Exact mapping of the quantum states in arbitraryN-level systems to the positions of classical coupled oscillators

Abstract

The quantum dynamics of arbitrary $N$-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary $N$-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension $is$ general for any $N$. The time evolution of an $N$-level system [generated by a complex element of the SU($N$) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in $n$ Euclidean dimensions [the rotation group SO($n$)] is then mapped directly to $n$ oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent $N$-level systems scales as ${N}^{2}$ for the density matrix formalism but increases only linearly with $N$ for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.