Photon blockade and the quantum-to-classical transition in the driven-dissipative Josephson pendulum coupled to a resonator

We investigate the driven quantum phase transition between the oscillating motion and the classical nearly free rotations of the Josephson pendulum coupled to a harmonic oscillator in the presence of dissipation. We refer to this as the Josephson-Rabi model. This model describes the standard setup of circuit quantum electrodynamics, where typically a transmon device is embedded in a superconducting cavity. We find that by treating the system quantum mechanically this transition occurs at higher drive powers than expected from an all-classical treatment, which is a consequence of the quasiperiodicity originating in the discrete energy spectrum of the bound states. We calculate the photon number in the resonator and show that its dependence on the drive power is nonlinear. In addition, the resulting multiphoton blockade phenomenon is sensitive to the truncation of the number of states in the transmon, which limits the applicability of the standard Jaynes-Cummings model as an approximation for the pendulum-oscillator system. We calculate the $n\mathrm{th}$-order correlation functions of the blockaded microwave photons and observe the differences between the rotating-wave approximation and the full multilevel Josephson-Rabi Hamiltonian with the counter-rotating terms included. Finally, we compare two different approaches to dissipation, namely the Floquet-Born-Markov and the Lindblad formalisms.

Figure 1

Electrical circuit and the corresponding eigenenergy spectrum. (a) Lumped-element schematic of a transmon-resonator superconducting circuit. The resonator and the transmon are marked with blue (left) and magenta (right) rectangles. (b) Numerically obtained eigenenergies of the resonator-pendulum Hamiltonian in Eq. (1) are shown in blue (gray) as a function of the resonator frequency. The bare pendulum eigenenergies $\hslash {\omega }_k$ are denoted with dashed horizontal lines and indicated with the label $k$. The eigenenergies of the uncoupled system, defined in Eqs. (2) and (3), are given by the dashed lines whose slope increases in integer steps with the number of quanta in the oscillator as $n\hslash {\omega }_{\mathrm{r}}$. We only show the eigenenergies of the uncoupled system for the case of pendulum in its ground state, but we note that one obtains a similar infinite fan of energies for each pendulum eigenstate. Note that in general ${\omega }_{\mathrm{q}}\ne {\omega }_{\mathrm{p}}$. We have used the parameters in Table 1 and fixed $n_{\mathrm{g}}=0$.

Figure 2

Classical dynamics of the resonator occupation $N_{\mathrm{r}}$ of the driven and dissipative resonator-transmon system. We show data for linear ($A/\kappa =0.005$, bottom line), chaotic ($A/\kappa =7$, middle line), and the bare-resonator ($A/\kappa =500$, top line) regimes. The bare-oscillator occupation in the steady state is given by Eq. (35) and indicated with dashed lines. We also show with dot-dashed lines the steady-state photon numbers for the linearized system, as given in Eq. (36). We have used the pendulum dissipation rate $\gamma /{\omega }_{\mathrm{r}}=2\times {10}^{-4}$. The other parameters are listed in Table 1.

Figure 3

Onset of the nonlinearities in the driven system. (a) The steady-state photon number $N_{\mathrm{r}}$ as a function of the drive amplitude. We compare the Floquet-Born-Markov (FBM) simulation with seven transmon states against the corresponding solutions for the Rabi Hamiltonian and the classical system. The classical region of bistability occurs between $A_{\min }/\kappa =0.97$ and $A_{\mathrm{crit}}/\kappa =1.2$, given by Eqs. (37) and (38), respectively. The classical simulation demonstrates switching between the two stable solutions at $A\approx A_{\mathrm{crit}}$. We also show the photon numbers of the linearized system and the bare resonator, as given by Eqs. (35) and (36), respectively. Note that both axes are shown in logarithmic scale. (b) Occupation probabilities $P_k$ of the transmon eigenstates calculated using FBM. We indicate the regime of classical response with the shaded region in both figures. Occupation probabilities for states with $k\ge 4$ are negligible for the used parameters. (c) Order parameter $\mathrm{\Xi }$ defined in Eq. (39). We have used $n_{\mathrm{g}}=0$ and the drive detuning ${\delta }_{\mathrm{d}}/{\omega }_{\mathrm{r}}=-0.02$. Other parameters are listed in Table 1.

Figure 4

Steady-state photon number as a function of drive detuning and amplitude for the resonant case with ${\delta }_{\mathrm{q}}=0.0$. Other parameters used in the simulation are listed in Table 1. We present the energy level diagrams of the coupled pendulum-resonator for the (a) two-state and (b) seven-state truncation for the transmon, and the corresponding simulations (c) and (d) for the average number of photons $N_{\mathrm{r}}$. In the diagrams (a) and (b), the states are labeled by the excitation number $N$ which is a good quantum number if the rotating-wave approximation is valid. We also highlight by blue (gray) rectangles the range of energies accessible by 1-, 2-, 3-, 4-, and 5-photon transitions, corresponding to the range of drive frequencies ${\delta }_{\mathrm{d}}/{\omega }_{\mathrm{r}}\in \{-0.06,0.06\}$ in panels (c) and (d). The vertically aligned arrows indicate the locations of transitions that correspond to the multiphoton blockades in panels (c) and (d), which are denoted with dashed lines. Absent transitions are denoted in red (gray).

Figure 5

Steady-state photon number $N_{\mathrm{r}}$ of the multilevel Josephson-Rabi model as a function of drive amplitude for different coupling strengths. The simulations are realized using the Floquet-Born-Markov approach with the seven-state truncation for the transmon. The drive detuning is ${\delta }_{\mathrm{d}}/{\omega }_{\mathrm{r}}=-0.02$ and also the other parameters are the same as in Table 1.

Figure 6

Correlations $G^{\left(n\right)}$ as a function of the drive amplitude. The system is driven at the two-photon blockade frequency ${\delta }_{\mathrm{d}}/{\omega }_{\mathrm{r}}=-0.018$. The calculations are done for a seven-state transmon (a) by performing the rotating-wave approximation, and (b) with the full Josephson-Rabi Hamiltonian. The solid lines correspond to zero temperature and the dashed lines to $k_{\mathrm{B}}T/\left(\hslash {\omega }_{\mathrm{r}}\right)=0.13$, while the other parameters are the same as in Table 1.

Figure 7

Onset of nonlinearity as a function of the gate charge. (a) The photon number $N_{\mathrm{r}}$ as a function of the drive amplitude. We compare the numerical data for $n_{\mathrm{g}}=0$ and $n_{\mathrm{g}}=0.5$ obtained with seven transmon states. (b) Corresponding occupations $P_k$ in the transmon eigenstates. The drive detuning is ${\delta }_{\mathrm{d}}/{\omega }_{\mathrm{r}}=-0.02$ and also the other parameters are the same as in Table 1.

Figure 8

Comparison between the Floquet-Born-Markov (FBM) and Lindblad models for dissipation in the two-level approximation for the transmon. The drive detuning is ${\delta }_{\mathrm{d}}/{\omega }_{\mathrm{r}}=-0.02$ and also the other parameters are the same as in Table 1.

Figure 9

Eigenvalues and eigenstates of the transmon obtained with the Mathieu equation (A5) for $\eta =30$. (a) Eigenenergies as a function of the superconducting phase difference $\phi$. The cosine potential is indicated with the blue (gray) line. Inside the well, the eigenenergies are discrete and denoted with dashed black lines. On top of each line, we show the absolute square of the corresponding Mathieu eigenfunction. The energy bands from (b) are indicated with gray. (b) Eigenenergies as a function of the gate charge. We compare the numerically exact eigenenergies $E_k$ (solid black) with those of the perturbative Duffing oscillator (dashed horizontal red) and the free rotor (dashed diagonal blue). The charge dispersion in the (nearly) free rotor states leads to energy bands, which are denoted with gray. We show the Duffing and free rotor solutions only inside and outside the potential, respectively.