1. Introduction

Circuit quantum electrodynamics (QED) system provides an excellent experimental platform for the study of quantum state engineering [1–3], quantum information processing [4,5], and quantum computing [6–9]. Compared with cavity QED, the superconducting qubits in the circuit QED system are more strongly coupled with the cavity on the chip, leading to more frequent photon exchange and much shorter acquisition times in the experiment [10]. In 2010, the first two experiments of ultrastrong coupling (USC) were realized in the circuit QED system [11,12]. Since then, many new physical processes have emerged in the ultrastrong coupling circuit QED system, as degeneracy of vacuum [13], modification of photon blockade [14], nonclassical radiation from the thermal cavities [15], and vacuum-induced symmetry breaking [16]. In the USC regime, the existence of counter-rotating wave terms makes the multi-photon Rabi oscillation possible, where multiple photons excite one qubit [17]. Similarly, a single photon can simultaneously excite multiple qubits [18], which has now been verified by experiments [19].

Recently, analogous processes have been implemented in different USC systems to yield frequency conversion [20], entanglement between photons [21], and spin squeezing [22]. In the scenarios above, the exploration of the avoided-crossing region is particularly important. At the avoided-crossing point, the single photon state and the multiple qubits state will be completely hybridized, resulting in a symmetric and anti-symmetric coupling state that does not exist under RWA (rotating-wave approximation). Surprisingly, different from the case of cavity-qubit near resonance [23,24], when the frequency of the cavity mode is twice that of the qubit [18], there will be two intermediate states containing only qubits and no photons. If we set the cavity loss to be much greater than the qubits decay, we can get two metastable states with slow decay rate. Can these two metastable states bring us any interesting phenomena? We think that we can use the electron shelving [25,26] of the intermediate states to realize the spectral narrowing.

Spectral narrowing can be applied to many research fields, such as laser spectroscopy [27] and quantum sensing [28]. As early as 1990, the sub-natural linewidth in the spontaneous emission spectrum of a three-level atom driven by two beams of light has been theoretically predicted by Narducci et al. [29], and a large number of experiments subsequently observed this phenomenon [30–33]. Recently, the ultranarrow linewidths have been observed in strongly coupled superconducting qubit systems [34,35]. In this paper, we investigate the ultranarrow spectrum of the qubit-cavity ultrastrong coupling system, which caused by the combined effect of quantum interference and electron shelving. The system contains two metastable states, both of which correspond to narrow peaks in the emission spectrum, thus the narrowing effect in this paper is stronger than the previous work [36].

In this paper, we study the ultranarrow spectra of double superconducting qubit-cavity coupling system in the regime of USC. Due to the counter-rotating wave terms in the quantum Rabi model (QRM), a splitting anti-crossing between levels three and four appears in the eigen-energy spectrum, as shown in Fig. 1. Therefore, we choose the frequency of the driving field to resonate with the transition from the ground state to the third energy level. And the circuit QED system is reduced to a four-level dressed state system. At the point of avoided-level crossing, an ultranarrow peak appears in the center of the cavity emission spectrum. Through careful analytical analysis, we find that the origin of ultranarrow linewidth is the slow decay of the incoherent terms in the density matrix and the quantum interference between two transition pathways. Moreover, at the point of level crossing, the extra inner sidebands of the emission spectrum will be highlighted compared to the Mollow-like triplet. Our results indicate that the spectral narrowing can be achieved by adjusting the coupling strength of the qubit-cavity ultrastrong coupling system.

This paper is organized as follows. In Section 2, we present the model and Hamiltonian of the circuit QED system in the USC regime, and then show the level crossing and avoided-crossing in the eigen-energy spectrum. The ultranarrow spectral line in the cavity emission spectrum is discussed in Section 3, including the five broad peaks and the ultranarrow peak imposed on the central peak. The main results and conclusions are summarized in Section 4.

2. Model and system

We consider two identical superconducting qubits with transition frequency $\omega _{q}$ placed in a cavity on chip with resonance frequency $\omega _{c}$, and coupled together with coupling strength $g$. The coupling strength between each qubit and the single-mode cavity is comparable to the cavity-qubit detuning $\Delta =\omega _{c}-\omega _{q}$, indicating the appearance of USC. Such a system has been implemented experimentally [19]. In addition, the cavity is driven by a coherent laser field with frequency $\omega _{l}$ and driving strength $\varepsilon$. In the qubit eigenbasis, the qubit Hamiltonian reads $H_q=\hbar \omega _{q}\sigma _{z}/2$, where $\hbar \omega _{q} =\sqrt {\Delta ^2 + (2I_{p}f)^2}$ is the qubit transition frequency with $f = \Phi -\Phi _0/2$. Here $I_{p}$ correspond to the persistent current in the qubit loop, $\Delta$ is the tunnel coupling between the two persistent current states. When the externally applied magnetic flux $\Phi \approx \Phi _0/2$ ($\Phi _0=h/2e$ is the magnetic-flux quantum), $\hbar \omega _{q} =\Delta$, the qubit behaves effectively as a two-level system [12].

The total Hamiltonian describing the circuit QED system can be written as [16] ($\hbar =1$)

In the regime of USC, the RWA breaks down and the Rabi Hamiltonian contains four counter-rotating terms of the form $\sigma _{+}^{(i)}a^{\dagger}$, $\sigma _{-}^{(i)}a$, $\sigma _{z}^{(i)}a$, and $\sigma _{z}^{(i)}a^{\dagger}$. In general, it is a challenge to obtain exact analytical solutions of energy levels for the USC systems. And several approximation methods have been developed to deal with this kind of Hamiltonian. Such as third-order perturbation theory [18] and adiabatic elimination method [37]. In order to get a more accurate expression of the effective Hamiltonian, we use the method in Ref. [38]. By diagonalizing $H_{0}$, the numerical eigenvalues $E_{n}$ and eigenstates $\vert \psi _{n} \rangle$ which satisfy the stationary Schrödinger equation $H_{0}\vert \psi _{n} \rangle =E_{n}\vert \psi _{n} \rangle$ can be obtained. Then the Rabi Hamiltonian in Eq. (2) can be replaced by

 

Fig. 1. (a) Normalized energy ladders of the Rabi Hamiltonian $H_{0}$ as a function of the coupling strength $g$ for $\omega _{c}/\omega _{q}=1.915$ and $\theta =\pi /6$. (b), (c) The energy level scheme of the cavity-qubits coupling system in dressed state representation for $g/\omega _{q}=0.2$ and $0.7056$, respectively. The modified Rabi frequency $\Omega$ stands for the standard amplitude of driving field from ground state $\vert \psi _{0} \rangle$ to excited state $\vert \psi _{3} \rangle$, $\Gamma _{mn}\ (m>n)$ are the relaxation coefficients between each state. For the standard damping rates $\kappa \gg \gamma$, the dissipation is dominated by $\Gamma _{30}$, which is marked in red.

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The driving Hamiltonian $H_{d}$ can be expanded by the dressed states $\vert \psi _{n} \rangle$, thus we have

In order to further investigate the phenomenon of avoided-crossing, we therefore need to set up the driving from the ground state $\vert \psi _{0} \rangle$ to the target state $\vert \psi _{3} \rangle$. Hence, we choose the frequency of diving field that resonates with the transition pathway $\vert \psi _{0} \rangle \rightarrow \vert \psi _{3} \rangle$ as

Therefore, we can make a four-state truncation of the Hilbert space. And the dressed state structures of the two cases we care about are shown in Fig. 1(b) and 1(c). Here we only consider that the system interacting with zero-temperature baths. Owing to the effects of the counter-rotating terms, the system can be driven out of the ground state, thus the approach of standard quantum-optical master equation breaks down. Assuming a weak coupling of the system and the baths, the dissipations can be treated by the Born-Markov approximation. Thus the modified master equation for the reduced density matrix can be obtained as [15,41]

The relaxation coefficients between each two dressed state for different coupling strengths are shown in Fig. 2. When the coupling between qubits and cavity is weak, the interaction Hamiltonian in Eq. (2) can be treated as perturbation, and the RWA is available. Consequently, the relaxation coefficients are reduced to the standard damping rates of the qubits and cavity in the case of weak coupling, where $\Gamma _{30}=\kappa$, $\Gamma _{10}=\Gamma _{20}=\gamma$. With the increase of $g$, the RWA breaks down and the counter-rotating terms are taken into account, which results in variations of the relaxation coefficients. In addition, at the point of avoided-level crossing that $g/\omega _{q}=0.2$, the eigenstates $\vert \psi _1\rangle$ and $\vert \psi _2\rangle$ are approximate to $( \vert g, e, 0\rangle \pm \vert e, g, 0\rangle )/\sqrt {2}$. For the case of $\kappa \gg \gamma$, $\vert \psi _1\rangle$ and $\vert \psi _2\rangle$ contain only qubits and no photons. These two metastable states are very important to us, and it is the electron shelving on these two states that causes the spectral narrowing. It needs to be emphasized that, different from the long-lived metastable states in Ref. [42], the cascaded transition in this paper differs by several orders of magnitude from the cascaded transition, resulting in a longer lifetime of metastable state here.

 

Fig. 2. (a)-(d) The relaxation coefficients $\Gamma _{mn}$ as functions of the coupling strengths $g$ for $\omega _{c}/\omega _{q}=1.915$ and $\theta =\pi /6$. The standard damping rates of the qubits and cavity are $\kappa =2\times 10^{-3}\omega _{q}$, $\gamma =2\times 10^{-5}\omega _{q}$.

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3. Ultranarrow spectral line in the cavity emission spectrum

According to the input-output relations proposed by Ridolfo et al.[14,15], the incoherent cavity emission spectrum $S(\omega )$ in the USC regime is defined as

Since $\rho _{01}^{ss}=\rho _{13}^{ss}=0$ (see Supplement 1, S1), the initial values of the cross-correlations in Eq. (11) are equal to zero, which leads to the disappearance of the cross-correlations. Therefore, the emission spectrum can be simplified as three auto-correlation functions of the transition operators in Eq. (12)

 

Fig. 3. The incoherent emission spectrum $S_{inc}(\omega )$ of the reduced nondegenerate four-level system in Fig. 1(b), with $g/\omega _{q}=0.2$, $\omega _{c}/\omega _{q}=1.915$, $\theta =\pi /6$, and $\varepsilon /\omega _{a}=8$, for (a) $\kappa =2$, $\gamma =0.1$; (b) $\kappa =2$, $\gamma =0.02$; (c) $\kappa =2$, $\gamma =0.01$; (d) $\kappa =1$, $\gamma =0.02$; (e) $\kappa =4$, $\gamma =0.02$; and (f) $\kappa =6$, $\gamma =0.02$ (units of $\omega _{a}$ where $\omega _{a}= 10^{-3}\omega _{q}$).

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In order to better explain the physical origin of spectral narrowing in the emission spectrum, we diagonalize the effective Hamiltonian $H_S$ under the condition that $\Omega \ll \kappa$ to get the dressed states

The emission spectrum consists of six parts: the central peak of the Mollow-like triple, the outer sidebands located at $\pm \Omega$, and the inner sidebands located at $\pm \Omega /2$. And most importantly, the additional narrow peak at line center which is significantly sharper than any other peaks. The physical origin of each component can be understood through the transition channels between the dressed states from two contiguous manifolds, as shown in Fig. 4.

 

Fig. 4. Diagram of dressed states and dressed state transition pathways of each component in the emission spectrum. The red arrows in this figure correspond to the central peak with line width $A$, the green arrows correspond to the outer sidebands with line width $(2A+\Gamma _{30})/2$, the blue and yellow arrows correspond to the inner sidebands with line width $(A+2\Gamma _{10})/2$.

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The central peak originates from transition channels $\vert +\rangle \rightarrow \vert +\rangle$ and $\vert -\rangle \rightarrow \vert -\rangle$. The corresponding Lorentzian with linewidth $A$ is related to the decay rate of the the population difference $\rho _{++}-\rho _{--}$. The outer sidebands located at $\pm \Omega$ come from transition pathways $\vert +\rangle \rightarrow \vert -\rangle$ and $\vert -\rangle \rightarrow \vert +\rangle$. The Lorentzians with linewidth $(2A+\Gamma _{30})/2$ are associated with the decay rates of the dressed state coherences $\rho _{+-}$ and $\rho _{-+}$, respectively. Since the stationary solutions $\rho _{++}^{ss}$ and $\rho _{--}^{ss}$ are equal, the heights of the two outer sidebands are the same. Moreover, the inner sidebands located at $\pm \Omega /2$ come from spectra $S_2(\omega )$ and $S_3(\omega )$. In the spectrum of $S_2(\omega )$, the right (left) side peak located at $\Omega /2$ ($-\Omega /2$) is resulting from the decay of $\vert \psi _{1}\rangle \rightarrow \vert -\rangle$ ($\vert +\rangle$), with the weight $\rho _{11}$, as shown in the blue arrows of Fig. 4. However, in the spectrum of $S_3(\omega )$, the right side peak is resulting from the transition channel $\vert +\rangle \rightarrow \vert \psi _{1}\rangle$, with the weight $\rho _{++}$. And the left side peak is resulting from the transition channel $\vert -\rangle \rightarrow \vert \psi _{1}\rangle$, with the weight $\rho _{--}$, as shown in the yellow arrows of Fig. 4. Owing to the same steady state population of $\rho _{++}$ and $\rho _{--}$, the heights of the inner sidebands are the same.

In summary, the transition channels corresponding to the five peaks mentioned above are either start from $\vert \pm \rangle$ or end to the $\vert \pm \rangle$, so their linewidths are related to $A$. Since $A$ contains a dominant dissipation process with relaxation coefficient $\Gamma _{30}$, these five peaks are all broad peaks.

Particularly, the extra narrow peak is an additional structure observed when we detecting the outer sidebands of the cavity emission spectrum. Since the stationary solution $\rho _{+-}^{ss}=\rho _{-+}^{ss}=0$ under secular approximation, i.e., $C_1^\pm =0$, the narrow peak will disappear, thus we cannot apply secular approximation approach in the process of solving the narrow peak. Through the analytical analysis of the ultranarrow peak in Supplement 1, S2B, we find that the narrow peak located at the line center comes from the correlation between the central peak and the side peaks of Mollow-like triple, and the corresponding motion equations are coupled to the equations of the sidebands through vacuum-induced quantum interference, as shown in Supplement 1, Eq. (S17). When we detecting the fluorescence radiated by the transition pathways $\vert \pm \rangle \rightarrow \vert \mp \rangle$, which corresponds to the observation operator $\langle \delta \sigma _{+-}-\delta \sigma _{-+}\rangle$, we can see not only the sidebands, but also an ultranarrow peak imposed on the central peak.

Comparing Eq. (17b) with Eq. (17a), we find that the time evolution of the population difference $\rho _{++}-\rho _{--}$ corresponding to the central peak is a rapid decay process with decay rate $A/2$, while the incoherent injection $\rho _{++}+\rho _{--}$ corresponding to the narrow peak is a slow decay process. If there is no electron shelving of the intermediate states $\vert \psi _1 \rangle$ and $\vert \psi _2 \rangle$, such as two-level atomic system, then the incoherent injection will be a constant that does not evolve with time, and there will be no narrow peak. Furthermore, the narrow peak here is different from the narrow peaks in the V-type three-level atomic systems [44,45]. In those systems, the narrow peaks at the line center are directly derived from the motion equation of the observation operator $\langle \delta \sigma _{++}-\delta \sigma _{--}\rangle$. However, here we find the narrow peak imposed on the central peak when detecting the fluorescence spectrum of the sidebands, which corresponds to observation operator $\langle \delta \sigma _{+-}-\delta \sigma _{-+}\rangle$. In previous works [44,45], in order to observe the quantum interference between different transition pathways through spectral narrowing, the two upper energy states must be degenerated or nearly degenerated. In this way, the frequencies corresponding to the transitions from the two upper states to the ground state are at the same level, and the quantum interference effect between these two transition channels is preserved after the secular approximation. However, in this paper, we can achieve spectral narrowing in three-level ($g/\omega _{q}=0.7056$) or four-level ($g/\omega _{q}=0.2$) systems without degenerate states, and dig out the hidden vacuum-induced quantum interference effects via spectral narrowing. This finding opens up a new possibility for studying the quantum interference effect in the USC system.

Since $\kappa \gg \gamma$, it is clear from Eq. (19) that the linewidths in $\lambda _0$, $\lambda _0^\pm$ and $\lambda _2^\pm$ which containing factor $A$ increases with $\kappa$. Meanwhile, these linewidths does not change with $\gamma$. But the narrow linewidths $\lambda _1^\pm$ are more sensitive to $\gamma$. As shown in Fig. 5, the relative linewidth of the narrow peak out of the central peak is proportional to $\gamma$ and inversely proportional to $\kappa$. From this figure, we can see that the minimum value of $\lambda _1^+$ (dashed red curve) is about $0.1\lambda _0$, which is not small enough to observe the narrowing of the spectrum. Actually, the occurrence of ultra-narrow spectrum is mainly due to the ultra-small relative linewidth $\lambda _1^-/\lambda _0$ (solid black curve). For $\kappa /\omega _a=2$ and $\gamma /\omega _a=0.1$, the linewidth $\lambda _1^-$ is $1/5$ of the center peak linewidth $\lambda _0$, while the height of the narrow peak is $1/16$ of the center peak, thus the spectral narrowing is not obvious in Fig. 3(a). However, when $\gamma \ll \kappa$, one can obtain a sharp ultranarrow peak with the linewidth of $0.01\lambda _0$, meanwhile, the height of the narrow peak is $1/4$ of the center peak. From Eq. (20), we can see that the amplitude $C_1^-$ of the narrow peak is multiplied by a factor $\Gamma _{30}/\Omega$ compared to the amplitude of the sidebands $C_0^\pm$. This factor is a small quantity, thus the narrow peak can be regarded as a correction to the outer sidebands. However, the linewidth $\lambda _1^-$ is also a small quantity, which makes the height of the narrow peak $C_1^-/\lambda _1^-$ comparable to the height of the central peak. At this point, the ultranarrow peak is highlighted in the emission spectrum, as shown in Figs. 3(c), 3(e) and 3(f).

 

Fig. 5. The relative linewidths of the narrow peak out of the central peak as functions of (a) $\gamma$ ($\kappa /\omega _a=2$); and (b) $\kappa$ ($\gamma /\omega _a=0.02$). The other parameters are the same as Fig. 3.

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We next consider the case of level crossing that $g/\omega _q=0.7056$, the incoherent emission spectra are shown in Fig. 6. As before, the Mollow-like triplet and the additional sidebands hardly changed with $\gamma$, but the linewidths of them grows and the height reduces with the increase of $\kappa$. However, different from the previous situation, the two inner sidebands are more pronounced, and the narrow peak is less obvious in this case.

 

Fig. 6. The incoherent emission spectrum $S_{inc}(\omega )$ of the reduced degenerate four-level system as shown in Fig. 1(c). The other parameters are the same as Fig. 3, except $g/\omega _q=0.7056$.

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By analyzing the components of dressed states $\vert \psi _1\rangle$ and $\vert \psi _3\rangle$, we find that when $g$ increases from $0.2\omega _{q}$ to $0.7056\omega _{q}$, the probability amplitudes of the bare states with photon population (such as $\vert e,g,1\rangle$ and $\vert g,g,2 \rangle$) increase. This leads to increased photon exchange between $\vert \psi _1\rangle$ and $\vert \psi _3\rangle$, i.e., the coefficient of $\kappa$ in $\Gamma _{31}$ increases. Since $\kappa \gg \gamma$, $\Gamma _{31}$ increases and the population accumulates on $\rho _{11}$. In this way, the transition pathways $\vert \psi _1\rangle \rightarrow \vert \pm \rangle$ and $\vert \pm \rangle \rightarrow \vert \psi _1\rangle$ that associated with $S_2(\omega )$ and $S_3(\omega )$ are enhanced, as shown in Fig. 4. Thus the proportion of $S_2(\omega )$ and $S_3(\omega )$ in the entire spectrum increases, and finally the inner sidebands are highlighted. Meanwhile, the increase of $\Gamma _{31}$ means that the decay rate of $\rho _{++}+\rho _{--}$ in Eq. (17b) increases, and the linewidth $\lambda _1^+$ corresponding to the narrow peak is getting bigger, as shown in Fig. 7(a). Whereas the ultra-small linewidth $\lambda _1^-/\lambda _0$ (solid black curve) corresponding to the ultra-narrow peak is basically the same as before. So what makes the narrow peak to become inconspicuous? We can revisit the energy level scheme in Fig. 1, the difference between Fig. 1(b) and 1(c) is that when $g/\omega _q=0.7056$, the energy states $\vert \psi _2\rangle$ and $\vert \psi _3\rangle$ degeneracy, which makes the relaxation coefficient $\Gamma _{32}=0$. Under this condition, one obtain that $\rho _{22}^{ss}=0$, and

 

Fig. 7. The relative linewidths of the narrow peak out of the central peak as functions of (a) $\gamma$ ($\kappa /\omega _a=2$); and (b) $\kappa$ ($\gamma /\omega _a=0.02$). The other parameters are the same as Fig. 3, except $g/\omega _q=0.7056$.

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In summary, for the case of energy level crossing that $g/\omega _q=0.7056$, the two states $\vert \psi _2\rangle$ and $\vert \psi _3\rangle$ are degenerate. When the photons are driven from the ground state to $\vert \psi _3\rangle$, the population will not be transferred to $\vert \psi _2\rangle$, so that the electron shelving only exists in $\vert \psi _1\rangle$, which makes the four-level system here equivalent to the three-level system in Ref. [36]. And the narrowing of the spectrum is not obvious. For the case of avoided-crossing that $g/\omega _q=0.2$, there are two metastable states $\vert \psi _1\rangle$ and $\vert \psi _2\rangle$ with electronic shelving, corresponding to two Lorentzians with narrow linewidths, one of which is an ultranarrow peak. Therefore, we claim that the emission spectrum with narrower linewidth can be obtained by adjusting the coupling strength of the qubit-cavity ultrastrong coupling system.

4. Conclusion

In this paper, we have investigated the cavity emission spectrum of a double qubit-cavity coupling system in the regime of USC. Owing to the ultrastrong coupling between qubits and cavity, the RWA is invalid, and the counter-rotating wave terms must be taken into account. In the energy spectrum of quantum Rabi Hamiltonian, we find level crossing between the second and third energy levels, and avoided-crossing between the third and fourth energy levels. In order to further investigate the fluorescence spectra in these two cases, we fix the frequency of the driving field to resonate with the transition from the ground state to the third energy level. Hence the atomic system is reduced to a four-level dressed state model driven by a light field with Rabi frequency. Different from the Mollow triplet in the two-level systems, we find an ultranarrow peak at the line center. The narrow linewidth is derived from the slowly decaying incoherent term $\rho _{++}+\rho _{--}$ and $\rho _{11}-\rho _{22}$ in the density matrix, and the physical origin is related to the quantum interference between two transition pathways in the $\Lambda$-type , V-type, and $\Xi$-type three-level structure. Furthermore, in the case of level crossing, the population on $\rho _{11}$ grows as the coupling strength increases, thus the transitions from $\vert \psi _{1}\rangle$ to $\vert +\rangle$ and $\vert -\rangle$ become more significant. Hence, a clear inner sidebands can be found in the fluorescence spectrum. Our results indicate that the spectral ultranarrowing can be achieved by adjusting the coupling strength of the USC system appropriately. This ultranarrow spectral feature can be applied to ultrahigh precision laser stabilization and spectroscopy [46]. And light source with such spectral feature has been proposed as a next-generation active atomic frequency reference, with the potential to enable high-precision optical frequency references to be used outside laboratory environments [47]. Beyond that, the process of exploring the physical origin for the narrow peak can be helpful for the study of quantum interference effect in the USC system.