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Chinese Physics B

Influence of pumping laser bandwidth on the quantum fluctuation chracteristic of a non-degenerate optical parametric amplifier

Chao Zhao, Xing-Zhen Ye, Cheng-Feng Yang, Liya Chen

Published Jul 1, 2012• 2 citations
Publisher

Abstract

Usually the quantum fluctuation characteristic of a non-degenerate optical parametric amplifier is analysed under the assumption of monochromatic pumping. However, in experiments, a driving beam with finite bandwidth is used to obtain the non-degenerate signal and idler beam amplifications. On account of that, we derive an analytical solution for the non-degenerate optical parametric amplification system with finite bandwidth laser pumping, and evaluate the associated quantum fluctuation. Finally, the application of the V1 criterion to bipartite entanglement is discussed.

DOI
10.1088/1674-1056/21/7/070308

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•Nov 21, 2007•0 citations
Usually the quantum fluctuation characteristic of a non-degenerate optical parametric amplifier is analysed under the assumption of monochromatic pumping. However, in experiments, a driving beam with finite bandwidth is used to obtain the non-degenerate signal and idler beam amplifications. On account of that, we derive an analytical solution for the non-degenerate optical parametric amplification system with finite bandwidth laser pumping, and evaluate the associated quantum fluctuation. Finally, the application of the V1 criterion to bipartite entanglement is discussed.
The solution of the time-dependent periodic pumping non-degenerate optical parametric amplifier (NOPA) is derived when the pump depletion is considered both above and below thresholds. Based on this solution, the quantum fluctuation calculated shows that a high entanglement and a good squeezing degree of the parametric light beams are achieved near and above thresholds. We adopt two kinds of pump fields: (i) a continuously modulated pump with a sinusoidal envelope; (ii) a sequence of laser pulses with Gaussian envelopes. We analyse the time evolution of continuous variable entanglement by analytical and numerical calculations, and show that the periodic driven pumping also improves the degree of entanglement. The squeezing and Einstein–Podolsky–Rosen (EPR) entanglement by using the two pumping driven functions are investigated from below to above the threshold regions, the tendencies are nearly the same, and the Gaussian driven function is superior to that of the sine driven function, when the maximum squeezing and the minimum variance of quantum fluctuation are considered. In the meantime, the generalization of frequency degenerate OPA to frequency non-degenerated OPA problem is investigated.
An analytical solution of the Fokker–Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).
In this paper, we first find out the analytic solution of the time-dependent Fokker-Planck equation of the non-degenerate optical parametric amplification (NOPA) system under the consideration of the dispersion, the loss and the pump depletion effects. Then, through the numerical calculation, we obtain the squeezing characteristic of the degenerate optical parametric amplification (DOPA) system with dispersion. the research indicates: the dispersion effect stems from the nonlinear susceptibility change from χ″ toχ″/{1+σ2/}/+2, with the increasing of the dispersion effect, the general feature of the squeezing curves beeps unchanged, and the curves contract toward left. The maximum squeezing approaches to the linear theory 1/(1+μ). Finally, we obtain the entanglement characteristic of the NOPA system with dispersion. We find out when σ is given, with the increasing of pump parameter μ, he corresponding phase makes a large change. The nonlinear susceptibility changes many times. When the polarity is positive, the system obtains the gain, when the polarity is negative, the system suffers the loss, but the gain is mainly dissipated by the loss, so the net gain is small, the squeezing is also small. The minimum variance V1 reduces gradually, and the whole curve moves to the right, approaches to the linear theory 1/(1+μ).
Using the quantum theory of a non-degenerate parametric amplifier, we discuss the phase correlations between signal and idler modes well above threshold. Particular attention is paid to the influence which fluctuation of the driving laser has on these correlations. We point out that the signal and idler mode phase correlation may be tuned by control of the driving laser field intensity and linewidth. For coherent driving, the mode phases are negatively cross-correlated. We stress the qualitative importance of these mode correlations for a class of two-photon resonant excitation experiments in atoms. In particular, the non-degenerate parametric oscillator represents a concrete model of a source with negatively cross-correlated phase fluctuations.
This thesis is primarily a theoretical study of degenerate parametric amplification as a means of generating squeezed-state light. i) A wideband traveling-wave formalism is developed for analyzing quantum mechanically a degenerate parametric amplifier. The formalism is based on spatial differential equations — spatial Langevin equations — that propagate temporal Fourier components of the field through the nonlinear medium. In addition to the parametric nonlinearity, the Langevin equations include absorption and associated fluctuations, dispersion, and pump quantum fluctuations. The dominant effects of dispersion and pump quantum fluctuations on the squeezing produced by a degenerate parametric amplifier are analyzed. ii) The wideband formalism of i) is used to carry out a more detailed analysis of the effects of phase mismatching. With the assumption of a lossless medium and a classical pump, we find that parametric amplification is capable of generating squeezed-state light over a wide band if materials with large χ(2) nonlinearities can be found, and that the squeezing bandwidth can be enhanced by phase mismatching away from degeneracy. iii) We consider again the effect of pump quantum fluctuations on the squeezing produced by parametric amplification. We perform discrete-mode calculations for a parametric amplifier with a quantum pump, and discuss some of the limitations of calculations of this sort in quantum optics. We derive stochastic differential equations (SDEs) for one- and two-mode parametric amplifiers, and from them obtain an iterative solution showing that pump quantum fluctuations impose a limitation on the degree of squeezing obtainable from a parametric amplifier. iv) A possible application of squeezing is considered; in particular, we study the effects of squeezing the intracavity noise in a laser oscillator. We solve the classical noise problem of a realistic laser model by making a bold — and possibly unrealizable — assumption, that the in-phase and quadrature Langevin sources which are responsible for the "noisiness" of the laser can be squeezed. We show that the effect of squeezing the in-phase quadrature is to reduce the phase noise, including the linewidth, of the laser but, due to amplitude-phase coupling, not to eliminate them altogether.