Full characterization of a highly multimode entangled state embedded in an optical frequency comb using pulse shaping

We present a detailed analysis of the multimode quantum state embedded in an optical frequency comb generated by a synchronously pumped optical parametric oscillator (SPOPO) [J. Roslund, R. Medeiros de Araújo, S. Jiang, C. Fabre, and N. Treps, Nat. Photon. 8, 109 (2014)]. The full covariance matrix of the state is obtained with homodyne detection where the local oscillator is spectrally controlled with pulse-shaping techniques. The resulting matrix reveals genuine multipartite entanglement. Additionally, the beam is comprised of several independent eigenmodes that correspond to specific pulse shapes. The experimental data is confirmed with numerical simulations. Finally, the potential to create continuous-variable cluster states from the quantum comb is analyzed. Multiple cluster states are shown to be simultaneously embedded in the SPOPO state, and these states can be revealed by a suitable basis change applied to the measured covariance matrix.

Figure 1

Parametric down-conversion of a femtosecond comb. The splitting of a single pump photon of frequency $2{\omega }_0$ by pathway 1 creates entanglement between the frequencies ${\omega }_a$ and ${\omega }_b$. An additional pump photon may down-convert by pathway 2 and correlate frequencies ${\omega }_a$ and ${\omega }_c$. A correlation is also established between frequencies ${\omega }_b$ and ${\omega }_c$ by virtue of their mutual link to ${\omega }_a$. In this manner, every frequency of the down-converted comb becomes correlated with every other member of the comb.

Figure 2

Four-mode linear, square, and T-cluster states (graphs and respective adjacency matrices $V_{\text{lin}}$, $V_{\text{square}}$, $V_{\text{T}}$).

Figure 3

Experimental layout for the creation and characterization of multimode frequency combs. A Ti-sapphire oscillator produces a 76 MHz train of $\sim$140 fs pulses centered at 795 nm. Its second harmonic synchronously pumps an OPO. The cavity output is analyzed with homodyne detection, where the spectral amplitude and phase of the local oscillator (LO) are shaped. The LO shaper is depicted here in a transmissive geometry for clarity. By varying the relative phase between the shaped LO and the SPOPO output, the $x$- and $p$-quadrature noises of the quantum state projected onto the LO mode are measured.

Figure 4

Experimentally measured quantum noise matrices for the (a) $x$ and (b) $p$ quadratures. The noise correlation matrix is defined as in Eq. (11). Each matrix reveals significant correlations among the frequency bands of the comb.

Figure 5

The PPT (blue) inseparability criteria for all 127 bipartite combinations of the eight spectral bands. All 127 bipartitions possess a PPT value below the entanglement boundary of 0.0, which indicates complete nonseparability for the state. The PPT criteria are also applied to a simulated single-mode squeezed state (red) with noise parameters corresponding to the first supermode. The single-mode PPT values are ordered according to the full PPT values. The black dotted line represents the mean single-mode PPT value. All full PPT bipartitions below this line are indicative of multimode character.

Figure 6

Mean noise levels and uncertainties (dB) for each of the orthogonalized Gram-Schmidt modes. The mean eigenspectra are shown for eight (red), six (blue), and four (green) unique frequency bands. The simulated eigenvalues corresponding to eight frequency bands are shown for comparison (black).

Figure 7

(a) Retrieved experimental supermodes with the spectral gaps removed. The field of each supermode is measured with spectral interferometry. (b) Noise traces corresponding to each of the experimental supermodes.

Figure 8

Amplitude spectra corresponding to each of the orthogonal supermodes retrieved from the covariance matrix shown in Fig. 4.