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Measurement-induced state transitions in a superconducting qubit: Beyond the rotating wave approximation
Daniel Sank, Zijun Chen, Mostafa Khezri, J. Kelly, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Mutus, M. Neeley, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, T. White, Alexander N. Korotkov, John M. Martinis
TL;DR
High-photon dispersive measurements can produce anomalous qubit transitions that limit measurement speed and fidelity, yet the Jaynes-Cummings interaction alone does not predict them. The paper develops and tests a level-crossing explanation involving normally neglected Hamiltonian terms and broken transmon-potential symmetry, finding resonant transitions beyond the two-level subspace.
Problem
Anomalous transitions at high resonator photon number are non-monotonic and are not predicted by the Jaynes-Cummings interaction, despite limiting measurement speed and fidelity.
Source: p. 1Method
The authors characterize qubit states after variable-power resonator drives and model the transitions as level crossings mediated by terms omitted under the rotating wave approximation.
Source: pp. 1, 3Results
Transitions from |0⟩ reach states above |3⟩, and measured resonance locations fit predicted |0,n⟩→|6,n−4⟩ and |3,n−2⟩ transitions without fitted parameters.
Source: pp. 2, 4Takeaways & Limitations
The most important transition mechanism involves unexpected broken symmetry in the transmon potential, and smaller |∆| should allow larger photon numbers before resonant transitions occur.
Source: p. 4Spurious two-level systems coupled to the transmon-resonator system can also participate in level crossings and produce similar features, possibly at lower photon numbers.
Source: p. 4
Abstract
from arXiv · showhide
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.
HAMILTONIAN
The Hamiltonian separates the bare qubit-resonator energies from their capacitive interaction, then distinguishes excitation-preserving RWA terms from non-RWA terms. Numerical Stark-shift calibration and non-RWA couplings connect the model to photon-dependent resonances.
- The bare Hamiltonian assigns each transmon-resonator state its transmon energy plus resonator-photon energy, while the capacitive interaction is divided into RWA and non-RWA parts.The RWA part conserves total excitation number; the non-RWA part contains the remaining terms.Source: HAMILTONIAN · p. 7
- Diagonalizing the bare plus RWA Hamiltonian yields eigenstates, eigenenergies, and photon-dependent transition frequencies within each RWA strip.Source: HAMILTONIAN · p. 7
- The numerical ac Stark curve departs from the conventional linear shift at large photon number, calibrating drive power to photon number for theory-experiment comparison.The calibration uses the photon-number-dependent shift computed from the RWA Hamiltonian.Source: HAMILTONIAN · p. 7, Figure S1 · p. 7
- Non-RWA terms connect RWA strips and can enable resonant transitions; the model considers the largest and least off-resonant couplings involving g_k,k+1 and g_k,k+3.Source: HAMILTONIAN · p. 7
- Although g_k,k+3 couplings are much smaller than g_k,k+1, they are numerically more important in this problem.The non-RWA Hamiltonian slightly shifts eigenenergies, an effect the analysis neglects.Source: HAMILTONIAN · pp. 7–8
- The symmetric transmon potential excludes g_k,k+2 terms, while broken symmetry adds nearest-neighbor strip couplings and enables processes such as |0,n⟩→|3,n−2⟩.Source: HAMILTONIAN · p. 8
EFFECTIVE COUPLING
Effective couplings connect resonant qubit–resonator states through virtual paths, but resonance alone does not ensure observable transitions. The analysis explains the measured features through these couplings, including a transition associated with transmon-potential asymmetry.
EFFECTIVE COUPLING
- Effective coupling links resonant initial and final states through intermediate off-resonant states, and multiple virtual paths can contribute.For example, |0,n⟩→|1,n−1⟩→|4,n−2⟩→|5,n−3⟩→|6,n−4⟩ is one path.Source: EFFECTIVE COUPLING · p. 8
- Alternating-sign terms in the coherent path sum can cancel; decoherence may suppress this cancellation, making the fully incoherent sum an upper bound.The coefficients alternate in sign for l < k, while those from the initial state are positive.Source: EFFECTIVE COUPLING · p. 8
- Feature B is explained by an effective coupling of about 1 MHz, consistent with the calculated values for |0,n⟩→|6,n−4⟩.The experiment has g/2π ≈87 MHz.Source: EFFECTIVE COUPLING · p. 8
- For feature A, a 1% selection-rule violation yields a few-MHz coupling; the measured g0,2/g ≃10−2 is an order-of-magnitude estimate.The calculation assumes gk,k+2 = 0.01 g√((k + 1)(k + 2)).Source: EFFECTIVE COUPLING · p. 9, Figure S2 · p. 9
- For the transition |0,n⟩→|3,n−2⟩, cancellation of terms in Eq. (S13) is less efficient than in Eq. (S11), because only two main terms contribute, involving g0,2 and g1,3.Source: EFFECTIVE COUPLING · p. 9
- The physical origin of the transmon’s broken symmetry remains speculative; SQUID asymmetry and magnetic-field gradients are proposed but not investigated.The mechanisms are left for future studies.Source: EFFECTIVE COUPLING · p. 9
TLS-ASSISTED TRANSITIONS
TLSs can mediate photon-dependent transmon transitions when the Stark-shifted qubit crosses a TLS. Virtual transitions through the Jaynes–Cummings ladder explain resonant features, including peaks at photon occupations near the critical value.
TLS-ASSISTED TRANSITIONS
- The resonator’s ac Stark shift can bring the transmon into resonance with TLSs, increasing relaxation and degrading measurement fidelity.The qubit frequency shift is approximately δω10 = −2|χ|n, and becomes substantial when n is comparable to nc.Source: TLS-ASSISTED TRANSITIONS · p. 9
- A virtual ladder process can excite |0⟩ to |2⟩ by transferring three resonator photons while exciting the TLS.The resonance condition is ωTLS ≈ ωr + 2|∆| + |η|; the example reaches n/nc ≈1, with a noticeable effective coupling.Source: TLS-ASSISTED TRANSITIONS · p. 10, Figure S3 · p. 10
- At n/nc ≈1.7, a weakly coupled TLS near 8.4 GHz can explain the observed peak in |1⟩ probability.The |2,n−2⟩ component is significant (amplitude ≈0.2); an incoherent TLS may yield a threshold-like rather than resonant condition.Source: TLS-ASSISTED TRANSITIONS · p. 10
- As n/nc approaches nc-scale occupations, more TLS-assisted processes become possible, which the authors suggest may contribute to measurement-fidelity deterioration.These processes may produce either transmon excitation or relaxation.Source: TLS-ASSISTED TRANSITIONS · p. 10