I have seen that both classical and quantum theories are developed for this phenomenon. So please explain me the need for this.
It depends on what you want to get from Spontaneous Parametric Down-Conversion.
If you want to perform Spontaneous Parametric Down-Conversion spectroscopy, then classical formalism is sufficient:
https://www.researchgate.net/publication/243159065_Measurement_of_the_extraordinary_refractive_index_dispersion_in_the_MIR_for_MgNdLiNbO_3_crystals_by_the_use_of_quasi-phase-matching_in_a_random_1D_domain_structure
If you want to use Spontaneous Parametric Down-Conversion to generate entangled photons for quantum information applications, then you'll have to treat generated photons using quantum approach, but you can still consider pump to be a classical wave:
https://www.researchgate.net/publication/264742557_Generation_of_nonclassical_biphoton_states_through_cascaded_quantum_walks_on_a_nonlinear_chip
If you want to explore interaction between pump, single and idler photons on a single-photon level, then you need to treat all photons on a quantum level:
https://www.researchgate.net/publication/261555438_Single-photon_spontaneous_parametric_down-conversion_in_quadratic_nonlinear_waveguide_arrays?ev=prf_pub
Article Measurement of the extraordinary refractive index dispersion...
Article Generation of Nonclassical Biphoton States through Cascaded ...
Article Single-photon spontaneous parametric down-conversion in quad...
It depends on what you want to get from Spontaneous Parametric Down-Conversion.
If you want to perform Spontaneous Parametric Down-Conversion spectroscopy, then classical formalism is sufficient:
https://www.researchgate.net/publication/243159065_Measurement_of_the_extraordinary_refractive_index_dispersion_in_the_MIR_for_MgNdLiNbO_3_crystals_by_the_use_of_quasi-phase-matching_in_a_random_1D_domain_structure
If you want to use Spontaneous Parametric Down-Conversion to generate entangled photons for quantum information applications, then you'll have to treat generated photons using quantum approach, but you can still consider pump to be a classical wave:
https://www.researchgate.net/publication/264742557_Generation_of_nonclassical_biphoton_states_through_cascaded_quantum_walks_on_a_nonlinear_chip
If you want to explore interaction between pump, single and idler photons on a single-photon level, then you need to treat all photons on a quantum level:
https://www.researchgate.net/publication/261555438_Single-photon_spontaneous_parametric_down-conversion_in_quadratic_nonlinear_waveguide_arrays?ev=prf_pub
Article Measurement of the extraordinary refractive index dispersion...
Article Generation of Nonclassical Biphoton States through Cascaded ...
Article Single-photon spontaneous parametric down-conversion in quad...
There is much discussion about where the boundary between classical and quantum phenomena lies. At some level, everything is quantum, but a classical description often suffices, depending on what we want to measure.
One operational measure of the boundary comes from phase-space stochastic integration. The truncated Wigner representation is equivalent to stochastic electrodynamics, a classical theory. The Positive-P representation captures all the quantum nature of the problem. When we look at SPDC, the truncated Wigner gives accurate answers, but not for everything we wish to calculate. Some high order correlations, and especially two-time correlation functions are only found accurately when we use the full positive-P.
My answer is then that it is a quantum phenomenon, but a classical analysis manages to describe most of what happens. I hope this helps.
Quantum description is needed for "small number of photons" phenomena (in the time or in the frequency domain). As mentioned by Alexander it is crucial for instance in entanglement, but also for detailed analysis of parametric amplifier, triple-photon, etc. Otherwise (semi)classical is sufficient (no tremendeous effect on macroscopic quantities).
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