Squeezing of thermal fluctuations in four-wave mixing in a $\mathrm{\Lambda }$ scheme

We theoretically investigated the mechanism of two-mode quadrature squeezing in a regime of four-wave mixing in a $\mathrm{\Lambda }$ scheme of three-level atoms embedded in a thermal reservoir. We demonstrated that the process of nonlinear transfer of noise from the low frequency of ground state splitting to the optical frequency is significant if the number of thermal photons at the low frequency is high. We have shown that correct calculation of the two-mode squeezing level taking into account both thermal noise and distortion of dissipative properties of the thermally excited medium resulted in a simple expression for the maximum squeezing level, which is defined by the ground-state coherence decay rate and the drive-field intensity. We found the optimal conditions for squeezing, in particular, the optimal density-length product of the active medium depending on the atomic relaxation parameters and the drive-field intensity.

Figure 1

The scheme of the resonant 4WM process in three-level $\mathrm{\Lambda }$ atoms. Here $\delta \omega ={\omega }_p-{\omega }_{31}$ is the one-photon resonance detuning of the “central” frequency of the quasimonochromatic probe wave, and $\delta \omega$ is chosen so that it corresponds to the strict four-wave spacial synchronism. $\nu$ is the detuning of the considered spectral component of the probe wave from the “central” frequency.

Figure 2

Two-mode noise level $Q_{\nu }$ (in dB) for the phase $\mathrm{\Psi }=\pi /2$ versus frequency detuning from the 4WM resonance $\nu$ normalized to the frequency band of the parametric instability for the case of negligible dissipation (${\kappa }_p=0$), zero temperature ($T=0$), and different lengths of interaction: (1) $l=0.5/\chi$, (2) $l=1/\chi$, (3) $l=1.5/\chi$, and (4) $l=3/\chi$.

Figure 3

Two-mode noise level $Q_{\nu }$ (in dB) for the phase $\mathrm{\Psi }=\pi /2$ versus normalized length of interaction for zero detuning ($\nu =0$), zero temperature ($T=0$), and different parameters ${\kappa }_p/\chi$: (0) ${\kappa }_p/\chi =0$, (1) ${\kappa }_p/\chi =0.02$, (2) ${\kappa }_p/\chi =0.1$, (3) ${\kappa }_p/\chi =0.5$, (4) ${\kappa }_p/\chi =1$, and (5) ${\kappa }_p/\chi =2$.

Figure 4

Two-mode noise level $Q_{\nu }$ (in dB) for the phase $\mathrm{\Psi }=\pi /2$ versus normalized detuning, for zero temperature ($T=0$) and different parameters ${\kappa }_p(\nu =0)/\chi$: (1) ${\kappa }_p/\chi =0.02$, (2) ${\kappa }_p/\chi =0.1$, (3) ${\kappa }_p/\chi =0.5$, (4) ${\kappa }_p/\chi =1$, and (5) ${\kappa }_p/\chi =2$. In panel (a) all spectra are obtained at the same normalized length of interaction, $l\chi =2$. In panel (b) each spectrum is obtained at the optimal length of interaction: (1) $l\chi =2.65$, (2) $l\chi =1.87$, (3) $l\chi =1.1$, (4) $l\chi =0.85$, and (5) $l\chi =0.62$.

Figure 5

Two-mode noise level $Q_{\nu }$ (in dB) for the phase $\mathrm{\Psi }=\pi /2$ for different temperatures (the corresponding averaged number of thermal photons at frequency ${\omega }_{21}$ is pointed out in every panel) and different parameters ${\kappa }_p(\nu =0)/\chi$: (1) ${\kappa }_p/\chi =0.02$, (2) ${\kappa }_p/\chi =0.1$, (3) ${\kappa }_p/\chi =0.5$, (4) ${\kappa }_p/\chi =1$, and (5) ${\kappa }_p/\chi =3$. In panels (a)–(c) the length dependence of the squeezing in the center of line is presented. In panels (d)–(f) the noise spectra are presented, and each spectrum is obtained at the optimal length of interaction: (1) $l\chi =2.65$, (2) $l\chi =1.87$, (3) $l\chi =1.1$, (4) $l\chi =0.85$, and (5) $l\chi =0.5$.