Rotation of the noise ellipse for squeezed vacuum light generated via four-wave mixing

We report the generation of a squeezed vacuum state of light whose noise ellipse rotates as a function of the detection frequency. The squeezed state is generated via a four-wave mixing process in a vapor of ${}^{85}$Rb. We observe that rotation varies with experimental parameters such as pump power and laser detunings. We use a theoretical model based on the Heisenberg-Langevin formalism to describe this effect. Our model can be used to investigate the parameter space and potentially to tailor the ellipse rotation in order to obtain an optimum squeezing angle, for example, for coupling to an interferometer whose optimal noise quadrature varies with frequency.

Figure 1

(a) Ball-and-stick representation for bright beams and vacuum states. (b) Energy levels of the D1 line in ${}^{85}$Rb and the optical frequencies arranged in the double-$\Lambda$ system. (c) Experimental setup to generate squeezed light and measure the squeezing level. AOM: acousto-optic modulator, TA: tapered amplifier, SA: radio-frequency spectrum analyzer, BS: nonpolarizing beam splitter, PBS: polarizing beam splitter, BB: beam block. (d) Typical vacuum squeezing noise measurement at 1-MHz analysis frequency [red (gray)] compared to the shot-noise limit (black).

Figure 2

(a) Diagram summarizing the various frequencies involved. ${\nu }_1$ and ${\nu }_2$ are the carrier frequencies of pumps 1 and 2, respectively. ${\nu }_{LO}$ is the carrier frequency of the local oscillator, and ${\nu }_p={\nu }_1+{\nu }_{HF}$. The two-photon detuning $\delta ={\nu }_{LO}-{\nu }_p$. The analysis frequency is denoted ${\omega }_a$, and the upper and lower sidebands are at ${\nu }_{LO}\pm {\omega }_a$. The gray area represents the gain region of the process (4WM bandwidth). (b) Measured noise level (normalized to shot noise) of the generated beam as a function of the two-photon detuning $\delta$ at an analysis frequency of ${\omega }_a=1$ MHz. Black squares corresponds to the choice of phase that maximizes the noise, and red circles corresponds to the phase that minimizes the noise. The state is squeezed for a noise level less than zero. The black line shows the shot-noise level.

Figure 3

Measured squeezing as a function of frequency for different values of the two-photon detuning $\delta$: (a) $\delta =0$ MHz, (b) $\delta =-4$ MHz, (c) $\delta =-12$ MHz, and (d) $\delta =-20$ MHz. Red traces (light gray): measured noise power when the phase of the LO is scanned. Blue traces (dark gray): measured noise power when the phase of the LO is locked (at 1 MHz). Green (gray) solid line: envelopes of the detected noise minima, calculated by applying a low-pass filter to the red (light gray) traces. Black line at 0 dB: shot-noise level. (i) to (iv) Representation of the noise ellipse at two analysis frequencies (5 and 15 MHz) for the corresponding value of $\delta$. For reference, the minor axis of the ellipse is vertical at an analysis frequency of 1 MHz. The green (light gray) circle represents a coherent state, and the ellipses are normalized to this radius. $\Delta =0.8$ GHz, the cell temperature $T_{\text{cell}}=90{}^{\circ }\text{C}$, and the power in pump 1 is 230 mW and in pump 2 is 160 mW.

Figure 4

Noise level at an analysis frequency of 0 MHz as a function of the two-photon detuning $\delta$. The red circles correspond to the measured noise power as a function of $\delta$ at 1 MHz. Blue (dark gray) solid line: calculated noise at an analysis frequency of 0 MHz as a function of $\delta$. Black solid line at 0 dB: shot-noise level.

Figure 5

Noise power as a function of the analysis frequency for, from top to bottom, $\delta =0$ MHz (red), $-4$ MHz (orange), $-12$ MHz (blue), $-20$ MHz (green). Dashed lines (regular, double, short, and long) are the envelopes corresponding to the minimum of noise traces shown in Fig. 3 for, respectively, $\delta =0$, $-4$, $-12$, $-20$ MHz. Solid lines are obtained numerically by summing the contributions of the two sidebands using the noise level at ${\omega }_a=0$ plotted in Fig. 4.

Figure 6

Noise power as a function of the analysis frequency. Red traces (light gray) in (a) and (c) are identical to the red traces in Fig. 3 and correspond to the measured noise power when the phase of the LO is scanned. The black solid lines in (a) and (c) are $N_{\min }({\omega }_a,\delta )$. The blue traces (dark gray) in (b) and (d) are identical to the blue traces in Fig. 3 and correspond to the measured noise power when the phase of the LO is locked for lowest noise at 1 MHz. The black solid lines in (b) and (d) are $N_1({\omega }_a,\delta )$. In (a) and (b) $\delta =-4$ MHz, and in (c) and (d) $\delta =-20$ MHz. For the numerical simulations, we used $\Delta \varphi =0.2\pi$.

Figure 7

Representation of the noise ellipse for (a) a phase-locked LO situation and (b) a rotated noise ellipse. $a$ and $b$ are, respectively, the major and minor axis radii of the ellipse. The vertical direction is the direction chosen by the phase-locking system at 1 MHz. $\Delta \varphi$ is the rotation angle, and $r\left(\Delta \varphi \right)$ is the projection of the ellipse on the direction of the phase-locking system at 1 MHz. The green (light gray) circle in (a) is a normalized coherent state.

Figure 8

(a) to (d) $r$ (black curves), $a$ [top red (gray) curves], and $b$ [bottom red (gray) curves] as a function of analysis frequency for the four different two-photon detunings. The red (gray) areas are the acceptable ranges for $r$. We normalized the diameter of a coherent state to 1, such that an axis radius below 0.5 corresponds to a squeezed state. (e) to (h) $\Delta \varphi$ as a function of analysis frequency. The two-photon detunings are the same as in Fig. 3. (a) and (e) 0 MHz, (b) and (f) $-4$ MHz, (c) and (g) $-12$ MHz, and (d) and (h) $-20$ MHz.

Figure 9

Wigner functions calculated assuming Gaussian states for $\delta =-12$ MHz. The normalization for a coherent state is shown on the left. As the analysis frequency is increased, in general, both $\Delta \varphi$ and the total uncertainty area in phase space increase. Three different analysis frequencies are shown: 6, 12, and 23 MHz.

Figure 10

Phase shift as a function of $\delta$. Numerical simulation parameters are $\Delta =0.8$ GHz, pump power of 200 mW, and an optical depth of 1000.