Technique for active stabilization of the relative phase between seed and pump in an optical parametric oscillator

We design and demonstrate a technique for the active stabilization of the relative phase between seed and pump in an optical parametric oscillator (OPO). We show that two error signals for the stabilization of the OPO frequency, based on Pound-Drever-Hall (PDH), and of the seed-pump relative phase can be obtained just from the reflected beam of the OPO cavity, without the necessity of two different modulation and demodulation stages. We also analyze the effect of the pump in the cavity stabilization for different seed-pump relative phase configurations, resulting in an offset in the PDH error signal, which has to be compensated. Finally, an application of our technique in the reliable generation of squeezed coherent states is presented.

Figure 1

Scheme of the OPO with the main elements. See text for details.

Figure 2

(top and center) Plots of $|E_{\mathrm{c}}{|}^2$ and $|E_R{|}^2$ (dimensionless units) with $E_{\mathrm{in}}=1$ as functions of the phase $\varphi$, proportional to the detuning $\nu$, and the pump phase ${\varphi }_{\mathrm{p}}$. The blue lines refer to the behavior of $|E_{\mathrm{c}}{|}^2$ and $|E_R{|}^2$ for a particular choice of the pump phase, from left to right ${\varphi }_p=-\pi , -\pi /2$, 0, $\pi /2$, and $\pi$. (bottom) Plot of the minimum value $|E_R^{\left(\min \right)}{|}^2$ of the reflected beam amplitude as a function of the pump phase ${\varphi }_{\mathrm{p}}$. The other involved parameters have been set to the realistic values $R_1=0.999, R_2=0.9, \mathrm{\Delta }=3.0\times {10}^{-3}$, and $\left|\gamma \right|=2.0\times {10}^{-2}$. See text for details.

Figure 3

Plots of the PDH error signal ${\epsilon }_{\mathrm{PDH}}$ as a function of $\varphi \propto \nu$ for fixed phase modulation ${\varphi }_{\mathrm{m}}=0.1$ and for different values of the pump phase ${\varphi }_{\mathrm{p}}$. (top) In the amplification (${\varphi }_{\mathrm{p}}=-\pi /2$) or in the de-amplification (${\varphi }_{\mathrm{p}}=\pi /2$) regime the PDH error signal vanishes at resonance ($\varphi =0$). (center) If ${\varphi }_{\mathrm{p}}\ne \pm \pi /2$ the PDH error signal becomes zero at $\varphi \ne 0$: there is an offset (arrows) due to the effect of the pump phase. (bottom) Plot of ${\epsilon }_{\mathrm{PDH}}$ as a function of the pump phase ${\varphi }_{\mathrm{p}}$ at resonance ($\varphi =0$). The other involved parameters have been set to the realistic values $R_1=0.999, R_2=0.9, \mathrm{\Delta }=3.0\times {10}^{-3}$, and $\left|\gamma \right|=2.0\times {10}^{-2}$.

Figure 4

Plots of $|E_{\mathrm{c}}{|}^2$ (solid blue lines), proportional to the OPO output, and $|E_R{|}^2$ (dashed red lines) with $E_{\mathrm{in}}=1$ (dimensionless units) as functions of the phase $\varphi \propto \nu$ for different values of the pump phase: (top) ${\varphi }_{\mathrm{p}}=-\pi /2$, amplification regime; (center) ${\varphi }_{\mathrm{p}}=\pi /2$, de-amplification regime; (bottom) ${\varphi }_{\mathrm{p}}=\pi$. The other involved parameters have been set to the realistic values $R_1=0.999, R_2=0.9, \mathrm{\Delta }=3.0\times {10}^{-3}$, and $\left|\gamma \right|=2.0\times {10}^{-2}$.

Figure 5

Plots of the $|E_c{|}^2$ as a function of $\varphi$ in the de-amplification regime (${\varphi }_{\mathrm{p}}=\pi /2$) for different values of $\left|\gamma \right|$ as reported in the figure. The dashed line refers to the empty cavity. The other parameters involved have been set to the realistic values $R_1=0.999, R_2=0.9$, and $\mathrm{\Delta }=3.0\times {10}^{-3}$.

Figure 6

Plots of $|E_{\mathrm{c}}{|}^2$ (solid blue lines), proportional to the OPO output, and $|E_R{|}^2$ (dashed red lines) with $E_{\mathrm{in}}=1$ (dimensionless units) as functions of the phase ${\varphi }_{\mathrm{p}}$ for different values of $\varphi \propto \nu$: (top) $\varphi =0$ (resonance), in this case $|E_R{|}^2$ corresponds to the minimum value of the reflected beam displayed in the bottom panel of Fig. 2; (bottom) $\varphi =0.02$. The other involved parameters have been set to the realistic values $R_1=0.999, R_2=0.9, \mathrm{\Delta }=3.0\times {10}^{-3}$, and $\left|\gamma \right|=2.0\times {10}^{-2}$.

Figure 7

Plots of the PDH error signal ${\epsilon }_{\mathrm{PDH}}$ (top panel) and the amplitude of the reflected beam $|E_R{|}^2$ (bottom panel) as functions of $\varphi \propto \nu$ for the pump phase ${\varphi }_{\mathrm{p}}=\pi$. The dashed lines refer to the original signals (see the center panel of Fig. 3 and the lower panel of Fig. 3), while the solid lines are obtained by setting to zero the ${\epsilon }_{\mathrm{PDH}}$ offset and the minimum of $|E_R{|}^2$, respectively. The other involved parameters have been set to the realistic values $R_1=0.999, R_2=0.9, \mathrm{\Delta }=3.0\times {10}^{-3}$, and $\left|\gamma \right|=2.0\times {10}^{-2}$.

Figure 8

Scheme of our setup. The main parts are three: a LASER which produces both the pump at 532 nm and the seed and the local oscillator (LO) at 1064 nm for the state detection; a state generation part (SG) used for quantum states generation and squeezing; a detection part based on homodyne detection (HD). An amplitude modulator (AM) is used to control the pump intensity. The OPO frequency is stabilized implementing the canonical PDH technique exploiting a high-frequency generator (HF) and a phase modulator (PM), while the seed-pump phase stabilization is performed by the seed-pump stabilizer (SPS) and it is based on our technique. See text for details.

Figure 9

Transmitted beam (lower red line), reflected beam (upper blue line) and PDH error signal (lower black line) as functions of the detuning $\nu$, obtained with a frequency scan of the OPO cavity. Solid lines are experimental data, while dashed lines are theoretical trends obtained from $R_1, R_2, \mathrm{\Delta }$, and $\left|\gamma \right|$. We plot curves in the amplification regime (${\varphi }_{\mathrm{p}}=-\pi /2$, upper left), de-amplification regime (${\varphi }_{\mathrm{p}}=\pi /2$, upper right), minus regime (${\varphi }_{\mathrm{p}}=0$, lower left), and plus regime (${\varphi }_{\mathrm{p}}=\pi$, lower right). For the theoretical previsions we used the following measured values of the experimental parameters $R_1=0.9988, R_2=0.917, \mathrm{\Delta }=2.4\times {10}^{-3}$, and $\left|\gamma \right|=1.85\times {10}^{-2}$.

Figure 10

Transmitted beam (lower red line) and reflected beam (upper blue line) for a scan of the cavity (here the time is related to the pump phase ${\varphi }_{\mathrm{p}}$). A triangular signal has been applied to the PIEZO in Fig. 8 acting on the pump phase. For comparison with the theory, see the plots of $|E_R{|}^2$ and $|E_c{|}^2$ as functions of $\varphi$ and ${\varphi }_{\mathrm{p}}$ in Figs. 2 and in 6, respectively.

Figure 11

PDH error signal (lower red line) and SPS error signal (upper blue line) during a scan of the cavity frequency with all the offsets optimized (here the time is related to the phase $\varphi \propto \nu$). The two signals can be electronically shifted along the vertical axis, as highlighted by the arrows: we can set to zero the signals without physically affecting the optical setup.

Figure 12

Example of OPO stabilization in different conditions as a function of time. Lines represent the transmitted beam in three cases: (i) pump turned off (lower red line), (ii) pump on and pump phase stabilized at ${\varphi }_{\mathrm{p}}=0$ with both integrator and proportional (upper black line), and (iii) pump on and pump phase stabilized at ${\varphi }_{\mathrm{p}}=0$ with the only integrator (upper light blue line). The relative intensity noise is 0.6%, 1.9%, and 6.2%, respectively. Notice that the black line has been rigidly shifted of $-0.4$ for a better visualization.

Figure 13

Examples of displaced squeezed states generation with different homodyne traces (left plots) and the corresponding tomographic reconstruction in the phase space (right plots). (top) Amplification regime with ${\varphi }_{\mathrm{p}}\approx -\pi /2\Rightarrow arg\left(\xi \right)\approx 0, \alpha =2.93$ and $N_{\mathrm{th}}=0.13$. (center) ${\varphi }_{\mathrm{p}}\approx 0\Rightarrow arg\left(\xi \right)\approx \pi /2, \alpha =1.97$, and $N_{\mathrm{th}}=0.12$. (bottom) De-amplification regime ${\varphi }_{\mathrm{p}}\approx \pi /2\Rightarrow arg\left(\xi \right)\approx \pi , \alpha =1.44$, and $N_{\mathrm{th}}=0.14$. In the left plots, the solid lines refer to the average value of the experimental trace, while the dashed lines represents the standard deviation, better highlighting the effect of squeezing. Note that, in the phase space (right plots), the angle between the major axis of the ellipses and the horizontal one is given by $arg\left(\xi \right)/2$. In the pictures we also report the corresponding squeezing level and total average number of photons. The vacuum variance is set to 1.