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We consider quantum dynamics of the order parameter in the discrete pairingmodel (Richardson model) in thermodynamic equilibrium. The integrableRichardson Hamiltonian is represented as a direct sum of Hamiltonians acting indifferent Hilbert spaces of single-particle and paired/empty states. Thisallows us to factorize the full thermodynamic partition function into acombination of simple terms associated with real spins on singly-occupiedstates and the partition function of the quantum XY-model for Andersonpseudospins associated with the paired/empty states. Using coherent-statepath-integral, we calculate the effects of superconducting phase fluctuationsexactly. The contribution of superconducting amplitude fluctuations to thepartition function in the broken-symmetry phase is shown to follow from theBogoliubov-de Gennes equations in imaginary time. These equations in turn allowseveral interesting mappings, e.g., they are shown to be in a one-to-onecorrespondence with the one-dimensional Schr\"odinger equation insupersymmetric Quantum Mechanics. However, the most practically useful approachto calculate functional determinants is found to be via an analyticalcontinuation of the quantum order parameter to real time, \Delta(\tau -> it),such that the problem maps onto that of a driven two-level system. Thecontribution of a particular dynamic order parameter to the partition functionis shown to correspond to the sum of the Berry phase and dynamic phaseaccumulated by the pseudospin. We also examine a family of exact solutions fortwo-level-system dynamics on a class of elliptic functions and suggest acompact expression to estimate the functional determinants on suchtrajectories. The possibility of having quantum soliton solutions co-existingwith classical BCS mean-field is discussed.

Non-perturbative Quantum Dynamics of the Order Parameter in the Pairing Model