Bloch-Messiah reduction for twin beams of light

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.

Figure 1

Schematic representation of a collinear type-I PDC in the pulsed regime. Pump pulse with the central frequency ${\omega }_p$ passes through a ${\chi }^{\left(2\right)}$ nonlinear crystal. For $z>0$ this pulse is accompanied by the signal and the idler pulses at frequencies close to ${\omega }_0={\omega }_p/2$. The point $z_0$ is the origin of the “interaction picture reference frame” and can be chosen anywhere inside the crystal. Relative delay between the pulses at the output is caused by the crystal positive dispersion: waves with a higher frequency travel at a lower group velocity.

Figure 2

The squeezing matrix $\mathrm{\Gamma }=-i{\mathbb{H}}_I$ as a function of detunings $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\prime }$ for frequency-nondegenerate PDC. The matrix elements are essentially nonzero at the intersection of the phase-matched area, where the sinc function in Eq. (72) is close to 1, and the pump-limited area, where $|\mathrm{\Omega }+{\mathrm{\Omega }}^{\prime }|$ is within the pump bandwidth. The two regions where the elements of $\mathrm{\Gamma }$ are close to maximum correspond to the signal and the idler bands.

Figure 3

Squeezing eigenvalues of twin beams generated in frequency-nondegenerate PDC. For well-separated signal and idler spectral regions all eigenvalues have multiplicity 2 and the logarithm of eigenvalue is a linear function of the mode pair number.

Figure 4

Modal functions of the squeezing eigenmodes of twin beams generated in frequency-nondegenerate PDC. The corresponding eigenvalues $\lambda$ of the squeezing matrix $\mathrm{\Gamma }$ are shown for each mode. The squeezing eigenvalues are given by $\left|\lambda \right|$ and have multiplicity 2, as prescribed by Theorem t1. Each modal function represents a concatenation of two Schmidt modal functions localized in the idler and the signal frequency bands.

Figure 5

JSA as a function of the detunings $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\prime }$ (contour map) and its double-Gaussian approximation (red dashed line).

Figure 6

Amplitude and phase of the modal function of the (properly delayed) signal Schmidt mode $C_1\left({\mathrm{\Omega }}_1\right)=h_1\left({\tau }_1\delta {\mathrm{\Omega }}_1\right)e^{i{\zeta }_1{\tau }_1^2\delta {\mathrm{\Omega }}_1^2}$ obtained by Gaussian modeling and by numerical SVD of JSA for two BBO crystal lengths: $L=0.5\mathrm{mm}$ in (a) and $L=2.0\mathrm{mm}$ in (b). A longer crystal corresponds to a narrower bandwidth. The chirp rate increases with length but for the considered example (BBO pumped at 397.5 nm) is negligible at any crystal length.

Figure 7

The squeezing matrix $\mathrm{\Gamma }$ as function of the signal and idler detunings $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\prime }$, respectively, for type-I PDC close to degeneracy. The signal and the idler bands are not separated in this case.

Figure 8

Squeezing eigenvalues of twin beams generated in PDC close to degeneracy in frequency. Only four first eigenvalues create pairs, as required by Theorem t1. At higher mode number the emission of two signal or two idler photons in one elementary act becomes significant and the Theorem t1 is not valid any more.

Figure 9

Modal functions of the squeezing eigenmodes of twin beams generated in PDC close to degeneracy in frequency. The corresponding eigenvalues of the squeezing matrix $\mathrm{\Gamma }$ are shown for each mode. The four first modal functions can be considered as concatenations of the Hermite-Gauss functions in the signal and the idler bands. However, the two last modal functions deflect from this rule around the origin. For these and higher order modes the twin beam analysis is not valid any more.