Macroscopicity of Mechanical Quantum Superposition States

We propose an experimentally accessible, objective measure for the macroscopicity of superposition states in mechanical quantum systems. Based on the observable consequences of a minimal, macrorealist extension of quantum mechanics, it allows one to quantify the degree of macroscopicity achieved in different experiments.

Figure 1

Lower bounds on the time parameter ${\tau }_e$, as set by various experiments. The calculations are done for the relevant range of critical length scales $\hslash /{\sigma }_q$, and at ${\sigma }_s=20\mathrm{pm}$. The solid line corresponds to the atom interferometer of Ref. 29; it rules out all time parameters ${\tau }_e$ below the curve. Future experiments may exclude a larger set, e.g., by interference of ${10}^5–{10}^7\mathrm{amu}$ gold clusters [30] (dashed lines) or of micromirror motion [31] (dash-dotted line). The dotted line corresponds to demonstrated persistent current superpostions in a SQUID loop [1]. The shaded region represents the excluded ${\tau }_e$ by a conceivable classical measurement of less than $1\mu \mathrm{K}/\mathrm{s}$ temperature increase in a Rb gas.

Figure 2

Timeline of macroscopicities reached in quantum superposition experiments [22]. The squares, the triangles, and the dots represent interference experiments with neutrons [37, 38], atoms [29, 39, 40, 41, 42] or atom Bose-Einstein condensates [3], and molecules [4, 43, 44, 45, 46, 47, 48], respectively. One notes that Bose-Einstein condensates do not substantially exceed the macroscopicities achieved with atom interferometers. This is due to the single-particle nature of the condensate wave function. The many-particle state is more involved in the case of superposition experiments with persistent supercurrent states in a large SQUID loop [1, 2], as represented by the stars. However, despite the large number of Cooper pairs contributing to the current superpositions in SQUIDs, such experiments lag behind in macroscopicity due to the small coherence times observed.