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Abstract
We propose a scheme to realize a two-photon Jaynes-Cummings model for a single atom inside an optical cavity. It is shown that the interplay of a laser detuning and atom (cavity) pump (driven) field gives rise to the strong single photon blockade, two-photon bundles, and photon-induced tunneling. With the cavity driven field, strong photon blockade occurs in the weak coupling regime, and switching between single photon blockade and photon-induced tunneling at two-photon resonance are achievable via increasing the driven strength. By turning on the atom pump field, quantum switching between two-photon bundles and photon-induced tunneling at four-photon resonance are realized. More interestingly, the high-quality quantum switching between single photon blockade, two-photon bundles, and photon-induced tunneling at three-photon resonance is achieved with combining the atom pump and cavity driven fields simultaneously. In contrast to the standard two-level Jaynes-Cummings model, our scheme with generating a two-photon (multi-photon) Jaynes-Cummings model reveals a prominent strategy to engineer a series of special nonclassical quantum states, which may pave the way for investigating basic quantum devices to implement in quantum information processing and quantum networks.
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1. Introduction
Manipulation of nonclassical states of light at the single photon or few photons levels is essential for advances in quantum information science that have been studied intensively because of its potential applications in quantum communication [1,2], quantum computation [3–6], and other emerging quantum technologies [7–9]. The driven-dissipative atom-cavity quantum electrodynamics (QEDs) system provides a versatile platform for generating such nonclassical states due to the enhancement of light-matter interaction induced by the anharmonic ladder in the dressed states [10]. This leads to the antibunching and sub-Poisson quantum statistics properties for cavity photon emission when the coherent driven field is resonant with one of the lowest doublet dressed states, which will suppress the two-photon excitations ascribed to the strong spectrum anharmonicity. The physical mechanism that a first generated photon will block the transmission of the second one in the far dispersive regime is known as conventional photon blockade (CPB). Meanwhile, an unconventional photon blockade (UPB) by constructing quantum interference for suppressing two-photon excitations has been studied [11].
Up to now, the mechanisms including CPB and UPB, as the key techniques for generating nonclassical light at the single-photon level, have led to tremendous advancements from quantum information to quantum metrology and exploration of fundamental physics [12,13]. The CPB relying on the strong energy spectrum anharmonicity has been experimentally observed in atom-cavity and circuit QEDs [10,14–17], and theoretically explored in a sequence of setups including atom-cavity QEDs [18–23], optomechanical resonators [24–26], and waveguide or circuit QEDs [27–29]. In contrast to the CPB with strong atom-cavity coupling, UPB utilizes the quantum destructive interference between different quantum transition paths from ground state to two-photon excitation state [30–39], which can be generated by using an auxiliary driving field [40–42], cavity field [43–45], and qubit [46,47]. Of particular interest, UPB in the weakly nonlinear regime has been experimentally observed in quantum dot-cavities [48] and coupled superconducting resonators [49].
Compared with single-photon blockade, the engineering $n$-photon blockade ($n$PB, $n\geq 2$) possessing inherently nonclassical correlations also provides a fundamental element in quantum information technology. The realization of $n$PB usually by employing a high-order $n$-photon resonance absorption is a challenge task due to the weak intrinsic nonlinear photon-photon interactions [50,51]. The state-of-the-art methods for realizing $n$PB are exploited including postselection [52], down-converted photons [53,54], squeezing [55,56], or using biexciton states in quantum dot systems [57–59]. Recently, 2PB has been experimentally achieved in a strongly coupled atom-cavity QEDs [60] and attracted much attention in a series of studies in Kerr-type system [61,62] and circuit QEDs [63–67]. Furthermore, the $n$-photon bundles state which is associated with $n$PB possesses a special quantum statistic property with a strong photon-antibunching for the separated bundles of photons [68]. Remarkably, the high-quality $n$-photon/phonon bundles state could offer new resources in a wide range of quantum science and technology [69–73].
In this work, we propose a scheme to construct a tunable two-photon Jaynes-Cummings model (JCM) in a single atom trapped in an optical cavity. We show that the two-photon Jaynes-Cummings (JC) type of interaction is conducive to generate strong PB at single-photon resonance due to the enhanced anharmonic dressed state splittings and the photon-induced tunneling (PIT) occurs at $n$-photon resonance ($n\geq 2$) with increasing the strength of cavity driven field. In the presence of atom pump field, two-photon bundles state with strong antibunching for the separated bundles of photons and bunching for single photons is generated, owing to the anharmonic ladder of the energy spectrum. Moreover, the PIT occurs at four-photon resonance corresponding to the enhanced two-photon atomic transition by the classical pump field. Furthermore, we show that rich nonclassical states can be observed with combining the two driven fields together in contrast to the results of separately turning on cavity or atom field. Remarkably, a high-quality quantum switching from strong PB to two-photon bundles and PIT is realized by tuning the cavity-light detuning. Our study with generation of two-photon JC-type interaction not only offers a facilitate platform to investigate intriguing nonclassical states but also provides a broad physics community for applications in quantum information science [74–77].
2. Model and Hamiltonian
Here we consider a single four-level atom trapped inside a high-finesse optical cavity with an intrinsic cavity decay rate $\kappa$. The laser configuration and energy level structure of the atom-cavity system are shown in Fig. 1(a). The atom consists of an electronic ground state $|g\rangle$, and three excited states $|e \rangle$, $|r \rangle$ and $|m\rangle$. The atomic transition $|g\rangle \leftrightarrow |e\rangle$ ($|e\rangle \leftrightarrow |m\rangle$) is coupled to the cavity with a single atom-cavity coupling $g_0$ and an atom-cavity detuning $\Delta _1$. As to the driven field, in order to generate an effective two-photon pump field, the atom is illuminated by two far-resonantly classical laser fields with Rabi frequencies $\Omega _1$ and $\Omega _2$, corresponding to the single-photon atom-pump detuning $\Delta _2$. In addition, the cavity is also driven by a weak laser field with driven amplitude $\eta$ and frequency $\omega _p$, which yields a cavity-light detuning $\Delta _c'=\omega _c-\omega _p$ with $\omega _c$ denoting the bare cavity frequency.
Fig. 1. (a) Scheme for creating single photons and two-photon bundles states in a single-atom trapped in an optical cavity. A single four-level atom is trapped in a high-finesse optical cavity. The atom and cavity are illuminated by the driven field with driven amplitude $\eta$ and pump field with Rabi frequency $\Omega$, respectively. The atomic transition $|g\rangle \leftrightarrow |e\rangle$ ($|e\rangle \leftrightarrow |m\rangle$) is coupled to the cavity with single atom-cavity coupling $g_0$ and the atom-cavity detuning $\Delta _1$. The transition $|g\rangle \leftrightarrow |r\rangle$ ($|r\rangle \leftrightarrow |m\rangle$) is coupled to the classical pump field with Rabi frequencies $\Omega _2$ and $\Omega _1$, corresponding to the single-photon atom-pump detuning $\Delta _2$. (b) The typical anharmonic energy spectrum for two-photon JCM with ignoring the weak driven and pump fields.
Under the rotating-wave approximation, the relevant Hamiltonian of the single atom-cavity system is given by
In the far-off resonance regime, i.e., $|g_0/\Delta _1|\ll 1$ and $|\Omega _2/\Delta _2|\ll 1$, the excited states $|e\rangle$ and $|r\rangle$ can be adiabatically eliminated, yielding an effective single two-level atom coupled to the single-mode cavity with two-photon transition. The relevant time-independent Hamiltonian for the single atom-cavity system is given by
Indeed, the Hamiltonian in Eq. (4) essentially describes a two-photon JCM [78]. As we will see below, a series of rich nonclassical correlated quantum states including strong PB, two-photon bundle states, and PIT are readily observed in our laser configuration. We should note that the effective atomic two-photon transition process ($\Omega$) will play an important role in generating the two-photon bundles emission.
3. Energy spectrum
To investigate the quantum properties of the proposed two-photon JCM, we first calculate the energy spectrum by diagonalizing the system Hamiltonian in Eq. (4). In the weak driving and atom pump fields regime, i.e., $|\eta /g| \ll 1$ and $|\Omega /g| \ll 1$, the total excitation number ${\hat {N}}= 2\hat {a}^{\dagger} \hat {a}+\hat {\sigma }_{mm}$ in our system can be regarded as a conserved quantity. Fix the cavity excitation number as $n$, the Hilbert spaces for the system are restricted to $|n, g \rangle$ and $|n-2, m \rangle$, then the relevant matrix ${\mathcal {M}}$ is readily calculated by solving the $Schr\ddot {o}dinger$ equation. Explicitly, the matrix ${\mathcal {M}}$ is expressed as
In the atom-photon resonance regime, i.e., $\Delta _a=2\Delta _c$, the typical anharmonicity energy spectrum of the two-photon JCM is shown in Fig. 1(b). As can be seen, when the first dressed state ($|1, \pm \rangle$) is resonant with the cavity driven field (red line), the energy levels of the second energy eigenstates $|2, \pm \rangle$ are off-resonance with an energy gap of $\sqrt {2} g$, which means that the two-photon excitation will be blockaded for a proper coupling strength with $g/\kappa \gg 1$. Naturally, the purity of single-photon excitation can be enhanced with an increasing $g$. Adjusting the atom pump field resonant to the second dressed states (blue line), the subsequent transitions from ($|2, \pm \rangle \leftrightarrow |4, \pm \rangle$) are suppressed due to the JC-like anharmonicity ladder, then 2PB is predicted. For nonzero atomic pump and cavity driven fields, we have checked that the level structure in Fig. 1(b) is slightly distorted with small $\Omega$ and $\eta$. We remark that the energy spectrum is immune to the weak atomic pump and cavity driven fields, i.e., $|\Omega /g| \ll 1$ and $|\eta /g| \ll 1$.
It should be noted that 2PB can also be obtained by employing a parametric driving term of the cavity [79], e.g., $H_d=\eta (\hat a^{{\dagger} 2} + \hat a ^2)$ (two-photon cavity driven, green line). However, this kind of squeezing-like nonlinear interaction in cavity is difficult to achieve in experiment. Moreover, we have checked that, compared with the direct atom pump field, the cavity parametric driven for generating 2PB requires a more larger atom-cavity coupling $g$. Therefore, our scheme will facilitate the experimental feasibility with mitigating the strong coupling regime. Interestingly, the single photon transition induced by the cavity field and the two-photon atomic transition driven by pump field can be highly controlled independently. Therefore the single PB and 2PB can be achieved at single-photon resonance and two-photon resonance respectively. Remarkably, the quantum switching between single photon and two-photon bundles emission can be realized by tuning the cavity-light detuning, as shown below.
4. Generalized quantum statistics
Considering dissipations of the single atom and cavity field, a complete description of quantum dynamics of the system is dominated by master equation. Evolution equation of the corresponding density matrix $\rho$ is described as
Here we investigate the quantum statistics and correlations of the emitted photons by introducing the generalized $k$th-order correlation function [68,73]
The single PB described the phenomenon that absorption of the first photon will suppress the transition of the following photons. Intuitively, single PB exhibits sub-Poissonian photon-number statistics and photon antibunching, which is usually characterized by second-order correlation function. Thus the emitted light field has to fulfill the following criteria to verify single PB: $g^{(2)}_1(0)<1$ and $g_1^{(2)}(0)<g_1^{(2)}(\tau )$. On the contrary, PIT shows the quantum effect that the first excited photon favors the transmission of subsequent photons. Therefore, PIT is photon bunching or super-Poissonian photon number statistics [15,44]. In fact, various criteria for PIT have been employed, i.e., $g_1^{(2)}(0)>1$ [83,84], where $g_1^{(2)}(0)$ is a local maximum [15]. Here we consider the criteria that $g_1^{(k)}(0)>1$ for $k=2,3,4$ to character the PIT effect [85].
Similar to single PB, absorption of $n$-photons ($n\geq 2$) can also prevent the absorption of subsequent photons, which is referred as $n$PB. The signatures for $n$PB are determined by the following relations: $g_1^{(n+1)}(0) < 1$ and $g_1^{(n)}(0) \ge 1$. This means that, the emitted light field exhibits the $(n + 1)$-photon sub-Poissonian photon statistics, and $n$-photon super-Poissonian if $g_1^{(n)}(0) >1$ or Poisson photon statistics if $g_1^{(n)}(0) =1$ [60,85]. As to the $n$-photon bundles emission ($n>1$), two additional conditions of $g_1^{(2)}(0)>g_1^{(2)}(\tau )$ and $g_n^{(2)}(0)<g_n^{(2)}(\tau )$ must be fulfilled to guarantee bunching of bundle photons and antibunching between the separated bundles of photons [53,68].
5. Numerical results
Now, we study the nonclassical states of photon emission in two-photon JCM by numerically solving the master equation in Eq. (6). We will show below that strong PB ($g_1^{(2)}(0)\leq 0.01$), PIT, two-photon bundles, and quantum switching between them are observed in different driven configurations for cavity and atom fields. As to experimental feasibility, our scheme for two-photon JCM could be applicable to Rydberg atoms by employing advantages of energy-level structures [86,87]. In our numerical simulations, we take the cavity decay rate $\kappa =2\pi \times 150$ kHz as energy unit, for which high-finesse optical cavity has been demonstrated the capability in current experiments for studying supersolid phase in Bose-Einstein condensates [88,89]. We consider the atom-photon resonance regime with fixing $\Delta _a = 2\Delta _c$ and an effective atomic spontaneous decay rate $\gamma /\kappa =0.1$. Therefore, the free parameters in our system are reduced to atom-cavity coupling $g$, cavity-light detuning $\Delta _c$, cavity driven field amplitude $\eta$, and atom pump field strength $\Omega$. Based on state-of-the-art cavity QEDs, it is noteworthy that the effective two-photon atom-cavity coupling strength can reach $g/\kappa =9.1$ for taking the cavity decay rate $\kappa =2\pi \times 147$ kHz [88,89], single-photon atom-cavity coupling strength $g_0=2\pi \times 20$ MHz [60], and an atom-cavity detuning $\Delta _1/g_0 =-15$. Then the numerical results exhibited below present parameter space of atom-cavity coupling $0.01\leq g/\kappa \leq 15$, which is in the same order of magnitude as the parameter in recent experiment and within reach current experimental capabilities [90–92]. In addition, our proposal could be realized in superconducting quantum circuits benefited from the experimental observations of ultrastrong light-matter coupling $g/\kappa \gg 1$ [93].
5.1 Quantum switching between strong PB and PIT
We first study the photon emission only in the presence of cavity driven field ($\Omega =0$). For the weak atom-cavity coupling and driven field limit, e.g., $g/\kappa = 1$ and $\eta /\kappa =0.1$, the second-order correlation function at zero time delay $g_1^{(2)}(0)$ and steady-state intracavity photon number $n_s$ as a function of cavity-light detuning $\Delta _c$ are plotted as shown in Fig. 2(a). As can be seen, the strong PB with $g_1^{(2)}(0)=7.4\times 10^{-3}$ and $n_s=3.7\times 10^{-2}$ at single-photon resonance with $\Delta _c/g=0$ is observed. We note that the realization of strong PB is ascribed to the strong energy anharmonicity ladder in the dressed energy spectrum, which is highly consistent with the result of energy spectrum as shown in Fig. 1(b) [red lines]. In Fig. 2(b), we plot the typical interval dependence of second-order correlation function $g_1^{(2)}(\tau )$ at $\Delta _c/g=0$, which demonstrates the sub-Poissonian and photon-antibunching statistics of photon emission with $g_1^{(2)}(0)<1$ and $g_1^{(2)}(0)<g_1^{(2)}(\tau )$.
Fig. 2. (a) The second-order correlation function $g_1^{(2)}(0)$ (red solid line) and the corresponding steady-state photon-number $n_s$ (blue dashed line) as a function of cavity-light detuning $\Delta _c$ with $g/\kappa =1$. (b) The time interval $\tau$ dependent second-order correlation function $g_1^{(2)}(\tau )$ with $\Delta _c/g=0$ and $g/\kappa =1$. (c) The steady-state photon-number distribution $\tilde {p}(q)$ with the same parameters in (b). (d) The optimal $g_{1,{\rm opt}}^{(2)}(0)$ (red solid line) and the corresponding steady-state photon-number $n_{s,{\rm opt}}$ (blue dashed line) as a function of atom-cavity coupling $g$ at the single-photon resonance. The other parameter is fixed at $\eta /\kappa =0.1$.
To further confirm the PB nature of the photon states, steady-state photon-number distribution $\tilde {p}(q)=qp(q)/n_s$, which characters the fraction of $q$-photon states among the total photon emissions. Here, $p(q)=\langle q | \hat {a}^{\dagger} \hat {a} |q \rangle$ denotes the probability of $p$-photon Fock state. Figure 2(c) displays the steady-state photon-number distribution $\tilde {p}(q)$ as a function of $q$-photon states. Of particular interest, the photon emission exhibits nearly 100$\%$ single-photon nature with $\tilde {p}(1) \approx 1$ and $\tilde {p}(q) \approx 0~ (q\neq 1)$, corresponding to the completely suppressed higher photon emission ($q\geq 2$). These key evidences unambiguously demonstrated that the strong PB is achieved even in the weak-coupling regime in our system.
Next we calculate the optimal $g_{1,{\rm opt}}^{(2)}(0)= {\rm min}[g_1^{(2)}(\Delta _c)]$ and the corresponding steady-state photon-number $n_{s,{\rm opt}}$ at the same value of the cavity-light detuning as a function of atom-cavity coupling $g$ to characterize the quantum statistics quantitatively, as shown in Fig. 2(d). Clearly, the optimal $g_{1,{\rm opt}}^{(2)}(0)$ decreases monotonically with an increasing coupling $g$. The mechanism for generation of strong PB is similar to the standard two-level JCM, where the two-photon transitions will be suppressed due to the enhanced energy spectrum anharmonicity with an increasing $g$. Moreover, with the increasing of $g$, the corresponding photon number $n_s$ has a slightly decreasing for small $g$, then maintains a stable and relatively large value, which is also an important index to character the high-quality single photon sources.
To have a further investigation of the photon emission, we increase the driven field amplitude from $\eta /\kappa =0.1$ to $0.5$ at a moderate atom-cavity coupling $g/\kappa = 5$. As shown in Fig. 3(a), with the increase of $\eta$, the peak values of cavity photon number $n_s$ at single-photon resonance are enhanced as expected. Interestingly, two additional peaks emerge at the two-photon resonance for a larger $\eta$. To character the quantum statistics properties of the system in large $\eta$ regime, the $n$-order correlation functions $g_1^{(n)}(0)$ ($n=2, 3, 4$) as a function of the cavity-light detuning $\Delta _c$ for $\eta /\kappa =0.5$ are displayed in Fig. 3(b). As can be seen, the strong PB with $g_1^{(2)}(0)=2.6\times 10^{-3}$ and a high photon number of $n_s=0.33$ is observed at the single-photon resonance with $\Delta _c/g=0$. As to the two-photon resonance, the correlation functions show $g_1^{(4)}(0) > g_1^{(3)}(0) > g_1^{(2)}(0)=8$, corresponding to the two peak values of steady-state photon number of $n_s$=0.07, which indicates the realization of PIT in contrast to the single-photon emission at the weak value of $\eta$.
Fig. 3. (a) The steady-state photon number $n_s$ as a function of cavity-light detuning $\Delta _c$ for different values of $\eta /\kappa =$0.1, 0.3 and 0.5. (b) The $n$-order correlation function $g_1^{(n)}(0)$ as a function of $\Delta _c$ for $\eta /\kappa =0.5$. In (a)-(b), the other parameter is fixed at $g/\kappa =5$. Distribution of (c) $g_1^{(n)}(0)$ and (d) the corresponding $n_s$ versus $g$ at the single-photon resonance and two-photon resonance with $\eta /\kappa =0.5$, respectively. Here, $n$PR denotes the $n$-photon resonance for shorthand notation. The black dotted lines indicating $g^{(n)}_{1}(0)=1$ are plotted to guide the eyes.
Figures 3(c) and 3(d) display $g_1^{(n)}(0)$ and $n_s$ as a function of $g$ at single-photon resonance and two-photon resonance. For an increasing $g$, single PB generated at single-photon resonance is enhanced with the values of $g_1^{(2)}(0)$ decreasing. Similar to the results of the weak cavity driven field [see Fig. 2(d) with $\eta /\kappa =0.1$], the corresponding $n_{s}$ for single PB decreases suddenly and then saturates to a large photon number of $n_s\approx 0.33$. For PIT occurring at the two-photon resonance, $g_1^{(n)}(0)$ with $n=2, 3, 4$ increase rapidly with an increasing $g$, indicating the enhancement of PIT effect although the corresponding $n_{s}$ decreases monotonously. We should note that the PIT is caused by the large cavity driven field favoring the atomic transition $|0 \rangle \leftrightarrow |2, \pm \rangle$. Therefore, the quantum switching between single PB and PIT is highly controlled by tuning the cavity-light detuning from single-photon resonance to two-photon resonance.
5.2 Quantum switching between two-photon bundles and PIT
We now turn to study the properties of photon states with turning on the atom pump field with $\eta =0$. Figures 4(a) and 4(b) display the steady-state photon number $n_s$ and correlation functions $g_1^{(n)}(0)$ ($n=2, 3$) versus cavity-light detuning $\Delta _c$ for atom-cavity coupling $g/\kappa =15$ and $\Omega /\kappa =0.3$. It is clear that 2PB can be obtained at the two-photon resonance with $g_1^{(2)}(0)=2.29>1$ and $g_1^{(3)}(0)=5\times 10^{-2} \ll 1$, corresponding to the peak value of photon number $n_s=0.22$. Moreover, the 2PB appears in a relatively large parameter regime as shown in the light-blue area in Figs. 4(a) and 4(b). This result indicates the 2PB is robust and therefore will facilitate the experimental feasibility. It should be noted that the single PB can not be occurred at the single-photon resonance with $\Delta _c/g=0$ due to the forbidden transition of $|0 \rangle \leftrightarrow |1, \pm \rangle$ in the absence of the cavity driven field, which is highly agreement with our analytical results for energy spectrum, as shown in Fig. 1(b) [blue lines].
Fig. 4. (a) The steady-state photon number $n_s$ and (b) correlation functions $g_1^{(n)}(0)$ as a function of cavity-light detuning $\Delta _c$ for $\Omega /\kappa =0.3$. (c) $n_s$ and (d) $g_1^{(n)}(0)$ as a function of $\Delta _c$ for $\Omega /\kappa =1.8$. The $g^{(n)}_{1}(0)$ and corresponding $n_s$ as a function of $\Omega$ at the two-photon resonance (e) and four-photon resonance (f), respectively. (g) Distribution of $g_1^{(2)}(\tau )$ (blue solid line) and $g^{(2)}_2(\tau )$ (red dashed line) at the two-photon resonance with $\Delta _c/g=\pm 0.71$ and $\Omega /\kappa =0.45$. The inset shows the steady-state photon-number distribution $\tilde {p}(q)$ with the same parameters in (g). The other parameters are fixed at $g/\kappa =15$ and $\eta /\kappa =0$. Here, $n$PR denotes the $n$-photon resonance for shorthand notation. The black dotted lines indicating $g^{(n)}_{1}(0)=1$ are plotted to guide the eyes.
To process further, we study the quantum statistics properties of the photon emission in a large atom pump field with fixing $\Omega /\kappa =1.8$. Figures 4(c) and 4(d) depict $n_s$ and $g_1^{(n)}(0)$ ($n=2, 3, 4$) versus $\Delta _c$, respectively. Besides the two-photon resonance, two new peaks appearing at the four-photon resonance with $\Delta _c/g= \pm 0.86$ are observed, which represent the PIT effect with satisfying $g^{(4)}_1(0) > g^{(3)}_1(0) > g^{(2)}_1(0)= 3.85$. For quantitative analysis the effect of the atom pump field, we plot $g^{(n)}_1(0)$ and the corresponding $n_s$ as a function of $\Omega$ at the two-photon resonance and four-photon resonance, as displayed in Figs. 4(e) and 4(f). With the increasing of $\Omega$, $g^{(2)}_1(0)$ reduces to less than 1 gradually, while $g^{(3)}_1(0)$ possesses a dip feature [Fig. 4(e)]. Interestingly, we find that the nonclassical photon states at the two-photon resonance are two-photon bundles state. In particular, the two-photon bundles emission emerges in a large parameter regime of $\Omega$, as shown in the light-blue area in Fig. 4(e). Moreover, the local minimum value of $g^{(3)}_1(0)=4\times 10^{-2}$ occurs at the two-photon emission regime with $\Omega /g=0.45$, corresponding to a large cavity photon number of $n_s=0.35$. Therefore, our proposal can be used as a high-quality two-photon emitter. As to four-photon resonance, the PIT effect occurs in the whole range of parameter $\Omega$, but is suppressed with an increasing $\Omega$, corresponding to $g^{(n)}_1(0)$ ($n=2, 3, 4$) decreasing monotonously [Fig. 4(f)]. Remarkably, the quantum switching between two-photon bundles emission and PIT can be realized by tuning the cavity-light detuning $\Delta _c$ within a large range of parameter $\Omega$ ($\Omega /\kappa <0.76$).
Then we plot the interval dependence of the correlation functions $g_1^{(2)}(\tau )$ (blue solid line) and $g_2^{(2)}(\tau )$ (red-dashed line) at two-photon resonance with $\Delta _c/g = 0.71$ for $\Omega /\kappa =0.45$ to further confirm nonclassical nature of the bundles emission, as shown in Fig. 4(g). Clearly, the conditions of $g_1^{(2)}(0)>g_1^{(2)}(\tau )$ and $g_2^{(2)}(0)< g_2^{(2)}(\tau )$ are satisfied, which demonstrating that the two-photon bundles state are realized. On the other hand, the two-photon bundles nature of the emitted photons is also confirmed by the photon-number distribution $\tilde {p}(q)$ shown in the inset of Fig. 4(g). As can be seen, the steady-state photon probabilities for $q=1$ and 2 exhibit almost the same values, which is highly close to the ideal two-photon bundles state [68]. Moreover, the higher photon emission ($q>2$) is completely suppressed, which clearly demonstrates the two-photon emission nature of the photon states. The observation of two-photon bundles state can be intuitively understood by calculating the energy spectrum [Fig. 1(b)]. Specifically, the transition $|0 \rangle \leftrightarrow |2, \pm \rangle$ is resonantly driven by atom pump field, while the far-off resonance transition $|2, \pm \rangle \leftrightarrow |4, \pm \rangle$ is suppressed. We should emphasize that the two-photon bundles emission is essentially different from the 2PB. In contrast to the quantum statistics of $n$PB characterized only by single photons, the additional condition for bundles emission requires antibunching for separated photon bundles [$g_2^{(2)}(0) < g_2^{(2)}(\tau )$].
As analyzed in energy spectrum, 2PB can also be obtained via a parametric cavity driven with the two-photon cavity driven term of $H_d=\eta (\hat a^{{\dagger} 2} + \hat a ^2)$, where the cavity field drives the atomic transition from vacuum state to $|2, \pm \rangle$ resonantly. Therefore, the realization of the 2PB can be ascribed to the JC-like anharmonicity ladder. To quantitatively distinguish the differences of the two driven cases, we calculate the correlation functions $g_1^{(2)}(0)$, $g_1^{(3)}(0)$, and the steady-state photon number $n_s$ for two-photon cavity driven with the optimal parameter of $\eta /\kappa =0.3, \Omega /\kappa =0$, and the direct atom pump with $\eta /\kappa =0, \Omega /\kappa =0.45$ as a function of cavity-light detuning $\Delta _c$ at a moderate atom-cavity coupling strength $g/\kappa =15$, as shown in Fig. 5. Evidently, $g_1^{(2)}(0)>1$ for both the two cases at the two-photon resonance, while the value of $g_1^{(3)}(0)$ for the direct atom pump is much less than the two-photon cavity driven case with a large range of $\Delta _c$, as displayed in Fig. 5(b). Interestingly, $g_1^{(3)}(0)=4\times 10^{-2}$ for the direct atom pump is almost an order of magnitude smaller than the two-photon cavity driven case ($g_1^{(3)}(0)=0.38$) at the two-photon resonance, with hosting a large steady-state photon number as well. This indicates that, the 2PB can be generated in the two cases, however, compared with the direct atom pump, the 2PB realized by the cavity parametric driven requires a more larger atom-cavity coupling $g$. Therefore, the proposal with the direct atom pump could facilitate the experimental feasibility for studying the high-quality two-photon source in the two-photon JCM.
Fig. 5. The correlation functions $g_1^{(2)}(0)$ (a), $g_1^{(3)}(0)$ (b) and the steady-state photon number $n_s$ (c) as a function of cavity-light detuning $\Delta _c$ for the two-photon cavity driven with $\eta /\kappa =0.3, \Omega /\kappa =0$ (the red solid lines) and the atom pump with $\eta /\kappa =0, \Omega /\kappa =0.45$ (the blue dotted-dashed lines), respectively. The other parameter is $g/\kappa =15$. The black dotted line indicating $g^{(3)}_{1}(0)=1$ is plotted to guide the eyes.
5.3 Quantum three-phase switching between PB, two-photon bundles, and PIT
Finally, we focus on the nonclassical correlated states of photon emission in the presence of cavity driven and atom pump fields concurrently. For fixing $g/\kappa =15$ and $\Omega /\kappa =0.45$, the steady-state photon number $n_s$ and the correlation functions $g_1^{(n)}(0)$ ($n=2, 3, 4$) are displayed in Figs. 6(a)–6(d) for $\eta /\kappa =0.2$ and 0.6, respectively. For a moderate cavity driven amplitude, e.g., $\eta /\kappa =0.2$, there exists three peaks for $n_s$ with respect to the single-photon resonance ($\Delta _c/g=0$) and two-photon resonance($\Delta _c/g = \pm 0.71$) [Fig. 6(a)]. Clearly, the singe-photon emission with $g_1^{(2)}(0)=6\times 10^{-2}$ and two-photon emission with $g_1^{(2)}(0)=1.43$, $g_1^{(3)}(0)=7\times 10^{-2}$ are observed in Fig. 6(b). With an increasing of $\eta /\kappa =0.6$, two additional peaks for $n_s$ appear at the three-photon resonance with $\Delta _c/g= \pm 0.82$ [Figs. 6(c)], which denotes the PIT effect with $g_1^{(4)}(0) > g_1^{(3)}(0) > g_1^{(2)}(0) = 12$ [Fig. 6(d)]. Compared to the results of $\eta /\kappa =0.2$, the single PB occurring at $\Delta _c=0$ can be further enhanced with $g_1^{(2)}(0)=6\times 10^{-3}$, while the 2PB occurring at $\Delta _c = \pm 0.71$ is weakened with $g_1^{(2)}(0)=1.41$ and $g_1^{(3)}(0)=0.25$.
Fig. 6. (a) The steady-state photon number $n_s$ and (b) the correlation functions $g_1^{(n)}(0)$ as a function of cavity-light detuning $\Delta _c$ for $\eta /\kappa =0.2$. (c) $n_s$ and (d) $g_1^{(n)}(0)$ as a function of $\Delta _c$ for $\eta /\kappa =0.6$. The correlation functions $g^{(n)}_{1}(0)$ and the corresponding $n_s$ as a function of $\eta$ at (e) single-photon resonance, (f) two-photon resonance, and (g) three-photon resonance, respectively. The other parameters are $g/\kappa =15$ and $\Omega /\kappa =0.45$. Here, $n$PR denotes the $n$-photon resonance for shorthand notation. The black dotted lines indicating $g^{(n)}_{1}(0)=1$ are plotted to guide the eyes.
In order to further reveal the nonclassical properties of photon emission, we calculate the optimal correlation functions $g^{(n)}_{1}(0)$ ($n=2, 3, 4$) and the corresponding $n_s$ as a function of $\eta$ at the different $n$-photon resonances for $\Omega /\kappa =0.45$, as displayed in Figs. 6(e)–6(g). It is obvious that $g^{(2)}_1(0)$ at the single-photon resonance decreases rapidly with an increasing $\eta$. This phenomenon is contrary to the case when the system is driven by the cavity field individually, where $g^{(2)}_1(0)$ increases with $\eta$ increasing. In addition, the 2PB can be realized in the whole range of parameter $\eta$ [Fig. 6(f)], despite the two-photon excitation is weakened with the enhanced $g^{(3)}_1(0)$ for an increasing $\eta$. Figure 6(g) depicts the PIT effect at three-photon resonance. We can see that the PIT always exists even for small driven amplitude $\eta$ with $n_s\sim 10^{-2}$. These results demonstrate that the quantum statistical properties for both atom and cavity driven fields are essentially different from the linear superposition of two independently controlled driven fields. Finally, we should note that a quantum three-phase switching between PB, two-photon bundles, and PIT by tuning the cavity-light detuning is realized. More important, the highly controlled quantum three-phase switching exists in a large parameter regime, which may provide versatile applications in quantum information processing [6].
6. Conclusion
We have investigated the nonclassical photon correlated emission in a trapped single atom coupled to a single-mode optical cavity system. With the two-photon JC type of interactions, strong PB is realized in the weak atom-cavity coupling regime by driving the cavity with a coherent field. Moreover, with an increasing driven strength, the PIT effect occuring at two-photon resonance is observed. Thus, the highly controllable switching between PB and PIT are achievable. When turn to the atom pump field, two-photon bundles emission is observed in strong atom-cavity coupling regime and switching between two-photon bundles and PIT at four-photon resonance is realized. On the other hand, by combining the cavity and atom pump fields together, a quantum three-phase switching between PB, two-photon bundles and PIT at three-photon resonance is realized and highly tunable by changing the amplitude of the driven fields benefiting from the two-photon JC type of interaction. Our proposal could be applied for generating nonclassical quantum states in cavity QED system, and could be implemented with current technology [94]. Our work can also be extended to multi-photon JCM to study more rich nonclassical quantum states, which are useful for applications in quantum information science, e.g., quantum light sources and other quantum devices [68].
Funding
National Natural Science Foundation of China (11804409).
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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