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Physical Review A

Photon number divergence in the quantum theory ofn-photon down conversion

Mark S. Hillery

Published Jul 1, 1990• 43 citations
Publisher

Abstract

The process of n-photon down conversion is considered both with a quantized pump and in the parametric approximation. In the parametric approximation it is formally found that the photon number diverges at a finite time for n>2. This divergence limits the times for which the parametric approximation is useful. The divergence does not occur if the pump is quantized. Furthermore, it is shown that the Hamiltonian with a quantized pump is essentially self-adjoint. This means that the questions that have been raised about the existence of the time-development operator in the parametric approximation do not arise when the pump is quantized.

DOI
10.1103/physreva.42.498

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The process of n-photon down conversion is considered both with a quantized pump and in the parametric approximation. In the parametric approximation it is formally found that the photon number diverges at a finite time for n>2. This divergence limits the times for which the parametric approximation is useful. The divergence does not occur if the pump is quantized. Furthermore, it is shown that the Hamiltonian with a quantized pump is essentially self-adjoint. This means that the questions that have been raised about the existence of the time-development operator in the parametric approximation do not arise when the pump is quantized.
Optical down-conversion is widely used for generating photon pairs, squeezed and entangled states of light, making it an indispensable tool in quantum optics and quantum information. In the regime where the pump is much stronger than the generated field, the standard parametric approximation treats the pump amplitude as a fixed parameter of the model. This approximation has a limited domain of validity since it assumes a nondepleted and nonentangled pump. By finding an approximate solution to the Schrödinger equation of the down-conversion process, we obtain an improved analytical model beyond the parametric one, which accounts for pump depletion and pump-signal entanglement. Our model is advantageous, first, because it allows one to compute averages of field operators far beyond the domain of validity of the parametric approximation, and second, because it allows one to establish error-specific limits of the latter domain. For a given pump amplitude, we find a maximum squeezing parameter, up to which the approximation remains valid within a specified acceptable error. Our results confirm that recent experiments on Gaussian boson sampling, with a squeezing parameter of r ≈ 1.8 and a coherent pump amplitude of α ≈ 2 × 10 6 , can still be accurately described by the standard parametric approximation. However, we observe a sharp decline in validity as the squeezing parameter increases. For pump amplitudes of α ≈ 2 × 10 6 , the parametric approximation breaks down when the squeezing parameter exceeds r ≈ 4.5 , whereas the new approximation remains valid up to r ≈ 6 with an acceptable error of 1%.
We study entanglement between field modes in the process of nondegenerate two-photon down-conversion with quantized pump. We show that due to the quantum nature of the dynamics, strong entanglement between the pump and the signal-idler subsystems can be observed. We find that the higher the initial intensity of the pump mode the stronger the entanglement between the pump and the signal-idler subsystem is established during the first instants of the time evolution. We also show that the signal and the idler modes are strongly entangled (correlated). This entanglement is much stronger than the entanglement between the pump and the signal-idler subsystem. Correlation between the signal and the idler modes leads to a high degree of two-mode squeezing, which can be observed during the first instants of the time evolution when the pump mode is still approximately in a pure state. On the other hand, the back action of the signal-idler subsystem on the pump mode leads to a strong single-mode squeezing of the pump mode. At the time interval during which squeezing of the pump mode can be observed the pump mode is far from being in the minimum uncertainty state. We also analyze the long-time behavior of the quantum-optical system under consideration and we show that the interesting collapse-revival effect in the time evolution of the mean photon number and of the purity parameters of field modes can be observed. Finally, we show that the degree of entanglement between modes in the nondegenerate quantum-optical down-conversion strongly depends on the initial state of the system.
A description of the parametric amplifier and frequency converter is presented without introducing the classical (i.e., parametric) approximation for the pumping field. Constants of the motion are found which reduce the solution of the Schr\"odinger equation to the diagonalization of a matrix. This diagonalization is accomplished numerically, and the eigenvalues and eigen-functions of a system with fixed energy are calculated. The time-dependent behavior of the mean number of photons in the amplified or frequency up-converted field is presented. The time evolution of the probability distributions is illustrated. The technique is extended to the problem of coherent spontaneous emission from a system of NNN two-level atoms interacting with the radiation field where both the atomic system and the radiation field are quantized.
We analyze the long-time quantum dynamics of degenerate parametric down-conversion from an initial sub-harmonic vacuum (spontaenous down-conversion). Standard linearization of the Heisenberg equations of motions fails in this case, since it is based on an expansion around an unstable classical solution and neglects pump depletion. Introducing a mean-field approximation we find a periodic exchange of energy between the pump and subharmonic mode goverened by an anharmonic pendulum equation. From this equation the optimum interaction time or crystal length for maximum conversion can be determined. A numerical integration of the 2-mode Schr"odinger equation using a dynamically optimized basis of displaced and squeezed number states verifies the characteristic times predicted by the mean-field approximation. In contrast to semiclassical and mean-field predictions it is found that quantum fluctuations of the pump mode lead to a substantial limitation of the efficiency of parametric down-conversion.
Abstract We analyze the long-time quantum dynamics of degenerate parametric down-conversion from an initial sub-harmonic vacuum (spontaenous down-conversion). Standard linearization of the Heisenberg equations of motion fails in this case, since it is based on an expansion around an unstable classical solution and neglects pump depletion. Introducing a mean-field approximation we find a periodic exchange of energy between the pump and subharmonic mode goverened by an anharmonic pendulum equation. From this equation the optimum interaction time or crystal length for maximum conversion can be determined. A numerical integration of the 2-mode Schrödinger equation using a dynamically optimized basis of displaced and squeezed number states verifies the characteristic times predicted by the mean-field approximation. In contrast to semiclassical and mean-field predictions it is found that quantum fluctuations of the pump mode lead to a substantial limitation of the efficiency of parametric down-conversion.