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arXiv:quant-ph/0006021 · Journal of Physics A Mathematical and General

Coherent-state path-integral calculation of the Wigner function

John H. Samson

Published Jul 17, 2000• 3 citations
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Abstract

We consider a set of operators = ( 1 ,..., N ) with diagonal representatives P ( n ) in the space of generalized coherent states | n ⟩: = ∫dµ( n ) P ( n )| n ⟩⟨ n |. We regularize the coherent-state path integral as a limit of a sequence of averages ⟨ ⟩ L over polygonal paths with L vertices n 1,..., L . The distribution of the path centroid = (1/ L )∑ l = 1 L P ( n l ) tends to the Wigner function W ( x ), the joint distribution for the operators: W ( x ) = lim L →∞ ⟨δ N ( x - )⟩ L . This result is proved in the case where the Hamiltonian commutes with . The Wigner function is non-positive if the dominant paths with path centroid in a certain region have Berry phases close to odd multiples of π. For finite L the path centroid distribution is a Wigner function convolved with a Gaussian of variance inversely proportional to L . The results are illustrated by numerical calculations of the spin Wigner function from SU (2) coherent states. The relevance to the quantum Monte Carlo sign problem is also discussed.

DOI
10.1088/0305-4470/33/29/306

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We consider a set of operators = ( 1 ,..., N ) with diagonal representatives P ( n ) in the space of generalized coherent states | n ⟩: = ∫dµ( n ) P ( n )| n ⟩⟨ n |. We regularize the coherent-state path integral as a limit of a sequence of averages ⟨ ⟩ L over polygonal paths with L vertices n 1,..., L . The distribution of the path centroid = (1/ L )∑ l = 1 L P ( n l ) tends to the Wigner function W ( x ), the joint distribution for the operators: W ( x ) = lim L →∞ ⟨δ N ( x - )⟩ L . This result is proved in the case where the Hamiltonian commutes with . The Wigner function is non-positive if the dominant paths with path centroid in a certain region have Berry phases close to odd multiples of π. For finite L the path centroid distribution is a Wigner function convolved with a Gaussian of variance inversely proportional to L . The results are illustrated by numerical calculations of the spin Wigner function from SU (2) coherent states. The relevance to the quantum Monte Carlo sign problem is also discussed.
The Wigner function W ( q , p ) is formulated as a phase-space path integral, whereby its sign oscillations can be seen to follow from interference between the geometrical phases of the paths. The approach has similarities to the path-centroid method in the configuration-space path integral. Paths can be classified by the midpoint of their ends; short paths where the midpoint is close to ( q , p ) and which lie in regions of low energy (low P function of the Hamiltonian) will dominate, and the enclosed area will determine the sign of the Wigner function. As a demonstration, the method is applied to a sequence of density matrices interpolating between a Poissonian number distribution and a number state, each member of which can be represented exactly by a discretized path integral with a finite number of vertices. Saddle-point evaluation of these integrals recovers (up to a constant factor) the WKB approximation to the Wigner function of a number state.
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics ("tails') of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.
The Wigner quasiprobability, along with some of its essentialproperties, is introduced and discussed in two versions, first covering real canonical variables such as W(q,p) and second a pair of complex conjugate coordinates such as W(α,α*). The reconstruction of the density operator ϱ of states is also given. Building upon the Susskind–Glogower concept of quantum phase operators, further aspects of phase operator algebras in the quantum optics of a harmonic oscillator are discussed in relation to the realization of the su(1,1) Lie algebra. Coherent phase states |ε⟩ are introduced in analogy to the common coherent states |α⟩ in two ways, as both eigenstates of certain operators and as states generated from a ground state |0⟩ by operators of the Lie group SU(1,1). The limiting transition to the non-normalizable Fritz London phase states |eiφ⟩ on the unit circle and an (over)-completeness relation for the coherent phase states are derived. The Wigner quasiprobability W(q,p) for the coherent phase states is calculated and graphically represented. From the Wigner quasiprobability, a phase distribution W(φ) is calculated by integrating over the radius, and its uncertainty is defined and presented. The Hilbert–Schmidt distance is discussed as a measure of the non-classicality of states, where most of our with Viktor Dodonov work was carried out.
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.
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.
Quantum interference and exchange statistical effects can affect the momentum distribution functions making them non-Maxwellian.Such effects may be important in studies of kinetic properties of matter at low temperatures and under extreme conditions.In this work we have generalized the path integral representation for Wigner function to strongly coupled three-dimensional quantum system of particles with Boltzmann and Fermi statistics.In suggested approach the explicit expression for Wigner function was obtained in harmonic approximation and Monte Carlo method allowing numerical calculation of Wigner function, distribution functions and average quantum values has been developed.As alternative more accurate single-momentum approach and related Monte Carlo method have been developed to calculation of the distribution functions of degenerate system of interacting fermions.It allows partially overcoming the well-known sign problem for degenerate Fermi systems.