Multiphoton quantum Rabi oscillations in ultrastrong cavity QED

When an atom is strongly coupled to a cavity, the two systems can exchange a single photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED was reached experimentally where the strength of the interaction between light and artificial atoms (qubits) becomes comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modify the concept of vacuum Rabi oscillations, enabling multiphoton exchanges between the qubit and the resonator. We find that experimental state-of-the-art circuit-QED systems can undergo two- and three-photon vacuum Rabi oscillations. These anomalous Rabi oscillations can be exploited for the realization of efficient Fock-state sources of light and complex entangled states of qubits.

Figure 1

(a) Sketch of the distribution of the first three resonator modes $m=1,2,3$ of a transmission-line $\lambda /2$ resonator. The resonance frequency of the first mode (blue solid curve) is set to ${\omega }_1^{\mathrm{r}}/2\pi =4$ GHz. The qubit is positioned at the center of the resonator so that it does not interact with the mode $m=2$ (red dashed curve) and is off-resonance with the mode $m=3$ (green dot-dashed curve) mode. The bare resonance frequencies of the second and third modes are ${\omega }_2^{\mathrm{r}}=2{\omega }_1^{\mathrm{r}}$ and ${\omega }_3^{\mathrm{r}}=3{\omega }_1^{\mathrm{r}}$, respectively. The qubit parameters are $\mathrm{\Delta }/h=2.25$ GHz and $2I_{\mathrm{p}}=630$ nA. (b) Frequency differences ${\omega }_{i,0}={\omega }_i-{\omega }_0$ for the lowest-energy dressed states as a function of the qubit transition frequency ${\omega }_{\mathrm{q}}$ (which can be tuned by changing the external flux bias $\delta {\mathrm{\Phi }}_x)$ for the JC model (red dashed curves) and the extended Rabi Hamiltonian (blue solid curves). We consider a normalized coupling rate $g_1/{\omega }_1^{\mathrm{r}}=0.15$ between the qubit and the resonator. In both cases the ground state level is not displayed. (c) Avoided level crossing (blue solid curves) resulting from the coupling between the states $|e,0\rangle$ and $|g,2\rangle$ due to the presence of counterrotating terms in the system Hamiltonian. The energy splitting reaches its minimum at ${\omega }_{\mathrm{q}}/2\pi \approx 7.97\mathrm{GHz}\approx 2({\omega }_1^{\mathrm{r}}/2\pi )$. The anticrossing is not present in the JC model (red dashed lines), since it arises from the coherent coupling between states with a different number of excitations.

Figure 2

Coupling between the bare states $|e,0\rangle$ and $|g,2\rangle$ via the intermediate states $|g,1\rangle$ and $|e,1\rangle$. The red dashed arrows and the solid black arrows indicate, respectively, the two different processes describing the two-photon resonant coupling between $|e,0\rangle$ and $|g,2\rangle$.

Figure 3

Time evolution of the cavity mean photon number $\langle {\widehat{X}}^{-}{\widehat{X}}^{+}\rangle$ (blue solid curve), the qubit mean excitation number $\langle {\widehat{C}}^{-}{\widehat{C}}^{+}\rangle$ (red dot-dashed curve), and the zero-delay two-photon correlation function $G^{\left(2\right)}\left(t\right)$ (dashed black curve) after the arrival of a $\pi$-like Gaussian pulse initially exciting the qubit (the black vertical line on the far left shows the wave-packet peak arrival time). The amplitude and the central frequency of the pulse are $A/{\omega }_1^{\mathrm{r}}=8.7\times {10}^{-2}$ and $\omega =({\omega }_{3,0}+{\omega }_{2,0})/2$, respectively. After the arrival of the pulse, the system undergoes vacuum Rabi oscillations showing the reversible excitation exchange of two photons between the qubit and the resonator. Initially, $\langle {\widehat{X}}^{-}{\widehat{X}}^{+}\rangle$ and $G^{\left(2\right)}\left(t\right)$ almost coincide. This perfect two-photon correlation is a signature that photons are actually emitted in pairs. The resonator and qubit damping rates are $\kappa /{\omega }_1^{\mathrm{r}}=1.8\times {10}^{-4}$ and $\gamma /{\omega }_1^{\mathrm{r}}=1.8\times {10}^{-4}$, respectively.

Figure 4

Effects of strong pure dephasing (dashed curves) on the dynamics of (a) the mean photon number $\langle {\widehat{X}}^{-}{\widehat{X}}^{+}\rangle$, (b) the effective qubit population $\langle {\widehat{C}}^{-}{\widehat{C}}^{+}\rangle$, and (c) the two-photon correlation function $\langle {\widehat{X}}^{-}{\widehat{X}}^{-}{\widehat{X}}^{+}{\widehat{X}}^{+}\rangle$. Calculations have been performed with the same parameters as in Fig. 3 with the addition of a pure dephasing rate ${\gamma }_{\varphi }=300\gamma$. Solid curves display numerical results obtained in the absence of pure dephasing, ${\gamma }_{\varphi }=0$. It can be observed that the physics of multiphoton vacuum Rabi oscillations is not significantly altered by the effects of pure dephasing.

Figure 5

(a) Sketch of the distribution of the first three resonator modes $m=1,2,3$ of a transmission-line $\lambda /4$ resonator. The resonance frequency of the first mode (blue solid curve) is ${\omega }_1^{\mathrm{r}}=4$ GHz. The qubit is positioned so that it does not interact with the mode $m=2$ (red dashed curve) with resonance frequency ${\omega }_2^{\mathrm{r}}=3{\omega }_1^{\mathrm{r}}$. The mode $m=3$ (green dot-dashed curve) has resonance frequency ${\omega }_3^{\mathrm{r}}=5{\omega }_1^{\mathrm{r}}$, which is much larger than ${\omega }_{\mathrm{q}}$, so only the interaction of the qubit with the fundamental resonator mode must be considered. The qubit parameters are $\mathrm{\Delta }/h=4.25$ GHz and $2I_{\mathrm{p}}=630$ nA. (b) Frequency differences ${\omega }_{i,0}={\omega }_i-{\omega }_0$ for the lowest-energy dressed states as a function of the qubit transition frequency ${\omega }_{\mathrm{q}}$ (which can be tuned by changing the external flux bias $\delta {\mathrm{\Phi }}_x)$ for the JC model (red dashed curves) and the extended Rabi Hamiltonian (blue solid curves) explicitly containing counterrotating terms. We consider a normalized coupling rate $g_1/{\omega }_1^{\mathrm{r}}=0.25$ between the qubit and the resonator. The spectrum shows two large-splitting anticrossings, which appear only in the continuous curves, plus a smaller avoided crossing, which is magnified in (c). (c) Three-photon vacuum Rabi splitting (blue solid curves) resulting from the coupling between the states $|e,0\rangle$ and $|g,3\rangle$ due to the presence of counterrotating terms in the system Hamiltonian. The energy splitting reaches its minimum at ${\omega }_{\mathrm{q}}/2\pi \approx 11.89\mathrm{GHz}\approx 3({\omega }_1^{\mathrm{r}}/2\pi )$. The anticrossing is not present in the JC model (red dashed curves), since it arises from the coherent coupling between states with a different number of excitations.

Figure 6

(a) Temporal evolution of the cavity mean photon number $\langle {\widehat{X}}^{-}{\widehat{X}}^{+}\rangle$ (blue solid curve) and the qubit mean excitation number $\langle {\widehat{C}}^{-}{\widehat{C}}^{+}\rangle$ (red dashed curve) after the arrival of a $\pi$-like Gaussian pulse exciting the qubit (the black vertical line on the far left shows the wave-packet center time). The amplitude and the central frequency of the pulse are $A/{\omega }_1^{\mathrm{r}}=9.4\times {10}^{-2}$ and $\omega =({\omega }_{4,0}+{\omega }_{3,0})/2$, respectively. After the arrival of the pulse, the system undergoes vacuum Rabi oscillations showing the reversible excitation exchange between the qubit and the resonator. The fact that the mean intracavity physical photon number at its first maximum is very close to 3 is an initial signature that, when the qubit is in its ground state, the resonator mode is in a three-photon state. (b) Time evolution of the zero-delay three-photon function $G^{\left(3\right)}\left(t\right)$ (dashed green curve) together with the intracavity photon number $\langle {\widehat{X}}^{-}{\widehat{X}}^{+}\rangle$ (solid blue curve). The first peak value of the three-photon correlation function is approximately two times higher than that of the mean photon number, a signature of an almost-perfect three-photon correlation. The parameters for the resonator and qubit losses are the same as in Fig. 3.

Figure 7

Comparison between the minimum energy splitting $2{\mathrm{\Omega }}_{\mathrm{eff}}^{\left(2\mathrm{ph}\right)}/{\omega }_{\mathrm{q}}$ obtained analytically (red solid curve) and numerically (blue points) as a function of $g_1/{\omega }_{\mathrm{q}}$ for $\theta =\pi /4$.