junction conditions. The design requires τsequence substantially shorter than the relevant measured τcoh, not merely shorter than an assumed bulk-metal relaxation time. Electronic coupling to the substrate, electron–phonon scattering, many-body excitations, and detector back-action must be included.
Use an open-system density matrix ρ with dρ/dt = −(i/ħ)[H(t),ρ] + D[ρ], where D captures experimentally constrained decoherence and dissipation. Compare a coherent model against a phase-incoherent rate or classical trajectory model on the same measured observables. A reduction in contrast as controlled dephasing is increased provides a direct coherence test.
Unitary propagation can, in principle, reverse a prepared wave-packet evolution if its phase structure and environment are controlled. Collection at an electrode, readout, and resetting the source are generally not unitary reversals of that same electron state. “Reversible” here refers to the coherent shaping stages; the full transport-and-reset apparatus is a repeatable driven cycle with dissipation.
7.2 Energy, Entropy, and Thermal Accounting
For each cycle measure or bound incident optical pulse energy, electrical work supplied by the bias and gating circuits, emitted or transferred electron energy, substrate/tip heating, and energy lost to inelastic channels. Report target charge per cycle and per joule, and target charge per injected electron; these answer different questions. An increase in normalized selectivity is not by itself an improvement in absolute efficiency.
Because Maier et al. identify laser-induced thermal effects as a serious artefact, phase modulation and laser-power stability must be preserved in any extension of the apparatus. Repeat measurements across duty cycles and repetition rates, and test whether apparent gains persist after matched temperature and average power are imposed. Reset cost, erasure of controller information, and detector dissipation belong in any complete thermodynamic efficiency claim. No energy-free amplification or entropy-free resetting is implied.
8. Falsification Criteria
The central engineering hypothesis would be weakened if the full protocol fails to exceed well-matched focusing-only and static-field controls; if any advantage disappears when total energy and accepted bandwidth are equalized; if target localization is insensitive to the predicted control-field phase; or if repeated cycling destroys coherence or produces prohibitive losses. A null result would still constrain the useful operating regime and clarify whether temporary delocalization offers a distinct engineering benefit.
9. Relationship to Boundary and Pathway Geometry
A narrow barrier or constricted transmission region can be treated as a selection element in a broader pathway-control problem. This creates a conceptual bridge to geometric descriptions of access, bottlenecks, and transmission. The bridge is methodological: engineered constraints select allowed pathways. It is not evidence that electronic tunnelling and spacetime topology are governed by the same geometry, nor that a physical spacetime throat is present in the apparatus.
10. Conclusion
The proposed process replaces a passive reading of uncertainty-related wave-packet spreading with an active design question: can a temporary broadening of accessible electronic states be followed by selective, phase-controlled reconcentration to improve a measurable transport task? Existing quantum dynamics permit spreading and refocusing in principle. Whether the complete five-stage cycle provides a practical advantage in ultrafast electron tunnelling remains an open experimental question. The decisive next step is a controlled comparison against simpler electron-optical alternatives, with full accounting for coherence, energy, and losses.
Selected Foundational References
Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 43, 172–198.
Landau, L. D., and Lifshitz, E. M. Quantum Mechanics: Non-Relativistic Theory. Pergamon Press.
Sakurai, J. J., and Napolitano, J. Modern Quantum Mechanics. Cambridge University Press.
University of Regensburg (2026). Research summary, republished by ScienceDaily November 7, 2026: https://www.sciencedaily.com/releases/2026/11/261106174537.htm
Maier, S., Spachtholz, R., Glöckl, K., et al. (2026). Tracking electrons at the space-time limit. Nature Photonics 20, 961–967. https://doi.org/10.1038/s41566-026-01932-0
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