50 GLYPH<c87> (a) (b) 300 K 4.2 K LO F digitizer 2.5 K C$_{1}$ C$_{2}$ hybrid ring - S + V S + V signal 2 S ancilla 2 V GLYPH<c99>$_{1}$ amplification and detection chain 1 amplification and detection chain 2 GLYPH<c99>$_{2}$ 300 mK circulator isolation 21 dB cryogenic HEMT amplifier mixer and local oscillator filter 12 bit resolution 4x10 samples/s 8 high frequency amplifier low frequency amplifier 300 K Figure 1: (a) Schematics of the dual path amplification and detection setup. The ancilla port of the 180 ◦ hybrid ring is terminated by a 50 Ω load at T = 300 mK. The split signals are amplified and detected in separate chains, which add the noise χ$_{1}$ and χ$_{2}$ to the signal. (b) Amplification and detection chain. The noise temperature is dominated by the cryogenic amplifiers anchored at 4 . 2 K. The isolation of the hybrid ring (40 dB) and the circulator prevents correlated noise contributions from the two chains. The phase difference between C$_{1}$ and C$_{2}$ is close to 180 $^{◦}$. the fourth port of the hybrid ring. Since we assume full knowledge of V , an obvious choice is the vacuum or, as in this work, a weak thermal state. Vacuum or thermal states at gigahertz frequencies can be prepared reliably in an experiment by controlling the temperature [20, 21]. In the next step, we compute suitable correlations of the form 〈 C ℓ $_{1}$C m 2 〉 , where ℓ, m ∈ N$_{0}$ and the brackets denote ensemble averaging. In contrast to Hanbury Brown and Twiss experiments based on intensity (power) correlations [20], our method is devoted to the correlations of field quadratures (voltages). For the first signal moment, the mean values 〈 χ$_{1}$ 〉 and 〈 χ$_{2}$ 〉 vanish and one obtains 〈 S 〉 = 〈 C$_{1}$ 〉 /G = −〈 C$_{2}$ 〉 /G . All higher moments of signal and noise can now be calculated by induction: 〈 S $^{n}$〉 = − 〈 C n − 1 1 C$_{2}$ 〉 /G n − n − 1 ∑ k =1 k ∑ j $_{=0}$( n − 1 k )( k j )$_{〈}$ S n − $^{k}$〉〈 V $^{j}$〉〈 χ k − j 1 〉 + n − 1 ∑ k =0 k ∑ j $_{=0}$( n − 1 k )( k j )$_{〈}$ S n − k − $^{1}$〉〈 V j $^{+1}$〉〈 χ k − j 1 〉 , (1) 3