On quantum light: Difference between revisions

    Just like vibrational modes in a solid (which can have both discrete and continuous components depending on the system), the lattice's vibrational modes could be quantized. Each discontinuity might correspond to a discrete energy state or vibration, with specific energy levels corresponding to particular vibrational frequencies.

    The smallest possible frequency would correspond to the minimum energy gap between discrete energy states, much like how energy levels are quantized in an atom or in a solid lattice (e.g., phonons).

    The frequency of these vibrations would be discrete if the geometry of the lattice imposes specific resonance frequencies (like modes of vibration in a crystal lattice, where only certain frequencies are allowed).

    Photons would not be able to take any arbitrary frequency but instead would only correspond to specific energy gaps or vibrational modes in the lattice.

    The energy gap ΔEΔE is related to the frequency gap ΔνΔν by Planck's relation:

    ΔE=hΔν

    ΔE=hΔν

    Where hh is Planck's constant (6.626×10−34 J\cdotps6.626×10−34 J\cdotps) and ΔνΔν is the frequency gap.

    If the energy gap is small (for example, on the scale of subatomic particles or very fine lattice structures), the required precision in measuring frequency would need to be extremely high to detect these small differences.

    The Planck scale itself sets a natural limit on the smallest energy gap. If the lattice elements are on the scale of Planck length (around 10−3510−35 meters), the energy gap could be on the order of Planck energy (about 1.22×1019 GeV1.22×1019 GeV).

    Smaller lattices with finer elements (larger discontinuities or more massive lattice particles) would have larger frequency gaps, which would reduce the precision required to observe the quantization.

    The energy of the smallest possible quantum in the lattice could be on the order of the Planck energy EPEP​. Using the relationship ΔE=hΔνΔE=hΔν, we can calculate the frequency resolution required to observe the Planck energy gap:

    Δν=EPh

    Δν=hEP​​

    Where EP≈1.22×1019 GeVEP​≈1.22×1019 GeV (or 1.96×10−10 joules1.96×10−10 joules).

    Substituting the values:

    Δν=1.96×10−106.626×10−34 Hz

    Δν=6.626×10−341.96×10−10​ Hz

    Δν≈2.96×1023 Hz

    Δν≈2.96×1023 Hz

    This is an extremely high frequency resolution (on the order of 1023 Hz1023 Hz), far beyond what current technology can measure. However, this scale sets the theoretical minimum frequency precision required to directly observe the quantization of energy levels in a lattice at the Planck scale.

    If the lattice has larger elements or is macroscopic in scale, the energy gaps between quantized vibrational modes could be much larger, reducing the required precision.

    The experimental setup (such as interferometers or spectrometers) would need to be sensitive enough to detect these fine frequency differences. With current technology, we can measure frequencies with precision on the order of 10−1510−15 to 10−1810−18 Hz in the case of ultra-high-frequency sources like lasers or resonators, but observing Planck-scale frequencies would require a new level of precision that goes far beyond today's capabilities.