Non-Abelian Geometric Quantum Energy Pump

• Non-Abelian Geometric Quantum Energy Pump is a protocol that leverages degenerate adiabatic states and non-commutative gauge structures to enable path-dependent, quantized energy transfer.
• It generalizes the Thouless pump from Abelian single-band systems to multiband degenerate regimes, utilizing the Berry curvature tensor and Wilczek–Zee holonomy.
• Experimental platforms such as photonic lattices, superconducting circuits, and cold atoms demonstrate its applications in quantum transduction, battery charging, and holonomic quantum processing.

A Non-Abelian Geometric Quantum Energy Pump is a quantum device or protocol leveraging the non-Abelian geometric aspects of adiabatically driven degenerate systems to transfer energy between distinct external drives or subsystems, where the resulting pumped energy is determined by non-Abelian gauge structures—specifically, the Berry curvature tensor and the associated Wilczek–Zee holonomy—along a trajectory in parameter space. This construction generalizes the topological charge pumping mechanism of the Thouless pump from the Abelian (single-band) regime to systems exhibiting degeneracies, leading to richer, path-dependent physics. The non-commutative geometry of parameter space manifests in observable energy transfer, robust quantization, and nontrivial dependence on the sequence and structure of control cycles.

1. Mathematical Framework for Non-Abelian Geometric Pumps

At the foundation of the non-Abelian geometric quantum energy pump is a Hamiltonian $H_0(\boldsymbol{\lambda})$ parameterized by controls $\boldsymbol{\lambda}=(\lambda^1,\ldots,\lambda^d)$ on a smooth manifold $M$. When $H_0(\boldsymbol{\lambda})$ possesses a degenerate eigenspace of energy $\epsilon_n(\boldsymbol{\lambda})$ (with orthonormal basis $$\{|n,\alpha(\boldsymbol{\lambda})\rangle\}_{\alpha=1}^g$$ and projector $\Pi^n_{\boldsymbol{\lambda}}$), the adiabatic evolution under a cyclic trajectory $\boldsymbol{\lambda}(t)$ generally leads to non-adiabatic transitions.

To enforce perfect adiabatic transport within the degenerate subspace, a counterdiabatic (CD) "Kato gauge" term $\mathcal{A}_t(\boldsymbol{\lambda}(t))$ is added: $$H(t)=H_0(\boldsymbol{\lambda}(t)) + \mathcal{A}_t(\boldsymbol{\lambda}(t))$$ with

$$\mathcal{A}_t = \sum_\mu \dot{\lambda}^\mu\,\mathcal{A}_\mu(\boldsymbol{\lambda}),\qquad \mathcal{A}_\mu = \frac{i}{2}\sum_n[\partial_\mu\Pi^n_{\boldsymbol{\lambda}},\,\Pi^n_{\boldsymbol{\lambda}}]$$

This ensures "transitionless" quantum driving: eigenstates initialized in the degenerate manifold remain there for all time.

The structure of the degenerate subspace is governed by the non-Abelian Wilczek–Zee connection,

$$[A_\mu(\boldsymbol{\lambda})]_{\alpha\beta} = i\langle n,\alpha(\boldsymbol{\lambda})|\partial_\mu n,\beta(\boldsymbol{\lambda})\rangle$$

with curvature (field strength)

$$[F_{\mu\nu}(\boldsymbol{\lambda})]_{\alpha\beta}= \partial_\mu A_\nu - \partial_\nu A_\mu - i[A_\mu, A_\nu]_{\alpha\beta}$$

The total evolution operator upon traversing a closed loop $\gamma$ in parameter space is the non-Abelian Wilson loop (holonomy),

$$U_\gamma = \mathcal{P}\exp\left(-\int_0^T A_{\mu} \dot{\lambda}^\mu dt\right) \in U(g)$$

Physically, $U_\gamma$ acts within the $g$-dimensional degenerate manifold, implementing a geometric unitary gate dictated entirely by the path $\gamma$.

2. Pumped Energy: Geometric Origin and Quantization

For a system evolving in the presence of independently controllable drives corresponding to each $\lambda^\mu$, the geometric component of the energy pumped into the $\mu$-th drive during a cyclic protocol is determined by the non-Abelian curvature: $$E_\mu = \int_0^T dt\,\dot{\lambda}^\mu(t)\langle\psi(t)|\partial_\mu \mathcal{A}_t|\psi(t)\rangle = \sum_\nu \int_0^T dt\,\dot{\lambda}^\nu(t)\dot{\lambda}^\mu(t)\langle\psi(t)|\mathcal{F}^n_{\nu\mu}(\boldsymbol{\lambda}(t))|\psi(t)\rangle$$ where $$\mathcal{F}^n_{\nu\mu} = i[\partial_\nu\Pi^n,\partial_\mu\Pi^n]$$. For a closed path in a two-dimensional subspace, this reduces to a (generalized) Stokes theorem involving the surface integral of the Berry curvature components: $$\Delta E_\mu = \oint d\lambda^\nu \wedge d\lambda^\mu \langle\psi|F_{\nu\mu}|\psi\rangle$$ This geometric contribution is independent of dynamical details or the protocol's speed, provided adiabatic conditions prevail.

In specific lattice models, such as ladder extensions of the Rice–Mele model with doubly degenerate bands, the pumped charge or energy per cycle is quantized and topologically protected, corresponding to the (non-Abelian) first Chern number. For a band with degeneracy $d=2$, the net shift or pumped energy is

$$\Delta E = \frac{1}{2\pi}\int_0^T dt\int_{-\pi}^\pi dk\,\operatorname{Tr}[E(k;\lambda(t))F_{k t}(k;\lambda(t))]$$

where $F_{kt}$ is the non-Abelian Berry curvature in $(k,t)$. The quantization follows from the integral of $\operatorname{Tr}[F_{k t}]$ over the $(k,\lambda)$ torus.

3. Non-Commutativity and Holonomy Structure

The central hallmark of non-Abelian pumping is the non-commutativity of geometric evolution: the outcome of sequential parameter cycles depends fundamentally on their order. For cyclic protocols $C$ and $C'$, with associated holonomies $U_C$ and $U_{C'}$,

$[U_C, U_{C'}] \neq 0$

This non-commutativity results in distinct physical shifts or state populations depending on the sequence of cycles, as observed in $Z_3$-type cycles in three-band lattice models, where the cycles effect permutation matrices in the degenerate subspace. The net effect can be a quantized shift within or across unit cells, with physical observables (population distributions, lattice site shifts, or energy transfer) encoding the non-Abelian group structure.

In quantum energy pumps implemented with $g$-dimensional degenerate manifolds, concatenation of non-commuting holonomies $C_1\circ C_2$ and $C_2\circ C_1$ leads to measurable differences in the final state—a direct signature of non-Abelian gauge geometry.

4. Physical Realizations and Implementation Platforms

Multiple experimental platforms realize the non-Abelian geometric quantum energy pump paradigm:

• Photonic and Acoustic Lattices: Experimental work has realized non-Abelian Thouless pumps in engineered acoustic waveguide arrays, where the propagation of eigenstate amplitudes under cyclic modulation is governed by a $3\times3$ (or higher) Bloch Hamiltonian. The corresponding couplings are engineered to create degeneracies and desired gauge structures. In photonic Lieb lattices with flat, degenerate bands, non-Abelian pumping is observable as quantized displacement or energy shifts of photonic wavepackets (You et al., 2021, Brosco et al., 2020).
• Superconducting Circuits: The three-SQUID Cooper-pair pump allows geometric manipulation of ground-state degeneracies, enabling quantized charge (and hence energy) pumping using adiabatic parameter cycles. The topologically protected swapped holonomy in the degenerate subspace is directly linked to the pumped charge per cycle (Pirkkalainen et al., 2010).
• Artificial Atoms and Cold Atoms: Trapped ions, superconducting qubits, or quantum dots configured in a tripod scheme (three ground states coupled to a common excited state) implement the non-Abelian geometric pump through modulation of Rabi frequencies. In such systems, modulation of microwave or optical fields traverses desired paths in parameter space, with energy transfer measured through external drives (Peng, 12 Nov 2025).
• Ultracold Atomic Ladders: Rice–Mele ladders engineered with ultracold atoms in optical lattices support twofold degenerate bands, allowing geometric $U(2)$ gate operations via non-Abelian adiabatic pumping cycles. Readout is achieved via center-of-mass measurements or population tomography (Danieli et al., 30 Sep 2024).

Core experimental signatures include quantized displacement, robust pumped energy per cycle, direct measurement of non-Abelian holonomies by projective state tomography, and path-order–dependent final outcomes in sequential pumping protocols.

5. Control of Pumping Power and Topological Classes

In practical realizations, control over the initial state within the degenerate manifold, as well as the Hamiltonian topology, enables continuous tuning of the pumping power and selection of robust topological classes. The averaged geometric pumping power, in generic cases where the initial phase is uncontrolled, can be expressed as

$$\overline{P}_{\nu\mu}(t) = \dot{f}^\nu(t)\dot{f}^\mu(t)\,c\,\sqrt{1-c^2}\,\sin(\phi_1 - \phi_2)\;\frac{\chi_{\nu\mu}}{\pi}$$

Here $\chi_{\nu\mu}$ is the Euler class of the underlying vector bundle, and the phase difference $\phi_1-\phi_2$ encodes initial-state coherence. By varying $(c,\phi_1-\phi_2)$ or switching the Hamiltonian topology (e.g., by changing paths that wind or do not wind around the Yang monopole in a Rice–Mele ladder), one can modulate the geometric pumping power and dynamically access distinct topological regimes (Peng, 12 Nov 2025, Danieli et al., 30 Sep 2024).

6. Applications and Functional Advantages

Non-Abelian geometric quantum energy pumps provide a suite of functionalities beyond their Abelian counterparts:

• Quantum Transduction: The ability to coherently transfer energy between independent drives or to convert between frequencies, utilizing the path- and phase-dependent nature of the pump. Quantum-driven frequency conversion in hybrid networks is a direct application (Peng, 12 Nov 2025).
• Quantum Battery Charging: Persistent, quantized energy transfer into a designated mode is possible without departing from the protected degenerate subspace, eliminating the need for dissipative resets.
• Holonomic Quantum Gates: Each pump cycle implements a geometric unitary transformation (a $U(d)$ gate) in the degenerate subspace, serving as a building block for holonomic quantum processing or fault-tolerant logic in optical or atomic platforms (Danieli et al., 30 Sep 2024).
• Metrological Tools: Dependence on phase coherence in the initial state enables interferometric protocols to measure phase differences and coherences in quantum states directly.
• Topological Metrology: The quantization of pumped energy, anchored in non-Abelian Chern or Euler classes, provides robust standards for quantum metrology, with insensitivity to moderate disorder or imperfections.

7. Distinctions from Abelian Pumps and Outlook

A fundamental distinction of non-Abelian geometric pumps is their non-commutative, multiband, and often degenerate-state topology. While the typical Thouless pump effects an integer-quantized transport governed by an Abelian Chern number of a nondegenerate band, non-Abelian pumps act within degenerate manifolds, where the pumping is encoded in the full holonomy (Wilson loop) and related topological invariants beyond the first Chern number (e.g., Euler or higher Chern classes, Yang monopole charge).

This structure allows for concatenation and selective sequencing of geometric gates, path-dependent operations not present in Abelian systems, and a hierarchy of robust quantized observables. The concept generalizes across fermionic, bosonic, photonic, and hybrid platforms and underlies experimentation at the intersection of topology, holonomic control, and energy manipulation at the quantum scale (Peng, 12 Nov 2025, Danieli et al., 30 Sep 2024, You et al., 2021).