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By Dumbrille J

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Critical Phenomena in Loop Models

While with reference to a continual section transition, many actual platforms can usefully be mapped to ensembles of fluctuating loops, which would symbolize for instance polymer jewelry, or line defects in a lattice magnet, or worldlines of quantum debris. 'Loop versions' offer a unifying geometric language for difficulties of this sort.

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The weak-coupling limit |U0 α| thus implies γabs,sca |α| κn .

73) 0 Inserting this into Eq. 74) 0 which depends on the whole past trajectory of Pρ. The integral represents the small correction to the particle state evolution in the absence of field fluctuations. A standard way of converting the above expression to a time-local differential equation is to expand it as Pρ = Pρ0 + Pρ1 + . . using the following iteration process: First one formally solves the time-local equation in the absence of the integral term to obtain the zeroth order solution Pρ0 (t) = exp [L P (t − t0 )] Pρ0 (t0 ).

The corresponding Lindblad term is then of random unitary type, Labs (ρ) = Γabs exp (ikz) ρ exp (−ikz) − ρ , and the quantum state of motion transforms as z|ρ|z → exp −n 0 1 − e ik (z−z ) z|ρ|z = e −n 0 ∞ n=0 ∞ ≡ z| n n0 z|einkz ρe−inkz |z n! pn Un ρU†n |z . 53) n=0 This expression is a probabilistic sum of unitary momentum kick transformations Un = exp (inkz), which also follows from a classical Poissonian ansatz. 2 Quantum Mechanics of Polarizable Point Particles in High-Finesse Cavities The preceding study of polarizable point particles interacting with coherent light fields has lead to a first assessment of cavity-induced slowing in Sect.

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