Coherent quantum states from classical oscillator amplitudes

In the first days of quantum mechanics Dirac pointed out an analogy between the time-dependent coefficients of an expansion of the Schrödinger equation and the classical position and momentum variables solving Hamilton's equations. Here it is shown that the analogy can be made an equivalence in that, in principle, systems of classical oscillators can be constructed whose position and momenta variables form time-dependent amplitudes which are identical to the complex quantum amplitudes of the coupled wave function of an $N$-level quantum system with real coupling matrix elements. Hence classical motion can reproduce quantum coherence.

Figure 1

Time dependence of quantum or $p$-and-$q$–coupled classical (blue, solid) and $q$-coupled classical (red, dashed) motion. The left column shows details for short times and the right column is for longer times. The upper row shows the absolute value squared and the middle row shows the real part of the amplitudes. The bottom panel shows $c_1^{*}{c}_2$ and $z_1^{*}{z}_2$. The time is given in units of $K/\left(2\omega \right)$ or, equivalently, $\mathcal{V}/\hslash$.

Figure 2

Same as Fig. 1 except that the coupling strength is $K=0.1$.