Macroscopic Superpositions

A macroscopic-superposition experiment tests coherence between alternatives distinguished by a specified observable. Large mass, large separation, many excitations, and long coherence time are different resources. None alone proves that a particular apparatus has prepared a coherent superposition rather than a mixture.

Coherence is the quantity to witness

For orthogonal branch states $\lvert L\rangle,\lvert R\rangle$, write

$$\rho= \begin{pmatrix} p&c\\c^*&1-p \end{pmatrix}, \qquad |c|^2\le p(1-p).$$

A phase-dependent recombination measures the off-diagonal element. For balanced recombination and an ideal readout, allowing unequal input populations,

$$P_+(\phi)=\frac12+\operatorname{Re}(c e^{i\phi}),\qquad V=\frac{P_{\max}-P_{\min}}{P_{\max}+P_{\min}}=2|c|.$$

The incoherent mixture has $c=0$. Observing both branch populations does not measure $c$; a phase-sensitive interference measurement or an appropriate state witness is necessary. Loss and imperfect contrast must be included in the detector model before interpreting a fitted visibility.

What makes the superposition macroscopic?

A superposition of two center-of-mass positions challenges spatial localization. Opposite currents challenge a different coarse observable. Two coherent states of a microwave mode may be well separated in phase space without displacing an entire device over a visible distance.

Nimmrichter and Hornberger define an experimental macroscopicity measure through the class of modifications to quantum dynamics an experiment excludes. This makes the comparison operational but also model dependent. Other measures use branch distinguishability, constituent number, or metrological sensitivity. A numerical ranking is meaningful only after the measure and degrees of freedom have been specified.

Consequently, a claim that a heavier object is “more quantum” is incomplete. A tiny spatial separation or very short duration may test a localization modification less strongly than a lighter object with a longer separated evolution.

Turning visibility into a constraint

For an illustrative additional Markovian dephasing rate $\Gamma$, suppose independently calibrated ordinary physics predicts visibility $V_{\rm env}$. The modified model predicts

$$V(t)=V_{\rm env}(t)e^{-\Gamma t}.$$

If the experiment establishes $V/V_{\rm env}>r_{\min}$ at a stated confidence level, then $\Gamma<-\ln(r_{\min})/t$ under that model. For $r_{\min}=0.9$ and $t=10$ ms, the bound is about $10.5\ {\rm s}^{-1}$.

This example assumes that the reference contrast is known and that the extra process simply multiplies coherence. An uncertain reference contrast can be degenerate with the proposed effect. Full collapse-model tests may require spatial, mass, and geometry dependence rather than one constant rate.

Interpreting successful interference

Interference establishes coherent alternatives within the tested preparation, evolution, and readout model. It disfavors modifications predicting excessive loss of that coherence in the regime tested. It does not establish unrestricted superposition at every scale, select one interpretation, or explain why an individual run has a particular outcome.

The practical routes include matter-wave interferometry, superconducting cat states, and optomechanical tests.

Exercise

For $p=0.9$, what is the largest possible visibility in the ideal balanced recombination defined above? Why can reduced visibility occur without dephasing?

Solution

Positivity gives $|c|\le\sqrt{0.9(0.1)}=0.3$, hence $V\le0.6$. A pure state with unequal branch weights saturates this bound. Thus low visibility can reflect population imbalance rather than an incoherent mixture.

References

• Bassi, A., Lochan, K., Satin, S., Singh, T. P., and Ulbricht, H., “Models of wave-function collapse, underlying theories, and experimental tests,” Reviews of Modern Physics 85, 471–527 (2013), doi:10.1103/RevModPhys.85.471 — modifications constrained by coherence experiments.
• Nimmrichter, S., and Hornberger, K., “Macroscopicity of mechanical quantum superposition states,” Physical Review Letters 110, 160403 (2013), arXiv:1205.3447 — an explicitly model-based macroscopicity measure.