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Physical Review A

Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides

M. J. Werner, P. D. Drummond

Published Aug 1, 1997• 23 citations
Publisher

Abstract

It is shown that coherent quantum simultons (simultaneous solitary waves at two different frequencies) can undergo quadrature-phase squeezing as they propagate through a dispersive ${\ensuremath{\chi}}^{(2)}$ waveguide. This requires a treatment of the coupled quantized fields including a quantized depleted pump field. A technique involving nonlinear stochastic parabolic partial differential equations using a nondiagonal coherent state representation in combination with an exact Wigner representation on a reduced phase space is outlined. We explicitly demonstrate that group-velocity matched ${\ensuremath{\chi}}^{(2)}$ waveguides which exhibit collinear propagation can produce quadrature-phase squeezed simultons. Quasi-phase-matched KTP waveguides, even with their large group-velocity mismatch between fundamental and second harmonic at 425 nm, can produce 3 dB squeezed bright pulses at 850 nm in the large phase-mismatch regime. This can be improved to more than 6 dB by using group-velocity matched waveguides.

DOI
10.1103/physreva.56.1508

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Journal of the Optical Society of America B•Nov 1, 1997•38 citations
It is shown that coherent quantum simultons (simultaneous solitary waves at two different frequencies) can undergo quadrature-phase squeezing as they propagate through a dispersive \ensuremathχ(2){\ensuremath{\chi}}^{(2)}\ensuremathχ(2) waveguide. This requires a treatment of the coupled quantized fields including a quantized depleted pump field. A technique involving nonlinear stochastic parabolic partial differential equations using a nondiagonal coherent state representation in combination with an exact Wigner representation on a reduced phase space is outlined. We explicitly demonstrate that group-velocity matched \ensuremathχ(2){\ensuremath{\chi}}^{(2)}\ensuremathχ(2) waveguides which exhibit collinear propagation can produce quadrature-phase squeezed simultons. Quasi-phase-matched KTP waveguides, even with their large group-velocity mismatch between fundamental and second harmonic at 425 nm, can produce 3 dB squeezed bright pulses at 850 nm in the large phase-mismatch regime. This can be improved to more than 6 dB by using group-velocity matched waveguides.
We present a unified analysis of the quantum fluctuations and quadrature squeezing of continuous wave (cw) light propagating in a single-mode quadratic (${\mathrm{\ensuremath{\chi}}}^{(2)}$) waveguide. We retain the effects of wave-vector and group-velocity mismatch, group-velocity dispersion, and depletion by linearizing the quantum fluctuations with respect to the the exact cw nonlinear semiclassical evolution. Each of the characteristic three-wave mixing phenomena, including parametric amplification, second-harmonic generation, and special eigenmode solutions are discussed and contrasted. Regimes of strong squeezing, and the influence of classical parametric modulational instabilities and spatial bifurcations on the spectrum of squeezed quantum fluctuations, are discussed. \textcopyright{} 1996 The American Physical Society.
We investigate temporal effects in pulsed squeezing by parametric amplification, including effects of group-velocity dispersion. Our calculations show that the local oscillator pulse used to detect the squeezed field cannot be made shorter than the inverse phase-matching bandwidth of the generation process without degrading the amount of squeezing detected. This result generalizes an earlier result that showed that in the absence of dispersion, the local oscillator pulse duration should approach zero for optimum squeezing detection. We further show that by using local oscillator amplitude and phase pulse shaping it should be possible to achieve more than 40 dB of detectable quadrature squeezing. This is applicable where it is possible to neglect transverse spatial dimensions and diffraction, such as in a waveguide. We derive the s-parametrized quasiprobability evolution equation for the traveling-wave parametric amplifier. As the Wigner representation results in third-order derivatives, we also use the positive-P representation as an exact representation with equivalent Ito stochastic differential equations. This allows us to compare approximate --- but easily simulated --- Wigner representation results with those using the positive-P representation.
Nonlinear optical waveguides hold promise as parametric amplifiers and as sources of quadrature-squeezed light. The confinement of the pump field drastically increases the efficiency of the nonlinear conversion. In addition, the control of the spatial mode profiles by the waveguide geometry should diminish the effect of gain-induced diffraction, which has been seen to limit the amount of attainable squeezing in bulk parametric crystals.1 Using subpicosecond laser pulses, we have observed parametric gain as well as quadrature squeezing in a quasi-phase matched KTP waveguide.
We analyze squeezing of both the fundamental and harmonic fields undergoing traveling-wave, second-harmonic generation (SHG) in second-order (${\mathrm{\ensuremath{\chi}}}^{(2)}$) nonlinear media. We take into account depletion of the fundamental field as well as the phase mismatch between the fundamental and harmonic fields. The behavior of the quantum noises on the propagating fields is studied by linearizing the nonlinear operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and harmonic fields under the conditions of perfect phase matching and large phase mismatch. We show that in the case of a large phase mismatch, the intensity-dependent self-phase shift of the fundamental field, arising due to cascading of the ${\mathrm{\ensuremath{\chi}}}^{(2)}$ nonlinearity, is responsible for the squeezing generation. We also numerically solve the linearized quadrature-operator equations together with the nonlinear mean-field equations. We find that in the case of a finite phase mismatch the harmonic field can be highly squeezed. This is in contrast to the perfectly phase-matched case where the maximum squeezing is limited to 50%. Finally, we analyze the nondegenerate case that applies to type-II phase-matching geometries. Here we show that the commonly used type-II phase-matched SHG process with the input harmonic field in the vacuum state is equivalent to a type-I SHG process in parallel with a degenerate optical-parametric process. The latter causes squeezing in the mode that is polarized orthogonal to the fundamental beam. In the perfect phase-matching case, the squeezing in the orthogonally polarized mode follows the simple expression S=1-\ensuremath{\gamma}, where \ensuremath{\gamma} is the harmonic conversion frequency.
We report traveling-wave quadrature squeezing at 1064 nm in a quasi-phase-matched LiNbO(3) waveguide, avoiding the gain-induced diffraction encountered in bulk squeezing experiments. The 10-mm-long singlemode waveguide parametric amplifier exhibited a gain of 1.9, averaged over the 20-ps mode-locked pulses, with only 0.5 W of peak pump power at 532 nm. Independent waveguides were employed for the degenerate parametric amplifier squeezer and for the second-harmonic generator that produced the pump for the parametric amplifier. We measured phase-sensitive amplification versus pump power and found close agreement with theory. The observed 14% squeezing correlates closely with the predicted value, based on the measured phase-sensitive amplification and 40% overall detection efficiency. Impedance-matched electrically resonant detection was used to boost our squeezing measurement signals substantially above the background level.