Dynamics of ideal quantum measurement of a spin 1 with a Curie-Weiss magnet

Abstract

Quantum measurement is a dynamical process of an apparatus coupled to a test system. Ideal measurement of the $z$-component of a spin-$\frac{1}{2}$ ($s_z=\pm\frac{1}{2}$) has been modeled by the Curie-Weiss model for quantum measurement. Recently, the model was generalized to higher spin and the thermodynamics was solved. Here the dynamics is considered. To this end, the dynamics for spin-$\frac{1}{2}$ case are cast in general notation. The dynamics of the measurement of the $z$-component of a spin-1 ($s_z=0,\pm 1$) are solved in detail and evaluated numerically. Energy costs of the measurement, which are macroscopic, are evaluated. Generalization to higher spin is straightforward.

Figures (4)

• Figure 1: Left: Evolution of the magnetization distribution $P_s(m_1;t)$ for $s=+\frac{1}{2}$ at times $0,1,\cdots,8$ in units of $1/\gamma T$. The paramagnetic state at $t=0$ is peaked around $m_1=0$; the coupling between S and A moves the peak towards $m_1=+\frac{1}{2}$. In doing so, it first broadens and later narrows significantly.
• Figure 3: Evolution of the dynamical free energy $F_{\rm dyn}^s(t)$, identical in both sectors $s=\pm\frac{1}{2}$, after coupling the apparatus to a spin-$\frac{1}{2}$ at time $t=0$. Its approach to the Gibbs state with $F_s(g)$ (bottom line), exponential in $t$, expresses the registration of the measurement.
• Figure 4: Snapshots of the distribution $P_s$ of the magnetization moments $m_{1,2}$ for registration of the spin-1 measurement. Left figure: $P_s\ge 10^{-3}$ data in the $s=0$ sector at times $t=(0,1,2,3)\times 2/\gamma T$ from right to left. Right figure: the $s=1$ sector at $t=(0,1,2,3)\times 5/\gamma T$ from left to right; it evolves slower. The parameters are listed in eq. (\ref{['l=1couplings']}).
• Figure 5: The spin-1 dynamical free energy $F_{\rm dyn} ^s$ of eq. (\ref{['Fdynspins']}) relaxes from its $t=0$ value to its thermodynamic value $F_s(g)$ of (\ref{['Fdyninf2']}), thereby registering the measurement. Parameters as in figs. \ref{['figonthemove']} a,b, and time in units of $1/\gamma T$. The relaxation for $s=\pm1$ is slower than for $s=0$ due to the occurrence of zero frequencies. The initial "shoulders" describe the initial broadenings in figs. \ref{['figonthemove']} a,b.