Quantum dynamics of a macroscopic magnet operating as an environment of a mechanical oscillator

We study the dynamics of a bipartite quantum system in a way such that its formal description keeps holding even if one of its parts becomes macroscopic; the problem is related to the analysis of the quantum-to-classical crossover, but our approach implies that the whole system stays genuinely quantum. The aim of the work is to understand (1) if, (2) to what extent, and possibly (3) how the evolution of a macroscopic environment testifies to the coupling with its microscopic quantum companion. To this purpose we consider a magnetic environment made of a large number of spin-$\frac{1}{2}$ particles, coupled with a quantum mechanical oscillator, possibly in the presence of an external magnetic field. We take the value of the total environmental spin $S$ constant and large, which allows us to consider the environment as one single macroscopic system, and further deal with the hurdles of the spin-algebra via approximations that are valid in the large-$S$ limit. We find an insightful expression for the propagator of the whole system, where we identify an effective “back-action” term, i.e., an operator acting on the magnetic environment only, and yet missing in the absence of the quantum principal system. This operator emerges as a time-dependent magnetic anisotropy whose character, whether uniaxial or planar, also depends on the detuning between the frequency of the oscillator and the level splitting in the spectrum of the free magnetic system, induced by the possible presence of the external field. The time dependence of the anisotropy is analyzed, and its effects on the dynamics of the magnet, as well as its relation to the entangling evolution of the overall system, are discussed.

Figure 1

Graphical representation of a magnetic system made of distinguishable particles, distributed on a ring-shaped lattice (referred to as a “spin ring” in the text). In panel (a) the system is isolated and its components interact with each other; in panel (b) the system is coupled with a quantum mechanical oscillator and its components are independent of each other.

Figure 2

Effective anisotropy $A_{\delta }\left(t\right)$ as a function of $t$, for $S=10$ and different values of negative $\delta$, as indicated.

Figure 3

Effective anisotropy $A_{\delta }\left(t\right)$ as a funcion of $t$, for $\delta =-0.5$ and different values of $S$, as indicated. The curve for $S=7$ fades when the validity of the results is not fully under control.

Figure 4

Effective anisotropy $A_{\delta }\left(t\right)$ as a function of $\delta$, for $t=0.1$ and different values of $S$, as indicated.

Figure 5

The back-action $g^2\left|G_{\delta }\left(t\right)\right|$ for $\delta =-0.5$ and different values of $S$; the inset shows the $S=3$ case in its proper plot-range. The curve for $S=3$ fades when the validity of the results is not fully under control.