Experimental characterization of the Gaussian state of squeezed light obtained via single passage through an atomic vapor

We show that the description of light in terms of Stokes operators in combination with the assumption of Gaussian statistics results in a dramatic simplification of the experimental study of fluctuations in the light transmitted through an atomic vapor: no local oscillator is required, the detected quadrature is easily selected by a wave-plate angle, and the complete noise ellipsis reconstruction is obtained via matrix diagonalization. We provide empirical support for the assumption of Gaussian statistics in quasiresonant light transmitted through an ${}^{87}\mathrm{Rb}$ vapor cell and we illustrate the suggested approach by studying the evolution of the fluctuation ellipsis as a function of laser detuning. Applying the method to two light beams obtained by parting squeezed light in a beam splitter, we have measured the entanglement and quantum Gaussian discord.

Figure 1

Experimental scheme for the measurement of the Stokes operator $S_{\theta }$. The angle $\theta$ is selected by the orientation of the half-wave-plate axis. PBS, polarization beam splitter; PD, photodetector.

Figure 2

Experimental setup used for the characterization of a two-mode state corresponding to the output of the $\sim 50\%$ nonpolarizing beam splitter (BS). On each output a Stokes operator measurement setup is implemented. The balanced detector outputs can be either added or subtracted for global operator measurements.

Figure 3

Normalized histograms of the filtered ($\mathrm{\Omega }=3$ MHz, $\mathrm{\Delta }=400$ kHz) time-domain fluctuations of the Stokes operators. Solid lines: Gaussian functions with the same variance than the corresponding data. Dashed line: Residual electronic-noise level. The laser is tuned near the ${}^{87}\mathrm{Rb} F_g=2\to F_e=2$ transition at the position of maximum squeezing.

Figure 4

Normalized statistical moments for the measurements of Stokes operators. The normalized moment of order $p$ is defined as $\langle x^p\rangle /[{\langle x^2\rangle }^{p/2}(p-1)!!]-{\varepsilon }_p$, where ${\varepsilon }_p=1$ if $p$ is even and zero otherwise. Here $x$ represents the result of a measurement. The normalized moments are zero for a Gaussian distribution.

Figure 5

(a) Reference-cell saturated-absorption signal used for frequency calibration. The hyperfine transitions of the ${}^{87}\mathrm{Rb} D1$ line are indicated. (b) Noise power at 2.7 MHz of the Stokes operator $S_{\theta }$ presenting maximum squeezing around the $F=2\to F^{\prime }=2$ transition in a cell containing pure ${}^{87}\mathrm{Rb}$ (circles). The triangles indicate the shot-noise level.

Figure 6

(a) Quadrature noise power normalized to the shot-noise level as a function of laser detuning. Squares, minimum quadrature noise; circles, maximum quadrature noise; solid line, reference saturated-absorption signal with relevant ${}^{87}\mathrm{Rb} D1$ transitions indicated. (b) Quadrature angle corresponding to the minimum noise. (See also [44].)

Figure 7

Two-mode state quantifiers for the two beam-splitter outputs as a function of laser detuning. Circles, minimum partial transpose eigenvalue MPTE ($\equiv {\tilde{\nu }}_{-}$); squares, logarithmic negativity LN, triangles, quantum Gaussian discord QGD.