Dynamics of Gaussian Wigner functions derived from a time-dependent variational principle

Abstract

By using a time-dependent variational principle formulated for Wigner phase-space functions, we obtain the optimal time-evolution for two classes of Gaussian Wigner functions, namely those of either thawed real-valued or frozen but complex Gaussians. It is shown that tunneling effects are approximately included in both schemes.

Quantum chaos, thermalization, and entanglement generation in real-time simulations of the Banks-Fischler-Shenker-Susskind matrix model

“…X, P | e iǫ P (0) Xa i (t) e −iǫ P (0) |X ′ , P ′ , (44) and similarly for the second commutator in (43). Calculating the expression for the matrix elements of X between the two Gaussian states and inserting it into (44) and (43), one can show that within the Gaussian state approximation the integral over X ′ and P ′ is saturated by the saddle point at X ′ = X (0), P ′ = P (0), at which the integrand is just the square of the expectation value of the single commutator X, P | Xa i (t) , P b j (0) |X, P .…”

“…X, P | e iǫ P (0) Xa i (t) e −iǫ P (0) |X ′ , P ′ , (44) and similarly for the second commutator in (43). Calculating the expression for the matrix elements of X between the two Gaussian states and inserting it into (44) and (43), one can show that within the Gaussian state approximation the integral over X ′ and P ′ is saturated by the saddle point at X ′ = X (0), P ′ = P (0), at which the integrand is just the square of the expectation value of the single commutator X, P | Xa i (t) , P b j (0) |X, P . Hence we can read off the quantum corrections to Lyapunov exponents from the X-and P -averaged norm of the Lyapunov distance vector δX a i (45) between the X a i coordinates for the two solutions of equations ( 9) and ( 13) with initial conditions which differ by an infinitely small coordinate shift ǫ:…”

“…In order to simulate the real-time dynamics of the BFSS model, we approximate the density matrix of the system by the most general Gaussian function with timedependent parameters which obey self-consistent equations of motion. Such an approach, which we will refer to as the Gaussian state approximation, is closely related to the semiclassical approximation [43,44], and is extensively used in the context of quantum chemistry [43,45]. For interacting many-body systems which admit a second-quantized QFT description, such as e.g.…”

Classical Wigner model based on a Feynman path integral open polymer

“…This principle of least action is equivalent to equations 3.6 and 3.8. In paper II 93 two new types of basis functions, φ, are suggested for building up the parametrized Wigner functions. As in the paper, the one-dimensional case will be shown here.…”

“…In 2011 Poulsen published a work where the Dirac-Frenkel variational principle, see section 2.4, was applied to Wigner functions. 92 In paper II 93 two new forms of basis functions to construct Wigner functions, for use with this variational principle are described and tested on a few model systems. The purpose is to evaluate if this could be a useful method for approximate quantum dynamics.…”