Antibunching and unconventional photon blockade with Gaussian squeezed states

Photon antibunching is a quantum phenomenon typically observed in strongly nonlinear systems where photon blockade suppresses the probability of detecting two photons at the same time. Antibunching has also been reported with Gaussian states, where optimized amplitude squeezing yields classically forbidden values of the intensity correlation, $g^{\left(2\right)}\left(0\right)<1$. As a consequence, observation of antibunching is not necessarily a signature of photon-photon interactions. To clarify the significance of the intensity correlations, we derive a sufficient condition for deducing whether a field is non-Gaussian based on a $g^{\left(2\right)}\left(0\right)$ measurement. We then show that the Gaussian antibunching obtained with a degenerate parametric amplifier is close to the ideal case reached using dissipative squeezing protocols. We finally shed light on the so-called unconventional photon blockade effect predicted in a driven two-cavity setup with surprisingly weak Kerr nonlinearities, stressing that it is a particular realization of optimized Gaussian amplitude squeezing.

Figure 1

(a) Ideally amplitude-squeezed Gaussian states for a squeeze parameter $r=r_{\mathrm{opt}}=0.13$ (blue), $r=0.3$ (green), and $r=0$ (yellow). (b) Realizations of a degenerate parametric amplifier with a pumped Kerr nonlinearity (top) and cavity frequency modulation (middle). Alternatively, highly pure intracavity squeezing can be generated using quantum bath engineering (QBE), where a cavity interacts with a structured reservoir (bottom). (c) Two-cavity setup where unconventional photon blockade is predicted. As pictured, the photon-photon interaction generates squeezing of the intracavity modes.

Figure 2

(a) Equal-time intensity correlation function $g^{\left(2\right)}\left(0\right)$ for a Gaussian state as a function of the displacement $\overline{\alpha }$ for different values of the state purity (as quantified by ${\overline{n}}_{\mathrm{eff}}$). For all curves the squeeze parameter has been set to its optimal value $r=r_{\mathrm{opt}}[\overline{\alpha },{\overline{n}}_{\mathrm{eff}}]$; $r_{\mathrm{opt}}$ is plotted in (c) as a function of $\overline{\alpha }$. The dotted black line corresponds to the degenerate parametric amplifier (DPA) (see Sec. 3) where ${\overline{n}}_{\mathrm{eff}}={sinh}^2{r}_{\mathrm{DPA}}$. The curve corresponding to ${\overline{n}}_{\mathrm{eff}}=0$ (solid blue) sets the minimum value of $g^{\left(2\right)}\left(0\right)$ possible for a Gaussian state with $|\langle \widehat{a}\rangle |=\overline{\alpha }$; any values lying in the darkest shaded region necessarily correspond to non-Gaussian states. For finite ${\overline{n}}_{\mathrm{eff}}$, $g^{\left(2\right)}\left(0\right)=2$ for $\alpha =0$ ($r=0$ also), as it should for a thermal state. (b) presents the minimal $g^{\left(2\right)}{\left(0\right)|}_{\min }$ (black full line) that can be achieved with a Gaussian state having a fixed value of ${\overline{n}}_{\mathrm{eff}}$. For these curves, $\alpha ={\overline{\alpha }}_{\mathrm{opt}}$ (red dot-dashed line) and $r=r_{\mathrm{opt}}$ (black dashed line) [cf. Eq. (8)].

Figure 3

Antibunching in the DPA versus QBE. (a) Two-time intensity correlation $g^{\left(2\right)}\left(\tau \right)$ versus $\tau$. Solid curves are for the DPA system, with different curves corresponding to different values of the strength of the squeezing interaction $\lambda$; for each curve, the coherent displacement $\overline{\alpha }={\overline{\alpha }}_{\mathrm{opt}}$. The dashed lines are for the QBE system. The intracavity squeeze parameter $r$ is the same in the two cases and set by the value of $\lambda /\kappa$. For QBE ${\Gamma }_{\mathrm{QBE}}=0.9\kappa$. The optimal displacement ${\overline{\alpha }}_{\mathrm{opt}}$ and the purity $P$ of the intracavity state [cf. Eq. (3)] is plotted in the inset as a function of $\lambda /\kappa$. For ${\Gamma }_{\mathrm{QBE}}\gg {\kappa }_{\mathrm{ext}}$, QBE generates a much better purity than the DPA. (b) Nonclassical nonmonotonic behavior of $g^{\left(2\right)}\left(\tau \right)$ for a DPA; parameters for both curves are indicated in the plot. Inset: ${\overline{\alpha }}_{\min }$ (green line) is the minimum value of $\overline{\alpha }$ for the given $\lambda /\kappa$ for which $g^{\left(2\right)}\left(0\right)<1$, while ${\overline{\alpha }}_{+}$ (red line) represents the minimum value of $\overline{\alpha }$ for which we have true antibunching [i.e., the slope of $g^{\left(2\right)}\left(\tau \right)$ at $\tau =0$ is positive]. Values of $\overline{\alpha }$ between these curves (shaded region) will yield $g^{\left(2\right)}\left(\tau \right)$ functions which are nonmonotonic.

Figure 4

$g^{\left(2\right)}\left(0\right)$ of the field inside cavity 1 of the two-cavity setup described in the main text as a function of the Kerr interaction strength $U$. The full line is the solution of the linearized Langevin equations and the dashed line is the solution of the quantum master equation (see Appendix pp4-s2). In the inset, we compare the corresponding squeeze parameter $r_1$ (blue dashed line) to the optimal value $r_{\mathrm{opt}}[{\overline{\alpha }}_1,{\overline{n}}_{\mathrm{eff},1}=0]$ (green dot-dashed line) and we see that full suppression of $g_1^{\left(2\right)}\left(0\right)$ corresponds to the case $r_1=r_{\mathrm{opt}}[{\overline{\alpha }}_1,{\overline{n}}_{\mathrm{eff},1}={sinh}^2{r}_1]$. For these curves, we use the same parameters as in [13], namely, ${\kappa }_1={\kappa }_2=\kappa$, $U_1=U_2=U$, ${\Delta }_1={\Delta }_2=-0.275\kappa$, $F=0.01\kappa$, and $J=3\kappa$. For such values ${\overline{n}}_{\mathrm{eff},1}\sim {10}^{-13}$ and the total photon number inside the cavity is ${\overline{n}}_{\mathrm{tot},1}\sim {10}^{-7}$ [cf. Eq. (4b)].

Figure 5

Steady-state solution of the quantum master equation Eq. (D13) for cavity 1: (a) Squeeze parameter $r_1$ and optimal value $r_{\mathrm{opt}1}$; (b) squeezing angle ${\theta }_1$ compared to twice the displacement argument ${\phi }_1$; (c) displacement ${\overline{\alpha }}_1$ and optimal value ${\overline{\alpha }}_{\mathrm{opt}1}$; (d) intensity correlation at equal times $g_1^{\left(2\right)}\left(0\right)$; (e) effective photon number ${\overline{n}}_{\mathrm{eff}1}$ and ${sinh}^2{r}_1$. The parameters are plotted against $U\equiv U_1=U_2$ for ${\kappa }_1={\kappa }_2\equiv \kappa$ and the same parameters as in Fig. 4. The vertical line is placed at the minimal value of the intensity correlation, and coincides with $r=r_{\mathrm{opt}}$, $\theta =2\phi$, and $\overline{\alpha }={\overline{\alpha }}_{\mathrm{opt}}$.