The geometric phase for a cyclic change of parameters

Berry's phase is the phase γₙ = i∮⟨n|∇_R n⟩⋅dR, over and above the dynamic phase −(1/ħ)∫Eₙ dt, that an eigenstate acquires when the parameters R of its Hamiltonian are carried slowly once round a closed loop; it depends on the loop, not on how long the trip takes. Use the supplied topic model Y_out = deg(((sign(rho-c)+1)/2*(rad(phi)+atan2(-c*sind(phi),rho+c*cosd(phi)))+(1-sign(rho-c))/2*atan2(rho*sind(phi),c+rho*cosd(phi))))+1e-9, where Y_out is MIXING ANGLE θ (°). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees; deg converts radians to degrees; rad converts degrees to radians; atan2(y,x) is the angle of the vector (x,y), in radians. rho is Loop radius ρ, measured in meV. Keep c (Loop centre, X = c) = 0.5 meV; phi (Progress round the loop φ) = 300 ° fixed. For this exercise use only 0.909375 meV ≤ rho ≤ 2.09063 meV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at rho = 1.16925 meV and 1.33462 meV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δrho across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.

Berry's phase is the phase γₙ = i∮⟨n|∇_R n⟩⋅dR, over and above the dynamic phase −(1/ħ)∫Eₙ dt, that an eigenstate acquires when the parameters R of its Hamiltonian are carried slowly once round a closed loop; it depends on the loop, not on how long the trip takes. Use the supplied topic model Y_out = deg(((sign(rho-c)+1)/2*(rad(phi)+atan2(-c*sind(phi),rho+c*cosd(phi)))+(1-sign(rho-c))/2*atan2(rho*sind(phi),c+rho*cosd(phi))))+1e-9, where Y_out is MIXING ANGLE θ (°). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees; deg converts radians to degrees; rad converts degrees to radians; atan2(y,x) is the angle of the vector (x,y), in radians. rho is Loop radius ρ, measured in meV. Keep c (Loop centre, X = c) = 0.5 meV; phi (Progress round the loop φ) = 300 ° fixed. For this exercise use only 0.909375 meV ≤ rho ≤ 2.09063 meV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 312.97091 °, with a bounded reading uncertainty ±0.37346 °. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown rho by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000118125 meV, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.