Quantized nonlinear Gaussian-beam dynamics: Tailoring multimode squeezed-light generation

We present a general, second quantization procedure for multitransverse-spatial-mode Gaussian-beam dynamics in nonlinear interactions. Previous treatments have focused on the spectral density and angular distribution of spatial modes. Here we go a layer deeper by investigating the complex transverse spatial mode in each angular spatial mode. Furthermore, to implement the theory, we simulate four-wave mixing and parametric down-conversion schemes, showing how one can elucidate and tailor the underlying multitransverse-spatial-mode structure along with its quantum properties.

Figure 1

A general unitary evolution generated in a ${\chi }^{\left(3\right)}$ material. In each spatial mode, there are actually many transverse spatial modes interacting not only within the mode, but potentially with many other modes in the adjacent spacial modes. Our theory can accommodate single- or dual-pump modes and single- or dual-vacuum input modes. Furthermore, a vacuum mode, for example, $\widehat{\mathit{b}}$, could be replaced with a seed beam and evolution could be calculated with our semiclassical-beam theory for nonlinear optics [45] and incorporated into this second quantization procedure.

Figure 2

Setup and energy diagrams describing the PSR effect in ${}^{87}\mathrm{Rb}$. In (a) we show a linear polarized pump beam interacting with vacuum fluctuations in the orthogonal polarization. Two tensor elements of the semiclassical susceptibility tensor survive: (b) depicts the single-photon resonance associated with the ${\chi }_{xyxy}$ element, that is, a response in $x$ due to stimulation in $yxy$, and (c) describes the two-photon resonance associated with the ${\chi }_{xyyx}$ element, that is, a response in $x$ due to stimulation in $yyx$. The input and output fields are copropagating.

Figure 3

The PSR noise matrices for the single-spatial-mode quadratures ${\widehat{\mathit{X}}}_1$ and ${\widehat{\mathit{X}}}_2$, in (a) the absence of TSM cross talk and (b) $p$-mode cross talk. The noise in the quadrature is depicted along the diagonal as the projection of the multimode Wigner function onto the particular TSM quadrature. The dashed black Gaussian is the projection of the vacuum state; thus the diagonal elements allow one to quickly observe how the noise suppression, if at all, is distributed among the spatial modes. The inset number is the amount of squeezing given in decibels. The off-diagonal elements represent the covariance between different spatial modes in the quadrature. In (a) there is no cross talk between the $x$-polarization modes, thus the covariance is zero. In (b) the $p$-mode cross talk is present and indicated by the nonzero off-diagonal elements.

Figure 4

The noise matrices for the joint quadratures ${\widehat{\mathit{X}}}_1$ and ${\widehat{\mathit{X}}}_2$, when full cross talk is present in a two-photon resonance scheme. The noise in the quadrature is depicted by the solid Gaussian along the diagonal, and the off-diagonal elements represent the covariance between different spatial modes in the quadrature. Red is negative, white is zero, and black is positive covariance. The inset gives the squeezing in decibels.

Figure 5

(a) Histogram of the photon-creation matrix elements (a.u.) given by Eq. (25) and (b) the average photon number per mode for degenerate four-wave mixing in a two-photon resonant scheme where full TSM cross talk is present. For the blue scale, $\max =1.25$. The strongest response is evident at the (0,0) mode.

Figure 6

(a) Schematic of our PDC simulation, (b) the energy-level diagram, (c) the noise ellipse of an ideal squeezed state with $\overline{n}=1$ (inset gives squeezing in dB), and (d) the density plot, discussed in the main text, which reveals a parameter island where (0,0) coupling is dominant and most of the down-converted photons are sent to the $u_{0,0}$ modes. This plot shows $|M_{00}|/{\sum }_{i,j}\left|M_{ij}\right|\times n_{00}/{\sum }_i{n}_{ii}$ as a function of the pump waist $w_{\mathrm{P}}$ and PDC detection waist $w_{\mathrm{DC}}$ in $\mu \mathrm{m}$.

Figure 7

(a) Noise matrices for the joint quadratures ${\widehat{\mathit{X}}}_1$ and ${\widehat{\mathit{X}}}_2$, when full cross talk is present in a type-II PDC scheme, and (b) the noise matrices when the crystal is pumped by the first eigenmode. The noise in the quadrature is depicted by the solid Gaussian along the diagonal, and the off-diagonal elements represent the covariance between different spatial modes in the quadrature. Red is negative, white is zero, and black is positive covariance. In (a), although $\overline{n}=1$, we can see a reduction in the maximum squeezing performance, compared with the ideal situation of only a single transverse spatial mode. In (b) we see that the squeezing changed marginally but the covariance elements experienced the most significant change. This suggests the mode coupling has been altered, and this can be seen clearly by comparing Figs. 8 and 9. We limit $-1\le \ell \le 1$ and $0\le p\le 2$ for display purposes.

Figure 8

(a) Histogram of the photon-creation matrix elements (a.u.) given by Eq. (25), (b) the average photon number per mode for our PDC scheme [corresponding to Fig. 7], and (c, d) the noise ellipses and average photon number for each of the eigenmodes of squeezing, progressing from left to right, starting with ${\lambda }_1$. In (c) the crystal is pumped with a $u_{0,0}$ mode and in (d) with the ${\lambda }_1$ eigenmode [corresponding to Fig. 7]. In (d) we observe an increase in the amount of noise suppression and a shift in the concentration of ${\overline{n}}_{\lambda }$ toward the first eigenmode. The blue scale is set to $\max =0.75$ for (b)–(d).

Figure 9

(a) Histogram of the photon-creation matrix elements (a.u.) given by Eq. (25) and (b) the average photon number per mode when the crystal is pumped with the ${\lambda }_1$ eigenmode. In (c) and (d) the pump mode is a $u_{0,0}$ mode but the waist has been increased to $400\mu \text{m}$. This configuration is ideal for heralding efficiency since it suppresses the off-diagonal coupling. In other words, it insures that the photon pairs are in the same TSM. The blue scale is set to $\max =0.5$ for (b) and $\max =0.1$ for (d).