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arXiv:1903.06578 · Physical Review A

Bloch-Messiah reduction for twin beams of light

D. B. Horoshko, Luca La Volpe, Francesco Arzani, Nicolas Treps, Claude Fabre, Mikhail I. Kolobov

Published Jul 18, 2019• 41 citations
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Abstract

We study the Bloch-Messiah reduction of parametric down-conversion of light in the pulsed regime with a nondegenerate phase matching providing generation of twin beams. We find that in this case every squeezing eigenvalue has multiplicity at least two. We discuss the problem of ambiguity in the definition of the squeezing eigenmodes in this case and develop two approaches to unique determination of the latter. First, we show that the modal functions of the squeezing eigenmodes can be tailored from the Schmidt modes of the signal and idler beams. Alternatively, they can be found as a solution of an eigenvalue problem for an associated Hermitian squeezing matrix. We illustrate the developed theory by an example of frequency nondegenerate collinear twin beams generated in beta barium borate crystal. In this example we demonstrate how the squeezing eigenmodes can be approximated analytically on the basis of the Mehler's formula, extended to complex kernels. We show how the multiplicity of the eigenvalues and the structure of the eigenmodes are changed when the phase matching approaches the degeneracy in frequency.

DOI
10.1103/physreva.100.013837

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We study the Bloch-Messiah reduction of parametric down-conversion of light in the pulsed regime with a nondegenerate phase matching providing generation of twin beams. We find that in this case every squeezing eigenvalue has multiplicity at least two. We discuss the problem of ambiguity in the definition of the squeezing eigenmodes in this case and develop two approaches to unique determination of the latter. First, we show that the modal functions of the squeezing eigenmodes can be tailored from the Schmidt modes of the signal and idler beams. Alternatively, they can be found as a solution of an eigenvalue problem for an associated Hermitian squeezing matrix. We illustrate the developed theory by an example of frequency nondegenerate collinear twin beams generated in beta barium borate crystal. In this example we demonstrate how the squeezing eigenmodes can be approximated analytically on the basis of the Mehler's formula, extended to complex kernels. We show how the multiplicity of the eigenvalues and the structure of the eigenmodes are changed when the phase matching approaches the degeneracy in frequency.
We discuss the Bloch-Messiah decomposition for the broadband squeezed light generated by type-I parametric down-conversion with a monochromatic pump. Using an exact solution for this process, we evaluate the squeezing parameters and the corresponding squeezing eigenmodes. Next we consider the Magnus expansion of the quantum-mechanical evolution operator for this process and obtain its first three approximation orders. Using these approximated solutions, we evaluate the corresponding approximations for the Bloch-Messiah decomposition. Our results allow us to conclude that the first-order approximation of the Magnus expansion is sufficient for description of the broadband squeezed light for squeezing values below 12.5 dB. For higher degrees of squeezing we show fast convergence of the Magnus series, providing a good approximation for the exact solution already in the third order. We propose a quantitative criterion for this ultrahigh-gain regime of parametric down-conversion when the higher-order terms of the Magnus expansion, known in the literature as the operator-ordering effects, become necessary.
We analyze the spectral properties of squeezed light produced by means of pulsed, single-pass degenerate parametric down-conversion. The multimode output of this process can be decomposed into characteristic modes undergoing independent squeezing evolution akin to the Schmidt decomposition of the biphoton spectrum. The main features of this decomposition can be understood using a simple analytical model developed in the perturbative regime. In the strong pumping regime, for which the perturbative approach is not valid, we present a numerical analysis, specializing to the case of one-dimensional propagation in a beta-barium borate waveguide. Characterization of the squeezing modes provides us with an insight necessary for optimizing homodyne detection of squeezing. For a weak parametric process, efficient squeezing is found in a broad range of local oscillator modes, whereas the intense generation regime places much more stringent conditions on the local oscillator. We point out that without meeting these conditions, the detected squeezing can actually diminish with the increasing pumping strength, and we expose physical reasons behind this inefficiency.
We analyze an optical-phase feed-forward scheme to produce a squeezed-light beam from intensity-correlated twin beams generated by nondegenerate parametric amplification. The quadrature measurement of the idler beam at the output of the parametric amplifier is linearly fed forward onto the output signal beam with an electro-optic phase modulator. The minimum available squeezing factor is shown to be 1/(2g - 1), where g is the parametric amplifier gain, and is equal to the nonclassical intensity-noise reduction factor of the original twin beams at the output of the parametric amplifier. By changing the frequency of the input signal beam, one can generate tunable squeezed light with this scheme for application in precision spectroscopy.
Optical parametric amplifiers (OPAs) configured as phase-sensitive amplifiers (PSAs) can be used for noiseless optical image amplification, generation of non-classical states of light, and, in particular, for multimode squeezed light generation. For all these applications and for effective use of the quantum properties of the multimode PSA, we need to know its independently squeezed (or amplified) eigenmodes. First we present the quantum theory of a spatially multimode traveling-wave PSA pumped by a high-power pump beam with arbitrary spatial profile. To quantitatively identify the Green's function of the PSA, we have developed a semi-analytical coupled-mode-theory of the PSA using 2D Hermite-Gaussian mode expansions of the signal- and pump-beam spatial distributions. By using Green's functions of the classical OPA, we calculate the normallyordered quadrature correlators at its output, which provide complete quantum description of the phase-sensitive OPA and enable determination of its independently squeezed eigenmodes. We find the number of the supported eigenmodes and their shapes for a spatially broadband frequency-degenerate optical parametric amplifier with elliptical Gaussian pump. We conclude by discussing our recent extensions of the coupled-mode-theory to study higher-order pump modes, compact representation of the PSA eigenmodes (especially convenient for experiments), and effect of non-zero phase mismatch on the PSA eigenmodes. We expect that the results from our model can be used for optimum mode matching in phase-sensitive image amplification and multimode squeezing generation.
Twin beams generated from a nondegenerate optical parametric amplifier (NOPA) of gaing possess1 an intensity correlation that falls below the quantum limit by (2g – 1). This occurs whenever the input idler to the NOPA is in the vacuum state. Such input is practically advantageous because it makes the NOPA output intensities phase insensitive. When a non-vacuum idler input state is chosen, the NOPA output intensities become phase sensitive. For a proper input state and a suitable pump phase the output intensity correlation falls below the quantum limit by ( g + g − 1 ) 2 , which is 3 dB superior to the vacuum-idler-input result for g >> 1 and is equivalent to the quantum-noise reduction obtainable from a squeezed state generated with the same gain. Thus the sensitivity improvement achievable with NOPA twin beams by using direct detection is equivalent to that obtainable with squeezed states by using homodyne detection. The above equivalence also provides a new scheme for the generation of pulsed squeezed light that, in addition to being simple, automatically generates a matched local oscillator2 for homodyne detection of the squeezed pulses.
Pulsed twin beams of light have been generated using an optical-parametric amplifier that is pumped by the second harmonic of a mode-locked and Q-switched neodymium:yttrium aluminum garnet laser. The intensity noise levels on the direct-detected signal and idler beams are found to be correlated by more than 15 dB, and the subtracted noise falls below the quantum limit by more than 6 dB (75%). To our knowledge, this is the first observation of pulsed twin beams of light, and yields the highest quantum-noise reduction observed in any experiment to date. Our measurements are in excellent agreement with the quantum theory of a nondegenerate optical-parametric amplifier.