Steady-state generation of negative-Wigner-function light using feedback

We propose a method of producing steady-state coherent light with negative Wigner functions in nonlinear media combined with feedback control. While the nonlinearities are essential to produce the Wigner negativities, this alone is insufficient to stabilize steady-state light with negativities. Using feedback control to control the phase in the cavity, we find that this produces significant total negativities for reasonable experimental parameters. The negative Wigner function is produced continuously and does not appear to be restricted to low-amplitude light. The technique is applicable to systems such as exciton-polaritons, where strong natural nonlinearities are present.

Figure 1

Experimental scheme considered in this paper. (a) The basic description of the ingredients necessary for steady-state negative-Wigner-function light generation. The parameters as defined in the master equation (1) are labeled. (b) The specific implementation with exciton-polaritons. A semiconductor microcavity structure with quantum wells (QWs) allows for the excitation of exciton-polaritons by the pump laser. The microcavity is formed by a combination of distributed Bragg reflectors (DBRs) on either side of the QWs, and a floating subwavelength grating (SWG) reflector. The resonant frequency of the cavity can be controlled by nanoelectromechanical actuators, in a way such as that realized in Ref. [41]. The phase of the output light is fixed with reference to a phase-stable local oscillator (LO) and is determined by homodyne detection.

Figure 2

Time evolution of the Wigner-function distributions for coherently excited cavity photons (polaritons) with no feedback. The used parameters are $\lambda =0, U/\gamma \hslash =0.3, A=3$, and $\theta ={60}^{\circ }$.

Figure 3

Time evolution of the Wigner-function distributions for coherently excited cavity photons (polaritons) with feedback. The used parameters are $\lambda =0.8, U/\gamma \hslash =0.3, A/\gamma =3$, and $\theta ={60}^{\circ }$.

Figure 4

Steady-state Wigner-function distributions for coherently excited cavity photons (polaritons). (a) The case with no feedback: $\lambda /\gamma =0, \mathrm{\Delta }/\gamma =0, U/\hslash \gamma =0.5, A/\gamma =3, \theta =0^{\circ }$, and $\eta =1$. (b) Typical distribution for a low-$Q$ cavity: $\lambda /\gamma =0.16, \mathrm{\Delta }/\gamma =0, U/\hslash \gamma =0.05, A/\gamma =3, \theta ={10}^{\circ }$, and $\eta =1$. (c) High-$Q$ cavity: $\lambda /\gamma =0.65, \mathrm{\Delta }/\gamma =0, U/\hslash \gamma =0.5, A/\gamma =3, \theta =-5^{\circ }$, and $\eta =1.0$.

Figure 5

Total integrated negativity $\mathcal{N}$ in the space of the feedback parameter $\lambda$ and pumping phase $\theta$, for various nonlinear interaction coefficients $U$. Common parameters are $\mathrm{\Delta }/\gamma =0, A/\gamma =3$, and $\eta =1.0$.

Figure 6

Total integrated negativity $\mathcal{N}$ as a function of various parameters. Parameters used are (a) $\mathrm{\Delta }/\gamma =0, A/\gamma =2, \eta =1$, and $\theta ={10}^{\circ }$; (b) $\mathrm{\Delta }/\gamma =0, A/\gamma =2, \eta =0.7$, and $\theta ={10}^{\circ }$; (c) $\mathrm{\Delta }/\gamma =0, A/\gamma =2, \eta =1$, and $U/\hslash \gamma =0.5$; and (d) $\lambda /\gamma =0.7, A/\gamma =3, \eta =1.0$, and $U/\hslash \gamma =0.5$.

Figure 7

Total integrated negativity $\mathcal{N}$ in the space of the feedback parameter $\lambda$ and pumping phase $\theta$, for various detunings $\mathrm{\Delta }$. Common parameters are $U/\gamma \hslash =0.3, A/\gamma =3$, and $\eta =1.0$.

Figure 8

Total integrated negativity $\mathcal{N}$ as a function of the effective detection efficiency $\eta$ for different parameters as marked. The common used parameters are (a) $A/\gamma =3, U/\gamma \hslash =0.3$, and $\theta ={55}^{\circ }$ and (b) $A/\gamma =3$ and $\mathrm{\Delta }=0.0$.

Figure 9

Modes involved in proposed experimental system. The first beam splitter with transmittivity $t$ acts to divert a portion $r=1-t$ of the light to be used for the feedback system. The homodyne system has a finite detector efficiency ${\eta }_0$ which can be equivalently considered to be the beam-splitter transmittivity [42].