Criteria for generalized macroscopic and mesoscopic quantum coherence

We consider macroscopic, mesoscopic, and “$S$-scopic” quantum superpositions of eigenstates of an observable and develop some signatures for their existence. We define the extent, or size $S$ of a superposition, with respect to an observable $\widehat{x}$, as being the range of outcomes of $\widehat{x}$ predicted by that superposition. Such superpositions are referred to as generalized $S$-scopic superpositions to distinguish them from the extreme superpositions that superpose only the two states that have a difference $S$ in their prediction for the observable. We also consider generalized $S$-scopic superpositions of coherent states. We explore the constraints that are placed on the statistics if we suppose a system to be described by mixtures of superpositions that are restricted in size. In this way we arrive at experimental criteria that are sufficient to deduce the existence of a generalized $S$-scopic superposition. The signatures developed are useful where one is able to demonstrate a degree of squeezing. We also discuss how the signatures enable a new type of Einstein-Podolsky-Rosen gedanken experiment.

Figure 1

(Color online) Probability distribution for outcomes $x$ of measurement $\widehat{x}$. If $x_1$ and $x_2$ are macroscopically separated, then we might expect the system to be described as the mixture (2), where ${\rho }_1$ encompasses outcomes $x<x_2$, and ${\rho }_2$ encompasses outcomes $x>x_1$. This means an absence of generalized macroscopic coherence, as defined in Sec. 2.

Figure 2

(Color online) Probability distribution for a measurement $\widehat{x}$. We bin results to give three distinct regions of outcome: 0,$-1,+1$.

Figure 3

(Color online) We consider an arbitrary probability distribution for a measurement $\widehat{x}$ that gives a macroscopic range of outcomes.

Figure 4

(Color online) $P\left(x\right)$ for a coherent state $|\alpha ⟩$: $\Delta x=\Delta p=1$.

Figure 5

(Color online) (a) $P\left(x\right)$ for a superposition of coherent states $\left(1/\sqrt{2}\right)\left\{e^{i\pi /4}|-\alpha ⟩+e^{-i\pi /4}|\alpha ⟩\right\}$ (here the scale is such that $\Delta x=1$ for the coherent state $|\alpha ⟩$).

Figure 6

Plot of $⟨x|\rho |x^{\prime }⟩$ for a coherent state $|\alpha ⟩$, where $\alpha =2.5$.

Figure 7

Plot of $⟨x|\rho |x^{\prime }⟩$ for the superposition state (51), where $\alpha =2.5$.

Figure 8

(a) $P\left(p\right)$ for a superposition (51) of two coherent states where $\alpha =2.5$ and (b) the reduced variance ${\Delta }^{2}p<1$, versus $\alpha$.

Figure 9

(Color online) (a) Probability distribution for a measurement $X$ for a momentum-squeezed state. The variance ${\Delta }^{2}x$ increases with squeezing in $p$, to give a macroscopic range of outcomes, and for the minimum uncertainty state (54) satisfies $\Delta x\Delta p=1$. (b) The $⟨x|\rho |x^{\prime }⟩$ for a squeezed state (54) with $r=13.4$ ($\Delta x=3.67$ which predicts $⟨a^{†}a⟩={2.5}^2$.

Figure 10

(Color online) Detection of underlying superpositions of size $S$ for the squeezed minimum uncertainty state (54) by violation of Eqs. (17) [dashed line of (b)] and (35) [full line of (b)]. $S_{\text{max}}$ is the maximum $S$ for which the inequalities are violated. Inset of (b) shows behavior of violation of Eq. (17) for general Gaussian-squeezed states. Inequality (35) depends only on $\Delta p$. The size of $S_{\text{max}}$ relative to $P\left(x\right)$ is illustrated in (a).