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Abstract
Photon blockade (PB) is one of the effective methods to generate single-photon sources. In general, both the PB effect with the significant sub-Poissonian statistics and a large mean photon number are desired to guarantee the brightness and the purity of single-photon sources. Here, we propose to obtain the PB effect at the cavity dark-state polariton (DSP) using a cavity Λ-type electromagnetically induced transparency (EIT) system with and without the two-photon dissipation (TPD). In the Raman resonance case, the PB effect at the DSP could by realized by using the TPD process in the weak or intermediate coupling regime, which accompanies with near unity transmission, i.e., very high photon occupation. In the slightly detuned Raman resonance case, the excited state is induced into the components of the DSP, and the atomic dissipation path is added into the two-photon excitation paths. Thus, the PB effect at the DSP can be obtained due to the quantum destructive interference (QDI) in the strong coupling regime, which can be further enhanced using the TPD process. Due to the slight detuning, the PB effect still remains high photon occupation and has highly tunability. This work provides an alternative way to manipulate the photon statistics by the PB effect and has potential applications in generating single-photon sources with high brightness and purity.
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1. Introduction
Single-photon sources with high purity and brightness play an important role in quantum information science [1,2]. As one of the effective methods for generating single photons, the realization of the PB effect has attracted much interest in the past two decays [1–42]. It refers to an effect where the absorption of the first photon blocks the transmission of the second photon. As a result, photons tend to orderly output one by one, which exhibits the strong photon antibunching and sub-Poissonian statistics. Currently, there are two mechanisms to achieve the photon blockade effect: the eigenenergy-level anharmonicity (ELA) and the quantum interference among different transition paths. The former mechanism results in the conventional PB (CPB) [4–9] and the latter one induces the unconventional PB (UPB) [10–17].
So far, a great deal of efforts have been made to realize PB effect, such as using the cavity quantum electrodynamics (QED) system [1–5], the optomechanical system [11,19–22], the superconducting circuits [23–26], the waveguide QED system [27–31]. Many of works about PB effect focus only on decreasing the second-order correlation function but ignore the mean photon number [9–13]. As one of crucial ways to generate singe-photon sources, a strong PB effect and high photon occupation are both desirable because the former ensures the high purity and the latter guarantees the brightness of single-photon sources. Thus, some works aim at realizing the PB effect accompanied with a large mean photon number [2,3,16,32,39–41].
It is known that the advances in the field of cavity QED have opened possibilities for the strong interactions between light and materials even at the single atom and single photon level [23,43–45]. It provides a powerful platform for wide applications, such as generation of single photons on demand [1–5], observing strong cavity-QED effects [46–48] and implementation of quantum communication and computation [49–51]. By far, the PB effect has been intensively studied in various cavity QED systems including cavity-driven [1–4] and atom-driven [4,5,39] systems. Most of those systems use J-C model with two-level emitters. However, due to the atomic spontaneous emission, the PB effect at the bright-state polariton (BSP) has small mean photon number [1,4,5,9,13]. In order to obtain high photon occupation, schemes for realizing the PB effect have been proposed via the cavity EIT based on the Kerr-type nonlinear-dispersive or nonlinear-dissipative effect created by the N-type atomic system [2,3,35,37,40,41]. To our knowledge, there is few research on realizing the PB effect using the traditional three-level EIT emitter driven by a probe field and a control field in a cavity [39]. In comparison with the PB effect at the BSP positions [1,4,5,9,13], the PB effect appearing at the DSP peak naturally possesses high photon occupation [2,3,35,37,40,41]. Furthermore, the cavity EIT in traditional three-level configurations creates a cavity DSP which features a tunable hybridization of photons and emitters, which is different from the EIT-Kerr system which eliminates the atomic degrees of freedom and behaves as an effective two-level system [35,41]. Thus, the realization of the PB effect in such traditional three-level EIT systems may have the potential for novel applications in quantum networks.
In this paper, we study the physical mechanism and the conditions of the PB effect with high cavity photon occupation in a cavity-driven $\Lambda$-type EIT system. When the Raman resonance condition is met, the PB effect can be obtained at the DSP position assisted by the two-photon dissipation (TPD) process in the weak or intermediate coupling regime. When the Raman resonance is slightly detuned, the atomic dissipation path is added into the two-photon excitation pathways. Thus, different from the PB effect based on the above mentioned nonlinear mechanisms [2,3,35,39–41], here the PB effect at the DSP can be obtained by the destructively quantum interference in the strong coupling regime even without the TPD process. Furthermore, due to slightly detuned resonance EIT, the PB effect accompanies with high photon occupation and highly tunability.
2. Model system
We consider a single cold EIT atom trapped in an optical cavity, as illustrated in Fig. 1(a). The relevant levels of the atom include two ground states $|g\rangle$ and $|s\rangle$ as well as one excited state $|e\rangle$. The single atom is initially prepared to the ground state $|g\rangle$. The cavity mode couples to the transition between $|g\rangle$ and $|e\rangle$, which is driven by a weak input probe field with frequency $\omega _p$. An external control field with Rabi frequency $\Omega _c$ drives the transition between $|s\rangle$ and $|e\rangle$, which can control the transition $|g\rangle \leftrightarrow |e\rangle$ through EIT process. Under the rotating-wave and electric dipole approximations, the Hamiltonian $\hat {H}$ of this system reads ($\hbar \equiv 1$)
Here, $\hat {a}$ is the annihilation operator of the cavity mode, and $\hat {\sigma }_{lm}=|l\rangle \langle m|$ ($l,m=g,s,e$) is the atomic raising or lowering operator for $l\neq m$ or the atomic population operator for $l=m$. $\omega _l$ ($l=g,s,e$) is the energy of the state $|l\rangle$ and we set $\omega _g\equiv 0$. $\omega _c$ and $\omega _a$ are the frequencies of the cavity and the control field, respectively. $g$ is the coupling strength between the cavity mode and the single atom. $E$ represents the driving strength of the input probe field.
Fig. 1. (a) Schematic of a three-level $\Lambda$-type atom coupled to an optical cavity. An input probe field $E$ drives the cavity mode which couples to the transition $|g\rangle \leftrightarrow |e\rangle$ with the coupling strength g. A control field $\Omega _c$ couples to the transition $|s\rangle \leftrightarrow |e\rangle$. $\Delta _a$ ($\Delta _c$) is the detuning between the cavity mode (control field $\Omega _c$) and the relative atomic transition. When the condition $\Delta _a=\Delta _c$ is met, we call this case the Raman two-photon resonance case or Raman resonance case for simplicity. (b) The single-photon dissipation process $\kappa _1$ absorbs only one single photon at once. The two-photon dissipation process $\kappa _2$ absorbs two overlapping photons simultaneously, which reduces the two-photon excitation with the one-photon excitation unchanged [1,13,41,42,52–55].
In the rotating frame defined by the unitary transformation
The dynamics of the system can be described by the master equation:
We focus on the zero-time delay second-order correlation function in the steady state (i.e., $t\rightarrow +\infty$) $g^{(2)}(0)=\frac {\langle \Psi |\hat {a}^{{\dagger }2}\hat {a}^2|\Psi \rangle _s}{\langle \Psi |\hat {a}^{\dagger }\hat {a}|\Psi \rangle _s^2}$ and the mean cavity photon number $\langle \Psi |\hat {a}^{\dagger }\hat {a}|\Psi \rangle _s$, which characterizes the purity and brightness of the single-photon source, respectively. The condition $g^{(2)}(0)>1$ ($g^{(2)}(0)<1$) indicates that the light field performs bunching (antibunching) effect with super-Poissonian (sub-Poissonian) photon-number statistics [4,5], and $g^{(2)}(0)=1$ represents a coherent light field. When $g^{(2)}(0)\ll 1$, it means PB effect is achieved and the intracavity photons tend to output one by one. Under the weak excitation (truncating the Hilbert space at the two quanta level), the wave function of the system is approximately expressed as
3. Photon blockade
The aim in this paper is to analyze the mechanism and the conditions for PB effect with high photon occupation through which the single-photon source with high purity and brightness can be generated in the $\Lambda$-type cavity EIT system. Thus, we investigate the photon statistics estimated by $g^{(2)}(0)$ and the mean cavity photon number indicated by the transmission $\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2$ at the same time in two cases: Raman resonance case ($\Delta _a-\Delta _c=0$) and Raman nonresonance case ($\Delta _a-\Delta _c\ne 0$) with and without considering the TPD process ($\kappa _2=0$ and $\kappa _2\ne 0$). For convenience, we set $\Delta _a=0$ throughout the paper.
Case of resonance. First, we focus on the Raman resonance case where the detunings $\Delta _a$ and $\Delta _c$ are taken to be zero ($\Delta _a=\Delta _c=0$). By numerically solving Eq. (4), the variation of $g^{(2)}(0)$ and $\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2$ are plotted as a function of the probe detuning $\Delta _p/\kappa _1$ in Fig. 2. We can see that there are three dips for the line of $g^{(2)}(0)$ corresponding to the three peaks for the line of $\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2$, which indicates the potential to obtain the output light with simultaneous sub-Poissonian distribution properties and a large mean photon number. The middle peak and the middle dip appear at the cavity DSP with frequency $\Delta _p=0$, while the two side peaks and the two side dips appear at the two cavity BSPs with $\Delta _p=\pm \sqrt {g^2+\Omega _c^2}$, respectively. In Fig. 2, it shows that the antibunching effect in this resonance case only exsits at the two BSP positions and is not available at the DSP position when the TPD process is not taken to be considered ($\kappa _2=0$). The physical mechanism for the antibunching effect is due to the well-known anharmonicity of the eigenenergy-level structure, and it can be further enhanced by using the larger coupling strength $g$ [4,5,57,58]. However, due to the coupling between the BSP and the excited state, the two peaks of the mean photon number are relatively small ($\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2\simeq 0.1$) for the spontaneous emission $\gamma _{eg}$ and $\gamma _{es}$.
Fig. 2. The second-order correlation function $g^{(2)}(0)$ (black solid line and yellow dash-dotted line) and the transmission $\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2$ (red dotted line and blue dashed line) of the system in the resonance case ($\Delta _a-\Delta _c=0$) with the TPD process $\kappa _2=20\kappa _1$ (yellow dash-dotted line and blue dashed line) and without the TPD process $\kappa _2=0$ (black solid line and red dotted line). Other parameters are $g=6\kappa _1$, $\Gamma =0.5\kappa _1$, $\gamma =10^{-3}\kappa _1$, $\Omega _c=10\kappa _1$, $\Delta _a=\Delta _c=0$, $E=0.05\kappa _1$, $\kappa =2\kappa _1$, $\gamma _{eg}=\gamma _{es}=2\Gamma$, $\gamma _{sg}=2\gamma$.
In contrast to the BSP, the DSP naturally possesses high photon transmission. Thus, in the following, we investigate the physical mechanism and the conditions for obtaining the PB effect at the DSP position in order to obtain the high photon occupation simultaneously. According to Fig. 2, assisted with the TPD process ($\kappa _2\ne 0$), the coherent output light ($g^{(2)}(0)=1$) transforms to the antibunching light with strong sub-Poissonian statistics ($g^{(2)}(0)\simeq 10^{-2.4}$) at the DSP position ($\Delta _p=0$). This is because the role of $\kappa _2$ is to reduce the occupation of the two-photon excitation while keeping the occupation of the single-photon excitation almost unchanged (red dotted line and blue dashed line in Fig. 2). Due to the EIT effect, the PB effect naturally accompanies with near unity transmission, i.e., very high photon occupation ($\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2\simeq 1$ as shown in Fig. 2).
Next we plot the processes and the interference paths of the two-photon excitation at the DSP position in detail to analyze the physical mechanism of the above photon statistics (Fig. 3). The process 1 in Fig. 3(a) represents that two photons from the driving field $E$ pass through the cavity without being absorbed, which corresponds to the path 1: $|0_c,0_e,0_s\rangle \xrightarrow {E}|1_c,0_e,0_s\rangle \xrightarrow {\sqrt {2}E}|2_c,0_e,0_s\rangle$ seen in Fig. 3(b). The process 2 in Fig. 3(a) indicates the two transmitted photons consist of one unabsorbed photon and one photon emitted by an excited atom in the state $|e\rangle$, which is related to the path 2: $|0_c,0_e,0_s\rangle \xrightarrow {E}|1_c,0_e,0_s\rangle \xrightarrow {g}|0_c,1_e,0_s\rangle \xrightarrow {E}|1_c,1_e,0_s\rangle \xrightarrow {\sqrt {2}g}|2_c,0_e,0_s\rangle$ in Fig. 3(b). The process 3 in Fig. 3(a) means one of the transmitted two photons is emitted from the atom in the state $|s\rangle$ and the other is directly from the one photon of the driving field without absorption, which corresponds to the path 3: $|0_c,0_e,0_s\rangle \xrightarrow {E}|1_c,0_e,0_s\rangle \xrightarrow {g}|0_c,1_e,0_s\rangle \xrightarrow {\Omega _c}|0_c,0_e,1_s\rangle \xrightarrow {E}|1_c,0_e,1_s\rangle \xrightarrow {\Omega _c}|1_c,1_e,0_s\rangle \xrightarrow {\sqrt {2}g}|2_c,0_e,0_s\rangle$ in Fig. 3(b). As mentioned above, we set $\Delta _a=0$ throughout the paper for convenience and focus on the DSP position ($\Delta _p=0$). Due to the almost completely DQI of EIT effect, the probabilities for the atom to be excited to the states $|0_c,1_e,0_s\rangle$ and $|1_c,1_e,0_s\rangle$ are extremely small for the near vanishing amplitudes $A_{010}$ and $A_{110}$ (blue dashed lines in Figs. 3(c-f)). Then, it can be deduced that the two-photon excitation path associated with the process 2 is nearly blocked as shown in Fig. 3(b). While $A_{100}$, $A_{001}$, $A_{200}$ and $A_{101}$ are relatively large under the control of $\Omega _c$ (red dotted lines and yellow dot-dashed lines in Figs. 3(c-d)), which indicates the path 1 and the path 3 are well opened. Due to the constructive interference between the path 1 and the the path 3, the PB effect can not exist and the output light behaves as a coherent field when the TPD process is not taken into account ($\kappa _2=0$). If the TPD effect is considered, comparing Figs. 3(c) and 3(e) with Figs. 3(d) and 3(f), we can see that the TPD process has almost no effect on all of the one-photon excitation probabilities $|A_{100}|^2$, $|A_{010}|^2$ and $|A_{001}|^2$, while significantly reduces the two-photon excitation probabilities $|A_{200}|^2$, $|A_{011}|^2$ and $|A_{101}|^2$. As a result, such nonlinear dissipation process (TPD) reduces the value of the second-order correlation function $g^{(2)}(0)\simeq 10^{-2.4}$, i.e., enhances the nonclassical sub-Poissonian statistics, and at the same time maintains the high photon occupation unchanged $\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2\simeq |A_{100}|^2\rightarrow 1$ (Fig. 2).
Fig. 3. In the Raman resonance case: Diagrammatic representation of the processes (1, 2, 3) which contribute to the two-photon transmission (a). The coupling schematic corresponds to the relative transitions. Due to the resonant coupling, the process 2 is almost blocked, which corresponds to the interference path $|0_c,0_e,0_s\rangle \xrightarrow {E}|1_c,0_e,0_s\rangle \xrightarrow {g}|0_c,1_e,0_s\rangle \xrightarrow {E}|1_c,1_e,0_s\rangle \xrightarrow {\sqrt {2}g}|2_c,0_e,0_s\rangle$ (b). The probabilities $|A_{100}|^2$, $|A_{010}|^2$ and $|A_{001}|^2$ ((c) and (d)) as well as $|A_{200}|^2$, $|A_{110}|^2$ and $|A_{101}|^2$ ((e) and (f)) versus the normalized detuning $\Delta _p/\kappa _1$. $\kappa _2=0$ for (c) and (e), and $\kappa _2=20\kappa _1$ for (d) and (f). Other parameters are the same as in Fig. 2: $g=6\kappa _1$, $\Gamma =0.5\kappa _1$, $\gamma =10^{-3}\kappa _1$, $\Omega _c=10\kappa _1$, $\Delta _a=\Delta _c=0$, $E=0.05\kappa _1$, $\kappa =2\kappa _1$, $\gamma _{eg}=\gamma _{es}=2\Gamma$, $\gamma _{sg}=2\gamma$.
As we all know that the cavity DSP operator can be written as $\hat {D}=\frac {\Omega _c}{\sqrt {\Omega _c^2+g^2}}\hat {a}-\frac {g}{\sqrt {\Omega _c^2+g^2}}\hat {\sigma }_{gs}$ [56]. The coupling strength $g$ and the control field strength $\Omega _c$ determine the components of the light and matter of the cavity DSP, which will also influence its photon statistical properties. In order to analyze the conditions of the PB effect, we next investigate the effect of the parameters $\Gamma$, $\kappa _2$ $g$, $\Omega _c$, and $\gamma$ on the photon statistics $g^{(2)}(0)$ and the probabilities $|A_{100}|^4$ and $|A_{200}|^2$ at the DSP position for the resonance case. According to Figs. 4(a) and 4(b), it can be seen that due to the EIT effect, the cavity DSP is uncoupled to the excited state $|0_c,1_e,0_s\rangle$ and $|1_c,1_e,0_s\rangle$, therefore, the atomic dissipation process ($\gamma _{eg}=\gamma _{es}=2\Gamma$) has no influence on the photon statistics with and without the TPD process. From Figs. 4(a-g), the TPD process only attenuates the two-photon excitation process and has no influence on the one-photon excitation as mentioned above. With the $\kappa _2$ increasing, the amplitude $A_{200}$ is reducing, which results in the vanishing value of $g^{(2)}(0)$ (Fig. 4(c)). To date, several systems have considered the large rate of $\kappa _2$ [41,52,53], and the related experimental technologies have also been reported [54,55]. It provides essential contribution for realizing the PB effect in the resonance case. Furthermore, the TPD process improves the potential tunabilities of the photon statistics by the coupling strength $g$ and the control field $\Omega _c$. When we close the TPD process, the influence of the parameters $g$ and $\Omega _c$ are almost vanishing on the amplitudes $A_{100}$ and $A_{200}$ due to the EIT effect. And the output light remains the statistical property of the coherent field unchanged in spite of the increasing value of $g$ and $\Omega _c$ (Figs. 4(d) and 4(f)). Once the TPD process is opened, it reduces the amplitude $A_{200}$ and $g^{(2)}(0)$ significantly (Figs. 4(e) and 4(g)). It is noticeable that the strong coupling strength $g$ destroys the reduction of the two-photon excitation by the TPD effect (Fig. 4(e)). Thus, we conclude that the strong PB effect accompanies with very high cavity photon occupation ($\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2\simeq 1$) can be realized via the caivty DSP assisted with the TPD process in a weak or intermediate coupling regime. Due to the hybridization of DSP, the realized antibunching photons simultaneously possess a tunable intracavity photon lifetime ($\tau _D=(\frac {\Omega _c^2}{\sqrt {\Omega _c^2+g^2}}\kappa )^{-1}$ [59,60]), which plays an essential role in the quantum storage, slowing the speed of light and so on [61,62]. In practice, the EIT may behave imperfect with a non-negligible decoherence rate $\gamma _{sg}$ due to the external magnetic field, or other non-idealities. In Figs. 4(h) and 4(i), the values of $g^{(2)}(0)$, $A_{100}$, and $A_{200}$ remains almost unchanged with the increasing of $\gamma$, which means the realized PB effect with high photon occupation shows the robustness to the decoherence $\gamma _{sg}$ within a certain range of values of the decoherence rate $\gamma _{sg}$ in this case. Actually, if we properly chose the atomic energy levels $|g\rangle$ and $|s\rangle$ between which the transition is electric-dipole forbidden, the small decoherence rate $\gamma _{sg}=2\gamma \simeq 10^{-3}\gamma _{eg}$ can be available as in Refs. [36–41].
The case of nonresonance. Once the Raman resonance condition is broken up, the excited state $|e\rangle$ is induced into the components of the DSP, which allows the atomic dissipation process (the process 2 in Fig. 3(a)). In order to maintain high photon occupation, we slightly detune the probe frequency from Raman resonance ($\Delta _a-\Delta _c\ne 0$). Different from the resonance case, the PB effect can be realized ($g^{(2)}(0)\simeq 10^{-2}$) at the DSP position even without the TPD process $\kappa _2=0$ (Fig. 5). Assisted with the TPD process ($\kappa _2\ne 0$), the PB effect ($g^{(2)}(0)\simeq 10^{-3.2}$) can be further enhanced. It is worth noting that the PB effect occurs exactly at the quasi-DSP peak with a small Raman two-photon detuning, therefore, it still accompanies with high intracavity photon occupation ($\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2\simeq 0.65$). The physical mechanism can be explained by interfering two-photon excitation paths. Due to the nonresonance case, the excited state $|3\rangle$ is induced into the DSP, which introduces the atomic dissipation process and open the path 2: $|0,0,0\rangle \xrightarrow {E}|1_c,0_e,0_s\rangle \xrightarrow {g}|0_c,1_e,0_s\rangle \xrightarrow {E}|1_c,1_e,0_s\rangle \xrightarrow {\sqrt {2}g}|2_c,0_e,0_s\rangle$ as shown in Figs. 6(a) and 6(b). The unnegligible population of the states $|0_c,1_e,0_s\rangle$ and $|1_c,1_e,0_s\rangle$ in Figs. 6(c-f) also proves the open path 2. Thus, by destructively interfering the three two-photon excitation paths, the PB effect is obtained even with the TPD process. When the TPD process is induced, the PB effect can be further enhanced due to the nonlinear dissipation, i.e., the property of reducing the two-photon excitation and meantime keeping one-photon excitation unchanged.
Fig. 4. The logarithm of $|A_{100}|^4$, $|A_{200}|^2$, and $g^{(2)}(0)$ versus the normalized parameters $\Gamma /\kappa _1$, $\kappa _2/\kappa _1$, $g/\kappa _1$, $\Omega _c/\kappa _1$, and $\gamma /\kappa _1$. $\Delta _p$ is taken at the DSP position, i.e., $\Delta _p=0$ in the two-photon resonance case. $\kappa _2=0$ for (a), (d), (f), and (h), and $\kappa _2=20\kappa _1$ for (b), (e), (g), and (i). Other parameters are the same as in Fig. 2: $g=6\kappa _1$, $\Gamma =0.5\kappa _1$, $\gamma =10^{-3}\kappa _1$, $\Omega _c=10\kappa _1$, $\Delta _a=\Delta _c=0$, $E=0.05\kappa _1$, $\kappa =2\kappa _1$, $\gamma _{eg}=\gamma _{es}=2\Gamma$, $\gamma _{sg}=2\gamma$.
Fig. 5. The second-order correlation function $g^{(2)}(0)$ (black solid line and yellow dash-dotted line) and the transmission $\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2$ (red dotted line and blue dashed line) of the system in the resonance case ($\Delta _a-\Delta _c\ne 0$) with the TPD process $\kappa _2=20\kappa _1$ (yellow dash-dotted line and red dashed line) and without the TPD process $\kappa _2=0$ (black solid line and blue dotted line). Other parameters are $g=15\kappa _1$, $\Delta _c=8\kappa _1$, $\Delta _a=0$, $\Gamma =0.5\kappa _1$, $\gamma =10^{-3}\kappa _1$, $\Omega _c=3\kappa _1$, $E=0.05\kappa _1$, $\kappa =2\kappa _1$, $\gamma _{eg}=\gamma _{es}=2\Gamma$, $\gamma _{sg}=2\gamma$.
Fig. 6. In the Raman nonresonance case: Diagrammatic representation of the processes (1, 2, 3) which contribute to the two-photon transmission (a). The coupling schematic corresponds to the relative transitions. Due to the off-resonant coupling, the process 2 is opened and the corresponding path $|0_c,0_e,0_s\rangle \xrightarrow {E}|1_c,0_e,0_s\rangle \xrightarrow {g}|0_c,1_e,0_s\rangle \xrightarrow {E}|1_c,1_e,0_s\rangle \xrightarrow {\sqrt {2}g}|2_c,0_e,0_s\rangle$ participates in the destructively interfering among different pathways (b). The probabilities $|A_{100}|^2$, $|A_{010}|^2$ and $|A_{001}/10|^2$ ((c) and (d)) as well as $|A_{200}|^2$, $|A_{110}|^2$ and $|A_{101}/10|^2$ ((e) and (f)) versus the normalized detuning $\Delta _p/\kappa _1$. $\kappa _2=0$ for (c) and (e), and $\kappa _2=20\kappa _1$ for (d) and (f). Other parameters are the same as in Fig. 5: $g=15\kappa _1$, $\Delta _c=8\kappa _1$, $\Delta _a=0$, $\Gamma =0.5\kappa _1$, $\gamma =10^{-3}\kappa _1$, $\Omega _c=3\kappa _1$, $E=0.05\kappa _1$, $\kappa =2\kappa _1$, $\gamma _{eg}=\gamma _{es}=2\Gamma$, $\gamma _{sg}=2\gamma$.
To find the conditions of the PB effect in the nonresonance case, we also analyze the effects of the parameters $\Gamma$, $\kappa _2$, $g$, $\Omega _c$ and $\gamma$ on the second-order correlation function $g^{(2)}(0)$ and the probabilities $|A_{100}|^4$, $|A_{200}|^2$ at the DSP position in the nonresonance case ($\Delta _a-\Delta _c\ne 0$). In Fig. 7, the parameter $\kappa _2$ as mentioned before plays a role of reducing the amplitude $A_{200}$ and has no effect on the amplitude $A_{100}$. The larger value of $\kappa _2$ results in the smaller value of $g^{(2)}(0)$. Due to the nonzero Raman two-photon detuning, the atomic dissipation process is induced, and the increasing parameter $\Gamma$ makes $A_{100}$ decrease significantly while keeping $A_{200}$ almost unchanged, which destroy the sub-Possionian statistics of the output light (Figs. 7(a) and 7(b)). According to Figs. 7(d-g), the coupling strength $g$ and the control field strength $\Omega _c$ have potentials to manipulate $g^{(2)}(0)$ with and without the TPD process, whose influences are very different from those in the resonance case and allows for more flexible manipulation of the photon statistics at the DSP position. According to Fig. 7(d), the PB effect can be obtained by the QDI in the strong coupling regime ($g^{(2)}(0)\simeq 10^{-2}$ for $g=15\kappa _1=7.5\kappa$) even without the TPD process ($\kappa _2=0$), which can be further enhanced by the TPD process ($\kappa _2\ne 0$ in Fig. 7(e)). Due to the small detuning, the transmission is still high enough to maintain the large mean photon number. In contrast to the resonance case, the transmission behaves more sensitive to the decoherence $\gamma _{sg}$ than the photon statistics as shown in Figs. 7(h) and 7(i). As $\gamma _{sg}$ increases, $A_{100}$ decreases faster than $g^{(2)}(0)$ and $A_{200}$. Within a certain range of values of $\gamma$ (<0.1$\kappa _1$), the PB effect and the high transmission can still be preserved ($g^{(2)}(0)<10^{-2}$, $\langle \hat {a}^{\dagger }\hat {a}\rangle _s/E^2\simeq |A_{100}|^2/E^2\approx 40{\% }$) as indicated in Figs. 7(h) and 7(i).
Fig. 7. The logarithm of $|A_{100}|^4$, $|A_{200}|^2$, and $g^{(2)}(0)$ versus the normalized parameters $\Gamma /\kappa _1$, $\kappa _2/\kappa _1$, $g/\kappa _1$, $\Omega _c/\kappa _1$, and $\gamma /\kappa _1$. $\Delta _p$ is taken at the DSP position in the Raman nonresonance case. $\kappa _2=0$ for (a), (d), (f), and (h), and $\kappa _2=20\kappa _1$ for (b), (e), (g), and (i). Other parameters are the same as in Fig. 5: $g=15\kappa _1$, $\Delta _c=8\kappa _1$, $\Delta _a=0$, $\Gamma =0.5\kappa _1$, $\gamma =10^{-3}\kappa _1$, $\Omega _c=3\kappa _1$, $E=0.05\kappa _1$, $\kappa =2\kappa _1$, $\gamma _{eg}=\gamma _{es}=2\Gamma$, $\gamma _{sg}=2\gamma$.
Comparison. According to the above analysis, the PB effect can be achieved with simultaneous high photon occupation in the cavity $\Lambda$-type EIT system. For clarity, we have listed the similarities, differences, advantages, and conditions of the schemes for realizing the PB effect via cavity EIT using a single atom in a traditional three-levle EIT configuration, a single Kerr-type nonlinear four-level atom [3,40] and Kerr-type nonlinear four-level atomic ensembles [2,35,41] in Table 1. The three major differences are summarized: (i) The QDI is the physical mechanism for realizing the PB effect in the nonresonance case in this paper, which distinguishes from the ELA mechanism or nonlinear dissipation mechanism in the cavity Kerr-type EIT systems [2,3,35,40,41]. (ii) The scheme in this paper possess more flexible manipulation of photon statistics by the parameters such as $g$ and $\Omega _c$ than that using atomic ensembles [2,35,41], whose effective Hamiltonian $\hat {H}^c$ (as shown in Table 1) eliminates the atomic degrees of freedom. (iii) In the resonance case, the PB effect can be realized with a weak coupling strength in this paper in contrast to the condition in Ref. [3]. Furthermore, in the nonresonance case, the requirement for the large value of $g$ to realize the PB effect in this paper can be appropriately relaxed by increasing the strength of the TPD $\kappa _2$.
Table 1. The similarities, differences, advantages, and conditions of the schemes for realizing PB effect via cavity EIT using a single three-level atom in a traditional EIT configuration, a single Kerr-type nonlinear four-level emitter, and Kerr-type nonlinear four-level atomic ensembles. $\hat {H}^a=A_1\hat {a}^{\dagger }\hat {a}+A_2\hat {\sigma }_{ee}+A_3\hat {a}^{\dagger }\hat {a}\hat {\sigma }_{ss}+(g\hat {a}\hat {\sigma }_{eg}+\Omega _c\hat {\sigma }_{es}+\textrm {H.c.})+E(\hat {a}^{\dagger }+\hat {a})$, $\hat {H}^b=B_1\hat {a}^{\dagger }\hat {a}+B_2\hat {\sigma }_{ee}+B_3\hat {a}^{\dagger }\hat {a}\hat {\sigma }_{gg}+B_4\hat {\sigma }_{ss}+(g\hat {a}\hat {\sigma }_{eg}+\Omega _c\hat {\sigma }_{es}+\textrm {H.c.})+E(\hat {a}^{\dagger }+\hat {a})$, $\hat {H}^c=C_1\hat {a}^{\dagger }\hat {a}+C_2\hat {a}^{\dagger }\hat {a}^{\dagger }\hat {a}\hat {a}+E(\hat {a}^{\dagger }+\hat {a})$, where the specific quantities related to the coefficients $A_i (i=1,2,3)$, $B_j (j=1,2,3,4)$, $C_k (i=1,2)$ can be found in Refs. [3], Refs. [40] and Refs. [2,35,41], respectively.
4. Conclusion
In summary, we investigate the physical mechanism and the conditions of the PB effect with high photon occupation in the cavity $\Lambda$-type EIT system with and without the TPD process. In the Raman resonance case, the PB effect can be obtained at the DSP position in the weak or intermediate coupling regime by using the TPD process, which accompanies with near unity transmission. Different from the resonance case, when the Raman resonance is slightly detuned, the excited state is induced into the components of the DSP, and the atomic dissipation path is opened, which participates in the interfering two-photon excitation paths. Thus, the PB effect with high photon occupation can be achieved by the QDI at the DSP position in the strong coupling regime even without the TPD process, which can be further enhanced by the TPD process. Furthermore, the tunability of the photon statistics via the parameters $g$ and $\Omega _c$ can be enhanced in the Raman nonresonance case. We also list out the similarities, differences, advantages, and conditions of the schemes for realizing PB effect via cavity EIT using a single three-level atom in a traditional EIT configuration, a single Kerr-type nonlinear four-level emitter, and Kerr-type nonlinear four-level atomic ensembles. Our results may provide a new approach to the single-photon sources with high brightness and purity via the PB effect.
Funding
Natural Science Foundation of Shandong Province (ZR2021LLZ006, ZR2021MA081); National Natural Science Foundation of China (12034007, 12204310, 61675118, 61773245); Taishan Scholar Foundation of Shandong Province (ts20190936); Shanghai Sailing Program (21YF1446900).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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