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Spectral Gap Protection of Qubit Coherence via a Fourth-Order Scalar Bath

Francis Procaccia

Published Jun 4, 2026• 0 citations
Publisher

Abstract

We study a qubit coupled to a bosonic bath described by a fourth-order scalar field Lagrangian, whose dispersion relation omega^2 = c^2(k^4 + alpha*k^2 + m^2) supports a tunable spectral gap. Below the gap, the local density of states vanishes exactly, suppressing both energy relaxation (T1) and pure dephasing (T2) at leading order. We derive the canonical quantization of the theory from first principles, establish an exact closed-form expression for the local density of states, and map out a phase diagram in (alpha, m^2) parameter space with a sharp gap-closing transition at m^2 = alpha^2/4. At the critical line, the momentum coupling (partial_t phi) uniquely yields an Ohmic bath, placing the system at the spin-boson quantum phase transition boundary, while direct field and gradient couplings produce infrared-divergent spectral functions due to the universal 1/sqrt(omega_k) mode normalization of soft critical modes. The framework provides a concrete design principle for spectral-gap-based qubit protection and connects fourth-order field theory to the spin-boson literature.

DOI
10.5281/zenodo.20548104

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We study a qubit coupled to a bosonic bath described by a fourth-order scalar field Lagrangian, whose dispersion relation omega^2 = c^2(k^4 + alpha*k^2 + m^2) supports a tunable spectral gap. Below the gap, the local density of states vanishes exactly, suppressing both energy relaxation (T1) and pure dephasing (T2) at leading order. We derive the canonical quantization of the theory from first principles, establish an exact closed-form expression for the local density of states, and map out a phase diagram in (alpha, m^2) parameter space with a sharp gap-closing transition at m^2 = alpha^2/4. At the critical line, the momentum coupling (partial_t phi) uniquely yields an Ohmic bath, placing the system at the spin-boson quantum phase transition boundary, while direct field and gradient couplings produce infrared-divergent spectral functions due to the universal 1/sqrt(omega_k) mode normalization of soft critical modes. The framework provides a concrete design principle for spectral-gap-based qubit protection and connects fourth-order field theory to the spin-boson literature.
We investigate the reduced dynamics of a qubit subject to correlated longitudinal and transverse noise arising from its coupling to a shared bosonic bath. The environmental fluctuations are characterized by a positive-semidefinite matrix-valued spectral density, whose complex off-diagonal elements encode correlations between dephasing and relaxation channels in the frequency domain. Within the second-order time-convolutionless framework, we derive closed time-local equations for the Bloch-vector components of the reduced density matrix. The numerical implementation is validated against the exact pure-dephasing solution and the established behavior of the transverse-coupling spin-boson model. When both noise channels are present, the cross-spectral terms couple the otherwise distinct dephasing and relaxation sectors, producing dynamics that cannot be reproduced by adding independent noise contributions. In particular, the correlations generate non-monotonic population relaxation and a transient revival of coherence following its initial decay. The strength, bandwidth, delay, and phase of the cross spectrum provide control parameters for the magnitude and temporal structure of these effects. Our results demonstrate that correlated multi-axis noise can redistribute coherence loss and energy relaxation in time, thereby providing finite temporal windows of enhanced coherence or suppressed relaxation within the weak-coupling regime.
We propose a general extended coherent state approach to the qubit (or fermion) and multimode boson coupling systems. The application to the spin-boson model with the discretization of a bosonic bath with arbitrary continuous spectral density is described in detail, and very accurate solutions can be obtained. The quantum phase transition in the nontrivial sub-Ohmic case can be located by the fidelity and the order-parameter critical exponents for the bath exponents s<1/2s<1/2s<1/2 can be correctly given by the fidelity susceptibility, demonstrating the strength of the approach.
We establish a unified theoretical framework for the conservation of quantum information in gapped dephasing baths. Building on the exact tripartite complementarity relation V^2 + C^2 = 1 and the non-circular crossover criterion \Gamma = \gamma_{pre} t_{ML} = 1, we classify the dynamics into distinct universality classes determined by the infrared (IR) structure of the bath. We prove the "Hard Gap Theorem": an abrupt hard gap (s=0) strictly preserves local coherence at zero temperature, while any soft gap (s>0) generates an IR divergence that forces complete local decoherence, even at infinitesimal coupling. Furthermore, we derive the exact finite-temperature phase diagram. We show that the critical coupling scales as J_0^{crit} \propto \Delta^2 / (k_B T) at high temperatures, a result that corrects previous conjectures and unifies the thermodynamic behavior across all gap geometries. This theorem is shown to be mathematically equivalent to the generation of pure topological constants in other gapped sectors, including the Casimir effect (c_0 = 0.8343), massive Unruh radiation (e^{-2\Delta/a}), and the Bekenstein entropy bound (C_d k_B). This preprint serves as the unifying Pillar 6 of a five-part series on gapped quantum systems.
This manuscript (Pillar 13 of the unified gapped-bath framework) presents an exact derivation of the quantum thermodynamic properties of a central spin undergoing pure dephasing in a hard-gap bosonic bath. By strictly avoiding the Lindblad master equation and any phenomenological approximations, we prove that the spectral gap acts as a thermodynamic resource. The gap protects a finite residual ergotropy at long times, E∞=(ℏω0/2)exp⁡(−J0/Δ)E∞=(ℏω0/2)exp(−J0/Δ), and caps entropy production strictly below the classical maximum. At the exact critical coupling J0=4Δ/π2J0=4Δ/π2, the residual ergotropy ratio is the pure mathematical number e−4/π2≈0.666787e−4/π2≈0.666787. We also demonstrate that pure dephasing induces zero heat exchange (Q=0Q=0), with the entropy saturating at exactly 0.9000460.900046 bits for J0=ΔJ0=Δ. The paper includes rigorous proofs, five falsifiable predictions, a comparison with standard frameworks (Lindblad, Caldeira-Leggett), and a fully reproducible Python validation script.
The spin-boson model is a paradigm for studying decoherence, relaxation, entanglement and other effects that arise in a quantum system coupled to environmental degrees of freedom. At zero temperature, a localization-delocalization phase transition is known to exist in the sub-Ohmic regime, where the standard density matrix renormalization group algorithm is inadequate due to the divergence in the number of low-frequency modes. This limitation is circumvented in this work by symmetrically optimizing the phonon basis and introducing an order parameter accounting for the U(1) symmetry for a two-bath spin-boson model, by which we are able to determine the classification and criticality of the phase transition explicitly. Compared with variational results, the critical phase is characterized by spontaneous vanishing of boson displacements in both the baths, resulting in an accurate phase diagram with three model parameters.
We study local phase estimation for a single quantum emitter coupled to a bosonic Su-Schrieffer-Heeger (SSH) bath within a microscopic lattice model. Dimerization opens a central gap supporting an in-gap emitter-bath bound state, which suppresses complete relaxation of the emitter coherence. A Dyson-equation analysis yields the local bath Green's function, the in-gap bound-state condition, and the emitter residue controlling the retained phase information. The phase quantum Fisher information links gap formation, detuning, and emitter-bath coupling to the post-transient metrological response. At resonance, stronger coupling enhances transient hybridization but reduces the retained signal by lowering the emitter weight in the bound state. Away from resonance, late-time averages, retention times, and useful interrogation windows track how phase-information protection weakens as the emitter is tuned toward and beyond the band edge. A uniform-chain control shows that the retained signal disappears when the gap closes. In the bulk local-coupling geometry considered here, the response is insensitive to the sign of the dimerization, so the protocol probes spectral-gap physics and bound-state support rather than the SSH winding sector.