A probe qubit locks only to an ordered bath

A qubit coupled to a quantum many-body bath through σᶻ_p Z acquires a coherence 𝓛(t) = Tr[ρ e^{iH₊t} e^{−iH₋t}], H± = H_B ± gZ, whose phase can lock to a fixed frequency set by the bath. We ask which states of a quantum bath can lock the probe, and what happens at finite temperature.

(1) Decomposition theorem: for any bath with a symmetry that reverses Z, the real part of the coherence depends only on the symmetric part of the bath state and the imaginary part only on its odd part. Symmetric states therefore give a real coherence and cannot lock, so a locked probe certifies order. ⟨σʸ_p⟩ ≠ 0 is a model-independent witness of symmetry breaking.

(2) Exact solution at all temperatures for a probe on the edge of the transverse-field Ising chain. Heating leaves Re 𝓛 unchanged and multiplies Im 𝓛 by the edge magnetization s_N(T) = ∏_k tanh(ε_k/2T), which we prove rigorously. Zero-temperature locking is inherited at all temperatures with the same frequency. In the infinite chain the coherence at any T > 0 equals that at infinite temperature.

(3) A mean-field (Lipkin–Meshkov–Glick) bath with a thermal transition. The probe stays locked below T_c, and its locked frequency is an order parameter that vanishes at T_c. Exact calculations up to N = 800 confirm this.

Exact diagonalization of a non-integrable chain shows that breaking integrability damps only the real part; the locked frequency is unchanged.

This record contains the Letter with its Supplemental Material and all numerical scripts.