Quantum correlations are crucial in modern quantum photonics research. The first discussions concerning quantum correlations were presented in the first half of the 20th century [1]. Nevertheless, currently, one can find many papers dealing with those topics. It is worth noting that some of these are devoted to optical and photonic models, and some concern proposals of new photonic systems.

Many papers deal with quantum correlations, especially in the context of applications of the considered quantum models in quantum information theory systems. At this point, one can mention correlations such as quantum entanglement, quantum steering, and Bell-type nonlocalities. These correlations are mostly related to spatial nonlocalities. On the other hand, one can ask about the existence of the counterparts of such nonlocalities in the time domain. The topics related to them are devoted to violating Legget–Garg inequalities (LGI) [2,3], which we consider here.

In this communication, we consider a system of two mutually coupled quantum nonlinear oscillators that are continuously driven by an external coherent field. For such a model, we discuss temporal correlations, examining LGI’s violation. We analyze various scenarios of measurements based on projection onto different Bell states, showing that the possibility of violating LGI inequalities is related to the choice of different projectors.

1. E. Schrodinger, Proc. Camb. Phil. Soc., 31:555 (1935); ibid 32, 446(1936).
2. A. J. Leggett, A. Garg, Phys. Rev. Lett. 54:857 (1985).
3. J. K. Kalaga, A. Kowalewska-Kudłaszyk, W. Leoński, J. Perina Jr., Opt. Express, 32:9946. (2024).