Exact steady state of a Kerr resonator with one- and two-photon driving and dissipation: Controllable Wigner-function multimodality and dissipative phase transitions

We present exact results for the steady-state density matrix of a general class of driven-dissipative systems consisting of a nonlinear Kerr resonator in the presence of both coherent (one-photon) and parametric (two-photon) driving and dissipation. Thanks to the analytical solution, obtained via the complex $P$-representation formalism, we are able to explore any regime, including photon blockade, multiphoton resonant effects, and a mesoscopic regime with large photon density and quantum correlations. We show how the interplay between one- and two-photon driving provides a way to control the multimodality of the Wigner function in regimes where the semiclassical theory exhibits multistability. We also study the emergence of dissipative phase transitions in the thermodynamic limit of large photon numbers.

Figure 1

Sketch of the considered class of systems. The picture represents a photon resonator subject to one-photon losses at rate $\gamma$ and coherently driven by a one-photon pump of amplitude $F$. The resonator is also subject to a coherent two-photon driving of amplitude $G$ and two-photon losses at rate $\eta$. The strength of the photon-photon interaction is quantified by $U$. At the right, we sketch the effects of these physical processes on the Fock (number) states $|n\rangle$.

Figure 2

Mean photon number $\langle {\widehat{a}}^{†}{\widehat{a}}^{}\rangle$ as a function of the pump-cavity detuning $\mathrm{\Delta }$ normalized by the photon-photon interaction strength $U$. Different curves and data sets correspond to different pump intensities (cf. the legend). Solid lines represent the analytic solution, while markers indicate the numerical results obtained by diagonalization of the Liouvillian superoperator of the master equation on a truncated Fock basis. Top: results in the absence of one-photon pumping, i.e. $F=0$ [Eq. (C3) for $i=j=1]$. Bottom: Results in the presence of both one- and two-photon pumping with $F=G$ [Eq. (15)]. In both panels, vertical dot-dashed red (dashed blue) lines mark the positions of odd (even) photonic resonances. One- and two-photon dissipation rates were set to $\gamma =\eta =0.03U$.

Figure 3

Left: Mean steady-state photon number $\langle {\widehat{a}}^{†}{\widehat{a}}^{}\rangle$ as a function of the dimensionless detuning parameter $\mathrm{\Delta }/U$. Green circles (red crosses) mark stable (unstable) semiclassical steady-state solutions. The black line is the analytic solution given by Eq. (15) ($i=j=1$). One- and two-photon dissipation rates were set to $\gamma =\eta =0.1U$. Right: Zoom-in on the region in which the almost-degenerate high-density semiclassical solutions become unstable.

Figure 4

Top: Steady-state photon number $\langle {\widehat{a}}^{†}{\widehat{a}}^{}\rangle$ as a function of the dimensionless detuning parameter $\mathrm{\Delta }/U$ for $F=U$, $G=10U$, $\gamma =\eta =0.1U$. Results were obtained through the exact solution, (15), for $i=j=1$. Vertical grid lines mark the values of $\mathrm{\Delta }/U$ for which we evaluated the steady-state Wigner function (cf. bottom plots). Bottom: Steady-state Wigner functions $W\left(z\right)$ calculated according to Eq. (20) for the same parameters as in the top panel and for different values of $\mathrm{\Delta }/U$ (see frame labels). Black dots mark the position of the corresponding stable semiclassical solutions.

Figure 5

Steady-state Wigner functions $W\left(z\right)$ calculated according to Eq. (20) for $\mathrm{\Delta }=28U$, $G=10U$, $\gamma =\eta =0.1U$ and for different complex values of $F$. In (e) we took $F=0$. In the others, $F/U=e^{i\varphi }$ and the phase $\varphi$ changes as sketched in the bottom-right plot.

Figure 6

Mean photon number $\langle {\widehat{a}}^{†}{\widehat{a}}^{}\rangle$ (top) and normalized second-order correlation function $g^{\left(2\right)}$ (bottom) as a function of the pump-cavity detuning $\mathrm{\Delta }$ normalized by the photon-photon interaction strength $U$ for a resonator subject only to one-photon coherent driving ($G=0$, $F\ne 0$). Different curves (and colors) correspond to different pumping intensities $F/U$, varied between 0.1 and 300, as indicated beside each curve in the top panel. One- and two-photon dissipation rates were set to $\gamma =\eta =0.03U$.

Figure 7

Same as Fig. 6, but in the presence of two-photon driving only, i.e., $F=0$ and $G\ne 0$. Different curves (and colors) correspond to different pumping strengths $G/U$, spanning from 0.1 to 300 as labeled beside each curve in the top panel. One- and two-photon dissipation rates were set to $\gamma =\eta =0.03U$.

Figure 8

Top: For the case $G=0$ (coherent driving only), the rescaled mean photon density $\chi =\langle {\widehat{a}}^{†}{\widehat{a}}^{}\rangle /{\left|f\right|}^{2/3}$ as a function of the dimensionless parameter $\tau =\mathrm{sgn}\left[\mathrm{\Delta }\right]\left|c\right|/{\left|f\right|}^{2/3}$ for different values of the dimensionless coherent drive intensity $\left|f\right|$ (see the legend). The smoothest curve corresponds to $\left|f\right|=1$, while for increasing values of $\left|f\right|$ the curve gets steeper, acquiring a triangular shape. Inset: Points mark the height and position of the peak in $\partial \chi /\partial \tau$ as a function of $\left|f\right|$. Solid lines are power-law fits with exponents $\pm 2/3$, performed on the last four points. Bottom: For the case $F=0$ (two-photon driving only), the rescaled mean photon density $\chi =\langle {\widehat{a}}^{†}{\widehat{a}}^{}\rangle /\left|g\right|$ as a function of $\tau =\mathrm{sgn}\left[\mathrm{\Delta }\right]\left|c\right|/\left|g\right|$ for different values of $\left|g\right|$ (see the legend). The smoothest curve corresponds to $\left|g\right|=1$, while $\chi$ progressively tends to a triangular-shaped curve for increasing $\left|g\right|$. Inset: The rapid growth in the derivative $\partial \chi /\partial \tau$ around $\tau =-1$. The smoothest behavior corresponds to $\left|g\right|=100$ and the curve progressively acquires a discontinuity with increasing $\left|g\right|$ (cf. inset legend). Overall, dissipations have been set to $\eta =0.1U$ and $\gamma =0.1\left|\mathrm{\Delta }\right|$.

Figure 9

Representation of the Pochhammer path on the complex plane $\left\{\mathrm{Re}\right(\xi ),\mathrm{Im}(\xi \left)\right\}$. The blue and red circles represent the poles of the integrand in Eq. (B3), located at $\xi =0,1$, together with the corresponding branch cuts (from each pole towards $+\infty$ on the real axis). The Pochhammer contour $\mathcal{C}$ crosses each of the cuts an equal number of times in one direction and in the opposite, hence, for any starting point, it begins and ends on the same Riemann sheet. The change in the path color emphasizes the passage to a different Riemann surface.

Figure 10

Numerical cutoff $\tilde{\mathcal{M}}$ as a function of the corresponding mean photon density ${\langle {\widehat{a}}^{†}{\widehat{a}}^{}\rangle }_{\tilde{\mathcal{M}}}$ [Eq. (C1)] for different system parameters. The plot was obtained setting the convergence criterion as explained in the text. Each point in the diagram corresponds to $\gamma =\eta =0.1U$, while we varied $\mathrm{\Delta }/U\in [-30,30]$ and $F/U,G/U\in [0,60]$