\section{Annexe}

The laser source is a titanium-sapphire mode-locked oscillator delivering $\sim 140 \textrm{fs}$ pulses ($\Delta \lambda \simeq 6 \textrm{nm}$ FWHM) centered at $795 \textrm{nm}$ with a repetition rate of $76 \textrm{MHz}$. Its second harmonic at $397 \textrm{nm}$ serves to synchronously pump an OPO, which consists of a 2mm BIBO crystal contained within a 4m long ring cavity exhibiting a finesse of 27 and escape efficiency of 95\%. In this configuration, the OPO reaches its threshold at a mean pump power of $\sim 75 \textrm{mW}$. Below threshold and in the absence of a seed, the device generates squeezed vacuum with a squeezing level of $\sim -6 \textrm{dB}$ (corrected) when projected onto the unshaped LO as seen in Fig.~\ref{fig-fullsqz}. 

\PDFfig{fig-fullsqz}{fullsqz-v2.pdf}{Noise trace observed with an unshaped LO pulse shape. The state exhibits squeezing and anti-squeezing values of $-5.9 \textrm{dB}$ and $7.8 \textrm{dB}$, respectively.}{80mm}


In order to lock the cavity length to the inter-pulse spacing, a near-infrared beam is phase-modulated at $1.7 \textrm{MHz}$ with an electro-optic modulator (EOM) and injected into the cavity in a direction counter-propagating to the pump and seed. Locking of the cavity is then accomplished with a Pound-Drever-Hall strategy. 

Frequency correlations of the state are investigated with homodyne detection where the local oscillator pulse form is manipulated with a pulse shaper. The $4 f$-configuration shaper is constructed in a reflective geometry with a programmable 512\,x\,512-element liquid-crystal modulator in the Fourier plane.  Application of a periodic spatial grating to the spatial light modulator induces diffraction of the spectrally-dispersed light. The amplitude and phase of the diffracted spectrum are independently controlled by the groove depth and position of the spatial grating, respectively \cite{nelson2005}.

The OPO output is projected by the homodyne detection technique onto the spectrally filtered LO pulse shape to assess the quadrature noise content of the corresponding spectral region. Light detection is performed with silicon photodiodes ($\sim 90\%$ detection efficiency, 100MHz detection bandwidth), and the noise level of the squeezed vacuum is examined at 1MHz. The homodyne visibility is $92 \%$. The cumulative loss of the system is taken to be $\sim 25 \%$ and the measured signals are corrected accordingly. 

\PDFfig{fig-pquad}{figure-pquad.pdf}{Noise correlation matrix defined as $C^{p}_{i,j} = \langle p_{i}p_{j} \rangle / \sqrt{\langle p_{i}^2 \rangle \langle p_{j}^2 \rangle} -  \delta_{i,j} \langle p_{i}^2 \rangle_{\textrm{vac}} / \langle p_{i}^2 \rangle$ for the $\hat{p}$ quadrature. The diagonal displays excess noise while the off-diagonal elements demonstrate quantum correlations among disparate spectral regions.}{75mm}

The optical spectrum of the local oscillator is divided into ten frequency bands of equal energy as seen in Fig.~1 of the main text, and spectral holes between the individual regions are imposed to ensure the absence of any spectral overlap. Importantly, the supplemental loss incurred from including these gaps is not accounted for when correcting the noise levels. 
The $\hat{x}$ quadrature is defined as the field quadrature of lowest noise for the unshaped LO pulse. 
The noise dependence on LO phase is then examined for all 55 possible pulse shapes and compared to that of the unshaped LO reference. Each individual trace is observed to follow the phase dependence of the reference, which confirms that the lowest noise level for every spectral combination is present in the $\hat{x}$ quadrature, i.e., there is no rotation of the squeezing ellipse between successive measurements. 


The noise content of both the $\hat{x}$- and $\hat{p}$-quadratures for each spectral region and all possible pairs of regions are measured as shown in Fig.~\ref{fig-sqzTraces}. The individual covariance elements are constructed according to the following equation:

\begin{eqnarray}
\langle x_{i} x_{j} \rangle &=& 
\left[ \langle (x_{i} + x_{j})^2 \rangle - \frac{P_{i}}{P_{i}+P_{j}} \langle x_{i}^2 \rangle - \frac{P_{j}}{P_{i}+P_{j}} \langle x_{j}^2 \rangle \right] \nonumber \\ && \times \, \frac{P_{i}+P_{j}}{2 \sqrt{P_{i} P_{j}}},
\end{eqnarray}
where $P_{i}$ and $P_{j}$ are the optical powers of frequency bands $i$ and $j$, respectively, which are measured with the homodyne photodiodes. 

\textbf{Data Analysis}. Noise fluctuations and correlations departing from the vacuum level for the $\hat{p}$-quadrature are represented in Fig.~\ref{fig-pquad}. Similar to the $\hat{x}$ quadrature, the vast majority of both the excess noise (diagonal peaks) and correlation (off-diagonal peaks) are localized in the spectral wings. As opposed to the $\hat{x}$ quadrature, however, the correlations within the $\hat{p}$ quadrature are positive. 

The purity $\mathcal{P}$, which is an intrinsic property of the state, is accessible from the covariance matrix with the relation $\mathcal{P} = 1 / \sqrt{\textrm{det}(\mathcal{C})}$, where $\mathcal{C}$ is the full covariance matrix of the state \cite{Adesso2007}. This relationship reveals that $\mathcal{P} \simeq 0.45 \pm 0.09$. 
The full covariance matrix is then eigendecomposed, where it is observed that although the individual $\hat{x}$ and $\hat{p}$ block eigenvectors are very similar, they are not identical. 
In general, two matrices are simultaneously diagonalizeable only if they commute. For instance, the commutator of the $\hat{x}$ and $\hat{p}$ covariance matrices for two-mode squeezed vacuum is zero even in the presence of spectrally-uniform loss, and a single eigenvector set perfectly decorrelates both quadratures. The introduction of a spectrally-dependent loss, however, spoils the commutation of the $\hat{x}$ and $\hat{p}$ blocks, and the magnitude of a commutator element is directly proportional to the spectral asymmetry of the loss. The output coupler of the SPOPO does not have a perfectly flat transmission window over the bandwidth of the pulse, and the magnitude of the commutator elements is largest in the spectral wings. Consequently, this asymmetry explains the fact that it is not possible to perfectly remove both the $\hat{x}$ and $\hat{p}$ correlation with a single basis change. 

\begin{table}
\begin{center}
\caption{Squeezing values and uncertainties (linear scale) for the modes retrieved from the twenty dimensional covariance matrix. Eight modes are clearly non-classical while a ninth mode is not included in a conservative estimate. \label{tab-sqz}}
\medskip
{
\begin{tabular}{c|c|c}
\hline
{Mode} & Squeezing Level & Anti-Squeezing Level \\
\hline
1 & $0.38 \pm 0.07$ & $3.86 \pm 0.12$ \\
2 & $0.48 \pm 0.06$ & $3.62 \pm 0.07$ \\
3 & $0.58 \pm 0.07$ & $2.74 \pm 0.10$ \\
4 & $0.74 \pm 0.06$ & $1.96 \pm 0.06$ \\
5 & $0.83 \pm 0.05$ & $1.41 \pm 0.06$ \\
6 & $0.90 \pm 0.03$ & $1.17 \pm 0.04$ \\
7 & $0.93 \pm 0.03$ & $1.11 \pm 0.03$ \\
8 & $0.95 \pm 0.02$ & $1.06 \pm 0.03$ \\
9 & $0.97 \pm 0.02$ & $1.03 \pm 0.03$ \\
10 & $0.98 \pm 0.02$ & $1.00 \pm 0.02$ \\
\end{tabular}
}
\end{center}
\end{table}

Yet, the commutator elements have a small magnitude.
Therefore, as a means of generating a mode set that efficiently removes the correlation of the original frequency bands, the ten most noise-robust eigenmodes (see statistical sampling procedure detailed below) are selected from the total eigenvector set. These vectors, which are drawn from both the $\hat{x}$ and $\hat{p}$ blocks, are orthogonalized with a Gram-Schmidt procedure, and the covariance matrix is subsequently re-expressed in this basis. The resulting matrix is nearly diagonal and contains the squeezing values for each orthogonalized mode on its diagonal. 


The error level associated with each of these squeezing values is subsequently estimated with a stochastic method. 
The data underlying every trace of Fig.~\ref{fig-sqzTraces} is collected for twenty seconds in order to accrue statistics about the individual squeezing and anti-squeezing values. The noise value for a given trace is then drawn from a normal distribution with a mean provided by the red (green) line, which represents the average of all identified peaks (valleys), and a variance specified by the variance of the peak (valley) amplitudes. After randomly sampling from these 110 normal distributions, the corresponding covariance matrix is constructed in the appropriate manner. A collection of $10^{4}$ individual matrices is assembled, and the Gram-Schmidt orthogonalization procedure is repeated for every matrix, which provides both a squeezing spectrum and orthonormal mode set. The mean squeezing spectrum from this procedure is shown in Fig.~\ref{fig-eigentrace} and the individual noise levels are detailed in Table~\ref{tab-sqz}. Likewise, the average mode structures corresponding to the six highest squeezing values are shown in Fig.~4 of the main text.  

\PDFfig{fig-eigentrace}{eigentrace-v2.pdf}{Mean noise levels and uncertainties (dB) for each of the orthogonalized Gramm-Schmidt modes. The eight modes that are clearly distinguished from the vacuum level are indicated in blue, and modes ascribed to vacuum fluctuations are indicated in red.}{85mm}


\PDFfigDouble{fig-sqzTraces}{sqz-Traces-v2.pdf}{Squeezing traces for determination of the ten frequency band covariance matrix. The traces are sequentially arranged according to the index scheme $\{i,j\}$, which indicates that the pulse shaper selects frequency bands $i$ and $j$ (e.g., \{1,1\}, \{1,2\},...,\{1,10\},\{2,2\},\{2,3\},...). In each trace, the black line represents the vacuum level while the green and red lines indicate the average squeezing and anti-squeezing levels, respectively. The LO phase is depicted with arbitrary units.}{180mm}

The mean spectrum is generally noise-robust and reveals that eight modes are statistically distinct from the vacuum level. Although a ninth mode may also be considered to exhibit non-classicality, its uncertainty is at the vacuum level, and is therefore not included in a conservative estimate. 


\

The research is supported by the ANR project \emph{Qualitime} and
the ERC starting grant \emph{Frecquam}. C. Fabre is a member of
the Institut Universitaire de France.

\bibliographystyle{apsrev}

