Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time

TL;DR

This paper proves that semiclassical approximations of quantum flows remain accurate over Ehrenfest timescales, with errors controlled by powers of Planck's constant and logarithmic factors, for holomorphic quadratic Hamiltonians.

Contribution

It provides a rigorous proof of the validity duration of semiclassical approximations for quantum flows generated by holomorphic quadratic Hamiltonians.

Findings

Error bounds grow polynomially with N and logarithmically with 1/ħ.

Semiclassical approximation remains valid up to Ehrenfest timescales.

Error can be controlled by choosing N(hbar) appropriately.

Abstract

Let ${\cal H}(x,\xi)$ be a holomorphic Hamiltonian of quadratic growth on $ R^{2n}$, $b$ a holomorphic exponentially localized observable, $H$, $B$ the corresponding operators on $L^2(R^n)$ generated by Weyl quantization, and $U(t)=\exp{iHt/\hbar}$. It is proved that the $L^2$ norm of the difference between the Heisenberg observable $B_t=U(t)BU(-t)$ and its semiclassical approximation of order ${N-1}$ is majorized by $K N^{(6n+1)N}(-\hbar ln\hbar)^N$ for $t\in [0,T_N(\hbar)]$ where $T_N(\hbar)=-{2 ln\hbar\over {N-1}}$. Choosing a suitable $N(\hbar)$ the error is majorized by $C\hbar^{ln|ln\hbar|}$, $0\leq t\leq |ln\hbar|/ln|ln\hbar|$. (Here $K,C$ are constants independent of $N,\hbar$).