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Quantum Theory of Stokes Pulse Energy Fluctuations

Michael G. Raymer, Ian A. Walmsley, Jan Mostowski, Božena Sobolewska

Published Jun 18, 1984• 0 citations
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Abstract

Recent experiments1,2 have raised questions concerning the effects of transverse propagation (diffraction) and collisional dephasing on the quantum fluctuations of the energy of Stokes pulses generated by stimulated Raman scattering (SRS). It was originally believed that, if the Fresnel number of the laser-excited volume was close to unity, a theory based on one-dimensional spatial propagation should be adequate. This theory3 predicted that in the transient, unsaturated regime the pulse-energy probability distribution function P(W) should be essentially a negative exponential function of pulse energy W, with the maximum value of P(W) at W ≅ 0. Experiments2 showed, however, a maximum in P(W) at about one tenth of the mean pulse energy, in disagreement with the one-dimensional theory.

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Recent experiments1,2 have raised questions concerning the effects of transverse propagation (diffraction) and collisional dephasing on the quantum fluctuations of the energy of Stokes pulses generated by stimulated Raman scattering (SRS). It was originally believed that, if the Fresnel number of the laser-excited volume was close to unity, a theory based on one-dimensional spatial propagation should be adequate. This theory3 predicted that in the transient, unsaturated regime the pulse-energy probability distribution function P(W) should be essentially a negative exponential function of pulse energy W, with the maximum value of P(W) at W ≅ 0. Experiments2 showed, however, a maximum in P(W) at about one tenth of the mean pulse energy, in disagreement with the one-dimensional theory.
We calculate the probability distribution P(W) for the energy W of pulses generated by transient stimulated Raman scattering and find that 100% macroscopic fluctuations are predicted. In the limit of large numbers of Stokes photons we find that P(W) is essentially exponential, implying that the most probable value for Stokes pulse energy is near zero. The macroscopic energy fluctuations, whose origin is quantum mechanical, may impose a fundamental limitation on the stability of the Raman generation process.
A quantum theory of stimulated Raman scattering is presented that takes into account three-dimensional propagation and collisional dephasing, allowing the study of the spatial and temporal coherence properties of the generated Stokes light. Maxwell-Bloch equations for the Stokes field operator and the collective atomic operators are solved analytically under low-signal-gain conditions, where the laser field and the atomic ground states remain undepleted. The intensity and the space-time autocorrelation function of the Stokes field are calculated. The Stokes field is expanded into a set of statistically independent ``coherence modes,'' which are determined explicitly for the case of a cylindrically shaped pumped volume. The Stokes pulse energy W is found to fluctuate from pulse to pulse. The probability distribution function for pulse energies P(W) is calculated for a range of Fresnel numbers of the excited volume and collisional dephasing rates. For small values of Fresnel number and dephasing rate, P(W) is a negative exponential distribution. For large values of either, P(W) narrows and approaches a Gaussian-shaped distribution. This occurs because many independent modes contribute to the Stokes emission, making it spatially and/or temporally incoherent.
A pencil shaped region of a Raman-active medium forms a source for Stokes-shifted light via stimulated Raman Scattering (SRS) when driven by an intense laser pulse. Stokes photons are scattered spontaneously at first, and then amplified during propagation through the medium. Since the scattering is initiated by quantum noise, the actual number of photons produced in any given pulse will be highly uncertain, even if the pulse energy of the driving laser is constant1. As the Stokes light propagates through such an extended region, it's coherence properties will be changed. The spatial coherence is determined by the size and shape of the region, which may be characterized by its Fresnel number F, defined as F= A/λSL, where A is the cross-sectional area of the interaction region, which has length L, and λS is the Stokes wavelength. The rate at which the microscopic scatterers (atoms or molecules) are dephased by collisions affects the temporal coherence of the Stokes light, characterised by the temporal coherence parameter Γτ /gL, where Γ is the collisional dephasing rate, τ is the laser pulse duration and g is the steady-state Raman gain coefficient.
The large-scale fluctuations of Stokes light generated by stimulated Raman scattering have been studied experimentally by measuring the probability density function of Stokes pulse energies. The size and shape of the scattering region and the rate of collisional dephasing of the scatterers determine the degree of coherence of the Stokes light, and this in turn determines the form of the probability distribution. The experiments were performed for several diameters of the scattering volume, and for various collisional dephasing rates. The results of the experiments are compared to a recently reported quantum theory, and qualitative agreement is found. The need for a more accurate theory is pointed out.
From a classical point of view the generation of Stokes light by stimulated Raman scattering (SRS) is not possible unless a Stokes input signal is supplied. On the other hand, the gain in SRS can be so large that without input the spontaneous build-up of strong Stokes waves can be observed. A proper quantum theoretical description1 shows that this spontaneous initiation of Stokes generation is due to microscopic random fluctuations originating from quantum noise associated with the dynamical variable Q(t) which is active in the Raman process, e.g., a molecular normal mode coordinate. In highly transient SRS -when the duration of the excitation pulse is shorter than the dephasing time- the Stokes field is proportional to the initial Heisenberg operator Q(0) describing the state of the system just before the excitation pulse is turned on (t=0). In this case the statistics of the final Stokes pulse energy is determined by quantum mechanical uncertainties associated with the random Raman polarization of the initial state2. On the other hand, in a steady state situation the Stokes pulse fluctuations are substantially reduced and the Stokes pulse energy is stabilized as a result of damping processes.