Multiphoton blockades in pulsed regimes beyond stationary limits

We demonstrate multiphoton blockades (PB) in the pulsed regime by using a Kerr-nonlinear dissipative resonator driven by a sequence of Gaussian pulses. It is shown that the results obtained for single-photon, two-photon, and three-photon blockades in the pulsed excitation regime differ considerably from analogous results obtained for the case of continuous-wave (cw) driving. We strongly demonstrate that for the case of cw pumping of the Kerr-nonlinear resonator there are fundamental limits on populations of lower photonic number states (with $n=0$, 1, 2, 3). Thus, such detailed comparison demonstrates that PB due to excitation with a suitable photon pulses is realized beyond the fundamental limits established for cw excitations. We analyze photon-number effects and investigate the phase-space properties of PB on the base of photon number populations, the second-order correlation functions, and the Wigner functions in phase space. The generation of Fock states due to PB in the pulsed regime is analyzed in details.

Figure 1

Schematic representation of Kerr-nonlinear resonator under a sequence of Gaussian pulses. In this scheme a emitter involving third-order susceptibility coupled to a resonator.

Figure 2

Schematic energy-level diagram with one-photon, two-photon, and $k$-photon resonant transitions between states of Kerr-nonlinear resonator. Selective excitations leading to multiphoton blockades are realized for the main frequencies of driving field: $k\hslash {\omega }_k=E_{k0}$ that are calculated as ${\omega }_k={\omega }_0+\chi (k-1)$.

Figure 3

Populations of photon number depending on the detuning. The parameters are $\Omega /\gamma =20$, $\chi /\gamma =15$.

Figure 4

Populations of photon numbers depending on the parameter of nonlinearity: (a) the curves of $P_0,P_1$ populations, (b) the curves of $P_2,P_3$ populations. The parameters are $\Omega /\gamma =20$, $\Delta /\gamma =0$.

Figure 5

The regime of single-photon blockade. The mean photon number versus time interval that repeats the Gaussian pulses and also indicates decay effects (a) for $k=1$; (b), (c) the time evolution of the second-order photon correlation function versus the dimensionless time for $k=1$; (d) the second-order photon correlation function, for time intervals at the peak values of the mean photon numbers versus the tuning parameter. The parameters are $\chi /\gamma =15$; the maximum amplitude of the driving field $\Omega /\gamma =6$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}$.

Figure 6

The regime of single-photon blockade. The populations $P\left(n\right)$ of $|n\rangle$ Fock states, for $n=0,1,2,3$, at the maximal peak values of the mean photon numbers versus the tuning parameter. The parameters are $\chi /\gamma =15$; the maximum amplitude of the driving field $\Omega /\gamma =6$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}.$

Figure 7

The regimes of single-photon and two-photon blockades. The mean photon numbers versus dimensionless time interval for (a) $k=2$ and (b) $k=1$ that repeat the Gaussian pulses and involve also decay effects; (c) time evolution of the population $P_2$ of the $|2\rangle$ Fock state, for $k=2$, versus dimensionless time; (d), (e) the second-order photon correlation function versus the dimensionless time, for $k=2$; (f) the second-order photon correlation function for time-intervals at the peak values of the mean photon numbers versus the tuning parameter. The parameters are $\chi /\gamma =30$; the maximum amplitude of the pump field $\Omega /\gamma =12$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}.$

Figure 8

The regimes of single-photon and two-photon blockades. The populations $P\left(n\right)$ for $n=0,1,2,3$, for time intervals at the peak values of the mean photon numbers versus the tuning parameter. The parameters are $\chi /\gamma =30$; the maximum amplitude of pump field $\Omega /\gamma =12$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}$.

Figure 9

The regime of three-photon blockades. The mean photon number versus dimensionless time for (a) $k=3$; (b) the maximal peak values of the mean photon numbers depending on the tuning parameter; (c) the second-order correlation function at the peak values of the mean photon numbers versus the tuning parameter; (d) the third-order correlation function at the peak values of the mean photon numbers versus the tuning parameter. The parameters are $\chi /\gamma =11$; the maximum amplitude of pump field $\Omega /\gamma =12$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}.$

Figure 10

The regime of three-photon blockades. The populations $P\left(n\right)$ for $n=0,1,2,3$, for time intervals at the peak values of the mean photon numbers versus the tuning parameter. The parameters are $\chi /\gamma =11$; the maximum amplitude of pump field $\Omega /\gamma =12$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}.$

Figure 11

The maximum values of the mean photon numbers depending on the tuning parameter. The parameters are (a) $\chi /\gamma =15$, $\Omega /\gamma =6$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}$; (b) $\chi /\gamma =30$, $\Omega /\gamma =12$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}$. The mean number of reservoir photons $N=0.01$.

Figure 12

The regime of selective two-photon excitation. (a) The Wigner function for the pure $|2\rangle$ state; (b) the Wigner function of the cavity mode for time interval corresponding to the maximal population of Fock state $|2\rangle$. The negative regions of the Wigner functions are indicated in black. The parameters are $\chi /\gamma =15$; the maximum amplitude of pump field $\Omega /\gamma =12$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}$.

Figure 13

The contour plots of the Wigner functions for the various selective excitations (a) $k=1$; (b) $k=1.24$, where the populations of $|0\rangle$ and $|1\rangle$ are crossing each other (see Fig. 8); (c) $k=2$; (d) $k=3$. The negative regions of the Wigner functions are indicated in black. The parameters are $\chi /\gamma =15$; the maximum amplitude of pump field $\Omega /\gamma =12$; the mean number of reservoir photons $N=0.01$, $\tau =5.5{\gamma }^{-1}$, $T=0.4{\gamma }^{-1}$.