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Journal of Modern Physics

Some New Features of Photon Statistics in a Fully Quantized Parametric Amplification Process

Joseph Akeyo Omolo

Published Jan 1, 2014• 1 citation

Abstract

An exact quantum treatment reveals that signal and idler photon number operators are not well-behaved dynamical operators for studying photon statistics in parametric amplification/down-conversion processes. Contrary to expectations, the mean signal-idler photon number difference varies with time, while the corresponding signal-idler photon number cross-correlation function is complex and experiences an interference phenomenon driven by the interaction parameters. The intensity operators and related polarization operators of the polarized signal-idler photon pair in positive and negative helicity states are identified as the appropriate operators specifying a conservation law and the dynamical symmetry group (SU(1, 1)) of the parametric amplification process. The conservation of the mean positive and negative helicity photon intensity difference and the purely real positive-negative helicity intensity cross-correlation function correctly account for the simultaneous production of polarized signal and idler photons in positive and negative helicity states.

DOI: 10.4236/jmp.2014.58082

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An exact quantum treatment reveals that signal and idler photon number operators are not well-behaved dynamical operators for studying photon statistics in parametric amplification/down-conversion processes. Contrary to expectations, the mean signal-idler photon number difference varies with time, while the corresponding signal-idler photon number cross-correlation function is complex and experiences an interference phenomenon driven by the interaction parameters. The intensity operators and related polarization operators of the polarized signal-idler photon pair in positive and negative helicity states are identified as the appropriate operators specifying a conservation law and the dynamical symmetry group (SU(1, 1)) of the parametric amplification process. The conservation of the mean positive and negative helicity photon intensity difference and the purely real positive-negative helicity intensity cross-correlation function correctly account for the simultaneous production of polarized signal and idler photons in positive and negative helicity states.

In this paper, a simple method for obtaining general analytical solutions of the time evolution equations for a fully quantized parametric oscillation process is developed. Heisenberg’s equations for the signal–idler photon annihilation operators are converted into a matrix equation equivalent to a two-state Jaynes–Cummings time evolution equation which has exact analytical solutions. The mean intensity inversion for the coupled signal–idler photon pair is found to undergo fractional revivals for pump photon in a Fock state, provided both signal and idler photons are in occupied Fock states. General collapses and revivals occur for interactions with pump photon in a coherent state, but now with both or either of signal and idler photons in occupied Fock states. An interpretation of the coupled signal–idler photon pair as a circularly polarized two-state system specified by positive and negative helicity states leads to an appropriate description of photon polarization state dynamics governed by the underlying Jaynes–Cummings interaction.

In nondegenerate parametric amplification of light, one pump photon is destroyed and two photons are simultaneously created, one in each of two modes with different frequencies (the signal and idler). If m signal photons are counted in time t, what is the state of the idler conditioned on this result? The projection postulate cannot be used to answer this question as photon-counting measurements are not ``first-kind'' measurements. We use a theory of continual quantum-counting measurements to show that in general the idler field is left in a classical mixture of number states with at least m quanta. In an appropriate limit this mixture reduces to an exact number state with m quanta.

Signal and idler waves produced by a nondegenerate parametric amplifier feature strong positive correlation between their photon numbers, or in-phase amplitudes. They feature equally strong negative correlation between their phases, or quadrature-phase amplitudes. These photon twins can produce arbitrarily squeezed states via state reduction by appropriate measurements of an idler wave. When the quadrature amplitude of an idler wave is measured, the signal wave is collapsed to a quadrature-amplitude squeezed state. When the two-quadrature amplitudes of an idler wave are measured simultaneously, the signal wave becomes a coherent state.

Measurement of photon-number statistics of fields composed of photon pairs generated in spontaneous parametric down-conversion pumped by strong ultrashort pulses is described. Finite detection quantum efficiencies, noises as well as possible loss of one or both photons from a pair are taken into account. Measured data provided by an intensified CCD camera are analyzed along the developed model. The joint signal-idler photon-number distribution is obtained using the expectation maximization algorithm. Covariance of the signal and idler photon-numbers exceeds 98%. Entanglement area of the correlated photon pairs is determined.

Noise spectra of the in-phase and quadrature-phase components of the signal and idler fields of a nondegenerate optical parametric amplifier are calculated. It is shown that, as is suggested by the operator Manley-Rowe relations, the external in-phase (amplitude) fluctuations of the signal and idler fields are correlated in the frequency region lower than the signal and idler cavity bandwidths. If the idler photon flux is continuously measured and the photocurrent fluctuations are fed back to control the pump intensity, the signal output will feature an amplitude noise spectral density reduced to below the standard quantum limit (amplitude squeezing) in the frequency region below the feedback loop and signal and idler cavity bandwidths. An all-optical realization of such a negative-feedback parametric oscillator is presented. It is also shown that a nondegenerate parametric amplifier with idler measurement feedback can add a fixed number of photons to a photon-number state.