Coherent Tunneling by Adiabatic Passage

• CTAP is an adiabatic quantum control protocol that transfers population between spatially separated sites via a dark eigenstate and counter-intuitive pulses.
• It demonstrates high robustness with >99% fidelity in platforms like atom chips, quantum dots, and photonic circuits under realistic parameter variations.
• Extensions of CTAP include many-body, topological, and graph-theoretical generalizations that expand its applications in quantum information processing.

Coherent Tunneling by Adiabatic Passage (CTAP) is an adiabatic quantum control protocol that enables high-fidelity, robust, and coherent population transfer between spatially separated quantum sites, notably without significant transient occupation of intermediate sites. Originally formulated as a spatial analog of the Stimulated Raman Adiabatic Passage (STIRAP) technique from quantum optics, CTAP exploits the presence of an instantaneous zero-energy “dark” eigenstate and a counter-intuitive pulse sequence to achieve deterministic quantum state transfer. The protocol has been realized and analyzed in a broad array of settings, including atom chips, solid-state donor arrays, quantum dots, photonic waveguides, many-body bosonic and fermionic systems, and high-dimensional lattices (Morgan et al., 2013, Rahman et al., 2010, Borovkova et al., 10 Dec 2025, Bradly et al., 2012, Groenland et al., 2019).

1. Fundamental Principles and Model Hamiltonian

CTAP operates in a three-site system (labelled |L⟩, |M⟩, |R⟩) governed by a time-dependent nearest-neighbor Hamiltonian: $$H(t) = \hbar \begin{pmatrix} 0 & -J_{LM}(t) & 0 \ -J_{LM}(t) & 0 & -J_{MR}(t) \ 0 & -J_{MR}(t) & 0 \end{pmatrix}$$ where $J_{LM}(t)$ and $J_{MR}(t)$ denote tunneling matrix elements, typically controlled dynamically (as a function of time, spatial position, or other parameters depending on the platform). The protocol requires a “counter-intuitive” pulse sequence: the coupling between the intermediate and final site ($J_{MR}(t)$) peaks before the coupling between the initial and intermediate sites ($J_{LM}(t)$), in direct analogy with STIRAP. This sequence ensures adiabatic following of the dark state.

The instantaneous eigenstates include a unique zero-energy dark state: $$|D(t)\rangle = \cos\theta(t)\;|L\rangle - \sin\theta(t)\;|R\rangle, \quad \tan\theta(t) = \frac{J_{LM}(t)}{J_{MR}(t)}$$ which is strictly orthogonal to |M⟩ ($\langle M|D(t)\rangle=0$) at all times. Adiabatic evolution along |D(t)⟩ leads to the transfer of population from |L⟩ to |R⟩ with negligible occupation of |M⟩ (Morgan et al., 2013, Bradly et al., 2012).

2. Adiabaticity Criteria and Robustness

The fidelity of CTAP relies on satisfying the adiabatic condition: $$\left|\langle E_{\pm}(t)|\frac{\partial H}{\partial t}|D(t)\rangle\right| \ll [E_{\pm}(t) - E_D(t)]^2$$ where $E_{\pm}(t)$ are the energies of the bright states and $E_D=0$ for the dark state. In practice, one enforces this by ensuring that the minimum energy gap $\Delta_{\text{min}}$ between the dark state and other eigenstates remains large compared to the timescale set by the protocol duration (T). The standard requirement is $\Omega_{\max} T \gg 1$ (for peak coupling $\Omega_{\max}$), but longer durations increase sensitivity to decoherence in open quantum systems (Morgan et al., 2013, Huneke et al., 2012).

CTAP protocols are notably robust against moderate timing errors and parameter variations. For instance, atom-chip simulation results show that a current fluctuation of order $10^{-3} A$ produces only percent-level changes in fidelity (for typical tunnel amplitudes), and transfer fidelities above 99% are possible with practical pulse shapes and device geometries (Morgan et al., 2013). Straddling and strong intermediate couplings enhance the energy gap, further mitigating non-adiabatic transitions (Groenland et al., 2019, Ban et al., 2018).

3. Physical Implementations and Extensions

Atom chips and quantum optical lattices: On atom chips, CTAP is implemented by spatially varying the separation between three parallel nanowires, mapping the spatial variation to an effective time-dependent protocol. Full 3D (GPU-accelerated) simulations confirm that >99% transfer fidelity is achievable with current experimental capabilities (Morgan et al., 2013). In Lieb-type and SSH-type optical lattices, manipulation of lattice depths and geometry allows spatial CTAP, including the transfer of topologically protected corner states in multidimensional models (Peng et al., 27 Nov 2025, Taie et al., 2017).

Solid-state donor and quantum dot chains: In buried triple phosphorus donor chains in silicon, CTAP forms the basis for solid-state qubit transport, with atomistic tight-binding modeling confirming that the three-state dark eigenstate persists even with realistic placement straggle and gate voltage fluctuations. Voltage-based compensation of straggle is effective; gate stabilities at the millivolt level suffice for near-ideal operation (Rahman et al., 2010). Quantum dot implementations benefit from similar physics, with CTAP protocols realized by controlling inter-dot tunneling via gate voltages. These approaches tolerate fabrication-related asymmetries if additional tuning elements are applied.

Photonic waveguides: CTAP has been adapted to photonic integrated circuits in both all-silicon nitride waveguides and hybrid Si₃N₄–Si–Si₃N₄ structures at 1.55 μm. Here, the propagation direction plays the role of effective time, and the coupling coefficients between waveguides are engineered to produce the CTAP protocol along the device length. High confinement and hybrid implementations suppress central guide occupancy to below 1%, with overall transfer efficiencies above 99% for optimized device parameters. Device performance is robust to ±20 nm fabrication tolerances (Borovkova et al., 10 Dec 2025).

Many-body and graph-theoretical generalizations: CTAP has been extended to the Bose–Hubbard model for N-boson transport across three wells, with adiabatic following of a multi-particle “null” state. Detuning and interaction strengths define windows of parameter space for high-fidelity CTAP (Bradly et al., 2012). More generally, the protocol has been generalized to a large class of bipartite graphs. The eigenstructure guarantees dark-state mediated transfer between designated nodes as long as the perfect matching criterion is met on subgraphs formed by removing sender or receiver nodes (Groenland et al., 2019).

4. Measurement, Noise, and Decoherence Effects

Measurement back-action can disrupt the perfect adiabatic following of the dark state. Continuous monitoring, for example by a quantum point contact (QPC) coupled to the intermediate site, introduces dephasing and alters the eigenstructure of the system, producing a secondary peak in measurement distributions correlated with leakage to the intermediate site. The effect is well-described by master-equation formulations with explicit inclusion of detector back-action (Rech et al., 2010).

CTAP is sensitive to environmental noise, particularly resonant noise from two-level fluctuators (TLFs) in solid-state platforms. Resonances between the fluctuator energy splitting and the system’s instantaneous Bohr frequency lead to sharply reduced fidelity. Bloch–Redfield theory is accurate for weakly coupled, fast-relaxing TLFs, while numerically exact approaches are required for highly coherent or structured environments. Strategies for mitigation include spectral engineering, dynamical decoupling, or avoiding resonance regions by pulse shaping and timing (Vogt et al., 2012).

Nevertheless, CTAP is robust to slow gate fluctuations, systematic detunings (when compensated), and even moderate levels of pure dephasing, especially in short-duration or shortcut-to-adiabaticity (STA) protocols that further suppress occupation of detrimental states (Ban et al., 2018).

5. Practical Conditions, Device Parameters, and Diagnostics

Successful CTAP operation relies on several practical and device-specific parameter regimes:

• Atom chips: Transverse trap frequencies of a few kHz and longitudinal frequencies (~5 Hz) allow adiabatic transfer in ~0.1 s, an order-of-magnitude below trap lifetimes. Stable current sources (<0.5% noise) and gentle wire curvatures ensure adiabaticity and resonance (Morgan et al., 2013).
• Donor and QD arrays: Donor placement accuracy of ~1 nm is tolerable with symmetry gate bias corrections. Barrier-gate voltages require stability at the few-millivolt level. For 15 nm spacing, tunnel splittings of 0.1–1 meV allow transfer times of ~10–100 ps for error rates <10⁻³ (Rahman et al., 2010).
• Photonics: Device lengths ~110–125 μm suffice in Si₃N₄ waveguides, with coupling gaps of ~0.3–1.4 μm. Peak transfer efficiencies >99% require smooth, adiabatic coupling profiles and weak overlap between coupling regions. Hybrid Si₃N₄–Si–Si₃N₄ structures further suppress central occupancy and enhance robustness to wavelength variations over ~20 nm around 1.55 μm (Borovkova et al., 10 Dec 2025).
• Solid-state transport: Suppressed shot noise and a low Fano factor $F \lesssim 0.2$ in steady-state CTAP (compared to $F \gtrsim 1$ without adiabatic passage) serve as an experimental fingerprint of successful, middle-dot-free transfer (Huneke et al., 2012).
• Diagnostics: In triple-dot CTAP, conditional measurement protocols can certify the vanishing occupation of the intermediate site in the weak measurement and adiabatic limits, providing a minimally invasive and theoretically sharp demonstration of CTAP operation (1901.10057).

6. Advanced Protocols, Extensions, and Outlook

CTAP protocols have been adapted to enable new functionalities and extended to nontrivial topological settings:

• Spin and entanglement transfer: Adiabatic passage protocols for spin qubit transport (spin-CTAP) exploit Heisenberg exchange modulation and magnetic field gradients. Conditional spin-CTAP enables GHZ state generation and quantum-controlled transfer, retaining high fidelities even under realistic noise (Gullans et al., 2020, Ferraro et al., 2014).
• Shortcuts to adiabaticity (STA): Counterdiabatic driving and inverse-engineered pulse sequences can accelerate CTAP protocols by orders of magnitude. These STA-based CTAP protocols achieve high fidelity with minimal intermediate-occupancy and exhibit robustness to larger dephasing rates compared to conventional adiabatic approaches (Ban et al., 2018).
• Multidimensional and topological scenarios: CTAP is extended to 2D SSH-type and Lieb-type lattices, where dark states are localized at topological corners or interface modes. These protocols enable high-fidelity, robust pumping of topologically protected states, outperforming Thouless-type adiabatic cycles both in speed and in avoidance of bulk excitations (Peng et al., 27 Nov 2025, Longhi, 2014).
• Graph-theoretic generalizations: The protocol generalizes to amplitude transfer on a wide class of semi-bipartite graphs, with successful transport guaranteed by a perfect matching combinatorial criterion. These results bridge quantum control, graph spectral theory, and potential applications in quantum networking (Groenland et al., 2019).

CTAP is now established as a universal quantum control primitive—platform-agnostic, robust to imperfections, and extensible to complex many-body and topological systems—underpinned by clear adiabatic eigenstructure and controllable dynamics. Ongoing research addresses scaling, integration, further noise resilience, and applications in quantum information processing, photonic routing, and topological quantum state manipulation.