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.
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.
Superconducting circuits consisting of a few low-anharmonic transmons coupled to readout and bus resonators can perform basic quantum computations. Since the number of qubits in such circuits is limited to not more than a few tens, the qubits can be designed to operate within the dispersive regime where frequency detunings are much stronger than coupling strengths. However, scaling up the number of qubits will bring the circuit out of the regime, and this invalidates current theories. We develop a formalism that allows to consistently diagonalize the superconducting circuit Hamiltonian beyond the dispersive regime. This will allow to study qubit-qubit interaction unperturbatively, therefore, our formalism remains valid and accurate at small or even negligible frequency detuning; thus, our formalism serves as a theoretical ground for designing qubit characteristics for scaling up the number of qubits in superconducting circuits. We study the most important circuits with single- and two-qubit gates, i.e., a single transmon coupled to a resonator and two transmons sharing a bus resonator. Surprisingly, our formalism allows to determine the circuit characteristics, such as dressed frequencies and Kerr couplings, in closed-form formulas that not only reproduce perturbative results, but also extrapolate beyond the dispersive regime and can ultimately reproduce (and even modify) the Jaynes-Cumming results at resonant frequencies.
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.
Electromagnetically induced transparency (EIT) has been extensively studied in various systems. However, it is not easy to observe in superconducting quantum circuits (SQCs) because the Rabi frequency of the strong-controlling field corresponding to EIT is limited by the decay rates of the SQCs. Here, we show that EIT can be achieved by engineering decay rates in a superconducting circuit QED system through a classical driving field on the qubit. Without such a driving field, the dressed states of the system, describing a superconducting qubit coupled to a cavity field, are approximately product states of the cavity and qubit states in the large-detuning regime. However, the driving field can strongly mix these dressed states. These doubly dressed states, here called polariton states, are formed by the driving field and dressed states, and are a mixture of light and matter. The weights of the qubit and cavity field in the polariton states can now be tuned by the driving field, and thus the decay rates of the polariton states can be changed. We choose the three lowest-energy polariton states with a $\mathrm{\ensuremath{\Lambda}}$-type transition in such a driven circuit QED system, and demonstrate how EIT and Autler-Townes splitting can be realized in this compound system. We believe that this study will be helpful for EIT experiments using SQCs.
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.
Spectral properties of a quantum circuit are efficiently read out by monitoring the resonance frequency shift it induces in a microwave resonator coupled to it. When the two systems are strongly detuned, theory attributes the shift to an effective resonator capacitance or inductance that depends on the quantum circuit state. At small detuning, the shift arises from the exchange of virtual photons, as described by the Jaynes-Cummings model. Here we present a theory bridging these two limits and illustrate, with several examples, its necessity for a general description of quantum circuits readout. Circuit Quantum Electrodynamics (cQED) is at the heart of most advanced superconducting quantum technologies. Different types of superconducting qubits can be strongly coupled to microwave resonators thus achieving regimes and phenomenena which cannot be reached within the realm of quantum optics [1]. More recently, strong coupling between microwave resonators and a variety of other quantum systems not necessarily involving superconductors has been achieved [2], extending further the realm of cQED. In all these applications the measurement of the qubit or the hybrid device state is achieved by monitoring the resonator properties. Theoretically , two regimes have been approached using disconnected descriptions [3]: the dispersive regime, where the qubit-resonator detuning is larger than the coupling strength yet small enough to allow the exchange of virtual photons, and the adiabatic regime, where the de-tuning is sufficiently large for virtual processes to be strongly suppressed. The dispersive regime, which describes level repulsion between those of the quantum circuit and of the resonator, is typically dealt with using a Jaynes-Cummings Hamiltonian within different levels of approximation [3-10]. In contrast, the adiabatic regime accounts for the renormalization of the resonator capaci-tance/inductance by the effective capacitance of the circuit , including its "quantum capacitance" [11, 12], or its effective inductance [13, 14], which modifies the resonator frequency [15, 16]. However, there is no actual border between these two regimes which could justify a separate treatment, as illustrated by recent experiments on hybrid cQED setups [17] that reveal features of both regimes for the same device. This situation claims for a unified description of quantum circuits readout, going beyond the standard Jaynes-Cummings model, which could be applied to different types of devices over a large range of parameters. * Corresponding author : a.l.yeyati@uam.es In the present Letter we derive a general expression for the resonator frequency shift when coupled to a generic quantum circuit. This expression naturally interpolates between the adiabatic and the dispersive regimes, thus allowing to clarify their origin from the same coupling Hamiltonian. In addition our formalism is not restricted to the usual two-level approximations but any multilevel situation can be described on the same footing. We illustrate the importance of the different terms in our expression by analyzing well-known models like a short single channel superconducting weak link hosting Andreev states, the RF-SQUID and the Cooper pair box. Resonator-quantum circuit coupling.-The system we consider comprises a resonant circuit and a quantum circuit coupled through phase or charge fluctuations as depicted in Figs. 1(a), 2(a) and in the inset of Fig. 3. The resonant circuit is represented as a lumped-element LC resonator with bare resonance frequency f r = ω r /2π, with ω r = 1/ √ L r C r. Introducing the photon annihilation (creation) operators a (a †), it can be described by the Hamiltonian H r = ω r a † a. On the other hand, the quantum circuit Hamiltonian,Ĥ qc (x), depends on a dimensionless control parameter x, corresponding to an excess charge on a capacitor or a flux through a loop. We denote by |Φ i (x) the eigenstates of the uncoupled quantum circuit,Ĥ qc (x)|Φ i (x) = E i (x)|Φ i (x). Flux (charge) fluctuations in the resonator lead to x → x 0 +x r , wherex r = λ(s a + s * a †) with a coupling constant λ, depending on a coupling scheme [19], and s = 1 (−i). We assume λ 1 in accordance with experiments. The resonator-quantum circuit coupling HamiltonianĤ c is obtained by expandingĤ qc (x 0 +x r) up to second order inx rĤ c (x 0) =x rĤ qc (x 0) +x 2 r 2Ĥ qc (x 0), (1) where the prime stands for the derivative with respect to x. The Hamiltonian describing resonator, quantum
Repeated quantum non-demolition measurement is a cornerstone of quantum error correction protocols. In superconducting qubits, the speed of dispersive state readout can be enhanced by increasing the power of the readout tone. However, such an increase has been found to result in additional qubit state transitions that violate the desired quantum non-demolition character of the measurement. Recently, the readout of a transmon superconducting qubit was improved by using a tone with frequency much larger than the qubit frequency. Here, we experimentally identify the mechanisms of readout-induced transitions in this regime. In the dominant mechanism, the energy of an incoming readout photon is partially absorbed by the transmon and partially returned to the transmission line as a photon with lower frequency. Other mechanisms involve the excitation of unwanted package modes, decay via material defects, and, at higher qubit frequencies, the activation of undesired resonances in the transmon spectrum. Our work provides a comprehensive characterization of superconducting qubit state transitions caused by a strong drive.