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arXiv:quant-ph/9809089 · Zeitschrift für Naturforschung A

Long-time Dynamics of Spontaneous Parametric Down-conversion and Quantum Limitations of Conversion Efficiency

Michael Fleischhauer, Oliver Veits

Published Jan 1, 1999• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1515/zna-1999-0108

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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.
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.
Ttie build - up of the third harmonic and the depletion of the pump mode are investigated d escribing both the fields quantum mechani- cally. The Heisenberg equationsof motion describing the dynamics of the pump and oF the third harmonic mode are solved in the short-time appro- ximation. It is found that the pump mode shows antibunching effect due to the nonlinear interaction between the modes. l'hird harmonic generation has been studied both theoretically as well as experimental 1 y by severa1 workersl-l1 in the f ield of nonl i- near optics. The third harmonic is being used to produce ultraviolet ra - diation not available from existing lasers by nonlinear optical mixing . The initial build-up of the third harmonic has been studied theoretically, in general, using the parametric approximation i.e. considering the pump depletion to be negligible. It is a reasonable approximation so long as the conversion efficiency is quite low, but recent experiments have shom increased irnprovement in the conversion ef f iciency l l. Under these c i r - cumstances, it is necessary to include pump depletion while studying the harmonic generation. The present communication aims at investigating the dynamics of the third harmonic generation including pump depletion and treating both the pump and the harmonic fields quantum mechanically. The model Hamiltonian of the system is written along the line of shen12 and ~alls'~. The Heisenberg equations of motion describing the time evolution of the pump and the harmonic mode are solved in the short - time approxima- tion14. Our analysis shows thatasaresultof nonlinear interaction, the pump photons manifest antibunching, if the pump mode is in the coherent state and the harmonic mode in the vacum state initially. The photons generated in the harmonic mode are uncorrelated for the quoted i n i tia l conditions.
The degenerate parametric oscillator above threshold is studied with phenomenological stochastic colored (nonwhite) pump noise for arbitrary pump to subharmonic relaxation rate. The current experimental limits of large intracavity threshold photon numbers (very small quantum noise) are considered, allowing for a semiclassical treatment of the system dynamics. A comparison between the effects of isotropic and squeezed pump noise on the internal transient and steady-state fluctuations is presented by simulation of the nonlinear semiclassical stochastic Langevin equations in the Wigner quadrature representation. It is found that the transient squeezing for the system starting in the unstable steady state (the vacuum) is not degraded by stochastic pump noise. A damped oscillatory behavior of the noise levels (periodic exchange of fluctuations between the squeezed quadrature of the signal and the pump) is observed for large damping of the signal in the turn on of pump depletion. Finally, it is shown that the limited squeezing above threshold in the steady state (50%) due to pump depletion can be enhanced if squeezed stochastic noise with sufficient significant spectral components (broadband squeezing) is fed to the pump. The above-threshold steady-state squeezing has been calculated analytically from the linearized stochastic equations and the effects of the time scale associated to the relaxation of the pump noise (the noise correlation time) compared to the dissipative time scale of the system and the pump to subharmonic loss ratio are presented.
The quantum fluctuations in a nondegenerate optical parametric oscillator are calculated without the standard linearization assumption. Analytical results are obtained with the help of a mean-field approximation, which is exact in the limit of a fast decaying subharmonic mode. This approach is equivalent to a linearization around a self-consistently determined pump field amplitude that takes into account fluctuations of the subharmonic mode. The range of validity of the mean-field approximation is analyzed, and the influence of the system size on field amplitudes and squeezing is discussed.
We study spontaneous parametric processes starting from the quantized field equations in the Heisenberg picture using the common approximation of treating the laser pump as classical. After a (standard) linearization procedure we pass to the Wigner representation and the process becomes formally equivalent to the parametric interaction of two incoming zero-point modes with a pumping laser wave coupled by an optically nonlinear medium. Then the treatment of parametric down conversion and parametric up conversion are described by the Maxwell equations and look like a parametric amplification of the vacuum field. The fourth-order correlation function between the signal and idler fields is obtained in a form that makes for easy comparison with experimental results.