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Bulletin of the Russian Academy of Sciences Physics

Quantum Features of Waves in the Parametric Down Conversion of Light

I. A. Tereshchenko, Olga V. Tikhonova

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Abstract A fully quantum approach is used to study the mutual influence of simultaneously occurring processes of second harmonic generation and parametric down conversion. The effect of quantum pump depletion is revealed. A balance is found between the considered processes and obtaining the equilibrium number of photons in field modes. The generation of a highly entangled two-mode field is demonstrated.

DOI
10.1134/s1062873824709127

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Abstract A fully quantum approach is used to study the mutual influence of simultaneously occurring processes of second harmonic generation and parametric down conversion. The effect of quantum pump depletion is revealed. A balance is found between the considered processes and obtaining the equilibrium number of photons in field modes. The generation of a highly entangled two-mode field is demonstrated.
A quantum theory of parametric amplification and frequency conversion of an optical image in coupled nonlinear optical processes that include one parametric amplification process at high-frequency pumping and two up-conversion processes in the same pump field is developed. The field momentum operator that takes into account the diffraction and group velocities of the waves is used to derive the quantum equations related to the spatial dynamics of the images during the interaction. An optical scheme for the amplification and conversion of a close image is considered. The mean photon number density and signal-to-noise ratio are calculated in the fixed-pump-field approximation for images at various frequencies. It has been established that the signal-to-noise ratio decreases with increasing interaction length in the amplified image and increases in the images at the generated frequencies, tending to asymptotic values for all interacting waves. The variance of the difference of the numbers of photons is calculated for various pairs of frequencies. The quantum entanglement of the optical images formed in a high-frequency pump field is shown to be converted to higher frequencies during the generation of sum frequencies. Thus, two pairs of entangled optical images are produced in the process considered.
We study the harmonic entanglement and squeezing in a two-mode radiation produced in a degenerate parametric down conversion process coupled to a two-mode vacuum reservoir employing the linearization procedure. It is found that there is a quadrature entanglement between the harmonically related fundamental and second-harmonic modes and the superimposed radiation exhibits a significant two-mode squeezing. The entanglement exits even when there is no two-mode squeezing, since the correlation leading to these phenomena are essentially different. In addition, the more the external coherent light is down converted, the more stronger the entanglement and mean photon number of the two-mode radiation would be which is not generally true for squeezing.
We investigate the process of entangled state of light generation while propagation along a one dimensional array of single-mode nonlinear waveguides. We consider a situation when entanglement is formed due to spontaneous parametric down-conversion of the pump which is present only in a signal waveguide. In the considered process the generated state of light is multi-mode squeezed. We demonstrate that starting from certain distance of light propagation only pairs of waveguides, located symmetrically with respect to the pumped one, occur to be entangled. Also there is an optimal pump amplitude for which the formed quantum correlations are most pronounced. Entanglement for multi-mode squeezed states may be very sensitive for phase fluctuations in the pump. We investigate the influence of such noise on the discussed process. We demonstrate that for situation of generation of few photon entangled states the influence of phase fluctuations is negligible. But it dramatically increase with the growth of average photon numbers in the formed quantum states.
Long ago, it was predicted that stimulating two-photon processes such as two-photon absorption (TPA) and second harmonic generation (SHG) with entangled photons produced by parametric down-conversion (PDC) could result in higher efficiencies compared to classical illumination [1]. This was experimentally demonstrated for SHG [2] and TPA [3], where, however, the classical comparison was performed by adding losses to the PDC state rather than employing a true classical reference. In addition, these results considered spatially single-mode PDC radiation and were conducted at low photon fluxes, where the PDC modes are, on average, populated by less than one photon. This resulted in an extremely low SHG/TPA signal, considered of limited practical use for metrological applications (such as two-photon fluorescence microscopy or SHG imaging - see, e.g., [4]). A recent theoretical investigation [5] highlighted that a proper analysis of quantum-enhanced two-photon processes requires a fully spatiotemporal model of the nonlinear interactions. It further showed that highly multimode (in space and time) entangled PDC photons can provide a quantum advantage compared to an equivalent classical pulse, even in regimes well beyond the “one-photon-per-mode” limit, considered the boundary of quantum behavior.
We investigate the quantum evolution of the pump field in second-harmonic generation under strong pump depletion. Starting from a coherent state, the pump develops a nonclassical phase-space structure resembling a Schrödinger cat state. This behavior originates from phase instability induced by vacuum fluctuations of the harmonic mode. A rigorous quantum analysis has been performed for mean photon numbers up to ⟨n^⟩=100\langle \hat n \rangle = 100⟨n^⟩=100 in pump mode. For larger photon numbers, up to ⟨n^⟩=107\langle \hat n \rangle = 10^{7}⟨n^⟩=107, the dynamics have been analyzed using a classical trajectory method with sampled initial conditions that reproduces the main features of the quantum evolution. The results indicate that nonlinear frequency conversion can generate macroscopic superposition-like states of the pump field. Although the resulting state is not pure due to correlations with the second-harmonic wave, it remains non-classical with negative zones of Wigner function. These results indicate that strongly nonlinear frequency conversion can provide a scalable route toward macroscopic nonclassical states of light.