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 coherent 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 deviations. 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 by using a hybrid phase--Fock-space description.
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 coherent 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 deviations. 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 by using a hybrid phase--Fock-space description.
Quantum computers have the capability to improve the efficiency and speed of many computational tasks. Among different candidates for physical implementation of quantum bits (qubits), superconducting qubits are currently one of the most promising candidates due to their accessible fabrication process and recent rapid developments. Measurement of these qubits is usually done in a circuit quantum electrodynamics (QED) setup, for which experimental and theoretical research is conducted to improve the accuracy and speed of the qubit readout. In this dissertation we study some aspects of the dispersive readout of superconducting qubits, and introduce tools and methods for studying these systems.In Chapter 2 we show that in presence of neighboring qubits, the system is typically measured in the basis of joint eigenstates of qubits, in contrast to what is expected from the textbook collapse postulate. In such setups, the excitation can switch between the eigenstates, leading to measurement error. In Chapter 3 we study the joint state of the qubit-resonator system during the measurement, and show that the qubit-induced nonlinearity of the resonator squeezes its state, and within the rotating wave approximation (RWA) the system mostly remains in the joint eigenladder that is associated with the qubit's initial state. In Chapter 4 we show that built-in energy resonances in the qubit-resonator Jaynes-Cummings ladder occur at specific resonator populations, and the couplings between these resonant levels are provided by the usually neglected non-RWA terms. Such resonances lead to measurement deterioration by exciting the qubit out of the computational subspace. In Chapter 5 we provide a hybrid phase-space-Fock-space approach for studying the evolution and squeezing of driven nonlinear resonators within Gaussian approximation, which is numerically efficient and sufficiently accurate. In Chapter 6 we study the propagating squeezed field that leaks out of the resonator, and write evolution equations for the correlators of the measured field quadrature. These equations are easy to simulate and can describe the squeezing of the propagating field during the transient, which can be used to optimize the fidelity and speed of the quadrature measurement in the dispersive readout of superconducting qubits.
In the circuit quantum electrodynamics architecture, both the resonance frequency and the coupling of superconducting qubits to microwave field modes can be controlled via external electric and magnetic fields to explore qubit-photon dynamics in a wide parameter range. Here, we experimentally demonstrate and analyze a scheme for tuning the coupling between a transmon qubit and a microwave resonator using a single coherent drive tone. We treat the transmon as a three-level system, with the qubit subspace defined by the ground and the second excited states. If the drive frequency matches the difference between the resonator and the qubit frequencies, a Jaynes-Cummings-type interaction is induced, which is tunable in both amplitude and phase. We show that coupling strengths of about 10 MHz can be achieved in our setup, limited only by the anharmonicity of the transmon qubit. This scheme has been successfully used to generate microwave photons with a controlled temporal shape [M. Pechal et al., Phys. Rev. X 4, 041010 (2014)] and can be directly implemented with superconducting quantum devices featuring larger anharmonicity for higher coupling strengths.
The potential of superconducting qubits as the medium for a scalable quantum computer has motivated the pursuit of improved interactions within this system. Two challenges for the field of superconducting qubits are measurement fidelity, to accurately determine the state of the qubit, and the efficient transfer of quantum states. In measurement, the current state-of-the-art method employs dispersive readout, by coupling the qubit to a cavity and reading the resulting shift in cavity frequency to infer the qubit's state; however, this is vulnerable to Purcell relaxation, as well as being modeled off a simplified two-level abstraction of the qubit. In state transfer, the existing proposal for moving quantum states is mostly untested against non-idealities that will likely be present in an experiment. In this dissertation, we examine three problems within these two areas.We first describe a new scheme for fast and high-fidelity dispersive measurement specifically designed to circumvent the Purcell Effect. To do this, the qubit-resonator interaction is turned on only when the resonator is decoupled from the environment; then, after the resonator state has shifted enough to infer the qubit state, the qubit-resonator interaction is turned off before the resonator and environment are recoupled. We also show that the effectiveness of this ``Catch-Disperse-Release'' procedure partly originates from quadrature squeezing of the resonator state induced by the Jaynes-Cummings nonlinearity.The Catch-Disperse-Release measurement scheme treats the qubit as a two-level system, which is a common simplification used in theoretical works. However, the most promising physical candidate for a superconducting qubit, the transmon, is a multi-level system. In the second work, we examine the effects of including the higher energy levels of the transmon. Specifically, we expand the eigenstate picture developed in the first work to encompass multiple qubit levels, and examine the resulting changes to the system. In particular, we analyze the population of the non-target eigenstates as a result of this expanded model, and provide an analytical form for these deviations from the simpler model in Catch-Disperse-Release (i.e., the dressed state approximation).Lastly, we assess the robustness of the existing quantum state transfer protocol, testing its performance under typical experimental deviations from the ideal case. We show that the procedure is resilient to almost all non-idealities, except frequency mismatch between the two cavities. We also demonstrate a method to compensate for one such error in frequency-matching.
A superconducting qubit coupled to a readout resonator is currently the building block of multiple quantum computing as well as quantum optics experiments. A typical qubit-resonator system is coupled in the dispersive regime, where the detuning between qubit and resonator is much greater than the coupling between them. In this work, we fabricated and measured a superconducting transmon-resonator system in the nondispersive regime. The dressed states formed by the mixing of the bare qubit and resonator states can be further mixed by applying a drive on the qubit, leading to the formation of polariton states. We report experimental studies of transitions between polariton states at varying driving powers and frequencies and show how the nondispersive coupling of the higher levels of the qubit-resonator system modifies the polariton eigenstates and the corresponding transition frequencies. We also report close agreement with numerical results obtained from a driven Jaynes-Cummings model beyond the dispersive regime.
In modern circuit QED architectures, superconducting transmon qubits are measured via the state-dependent phase and amplitude shift of a microwave field leaking from a coupled resonator. Determining this shift requires integrating the field quadratures for a nonzero duration, which can permit unwanted concurrent evolution. Here we investigate such dynamical degradation of the measurement fidelity caused by a detuned neighboring qubit. We find that in realistic parameter regimes, where the qubit ensemble-dephasing rate is slower than the qubit-qubit detuning, the joint qubit-qubit eigenstates are better discriminated by measurement than the bare states. Furthermore, we show that when the resonator leaks much more slowly than the qubit-qubit detuning, the measurement tracks the joint eigenstates nearly adiabatically. However, the measurement process also causes rare quantum jumps between the eigenstates. The rate of these jumps becomes significant if the resonator decay is comparable to or faster than the qubit-qubit detuning, thus significantly degrading the measurement fidelity in a manner reminiscent of energy relaxation processes.
The exceptionally strong coupling realizable between superconducting qubits and photons stored in an on-chip microwave resonator allows for the detailed study of matter–light interactions in the realm of circuit quantum electrodynamics (QED). Here we investigate the resonant interaction between a single transmon-type multilevel artificial atom and weak thermal and coherent fields. We explore up to three photon dressed states of the coupled system in a linear response heterodyne transmission measurement. The results are in good quantitative agreement with a generalized Jaynes–Cummings model. Our data indicate that the role of thermal fields in resonant cavity QED can be studied in detail using superconducting circuits.