Deconstructing squeezed light: Schmidt decomposition versus the Whittaker-Shannon interpolation (2024)

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  Figure 1

  Schematic of a joint intensity represented in time on the left and frequency on the right. The horizontal width of the joint temporal (spectral) amplitude is denoted by ${\mathcal{T}}_p$ (${\mathcal{B}}_c$), which is the effective pulse duration (bandwidth). The narrow horizontal width at $t_2=0$ is denoted by ${\mathcal{T}}_c=1/{\mathcal{B}}_c$ and is the coherence time of photon pairs.

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  Figure 2

  For the double-Gaussian, from left to right we plot the joint temporal intensity divided by its maximum value with the axes normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$, the joint spectral intensity divided by its maximum value with the axes normalized by $2\pi {\mathcal{B}}_c^{\mathrm{DG}}$, and the Schmidt amplitudes $p_n$ up to $n=39$.

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  Figure 3

  For the double-Gaussian, from left to right we plot ${\overline{G}}^{\left(1\right)}\left(t\right)/\mathrm{\Phi }$ and a few contributions from different Schmidt modes in Eq.(3.20) with the horizontal axis normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$ for $\beta =0.1$, 5, and 10. In each plot the $n=0$ term is the largest Schmidt mode contribution and they get smaller as $n$ increases.

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  Figure 4

  For the double Gaussian, from left to right we plot ${\overline{G}}^{\left(2\right)}(t_1,t_2)/{\mathrm{\Phi }}^2$ (top) and the coherent and incoherent contribution to ${\overline{G}}^{\left(2\right)}(t/2,-t/2)/{\mathrm{\Phi }}^2$ (bottom) with the axes normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$ for $\beta =0.1$, 5, and 10.

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  Figure 5

  For the sinc hat, from left to right we plot the joint temporal intensity divided by its maximum value with the axes normalized by ${\mathcal{T}}_p^{\mathrm{SH}}$, the joint spectral intensity divided by its maximum value with the axes normalized by $2\pi {\mathcal{B}}_c^{\mathrm{SH}}$, and the Schmidt amplitudes $p_n$ up to $n=39$.

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  Figure 6

  For the sinc hat, from left to right we plot ${\overline{G}}^{\left(1\right)}\left(t\right)/\mathrm{\Phi }$ and a few contributions from different Schmidt modes in Eq.(3.20) with the horizontal axis normalized by ${\mathcal{T}}_p^{\mathrm{SH}}$ for $\beta =0.1$, 5, and 10.

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  Figure 7

  For the sinc hat, from left to right we plot ${\overline{G}}^{\left(2\right)}(t_1,t_2)/{\mathrm{\Phi }}^2$ (top) and the coherent and incoherent contribution to ${\overline{G}}^{\left(2\right)}(t/2,-t/2)/{\mathrm{\Phi }}^2$ (bottom) with the axes normalized by ${\mathcal{T}}_p^{\mathrm{SH}}$ for $\beta =0.1$, 5, and 10.

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  Figure 8

  Plot of the approximate joint spectral intensity divided by its maximum value with the axes normalized by $2\pi {\mathcal{B}}_c^{\mathrm{SH}}$. The “false” contributions highlighted by the black dashed circles are due to the periodicity of the function $\widehat{u}\left(\omega \right)$ with a period $T_c$; see the discussion in the paragraph above Eq.(4.14).

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  Figure 9

  From left to right we plot the sinc-hat joint temporal intensity divided by its maximum value, the approximate joint temporal intensity divided by its maximum value, and two contributions to the approximate joint temporal intensity divided by its maximum value when $n=0$ and $n=5$ with the axes all normalized by ${\mathcal{T}}_p^{\mathrm{SH}}$.

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  Figure 10

  For the sinc hat calculated from the pseudo-Schmidt decomposition, from left to right we plot ${\overline{G}}^{\left(1\right)}\left(t\right)/\mathrm{\Phi }$ and a few contributions from different pseudo-Schmidt modes in Eq.(4.16) with the horizontal axis normalized by ${\mathcal{T}}_p^{\mathrm{SH}}$ for $\beta =0.1$, 5, and 10. In each plot the $n=-6$ pseudo-Schmidt mode is the leftmost contribution and they move towards to right as $n$ increases.

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  Figure 11

  For the sinc hat calculated from the pseudo-Schmidt decomposition, from left to right we plot ${\overline{G}}^{\left(2\right)}(t_1,t_2)/{\mathrm{\Phi }}^2$ (top) and the coherent and incoherent contribution to ${\overline{G}}^{\left(2\right)}(t/2,-t/2)/{\mathrm{\Phi }}^2$ (bottom) with the axes normalized by ${\mathcal{T}}_p^{\mathrm{SH}}$ for $\beta =0.1$, 5, and 10.

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  Figure 12

  Using the analytical result (4.27), from left to right we plot ${\overline{G}}^{\left(2\right)}(t_1,t_2)/{\mathrm{\Phi }}^2$ (top) and the coherent and incoherent contribution to ${\overline{G}}^{\left(2\right)}(t/2,-t/2)/{\mathrm{\Phi }}^2$ (bottom) with the axes normalized by ${\mathcal{T}}_p^{\mathrm{SH}}$ for $\beta =0.1$, 5, and 10.

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  Figure 13

  For the double-Gaussian and using the Whittaker-Shannon decomposition, from left to right we plot the joint temporal intensity divided by its maximum value with the axes normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$, the joint spectral intensity divided by its maximum value with the axes normalized by $2\pi {\mathcal{B}}_c^{\mathrm{DG}}$, and the amplitudes $r_{nm}$.

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  Figure 14

  For the double-Gaussian, from left to right we plot ${\overline{G}}^{\left(1\right)}\left(t\right)/\mathrm{\Phi }$ calculated using the Whittaker-Shannon decomposition and a few contributions from different packets in Eq.(6.8) compared with the exact calculation (3.20), with the horizontal axis normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$ for $\beta =0.1,5$, and 10. In each plot the $n=-32$ packet is the leftmost contribution and they move towards to right as $n$ increases.

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  Figure 15

  For the double-Gaussian, from left to right we plot ${\left(\text{sinh}\mathbf{P}\right)}_{nm}$ normalized by the respective maximum values for $\beta =0.1,5$, and 10 with the horizontal axis normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$.

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  Figure 16

  For the double-Gaussian and using Whittaker-Shannon decompositions, from left to right we plot ${\overline{G}}^{\left(2\right)}(t_1,t_2)/{\mathrm{\Phi }}^2$ (top) and the coherent and incoherent contribution to ${\overline{G}}^{\left(2\right)}(t/2,-t/2)/{\mathrm{\Phi }}^2$ (bottom) with the axes normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$ for $\beta =0.1$, 5, and 10.

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  Figure 17

  Schematic of the matrix ${\mathbf{R}}^{\mathrm{I}}$ which has nonzero entries centered at ${\beta }_{n_{\mathrm{I}}{n}_{\mathrm{I}}}$ with a width $d$ and zeros everywhere else. The matrix $\mathbf{K}=\mathit{\beta }-{\mathbf{R}}^{\mathrm{I}}$ and consists of every nonzero element that we set to zero in ${\mathbf{R}}^{\mathrm{I}}$.

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  Figure 18

  For the double-Gaussian, from left to right we plot ${\overline{G}}^{\left(1\right)}\left(t\right)/\mathrm{\Phi }$ calculated using the Whittaker-Shannon decomposition with the full ${\beta }_{nm}$ compared with the approximate calculation using $R^{\mathrm{I}}$ near $t_{\mathrm{I}}=0$ for $d=7,9$ and 11, with the horizontal axis normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$ for $\beta =0.1$, 5, and 10.

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  Figure 19

  Schematic of the matrix $\mathit{\beta }$ partitioned into a set of nonoverlapping matrices ${\mathbf{R}}^J$, each with nonzero values centered at ${\beta }_{n_J{n}_J}$ of size $d_J$ denoted by the red squares. The matrix $\mathbf{K}=\mathit{\beta }-{\mathbf{R}}^{\text{I}}-{\mathbf{R}}^{\text{II}}-\cdots$ and consists of every other nonzero element contained in $\mathit{\beta }$.

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  Figure 20

  For the double-Gaussian joint amplitude, we plot ${\overline{G}}^{\left(1\right)}\left(t\right)/\mathrm{\Phi }$ calculated using the Schmidt decomposition (top) and Whittaker-Shannon decomposition (bottom) as well as a few contributions from each calculation, for $\beta =150$ with the horizontal axis normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$. In the bottom plot, the smallest and leftmost contribution is from the $n=-4$ packet and the largest is the $n=0$ packet.

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  Figure 21

  For the double-Gaussian joint amplitude, we plot ${\overline{G}}^{\left(2\right)}(t_1,t_2)/{\mathrm{\Phi }}^2$ (top), and the coherent and incoherent contribution to ${\overline{G}}^{\left(2\right)}(t/2,-t/2)/{\mathrm{\Phi }}^2$ (bottom) calculated using the Schmidt decomposition, for $\beta =150$ with the horizontal and vertical axis normalized by ${\mathcal{T}}_p^{\mathrm{DG}}$.

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  Figure 22

  From left to right we plot the joint temporal intensity, the joint spectral intensity, and the Schmidt amplitudes generated from a dual-pump spontaneous four-wave mixing process. The two pump functions are centered at the wavelengths ${\overline{\lambda }}_{\text{P1}}=1.556\mu \mathrm{m}$ and ${\overline{\lambda }}_{\text{P2}}=1.547\mu \mathrm{m}$ and each have temporal FWHM of 100ns and an energy of ${10}^3$ pJ. The generated photons are centered at ${\overline{\lambda }}_{\text{S}}=1.552\mu \mathrm{m}$ (${\overline{\omega }}_{\text{S}}/2\pi =193.164$ THz) and have a bandwidth on the order of a GHz. The ring resonator has quality factors $Q_{\text{P1}}=1529378,Q_{\text{P2}}=3844257$, and $Q_{\text{S}}=2704405$ for the three modes and a nonlinear coupling $\mathrm{\Lambda }=5$ THz [4].

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  Figure 23

  Joint temporal intensity calculated using the Whittaker-Shannon decomposition.

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  Figure 24

  Plot of ${\overline{G}}^{\left(1\right)}\left(t\right)$ calculated using the Schmidt decomposition (top) and Whittaker-Shannon decomposition (bottom). In the top plot the $n=10$ Schmidt corresponds to the larger contribution. In the bottom plot the $n=-20$ is the leftmost packet.

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  Figure 25

  Plot of $G^{\left(2\right)}(t_1,t_2)$ (top) and the coherent and incoherent contribution to $G^{\left(2\right)}(t/2,-t/2)$ calculated using the Whittaker-Shannon decomposition.

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  Figure 26

  Schematic of the sinc-hat joint intensities showing the extra contributions to the horizontal widths in the respective corners.

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  Figure 27

  Schematic of a general joint temporal amplitude with a pulse duration and coherence time denoted by ${\mathcal{T}}_p$ and ${\mathcal{T}}_c$, respectively, in the original and rotated coordinate system.

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