Measurement-Induced State Transitions in a Superconducting Qubit: Beyond the Rotating Wave Approximation

Many superconducting qubit systems use the dispersive interaction between the qubit and a coupled harmonic resonator to perform quantum state measurement. Previous works have found that such measurements can induce state transitions in the qubit if the number of photons in the resonator is too high. We investigate these transitions and find that they can push the qubit out of the two-level subspace, and that they show resonant behavior as a function of photon number. We develop a theory for these observations based on level crossings within the Jaynes-Cummings ladder, with transitions mediated by terms in the Hamiltonian that are typically ignored by the rotating wave approximation. We find that the most important of these terms comes from an unexpected broken symmetry in the qubit potential. We confirm the theory by measuring the photon occupation of the resonator when transitions occur while varying the detuning between the qubit and resonator.

Figure 1

Transmon-resonator system. (a) Circuit and potential diagrams. The transmon (violet) is capacitively coupled to the resonator (orange). The resonator is inductively coupled to a bandpass Purcell filter with $Q\approx 30$ [20]. The resonator is driven by an arbitrary waveform generator connected to the filter, and the dispersed photons are measured by a low noise, impedance matched parametric amplifier [25] also connected to the filter. (b) In-phase and quadrature (IQ) components of the dispersed signal measured with the transmon prepared in the first four states, with each state forming an IQ “cloud.” The circles represent $3\sigma$ from fitting a Gaussian distribution to each cloud’s projection onto lines connecting the clouds’ centers.

Figure 2

(a) Control sequence for probing the effect of resonator photons on the transmon. The spectroscopy pulse is used only in the ac Stark measurement. (b) IQ data for drive powers 0.02, 0.2, and 0.8 (arbitrary units), with ${\omega }_{10}=5.38\mathrm{GHz}$. The circles represent $3\sigma$ for the four resolvable transmon states as calibrated in Fig. 1. At high power, the transmon is clearly driven to states higher than $|3⟩$. (c) Transmon state probabilities versus stimulation power. In addition to the four calibrated transmon states, we show the probability that the measurement was $>3\sigma$ from any of the resolved states, labeled “outliers.” Note the two large resonance shaped peaks labeled $A$ and $B$. (d) Stark shifted transmon frequency ${\omega }_{10}$ versus stimulation pulse power. We convert the shifted ${\omega }_{10}$ to $\overline{n}$ using a numerical theory (right vertical axis) [28].

Figure 3

JC ladder for large values of $n$. Bare states are shown as solid lines and two of the eigenstates are shown as dashed lines. Dark curved arrows indicate coupling within an RWA strip with corresponding RWA coupling strengths shown below. The ladder has an energy resonance between $|0,n⟩$ and $|6,n-4⟩$ (long black arrow). Non-RWA couplings (short straight arrows) allow for interstrip transitions. The couplings to $|1,n+1⟩$ (red) and $|3,n-1⟩$ (yellow), along with those within the RWA strip, mediate the transition between the resonant levels. The coupling to $|2,n-1⟩$ (green), which mediates additional resonant transitions, requires a Hamiltonian term coupling transmon states of equal parity; this is forbidden if the transmon potential is symmetric. Note the energy spacing between states $|k,n⟩$ and $|k+1,n-1⟩$ is $\mathrm{\Delta }$ as indicated in the top left.

Figure 4

(a) Fan diagram of the energy levels within an RWA strip. Solid: Frequencies ${\overline{\omega }}_k(n)\equiv E_{\overline{|k,n-k⟩}}/\hslash -n{\omega }_r$ versus photon number $n$ for $|\mathrm{\Delta }|=1.4\mathrm{GHz}$. As $n$ increases, the levels repel more strongly and fan out. Dashed: Same frequencies shifted by $2{\omega }_r$, which represent the next-nearest neighboring RWA strip. The red dots show energy resonances with the qubit state $|0⟩$ occurring at specific values of $n$. The left dot corresponds to the resonance shown in Fig. 3. (b) Photon number at level crossing versus ${\omega }_{10}$, compared between experiment and theory. Black circles and blue squares show experimental features $A$ and $B$ from Fig. 2, respectively, and the error bars represent the apparent widths of the features. The solid red line is the theory prediction for level crossing between eigenlevels of $|0,n⟩$ and $|6,n-4⟩$. The dashed blue line is the theory prediction for an asymmetric transmon that breaks the selection rule by at least 1%, yielding level crossings between eigenlevels of $|0,n⟩$ and $|3,n-2⟩$.