PHY.9:5.3 - Prepare a probe whose phase difference can be read

An ideal two-level probe couples to a constant classical field B through

H=(hbar gamma B/2) sigma_z,

where gamma is a known coupling coefficient and sigma_z has eigenvalues +1 and -1 for states |0> and |1>. Treat the field as unchanged by this probe during the interrogation time t. The wanted quantity is B within a supplied range.

Prepare |0>. Evolution multiplies it by a global phase; measurements of this state cannot reveal that phase. Increasing t alone does not create a readable field dependence.

Instead prepare |+>=(|0>+|1>)/sqrt(2). Evolution produces a relative phase phi=gamma B t. Reading sigma_z still gives equal probabilities and is uninformative about phi. A controlled rotation before detection permits a sigma_x or sigma_y measurement. Their means are cos(phi) and sin(phi), so the two preparations for readout can recover phase modulo 2 pi. Each mean requires the corresponding repeated preparation and measurement.

For gamma=1 rad/(s·field-unit), t=1 s and B=pi/3 field-units, the ideal means are 1/2 and sqrt(3)/2. They select phi=pi/3 modulo 2 pi. A supplied range -pi < gamma B t <= pi makes that field value unique.

If the field may instead lie between -4 pi and 4 pi field-units, the same records admit several values. Shorten the initial interrogation, for example to t=0.1 s with the same gamma: the whole supplied range then lies inside the unambiguous phase interval. A later longer interrogation can refine a value once the remaining range supports its interpretation.

Coherence loss, imperfect rotations or an uncertain gamma change the response law. They can be included or constrained where the required estimate needs them. The local phase response alone is not a claim that arbitrarily long interrogation or another quantum resource improves every sensing task.