Measuring a transmon qubit in circuit QED: Dressed squeezed states

Measuring a transmon qubit in circuit QED: dressed squeezed states Mostafa Khezri, 1, 2, ∗ Eric Mlinar, 1 Justin Dressel, 3, 4 and Alexander N. Korotkov 1 arXiv:1606.04204v2 [quant-ph] 1 Aug 2016 Department of Electrical and Computer Engineering, University of California, Riverside, California 92521, USA. Department of Physics, University of California, Riverside, California 92521, USA. Institute for Quantum Studies, Chapman University, Orange, California 92866, USA. Schmid College of Science and Technology, Chapman University, Orange, California 92866, USA. (Dated: August 2, 2016) Using circuit QED, we consider the measurement of a superconducting transmon qubit via a coupled microwave resonator. For ideally dispersive coupling, ringing up the resonator produces co- herent states with frequencies matched to transmon energy states. Realistic coupling is not ideally dispersive, however, so transmon-resonator energy levels hybridize into joint eigenstate ladders of the Jaynes-Cummings type. Previous work has shown that ringing up the resonator approximately respects this ladder structure to produce a coherent state in the eigenbasis (a dressed coherent state). We numerically investigate the validity of this coherent state approximation to find two primary de- viations. First, resonator ring-up leaks small stray populations into eigenstate ladders corresponding to different transmon states. Second, within an eigenstate ladder the transmon nonlinearity shears the coherent state as it evolves. We then show that the next natural approximation for this sheared state in the eigenbasis is a dressed squeezed state, and derive simple evolution equations for such states using a hybrid phase-Fock-space description. I. INTRODUCTION Qubit technology using superconducting circuit quan- tum electrodynamics (QED) [1, 2] has rapidly developed over the past decade to become a leading contender for realizing a scalable quantum computer. Most recent qubit designs favor variations of the transmon [3–9] due to its charge-noise insensitivity, which permits long co- herence times while also enabling high-fidelity quantum gates [10–12] and high-fidelity dispersive qubit readout [13–15] via coupled microwave resonators. Transmon- based circuit operation fidelities are now near the thresh- old for quantum error correction protocols, some versions of which have been realized [16–19]. The quantized energy states of a transmon are mea- sured in circuit QED by coupling them to a detuned mi- crowave resonator. For low numbers of photons popu- lating the readout resonator, the coupling is well-studied [1, 3, 20] and approximates an idealized dispersive quan- tum non-demolition (QND) measurement [21]. Each transmon energy level dispersively shifts the frequency of the coupled resonator by a distinct amount, allowing the transmon state to be determined by measurement of the microwave field transmitted through or reflected from the resonator. However, nondispersive effects become im- portant when the number of resonator photons becomes comparable to a characteristic (“critical”) number set by the detuning and coupling strength [1, 22, 23]; present- day experiments often operate in this nondispersive (or nonlinear dispersive) regime [15, 24–26]. In this paper, we analyze and model the nondispersive effects that occur during the ring-up of a readout res- onator coupled to a transmon. These effects arise from email: mostafa.khezri@email.ucr.edu the hybridization of the resonator and transmon states into joint resonator-transmon eigenstates. While ringing up the resonator from its ground state, the joint state remains largely confined to a single Jaynes-Cummings eigenstate ladder that corresponds to the initial transmon state. As pointed out in Refs. [27–29], this joint state can be approximated by a coherent state in the eigenbasis (recently named a dressed coherent state [29]). Here we refine this initial approximation and provide a more accu- rate model for the hybridized resonator-transmon state. We numerically simulate the ring-up process for a res- onator coupled to a transmon, then use this simulation to develop and verify our analytical model. We find two dominant deviations from a dressed coherent state. First, we show that the ring-up process allows a small popula- tion to leak from an initial transmon state into neighbor- ing eigenstate ladders, and find simple expressions that quantify this stray population. Second, we show that the transmon-induced nonlinearity of the resonator distorts the dressed coherent state remaining in the correct eigen- state ladder with a shearing effect as it evolves, and show that this effect closely approximates self-squeezing of the dressed field at higher photon numbers. We then use a hybrid phase-Fock-space method to find equations of mo- tion for the parameters of an effective dressed squeezed state that is formed during the ring-up process. Our im- proved model is satisfyingly simple yet quite accurate. To simplify our analysis and isolate the hybridization effects of interest, we restrict our attention to a trans- mon (modeled as a seven-level nonlinear oscillator) cou- pled to a coherently pumped but non-leaking resonator (using the rotating wave approximation). The simplifi- cation of no resonator leakage may seem artificial, but it is still a reasonable approximation during the resonator ring-up and it is also relevant for at least two known protocols. First, the catch-disperse-release protocol [27]