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arXiv:quant-ph/0503213 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Coherent state path integral for linear systems

Charles G Torre

Published Jul 12, 2005• 4 citations
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Abstract

We present a computation of the coherent state path integral for a generic linear system using ``functional methods'' (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary---but not sufficient---condition for existence of the path integral is that the symplectic transformation generated by the Hamiltonian is (unitarily) implementable on the Fock space for the system.

DOI
10.1103/physrevd.72.025004

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We present a computation of the coherent state path integral for a generic linear system using ``functional methods'' (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary---but not sufficient---condition for existence of the path integral is that the symplectic transformation generated by the Hamiltonian is (unitarily) implementable on the Fock space for the system.
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We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.
In this article we present two phase space path integrals in terms of coherent states. The first one is derived in a standard fashion but using a nonstandard resolution of the identity in terms of coherent states with different width parameters. The second path integral emerges from a novel phase space representation in terms of coherent states distributed on n-dimensional manifolds embedded in the 2n-dimensional phase space of an n-degree-of-freedom system. These states are shown to form locally complete basis sets since we show that fairly smooth and localized functions can be expanded in terms of them in a unique way. In this representation the time evolution operator can be cast in the form of a phase space path integral. Both path integrals can be evaluated by straightforward implementation of Monte Carlo methods. In both cases the probability amplitude between two phase points turns out to be proportional to the average of the phase, eih∫(p dq−H dt), over a Markov process of phase space paths consisting of classical trajectories interrupted at intervals by Gaussian ‘‘quantum jumps.’’ The numerical evaluation of these expressions through importance sampling is demonstrated.