Unconventional photon blockade with non-Markovian effects in driven dissipative coupled cavities

The photon blockade based on destructive quantum interference is called the unconventional photon blockade (UPB), which has been intensively studied in Markovian systems but barely explored in the non-Markovian ones. In this paper, we construct a coupled-cavities system to achieve UPB with the non-Markovian effect, where the dissipationless left cavity and Markovian dissipative right cavity are respectively mediated by two-photon pump and single-photon driving field. Through the equivalence between the Markovian master equation and Heisenberg-Langevin equation for the environment initialization in the vacuum state, we can derive the exact non-Markovian Heisenberg-Langevin equation and reduced master equation for the left cavity, which contains the two-photon pump and effective single-photon driving field. In the non-Markovian regime (the dissipation falling below a threshold), the effective single-photon driving field holds nonzero, which can lead to UPB occurring due to a closed quantum interference path forming. When the dissipation exceeds the threshold, the system enters the Markovian regime, where UPB weakens. Especially, if the dissipation approaches infinity, UPB for the left cavity disappears due to the effective single-photon driving field tending to zero. We analytically derive an optimal condition for UPB, which is in good agreement with that obtained by the numerical simulation. We also discuss the situation where both cavities have dissipations. Finally, the above model is extended to a general system involving a dissipationless left cavity (mediated by two-photon pump) coupling with noninteracting dissipative right cavities (driven by single-photon driving fields). Our scheme might pave an avenue towards applications on photon statistics and quantum optics with the non-Markovian effect.

Figure 1

Setup for UPB with the non-Markovian effect. The left cavity (eigenfrequency ${\omega }_a$) and right cavity (eigenfrequency ${\omega }_b$) with the coupling strength $g$ are mediated by the two-photon pump (strength $G$ and frequency ${\omega }_p$) [13, 90, 120, 121, 122, 191, 192, 193, 194, 195, 196, 197, 198] and single-photon driving field (strength $F$ and frequency ${\omega }_l$), respectively. The left cavity is assumed to have no photon leakage (also see discussions for the dissipation to the left cavity in Sec. 6), while the right cavity has the dissipation $\gamma$ with the Markovian approximation.

Figure 2

The influences of the dissipation $\gamma$ on UPB with the second-order correlation function [$g^{\left(2\right)}\left(0\right)$ in log scale] as a function of the phase $\varphi$ (in units of $\mathrm{rad}$) by numerically solving master equation (3). $\mathrm{\Delta }$ takes its negative optimal value, i.e., ${\mathrm{\Delta }}_{\mathrm{opt}}=(-1.59782{\omega }_a, -1.59149{\omega }_a, -1.58092{\omega }_a, -1.56697{\omega }_a)$ respectively corresponding to $\gamma =(0.2{\omega }_a, 0.4{\omega }_a, 0.6{\omega }_a, 0.79{\omega }_a)$. The other parameters chosen are $g=0.2{\omega }_a, G=0.0001{\omega }_a$, and $f=0.1{\omega }_a$.

Figure 3

The figure [$g^{\left(2\right)}\left(0\right)$ in log scale] shows the case of taking the positive optimal detuning ${\mathrm{\Delta }}_{\mathrm{opt}}=(1.59782{\omega }_a,1.59149{\omega }_a, 1.58092{\omega }_a, 1.56697{\omega }_a)$ vs $\gamma =(0.2{\omega }_a, 0.4{\omega }_a, 0.6{\omega }_a, 0.79{\omega }_a)$. The phase $\varphi$ is in units of $\mathrm{rad}$. The other parameters chosen are the same as those in Fig. 2.

Figure 4

The influences of the large dissipation $\gamma =(0.81{\omega }_a,1{\omega }_a,10{\omega }_a,50{\omega }_a,130{\omega }_a,199{\omega }_a)$ on UPB [$g^{\left(2\right)}\left(0\right)$ in log scale], where the negative optimal detuning ${\mathrm{\Delta }}_{\mathrm{opt}}=(-1.56528{\omega }_a, -1.54712{\omega }_a, -0.15964{\omega }_a, -0.00620{\omega }_a, -0.00072{\omega }_a, -0.00004{\omega }_a)$. The phase $\varphi$ is in units of $\mathrm{rad}$. The other parameters chosen are the same as those in Fig. 2.

Figure 5

The figure [$g^{\left(2\right)}\left(0\right)$ in log scale] shows the case of the positive optimal detuning ${\mathrm{\Delta }}_{\mathrm{opt}}=(1.56528{\omega }_a, 1.54712{\omega }_a, 0.15964{\omega }_a, 0.00620{\omega }_a, 0.00072{\omega }_a, 0.00004{\omega }_a)$ vs $\gamma =(0.81{\omega }_a,1{\omega }_a,10{\omega }_a,50{\omega }_a,130{\omega }_a,199{\omega }_a)$. The phase $\varphi$ is in units of $\mathrm{rad}$. The other parameters chosen are the same as Fig. 4.

Figure 6

Energy-level diagram with the zero-, one-, and two-photon states and transition paths leading to the quantum interference responsible for the strong photon antibunching. For the left cavity, the destructive quantum interference processes occur in the following two paths: (i) the direct excitation via two-photon pump in Eq. (8) and (ii) the indirect transitions by single-photon driving field and tunneling coupling in Eqs. (9) and (10).

Figure 7

(a) The two-photon pump and single-photon driving field respectively mediating the dissipationless left cavity and Markovian dissipative right cavity are equivalent to the fact that the two-photon pumped left cavity couples with a driven non-Markovian environment [see Eq. (20) and Appendix pp5 for more details]. In other words, the parts outside the left cavity can be considered as a non-Markovian composite environment [right cavity plus its Markovian environment, see Eq. (13)], which corresponds to the pseudomode theory [201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213]. (b) Energy-level diagram of the left cavity with the zero-, one-, and two-photon states and transition paths leading to the quantum interference responsible for the strong photon antibunching. There are two interference paths occurring in the left cavity from ${|0\rangle }_a$ to ${|2\rangle }_a$ in the non-Markovian regime: (i) ${|0\rangle }_a\stackrel{G}{\to }{|2\rangle }_a$ excited by two-photon pump with strength $G$ and (ii) ${|0\rangle }_a\stackrel{F_{\mathrm{eff}}}{\to }{|1\rangle }_a\stackrel{F_{\mathrm{eff}}}{\to }{|2\rangle }_a$ driven by effective single-photon driving field with strength $F_{\mathrm{eff}}$. However, there is only one path from ${|0\rangle }_a$ to ${|2\rangle }_a$ mediated by two-photon pump when the dissipation tends to infinity under the Markovian limit, where $F_{\mathrm{eff}}$ is close to zero.

Figure 8

$\left|u\right(t\left)\right|$ in Eq. (18) as a function of the time $t$ with different dissipation $\gamma$ from $0.2{\omega }_a$ to $0.7{\omega }_a$, which lies in the non-Markovian regime. The solid lines and circles correspond to $G=0.0001{\omega }_a$ and 0, respectively. The other parameters chosen are the same as those in Fig. 2.

Figure 9

$\left|u\right(t\left)\right|$ as a function of the time $t$ with different dissipation $\gamma$ taking $0.7{\omega }_{a}0.75{\omega }_a, 0.79{\omega }_a$, and $0.81{\omega }_a$ near the threshold $\gamma =0.8{\omega }_a$. The circles and solid lines correspond to Eq. (23) for $G=0$ and Eq. (18) for $G=0.0001{\omega }_a$, respectively. With Figs. 8 and 9, we find that $\left|u\right(t\left)\right|$ exhibits the backflow oscillation of the photon from the non-Markovian composite environment for $\gamma <0.8{\omega }_a$, while it displays decay in the Markovian regime when $\gamma >0.8{\omega }_a$. The other parameters chosen are the same as those in Fig. 2.

Figure 10

The figure corresponds to the Markovian regime, where the optimal detuning ${\mathrm{\Delta }}_{\mathrm{opt}}$ takes negative values. The other parameters chosen are the same as those in Fig. 4.

Figure 11

The dissipation rate ${\kappa }_1\left(t\right)$ in Eq. (21) of the left cavity as a function of the time $t$ with different dissipation $\gamma$ taking $0.2{\omega }_a, 0.5{\omega }_a$, and $0.78{\omega }_a$, which correspond to the non-Markovian regime. The other parameters chosen are the same as those in Fig. 2.

Figure 12

The figure shows the dissipation rate ${\kappa }_1\left(t\right)$ in Eq. (21) in the Markovian regime, where the optimal detuning ${\mathrm{\Delta }}_{\mathrm{opt}}$ takes negative values. The other parameters chosen are the same as those in Fig. 4.

Figure 13

In order to further understand the origin of UPB for the left cavity with the non-Markovian effect, we in this figure plot the time-dependent coefficients in the exact non-Markovian master equation (21): effective transition frequency $\left|X\right(t\left)\right|$, two-photon term $\left|Y\right(t\left)\right|$, coherent driving term $\left|Z\right(t\left)\right|$, decay rate $|{\kappa }_1\left(t\right)|$, fluctuation (noise) coefficient $|{\kappa }_2\left(t\right)|$, and squeezing rate $|{\kappa }_3\left(t\right)|$ as a function of the time $t$. Figure 13 takes ${\mathrm{\Delta }}_{\mathrm{opt}}=-1.59782{\omega }_a, {\varphi }_{\mathrm{opt}}=-1.50742\mathrm{rad}$, and $\gamma =0.2{\omega }_a$ in Fig. 2, while ${\mathrm{\Delta }}_{\mathrm{opt}}=-0.00004{\omega }_a, {\varphi }_{\mathrm{opt}}=-2.40619\mathrm{rad}$, and $\gamma =199{\omega }_a$ in Fig. 4 correspond to Fig. 13. The other parameters chosen are $g=0.2{\omega }_a, G=0.0001{\omega }_a$, and $f=0.1{\omega }_a$.

Figure 14

The second-order correlation function [$g^{\left(2\right)}\left(0\right)$ in log scale] of the left cavity as a function of the phase $\varphi$ (in units of $\mathrm{rad}$) with the different dissipations $\gamma =0.2{\omega }_a, 0.4{\omega }_a, 0.6{\omega }_a$, and $0.79{\omega }_a$ for (a)–(d) in the non-Markovian regime. The red circles indicate the approximate analytical result (26), while the blue stars correspond to the numerical simulation by solving master equation (3). The other parameters chosen are the same as those in Fig. 2.

Figure 15

UPB with the non-Markovian effect for the quantum network can be realized in the dissipationless left cavity (eigenfrequency ${\omega }_a$) mediated by two-photon pump with amplitude $G$ and frequency ${\omega }_p$, where the left cavity couples to several dissipative right cavities (eigenfrequency ${\omega }_n$ and dissipation ${\gamma }_n$) with the coupling strength $g_n$. The right cavity $b_n$ is driven by single-photon driving field with frequency ${\mathrm{\Omega }}_n$ and amplitude $F_n$.

Figure 16

Energy-level diagram of the quantum network with the zero-, one-, and two-photon states and transition paths leading to the quantum interference responsible for the strong photon antibunching. For the left cavity, the quantum interference processes occur in $n+1$ paths as the figure shows.