Quantum macroscopicity measure for arbitrary spin systems and its application to quantum phase transitions

We explore a previously unknown connection between two important problems in physics, i.e., quantum macroscopicity and the quantum phase transition. We devise a general and computable measure of quantum macroscopicity that can be applied to arbitrary spin states. We find that a macroscopic quantum superposition of an extremely large size arises during the quantum phase transition of the transverse Ising model in contrast to some seeming macroscopic quantum phenomena such as superconductivity, superfluidity, and Bose-Einstein condensates. Our result may be an important step forward in understanding macroscopic quantum properties of many-body systems.

Figure 1

Wigner distributions of $(|S,S\rangle +|S,-S\rangle )/\sqrt{2}$ for (a) $S=2$ and (b) $S=5$. Each distribution consists of two peaks along the $\pm z$ direction and interference fringes around the $z$ axis. When the pure superposition with $S=5$ becomes a mixed state, $\{\left|5,5\right.〉\left.〈5,5\right|+\left|5,-5\right.〉\left.〈5,-5\right|+\gamma (\left|5,5\right.〉\left.〈5,-5\right|+\left|5,-5\right.〉\left.〈5,5\right|)\}/2$, the interference fringes are obviously reduced for (c) $\gamma =1/2$ and they completely disappear for (d) $\gamma =0$.

Figure 2

Dynamics of $\mathcal{I}\left(\rho \right)$ and $\mathcal{F}\left(\rho \right)$ under the dissipation given by master equation Eq. (11) for the initial GHZ state with $N=50$.

Figure 3

Quantum macroscopicity in terms of $\mathcal{I}\left({\rho }_L\right)$ (left) and $\mathcal{F}\left({\rho }_L\right)$ (right) for a partial block of $L$ contiguous particles in the ground state vs the interaction strength $\lambda$ of the transverse Ising model. The QPT occurs at the critical point $\lambda =1$. Insets: Values of each measure as a function of $L$ at the critical point of $\lambda =1$.

Figure 4

Calculation times (in seconds) for obtaining $\mathcal{F}$ and $\mathcal{I}$ for 100 random mixed states. The dashed curves show the elapsed times only for constructing the matrix $V$ or $W$, whereas the solid curves show the results including the elapsed time to get an optimized value. The upper two curves (results for $\mathcal{F}$) are steeper than the lower curves (results for $\mathcal{I}$).