Spatial multimode structure of atom-generated squeezed light

We investigated the spatial distribution of quantum fluctuations in a squeezed vacuum field, generated via polarization self-rotation (PSR) interaction of an ensemble of Rb atoms and a strong near-resonant linearly polarized laser field. We found that the noise suppression is greatly effected by the transverse profile of a spatial mask, placed in both the squeezed field and the local oscillator, as well as its position along the focused beam near the focal point. These observations indicate the spatial multimode structure of the squeezed vacuum field. We have developed a theoretical model that describes the generation of higher-order Laguerre-Gauss modes as a result of PSR light-atom interaction. The prediction of this model is in a good qualitative agreement with the experimental measurements.

Figure 1

Experimental setup. SMPM is a single-mode polarization-maintaining fiber, $\lambda /2$ is half-wave plate, GP is Glan-laser polarizer, PBS is a polarizing beam splitter, PhR is a phase-retarding wave plate, and BPD is a balanced photodetector. In the experiment a mask was inserted either in the collimated beam (a) or in the focused beam (b).

Figure 2

Measured minimum (diamonds) and maximum (circles) quadrature noise when the laser beam is partially blocked by (a) an iris mask, (b) a disk mask, (c),(d) the combined masks, formed by both iris and the disk. For all measurements the horizontal axis indicates the fraction of LO intensity, transmitted through the mask. In (c) the central disk alone blocked 8% of the LO power and in (d) the power loss was 25%. Dashed and dotted lines indicate the prediction of the single-mode model, described by Eq. (1). The zero of the vertical axis corresponds to the shot-noise level.

Figure 3

Minimum quadrature noise vs a variable iris position. The iris size was adjusted to maintain constant transmission of the LO beam for each trace as depicted in the legend: 55% [(a), circles], 70% [(b), squares], 75% [(c), crosses], and 95% [(d), plus signs].

Figure 4

Comparison of the experimental and theoretical (left and right column) dependencies of the LO beam transmission (top row) and the squeezed field minimal noise power (bottom row) on the iris position for several fixed iris sizes. The legend denotes the peak transmission for each iris. The Rayleigh range is $Z_R=2.5$ cm. To calculate transmission and noise power, we used Eqs. (11) and (1).

Figure 5

Wigner function of (a) the experimentally realized squeezed state and (b) a hypothetical minimum uncertainty state with 1.9 dB of squeezing along the $X_1$ quadrature, shown here in three dimensions and contour. The simple second quantized theory predicts the multimode structure (Fig. 6) that results in the same level of measured squeezing via homodyne detection. The axis labels $x$ and $y$ are proportional to the $X_1$ and $X_2$ quadratures respectively. The Wigner function has been rescaled by the peak amplitude of the vacuum Wigner function.

Figure 6

The Wigner functions for the $p=0–2$ modes [(a)–(c) respectively] which, when measured simultaneously via homodyne detection, recreate the example Wigner function depicted in Fig. 5. The $p=3$ and greater modes are omitted since they appear essentially as vacuum modes. The axis labels $x$ and $y$ are proportional to the $X_1$ and $X_2$ quadratures respectively. The Wigner functions have been rescaled by the peak amplitude of the vacuum Wigner function.