Nonclassicality of the Harmonic-Oscillator Coherent State Persisting up to the Macroscopic Domain

Can the most “classical-like” of all quantum states, namely the Schrödinger coherent state of a harmonic oscillator, exhibit nonclassical behavior? We find that for an oscillating object initially in a coherent state, merely by observing at various instants which spatial region the object is in, the Leggett-Garg inequality (LGI) can be violated through a genuine negative result measurement, thereby repudiating the everyday notion of macrorealism. This violation thus reveals an unnoticed nonclassicality of the very state which epitomizes classicality within the quantum description. It is found that for any given mass and oscillator frequency, a significant quantum violation of LGI can be obtained by suitably choosing the initial peak momentum of the coherent state wave packet. It thus opens up potentially the simplest way (without coupling with any ancillary quantum system or using nonlinearity) for testing whether various recently engineered and sought after macroscopic oscillators, such as feedback cooled thermal trapped nanocrystals of $\sim {10}^6–{10}^9\mathrm{amu}$ mass, are indeed bona fide nonclassical objects.

Figure 1

Setup for testing the violation of macrorealism for an object in a coherent state in a harmonic well. A probe beam, shown as applied to the greyed circular region, illuminates one half of a harmonic well. If no light is scattered using this probe beam, it is concluded that the object must be on the unilluminated half of the well. These outcomes constitute an ideal negative result measurement as the probe has not interacted with the object corresponding to these outcomes. Two-time correlation functions computed using such outcomes then provide a loophole-free violation of macrorealism.