Quantum state reconstruction of spectral field modes: Homodyne and resonator detection schemes

We revisit the problem of quantum state reconstruction of light beams from the photocurrent quantum noise. As is well known, but often overlooked, two longitudinal field modes contribute to each spectral component of the photocurrent (sideband modes). We show that spectral homodyne detection is intrinsically incapable of providing all the information needed for the full reconstruction of the two-mode spectral quantum state. Such a limitation is overcome by the technique of resonator detection. A detailed theoretical description and comparison of both methods is presented, as well as an experiment to measure the six-mode quantum state of pump-signal-idler beams of an optical parametric oscillator above the oscillation threshold.

Figure 1

Scheme to measure electronic quadrature components of each photocurrent signal. The photocurrent is mixed with two electronic references in quadrature.

Figure 2

Schematic view of the balanced HD. Prior to detection, LO field in state $|{\alpha }_{{\omega }_0}\rangle$ is added to the quantum field modes of interest with a controlled phase $\phi$, using a 50:50 beam splitter. Information about the quantum field is retrieved after subtraction of the photocurrents.

Figure 3

Schematic view of RD. The state of interest and the carrier mode are reflected off an optical resonator prior to photodetection. Frequency-dependent losses and phase shifts, controlled by resonator detuning $\Delta$, allow quantum state reconstruction.

Figure 4

Coefficients of Eq. (31) as functions of $\Delta$. Resonator parameters are $d=0.9$ and $\Omega /\gamma =5$.

Figure 5

Scheme to measure electronic quadrature components of two photocurrent signals produced by a pair of light beams. Photocurrents are mixed with two electronic references in quadrature.

Figure 6

Setup for the reconstruction of the OPO beams' covariance matrix. PBS, polarizing beam splitter; BS, 50:50 beam splitter; HS, harmonic separator; IC, input coupler; OC, output coupler (OPO cavity); FR, Faraday rotator.

Figure 7

Pump (green squares), signal (red open circles), and idler (blue solid circles) photocurrent noise power $S_{\mathrm{HD}}^{\left(j\right)}$ as functions of respective resonator detuning ${\Delta }_j$. Lines represent theoretical fits of Eq. (31) either considering (solid) or disregarding (dashed) the respective hidden moment ${\delta }^{\left(j\right)}$.

Figure 8

Quantum correlations between photocurrent components of different beams as functions of resonator detuning. Correlations in phase $C_{\mathrm{p}}^{\left(12\right)}$ (red open circles) and in quadrature $C_{\mathrm{q}}^{\left(12\right)}$ (blue solid circles) are depicted. Theoretical fits of Eqs. (45) and (46) to the data either consider (solid lines) or disregard (dashed lines) hidden four-mode correlations.

Figure 9

Same as Fig. 8, for pump and idler modes.

Figure 10

Same as Fig. 8, for pump and idler modes.