Quasienergy description of the driven Jaynes-Cummings model

We analyze the driven resonantly coupled Jaynes-Cummings model in terms of a quasienergy approach by switching to a frame rotating with the external modulation frequency and by using the dressed atom picture. A quasienergy surface in phase space emerges whose level spacing is governed by a rescaled effective Planck constant. Moreover, the well-known multiphoton transitions can be reinterpreted as resonant tunneling transitions from the local maximum of the quasienergy surface. Most importantly, the driving defines a quasienergy well which is nonperturbative in nature. The quantum-mechanical quasienergy state localized at its bottom is squeezed. In the Purcell-limited regime, the potential well is metastable and the effective local temperature close to its minimum is uniquely determined by the squeezing factor. The activation occurs in this case via dressed spin-flip transitions rather than via quantum activation as in other driven nonlinear quantum systems such as the quantum Duffing oscillator. The local maximum is, in general, stable. However, in presence of resonant coherent or dissipative tunneling transitions the system can escape from it and a stationary state arises as a statistical mixture of quasienergy states being localized in the two basins of attraction. This gives rise to a resonant or an antiresonant nonlinear response of the cavity at multiphoton transitions. The model finds direct application in recent experiments with a driven superconducting circuit quantum electrodynamics setup.

Figure 1

Quasienergy spectrum (in unit of ${\omega }_{\text{r}}$) of the driven Jaynes-Cummings-Hamiltonian, Eq. (2), for the parameters $g=0.026{\omega }_{\text{r}}$ and $f=0.004{\omega }_{\text{r}}$.

Figure 2

(Color online) 3D plot of the quasienergy surfaces $Q\left(\mathcal{X},\mathcal{P}\right)$ for $\mathcal{P}>0$ and $f/g=0.154$. (Note that this yields an effective temperature of $T_{\text{eff}}=0.65$, see text.) The quasienergy surface for the dressed spin orientation $e\left({\sigma }_z=+1\right)$ is a monostable reversed cone. The one for the opposite dressed spin $g\left({\sigma }_z=-1\right)$ is bistable and is shown as a density plot as well. In the density plot, the oriented black solid line indicates the separatrix between the two domains of attraction.

Figure 3

(Color online) Rescaled quasienergy spectrum of the driven Jaynes-Cummings Hamiltonian in Eq. (2) for the parameters $f/g=0.154$ (solid black lines). The green dotted line indicates the saddle-point quasienergy. The red dashed lines correspond to the approximate $\left(\lambda /\delta \omega \right)E_n$, see text. The blue dotted-dashed line marks the rescaled quasienergy ${\epsilon }_0$ of $|0\left(f\right)⟩$. The orange circle highlights avoided quasienergy level crossings (multiphoton transitions) for large photon numbers $N$.

Figure 4

(Color online) Nonlinear response of the driven JC model: amplitude $A$ as a function of the driving frequency ${\omega }_{\text{ex}}$ for (a) $f=0.035g$, (b) $f=0.071g$, and (c) $f=0.17g$. Moreover, $\kappa =6.1\times {10}^{-3}g$, ${\gamma }_1=1.4\times {10}^{-3}g$, ${\gamma }_{\varphi }=1.4\times {10}^{-4}g$, and $g=0.007{\omega }_{\text{r}}$. A realistic value for ${\omega }_{\text{r}}/\left(2\pi \right)$ is 7 GHz (Refs. 25, 38).

Figure 5

(Color online) Solid lines: nonlinear response of resonator in the circuit QED setup based on the transmon architecture as realized in Ref. 25, i.e., $\omega /\left(2\pi \right)=6.920\text{GHz}$, $g/\pi =94.4\text{MHz}$, $\kappa /\left(2\pi \right)=300\text{kHz}$, $f/\left(2\pi \right)=8.304\text{MHz}$, ${\gamma }_1/\left(2\pi \right)=98\text{kHz}$, and ${\gamma }_{\varphi }=0$. Dashed lines: same parameters but without direct qubit relaxation $\left({\gamma }_1=0\right)$. Shown are (a) the amplitude $A$, (b) the phase $\varphi$, (c) the quasienergy spectrum, and (d) the population $P^{\ast }$ of the lowest quasienergy state $|{\psi }_0^{\ast }⟩$. The red dottted-dashed line in (a) shows the result for the population of the zero-photon state being assumed as close to one, see text.

Figure 6

(Color online) Nonlinear response of the driven JC model: (a) amplitude, (b) phase, (c) quasienergies, and (d) population of the lowest quasienergy state for $f=0.057g$, $\kappa =0.0009g$, and $g=0.026{\omega }_{\text{r}}$. Dashed red (dotted blue) line in (a)–(c): lowest-order result for $|0\left(f\right)⟩$ $\left(|1\left(f\right)⟩\right)$.

Figure 7

(Color online) Nonlinear response of the driven JC model: (a) amplitude, (b) phase, (c) quasienergies, and (d) population of the lowest quasienergy states $|{\psi }_n^{\ast }⟩\left(n=0,1,2\right)$ as a function of the rescaled Planck constant $\lambda$ for the parameters $f/g=0.171$, $\kappa /g={6.110}^{-3}{\omega }_{\text{r}}$, and ${\gamma }_1,{\gamma }_{\varphi }=0$ (Purcell limit). Green dotted line in (c): quasienergy of the saddle point.

Figure 8

(Color online) Solid line: smallest eigenvalue of the Lindblad master equation. Dashed red line: the same without dissipative spin flips. Dashed-dotted orange line: decay rate for $|{\psi }^{\ast }⟩\to |{\varphi }_0^f⟩$. Dotted lines correspond to $\kappa /2$ and $\kappa /4$.