Nonlinear response of the vacuum Rabi resonance

METHODS

THEORY The modelling of the transmon follows ref. koch_charge-insensitive_2007, . While some expressions given in the main text are asymptotic results valid for $E_{J}/E_{C}\gg 1$, our calculations are based on a full diagonalization of the Cooper pair box Hamiltonian.

The description of the transmon–cavity system in terms of a master equation requires a model for the relaxation and dephasing of higher transmon levels. As detailed studies of the microscopic origin of the dominant relaxation and dephasing channels are still outstanding, we have chosen plausible superoperators for our master equation (Nonlinear response of the vacuum Rabi resonance). Assuming that relaxation of higher transmon levels may arise due to a coupling of external degrees of freedom to the charge on the superconducting island, we take the relative strengths of relaxation to be related to the coupling parameters as $\alpha_{j}=g_{j}/g_{0}$. Dephasing of higher levels is likely to be due to charge noise. Denoting the charge dispersionkoch_charge-insensitive_2007 of level $j$ by $\epsilon_{j}=\omega_{j}(n_{g}=0)-\omega_{j}(n_{g}=1/2)$, we therefore take the relative dephasing strengths to be $\beta_{j}=2\epsilon_{j}/(\epsilon_{1}-\epsilon_{0}$). (The normalisation of $\alpha_{j}$ and $\beta_{j}$ is chosen to allow the usual interpretation of $\gamma_{1}$ and $\gamma_{\varphi}$ as relaxation and dephasing rates for the first two levels of the transmon spectrum.) In fact, the pure dephasing rate is sufficiently small for our qubits that we set $\gamma_{\varphi}=0$. A comparison of additional simulations with auxiliary experimental results at increased temperatures allows us to place an approximate upper bound of $0.003$ on the number of thermal photons in the cavity.

For the steady-state solution of Eq. (Nonlinear response of the vacuum Rabi resonance), the Hilbert space is truncated to a subspace with maximum number of excitations $N$, using the projector $P_{N}=\sum_{0\leq n+j\leq N}\left\lvert n,j\right\rangle\left\langle n,j\right\rvert$. In our simulations, we keep up to $N=7$ excitations. To reach agreement with the experimentally measured signal for the strongest drive powers, it is necessary to account for a small amount ($\sim-55\,\text{dB}$) of leakage of the drive past the cavity. In addition, there is a small bias introduced by measuring the transmission as the square of the $I$ and $Q$ quadratures, each of which is subject to noise. Accordingly, the quantity that corresponds to the experimental signal is $A^{2}=\lvert 2\sqrt{\kappa}\operatorname{tr}(\rho_{s}a)+b\xi\rvert^{2}+2\sigma_{n}^{2}$, where $b$ describes the leakage of the drive bypassing the cavity, and $\sigma_{n}$ is the measurement noise in each of the $I$ and $Q$ channels.

Fits are obtained by minimizing the mean squared deviation between experiment and calculation over the full power range, with fit parameters being $b$ and the two scaling factors describing the signal attenuation and amplification for input and output signals. To obtain optimal agreement, we also make small adjustments to the system parameters $\gamma_{1}$, $\kappa$, $g_{0}$, $\omega_{\text{r}}$, $\omega_{01}$, and $E_{C}$. These parameters are confined by separate experiments to narrow ranges, and all values used in fits are consistent within the experimental uncertainties. Once obtained, the same set of parameters was used in generating Figs. 3, 4, and the Supplementary Movies.

EXPERIMENT Measurements are performed in a dilution refrigerator at $15\,\text{mK}$. The sample consists of two superconducting transmon qubitskoch_charge-insensitive_2007 ; schreier_suppressing_2008 , coupled to an on-chip coplanar waveguide (CPW) cavity. Fabrication of the sample followed the description given in ref. schreier_suppressing_2008, . The CPW resonator has a half-wavelength resonant frequency of $\omega_{\text{r}}/2\pi=6.92\,\text{GHz}$ and a photon decay rate of $\kappa/2\pi=300\,\text{kHz}$. Transmission measurements are performed using a heterodyne detection scheme. The transmitted RF voltage signal through the cavity is mixed down to a 1 MHz carrier signal, and then digitally mixed down to dc to obtain the transmitted voltage amplitude as a function of frequency. The vacuum Rabi coupling strengths for the two qubits are obtained as $g_{0}/\pi=347\,\text{MHz}$ (qubit 1) and $g_{0}/\pi=94.4\,\text{MHz}$ (qubit 2). Time domain measurements of the qubits show that they are Purcell-limited and completely homogenously broadened at their flux sweet spotshouck_controlling_2008 . The coherence times are $T_{1}=1.7\,\mu\text{s}$ and $T_{2}=0.7\,\mu\text{s}$ (qubit 1, away from the flux sweet spot) and $T_{1}=1.4\,\mu$s and $T_{2}=2.8\,\mu$s (qubit 2, at flux sweet spot). The charging energies of the two qubits are measured to be $E_{C}/h=400\,\text{MHz}$ and $E_{C}/h=340\,\text{MHz}$. (See Supplementary Information, Methods, for further details.)

References

Acknowledgments

This work has been supported by Yale University via a Quantum Information and Mesoscopic Physics Fellowship (AAH, JK), the LPS/NSA-ARO grant W911NF-05-1-0365, NSF grants DMR-0653377, DMR-0603369, and PHY-0653073, and by Academy of Finland. We thank J. Gambetta, A. Blais, and A. Wallraff for helpful discussions, and L. Frunzio and B. Johnson for fabrication of the sample.

Author contributions

JMC led the experimental effort. LSB and JK performed the calculations and did most of the writing. AAH gave technical support and conceptual advice. ET contributed to the early theory. MHD, SMG, and RJS provided support and supervised the project.

Author information

Correspondence and requests for materials should be addressed to RJS.