1. Introduction

Ultrastrong coupling (USC) regime has transitioned from theoretical ideas to the experimental realization over the past few years [1–6] and has attracted much attention in many fields [7–16]. The tremendous difference between the USC regime and the non-USC regime is the nature of the ground state of the hybrid system [17–22]. For the weak and strong coupling of the light and the matter, i.e. $g/\omega _0 <0.1$, where $g$ is the coupling strength between the cavity and the atom, $\omega _0$ is the frequency of the cavity, the ground state is the empty cavity and the atom stays in the ground state. However, as the coupling strength increases into the USC regime ($g/\omega _0>0.1$), the properties of the ground state change greatly. One of the most amazing physical phenomena in nature is that the vacuum is not empty, but contains virtual particles [19,20,23]. This is because, in the USC regime, the rotating wave approximation is no longer applicable [8,24], and we have to consider the influence of the counter-rotating terms in the Hamiltonian [25]. The ground state of the USC regime can be expressed as a linear superposition of bare states [26,27]. Therefore, the detection and extraction of excitations in the ground state in the USC regime are important research topics [19,20,28,29].

The USC regime is difficult to achieve in conventional quantum-optical cavity QED but has been implemented in superconducting circuits [3,25,30], Landau polarons [31–35], etc. Recently, the USC between plasmonic modes and SiO$_2$ phonons has been achieved through epsilon-near-zero (ENZ) nanocavities [5], which greatly reduces the size of the system and thus reduces the amount of material involved in realizing mid-infrared USC. The mechanism of the SiO$_2$ phonon and the ENZ cavity coupling is different from the conventional optomechanical coupling in that the SiO$_2$ films are grown on the naked surface and sidewalls by atomic layer deposition (ALD), followed by metal cladding deposition and planarization by glancing-angle ion milling. The use of the ENZ cavity here lies in its unique advantages, the first being the control of light propagation and the improvement of the Kerr coefficient and other nonlinearities to increase the emission rate and energy transfer rate [36], the second being that the ENZ mode is independent of the cavity length and therefore insensitive to changes in Au film thickness [5]. This new nanocavity can achieve a ratio of the coupling strength of the ENZ cavity mode with the SiO$_2$ phonon to the frequency of the SiO$_2$ vibrating phonon greater than 0.25. The hybrid ENZ mode and vibration strong coupling mechanisms have been studied in many works [37–40]. We also investigated the photon/phonon statistics of the hybrid system with the ENZ nanocavity and a trapped two-level atom [26]. Considering the realizable USC regime in the hybrid ENZ system, it provides a new platform to detect the spontaneous release of the virtual photon/phonon pairs in the ground state.

In this paper, we consider the qubit-plasmon-phonon ultrastrong coupling system where the three-level atom is ultrastrongly coupled to the plasmon mode and SiO$_2$ phonon via its upper two energy levels (qubit). It is shown that spontaneous emission of the atom from intermediate energy levels to the ground state is accompanied by the generation of photon pairs and phonon pairs and does not require any external force. In the USC regime, the standard quantum optical master equation cannot describe the dynamic evolution of the system [8,30,41–43] and we have to employ the global master equation where the dissipations are characterized by the eigenoperators related to the full eigenstates of the hybrid qubit-plasmon-phonon system [44–47]. In addition, the conventional quantum optical correlation functions of photon/phonon fail to represent the photon/phonon detection experiments for such a hybrid system, instead one will have to use $\left \langle X^-_{c}(t)X^+_{c}(t)\right \rangle$ where $X^{+}_{c}(t)=\sum _{j,k>j}\left \langle j\right |c+c^{\dagger }| k \rangle \left |j\right \rangle \left \langle k \right |$ $(c=a,b)$ [19] with $a$ ($b$) and $a^{\dagger }$ ($b^{\dagger }$) denoting the annihilation operator and creation operator of the photon (phonon). Accordingly, the input-output relations of the ENZ nanocavity will also be changed [48,49]. Therefore, the virtual photon/phonon in the ground state $\left | \tilde {0} \right \rangle$ cannot be directly detected due to $\left \langle \tilde {0} \right | X^-_{a(b)} X^+_{a(b)}\left | \tilde {0}\right \rangle =0$. We consider the qubit-plasmon-phonon ultrastrong coupling system including a three-level atom that helps convert the virtual photon/phonon into the practically detectable photon/phonon. We study the photon/phonon number rate $\left \langle X_c^-(t)X_c^+(t)\right \rangle$ detected in experiments outside the ENZ nanocavity and find that the greater the coupling strength is, the greater the number of photons and phonons one can detect in the experiments. In particular, we show that the rates of photons/phonons are much larger than previously setting [19], which makes it easier to detect in experiments. In addition, we also analyze the influence of different coupling mechanisms in the ENZ nanocavity on the output photon/phonon number rates [5]. We find that the coupling between the atom and phonon will promote the output photon/phonon number rates, but the vibrational coupling in the ENZ nanocavity weakens them slightly. By analyzing various high-order correlation functions on photons/phonons [8,50], we show that the photons and phonons emitted out of the ENZ cavity appear in pairs. The study offers some important insights into the photon and phonon detection in the ground state of the USC regime [51–53]. The remainder of this paper is organized as follows. The model and the energy levels are briefly introduced in Sec. II. The photon/phonon number rates are mainly studied in Sec. III. The correlation of the photons/phonons is investigated by using modified correlation functions in Sec. IV. We draw our conclusion in Sec. V.

2. Model

The model of interest sketched in Fig. 1 is a hybrid qubit-plasmon-phonon system that includes a three-level artificial cascade atom interacting with an ENZ nanocavity with vibrational ultrastrong coupling between the plasmon mode and SiO$_2$ phonon. The atom is a qubit based on phase-biased flux. Two excited states and one ground state of the three-level atom are labeled by $\left |e\right \rangle$, $\left |g \right \rangle$, and $\left |l\right \rangle$, respectively. It should be noted that the operating frequency of our system is the mid-infrared frequency [5]. The free Hamiltonian of the cavity and the atom reads

 

Fig. 1. Schematic diagram of the hybrid qubit-plasmon-phonon system. The energy levels of the three-level atom are marked $\left \vert e \right \rangle$, $\left \vert g \right \rangle$, and $\left \vert l \right \rangle$ from high to low. The first and second excited states of the atom are coupled to the plasmon mode and SiO$_2$ phonon. Spontaneous decay process $\left | g \right \rangle \to \left | l \right \rangle$ is accompanied by the generation of photon and phonon pairs.

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Fig. 2. Lower energy levels of the full hybrid qubit-plasmon-phonon system as a function of the coupling strength $g_1$. Y-axis represents the eigenvalues of the $H$. The bending lines correspond to the eigenvalues of $H_1$ and the horizontal lines illustrate the eigenvalues of $H_l$. Here we take $g_1=g_2$ for simplicity. $\left |0\right \rangle$ to $\left |5\right \rangle$ denote the six lowest energy eigenstates of the full hybrid system, respectively.

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Here we consider that the system starts with the lowest state $\left | \tilde {0} \right \rangle$ of the Hamiltonian $H_1$, which is close to the bare state $\left |0,0,g\right \rangle$. This state can be prepared by exciting the atom with $\pi$ pulse and the driving Hamiltonian is given by $H_p=\frac {\sqrt {3\pi }}{\sigma } e^{-t^2/(2\sigma ^2)}\cos \omega t (\sigma _{gl}+\sigma _{lg})$ with $\sigma =5/\omega _0$ and $\omega =\omega _s$, which is marked in Fig. 6. An intuitive comparison of the evolutions of $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle (c=a,b)$ from the $\pi$-pulse prepared initial state and from the initial state $\left | \tilde {0} \right \rangle$ is plotted in Fig. 3.

 

Fig. 3. Comparison of the evolutions from the $\pi$-pulse prepared initial state (red line) and from the initial state $\left | \tilde {0} \right \rangle$ (blue dots). (a) represents the time evolution of the mean photon number $\left \langle X_a^{-}(t)X_a^{+}(t)\right \rangle$. (b) represents the time evolution of the mean phonon number $\left \langle X_b^{-}(t)X_b^{+}(t)\right \rangle$. The internal dissipations of the system are taken as $\gamma _a=\gamma _b=\gamma _{eg}=\gamma _{gl}=0.02\omega _0$, $\omega _{gl}=3.5\omega _0$, $\omega _p=0.25\omega _0$, $g_1=g_2=0.6\omega _0$.

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3. Creation of photon and phonon pairs

In the USC regime, various components of the system ultrastrongly interact with each other and are inseparable [59]. To consider the dynamics of the system which includes a variety of dissipation channels, one will have to employ the global master equation [19,26,60,61]

To detect the emitted photons and phonons from the ENZ nanocavity in experiments, one will have to study the corresponding average emitted particle number $\left \langle c_{out}^{\dagger }c_{out}\right \rangle (c=a,b)$, which are proportional to $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle (c=a,b)$ if the input is in the vacuum state, based on the input-output relations $c_{out}(t)=c_{in}(t)-\sqrt {\gamma _c}X_c^{+}(t) (c=a,b)$ in the USC regime [49,62]. One can note that the coupling strength of the plasmon mode and SiO$_2$ phonon mode $g_C$, and the two-photon coupling strength $g_D$ depend on the vibrational coupling constant $\omega _p$. It is easy to show that $g_C$ is greater than $g_D$, which means the coupling effect between the plasmon mode and the phonon is stronger in these two coupling mechanisms. In this sense, the difference between the dynamics of photon and phonon could be quite subtle for the weak vibrational coupling $\omega _p$, if other parameters have no obvious difference. In Fig. 4(a) and (b), we show the number rates of output photons $\left \langle X_a^{-}(t)X_a^{+}(t)\right \rangle$ and phonons $\left \langle X_b^{-}(t)X_b^{+}(t)\right \rangle$ corresponding to different vibrational coupling constants $\omega _p=0.25\omega _0, 0.4\omega _0, 0.8\omega _0$, respectively. All subsequent calculations are in the case of resonance. Other parameters are taken as $\gamma _a=\gamma _b=\gamma _{eg}=\gamma _{gl}=0.02\omega _0$, $\omega _{gl}=3.5\omega _0$. The initial state is the lowest state $\left | \tilde {0} \right \rangle$. It is also shown that with the increase of the vibrational coupling constant, the maximum values of the output photon and phonon number rates will decrease slightly, which shows the negative effect of the vibrational coupling and is a noticeable phenomenon. In addition, comparing Fig. 4(a) and (b) shows that for $\omega _p=0.25\omega _0$, the evolutions of photon and phonon are almost the same, and with $\omega _p$ increasing, their difference increases, which coincides exactly with our analysis. Reference [5] shows that the plasmon-phonon coupling in the ENZ nanocavity can be realized in the ultrastrong regime in experiments, which corresponds to the vibrational coupling constant ($\omega _p$) of this nanocavity around $0.25\omega _0$. However, only when the vibrational coupling constant $\omega _p$ increases to such an extent that the coupling between the photon and the photon associated with $g_D$ also enters into an ultrastrong coupling regime or even deep coupling regime that the results for the photon and phonon become significantly different, which could be difficult to achieve in current experiments. Therefore, in the following, we will only take the vibrational coupling constant $\omega _p$ to be $0.25\omega _0$ if not specified. That is, $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$ in the latter figures can be roughly considered to simultaneously represent photon and phonon numbers.

 

Fig. 4. The evolution of the intra-cavity mean photon and phonon numbers $\left \langle X_a^{-}(t)X_a^{+}(t)\right \rangle$ (a) and $\left \langle X_b^{-}(t)X_b^{+}(t)\right \rangle$ (b) for different vibrational coupling constants $\omega _p=0.25\omega _0$ (red line), $\omega _p=0.4\omega _0$ (blue line), $\omega _p=0.8\omega _0$ (black line), $\gamma _a=\gamma _b=\gamma _{eg}=\gamma _{gl}=0.02\omega _0$, $g_1=g_2=0.6\omega _0$. (c) The population dynamics of the initial state $\left \vert \tilde {0} \right \rangle$. (d) The evolution of the intra-cavity mean photon and phonon numbers $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$, $(c=a,b)$ for different coupling strengths $g_1/\omega _0 = 0.4$(red line), $0.6$(blue line), $0.8$(black line), with $g_1=g_2$, $\omega _p=0.25\omega _0$. (e) $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$, $(c=a,b)$ versus $\gamma t$ corresponds to different atom-phonon coupling $g_2=0.4\omega _0$ (red line), $g_2=0.6\omega _0$ (blue line), $g_2=0.8\omega _0$ (black line), where $g_1=0.6\omega _0$, $\omega _p=0.25\omega _0$. (f) $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$, $(c=a,b)$ versus $\gamma t$ for different spontaneous emission rates $\gamma _{gl}=0.01\omega _0$ (red line), $\gamma _{gl}=0.015\omega _0$ (blue line), $\gamma _{gl}=0.03\omega _0$ (black line) and $\gamma _{gl}=0.04\omega _0$ (yellow line) with $g_1=g_2=0.6\omega _0$, $\gamma _a=\gamma _b=0$, $\gamma _{eg}=0.02\omega _0$, $\omega _p=0.25\omega _0$.

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The output photon and phonon beam comes from spontaneous decay of the state $\left | \tilde {0} \right \rangle$. Taking into account dissipations, we have studied the population evolution of the $\left \vert \tilde {0} \right \rangle$ in Fig. 4(c), which indicates the decay of the population. However, one can see that the output phonon and photon beams reach a maximum value before the exponential decay to the tail by comparing Fig. 4(c) with Fig. 4(a) and (b). This allows virtual photons and phonons in the $\left | \tilde {0} \right \rangle$ to be well detected. Since the three-level atom plays the key role in the detection of emitted photons/phonons, we also studied the evolution of $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$ in Fig. 4(d), where we take the couplings of the photon and phonon to the atom to be equal. The red, blue, and black lines represent different coupling strengths $g_1=g_2$ between photon/phonon and the atom $0.4\omega _0$, $0.6\omega _0$, and $0.8\omega _0$, respectively. Figure 4(d) indicates that as the coupling strength increases, the maximum number of the detected photons and phonons increases. When the coupling strength $g_1=0.8\omega _0$, the maximum value of $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$ can reach $0.09$, which is obviously larger than that in the previous scheme [19]. So the current technology should be easy to experimentally detect the photons and phonons using quadrature amplitude detectors [63–65]. The coupling of the atom and the SiO$_2$ phonon mode is another important feature of the current system. The coupling of the atom and the phonon has also a significant influence on the released photons and phonons from the ENZ cavity, which is illustrated in Fig. 4(e), where the coupling of the atom and the photon is taken as $g_1=0.6\omega _0$ and the coupling strength between phonon and atom is taken as $g_2=0.4\omega _0, 0.6\omega _0, 0.8\omega _0$ corresponding to red, blue and black lines, respectively. Apparently, the output phonon and photon number rates increase with the increase of the coupling between the phonon and atom, which indicates that the phonon-atom coupling promotes the experimentally detectable phonon and photon number rates. We also investigate the atomic spontaneous decay on the mean photon (phonon) number $\left \langle X_{c}^{-}(t)X_{c}^{+}(t)\right \rangle$, which is plotted in Fig. 4(f). To illustrate the effect of atomic decay well, we let $\gamma _a=\gamma _b=0$. It can be found that the maximum value of the mean photon and phonon numbers does not hinge on $\gamma _{gl}$ in the absence of the photon and phonon losses. $\gamma _{gl}$ only affects the speed of increase, the larger its value is, the faster $\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$ rises.

Meanwhile, for the completeness of the analysis, we also investigated the effect of the loss of the optical cavity, and the results are shown in Fig. 5. When optical cavity and vibration mode losses are not considered, the average photon number and phonon number remain constant after reaching the maximum value, and when optical cavity and vibrational mode losses are considered, the maximum values of the average photon number and phonon number decrease, and the larger the loss, the smaller the maximum value, and decay rapidly after reaching the maximum values.

 

Fig. 5. (a)$\left \langle X_c^{-}(t)X_c^{+}(t)\right \rangle$, $(c=a,b)$ versus $\gamma t$ for different the loss of the cavity $\gamma _{c}=0$ (red line), $\gamma _{c}=0.01\omega _0$ (blue line), and $\gamma _{c}=0.02\omega _0$ (black line). (b) Purity versus time. The black, blue, and red lines correspond to the case of Fig. 4(d) in the main text, respectively.

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To further confirm the validity of our scheme for releasing photon-phonon pairs, we add the analysis of the purity, which is defined as:

Purity is used to measure the degree of mixing of quantum states [66]. The lower the purity, the greater the mixing of the states, the greater the coupling effect achieved between the system and the environment. As shown in Fig. 5(b) that the point where the maximum number of photon-phonon pairs is released corresponds to the lowest point of purity, which means that the mixing between the system and the environment is maximized when the maximum number of photon-phonon pairs are released. Comparing Fig. 4(d) and Fig. 5(b), we can see that when the coupling strength is relatively small and the released photon-phonon pairs are relatively small, the purity of the photon-phonon pairs increases accordingly, which illustrates the validity of our scheme.

4. Correlation of the photon and phonon pairs

As mentioned previously, phonons and photons emitting from the nanocavity are caused by the spontaneous emission of the state $\left | \tilde {0} \right \rangle$. To explicitly demonstrate the mechanism, we investigate the spectrum of photons and phonons due to the spontaneous emission of the atom

 

Fig. 6. Spectrum $S(\omega )$ of spontaneously emitted photons and phonons, and the energy level transition diagram. (a) The red line represents the absence of interaction between the atom and the ENZ cavity, the blue line represents the lack of coupling between the SiO$_2$ phonon mode and the atom in the cavity, and the black line represents the consideration of the interaction of the full cavity with the atom. (b) The coupling strength increases sequentially from the red line ($g_1=g_2=0.4\omega _0$) , the black line($g_1=g_2=0.6\omega _0$) to the blue line ($g_1=g_2=0.8\omega _0$). (c) Energy level transition diagram, the situation considered here is $g_1=g_2=0.6\omega _0$. In the diagram on the left Energy levels that involve transitions are shown by solid lines, and those that do not are shown by dashed lines. The bare states corresponding to the main components of each energy level of the whole system are given on the right hand side.

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To better study the statistics of photon and phonon pairs emitted from the cavity, we further analyze the equal-time high-order ($n$th-order) correlation functions [8,70], which are defined at the moment $t$ as

 

Fig. 7. Statistics of the output photons and phonons. (a) The second-order correlation function of photons and phonons $g^{(2)}(t,t)$, where the red line represents the correlation function for photons, the black dotted line for phonons, and the blue line for the second-order cross-correlation function between photons and phonons. (b) The third-order correlation function of photons and phonons $g_{a(b)}^{(3)}(t,t)$, where the red line represents the correlation function for photons and the blue line for phonons. (c) The third-order cross-correlation function between photons and phonons $g_{ab}^{(3)}(t,t)$, the red line shows the correlation of a photon and two phonons, while the blue line shows the correlation of a phonon and two photons. (d) The third-order cross-correlation functions. The curves correspond to the probability of detecting one atomic spontaneous emitted photon accompanied by photon pairs (red), phonon pairs (blue circle), and photon-phonon pairs (black plus). (e) The second-order delay correlation function $g^{(2)}(t,t+\tau )$. (f) The third-order delay correlation function $g^{(3)}(t,t,t+\tau )$.

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To further illustrate our results, we give a comparison of the equal-time and delay correlation functions [72]. The standard form of second-order delay correlation function is defined as [73,74]:

As shown in Fig. 7(e)-(f), we give a comparison between the delay correlation function and the equal-time correlation function. We found that $g^{(2)}(t,t)>g^{(2)}(t,t+\tau )$, $g^{(3)}(t,t,t)<g^{(3)}(t,t,t+\tau )$, which is further shown that photon-phonon pairs are antibunching with each other and photons and phonons are produced in pairs. Here we take the initial time is $t=\gamma _a$. Meanwhile, we also calculated the three-order cross-correlation functions of atomic spontaneous emission and photon/phonon pairs in Fig. 7(d). The three correlation functions are apparently much larger than 1, which further means the large probability of detecting one atomic spontaneous emitted photon (corresponding to the three lower peaks in Fig. 6(a) accompanied by photon pairs, phonon pairs, and photon-phonon pairs.

Finally, we’d like to emphasize that the ultrastrong coupling of the plasmon mode and the phonon has been experimentally achieved [5]. With the development of the experiment, the coupling strength of phonons, photons, and atoms keeps increasing [75–78], entering the USC regime. Meanwhile, the experimental detection of phonon pairs and photon pairs can be performed by homodyne detection [79], optical interference detection [80], Raman spectroscopy [81], etc. So our scheme should be feasible for generating phonon pairs and photon pairs in this new ENZ nanocavity [5]. The parameters selected in our calculations are experimentally feasible: vibration constant $\omega _p=0.25\omega _0$ [5], $g\approx \omega _0, \omega _b(0.2<g/\omega _0,\omega _b<1)$ in the USC regime [25], and dissipation parameters $\gamma _a, \gamma _b, \gamma _{eg}, \gamma _{gl}/\omega _0 \approx 10^{-2}$ are shown in the Methods section on page 7 of this work [82]. In our work we use the frequency $\omega _0/2\pi =2.782$GHZ used in Fig. 1 of this paper [25], and the adjustment of the coupling constant and dissipation frequency in our study are in the range of the above experimental papers. We have verified that the physical properties obtained are independent of the specific values of the parameters when the parameter ratios are the same. In addition, all the results in this article are calculated numerically using Qutip [83], where we truncate the numbers of photons and phonons up to 4. We find that the lowest state of energy with interactions and the energy levels below are almost independent of the truncation dimension, so the physical processes we study are not subject to change with the truncation dimension. We have verified that the final results of this study are almost not affected by the higher dimension of Hilbert space.

5. Conclusion

We have investigated the detection of the photon and phonon pairs in the ground state of the qubit-plasmon-phonon hybrid system in the USC regime. The system includes an artificial three-level atom with only the upper two energy levels coupled to plasmon mode and SiO$_2$ phonon, and the transition frequency of the first and the second excited state is resonant with the photon and phonon. In this way, the upper and lower energy levels can be regarded as a two-level system. It can be found that the spontaneous emission of the atom from the intermediate state to the ground state is accompanied by the generation of photon pairs and phonon pairs. The detected photon and phonon number rates have increased compared to before, which is easier to detect in experiments. It is worth noting that the coupling of the phonon and the atom in the ENZ nanocavity we studied can promote the maximum value of the output photon and phonon number rates. In addition, we also analyzed the spectrum $S(\omega )$ of the emitted photons and phonons and compared the spectrum with and without interaction. Finally, by calculating the correlation function of photons and phonons, we confirmed that photons and phonons are emitted in pairs. However, the experimental realization of the ultrastrong coupling between the three-level atom and the ENZ cavity is also briefly discussed. Our research will advance the study of the ground state properties in the USC regime.