Unifying Floquet Theory of Longitudinal and Dispersive Readout
Alessandro Chessari, Esteban A. Rodríguez-Mena, J. C. Abadillo-Uriel, V. Champain, Simon Zihlmann, Romain Maurand, Yann‐Michel Niquet, Michele Filippone
We devise a Floquet theory of longitudinal and dispersive readout in circuit quantum electrodynamics (cQED). By studying qubits coupled to cavity photons and driven at the resonance frequency of the cavity ${\ensuremath{\omega}}_{\mathrm{r}}$, we establish a universal connection between the qubit ac Stark shift and the longitudinal and dispersive coupling to photons. We find that the longitudinal coupling ${g}_{\ensuremath{\parallel}}$ is controlled by the slope of the ac Stark shift as function of the driving strength ${A}_{\mathrm{q}}$, while the dispersive shift $\ensuremath{\chi}$ depends on its curvature. The two quantities become proportional to each other in the weak drive limit (${A}_{\mathrm{q}}\ensuremath{\rightarrow}0$). Our approach unifies the adiabatic limit (${\ensuremath{\omega}}_{\mathrm{r}}\ensuremath{\rightarrow}0$)---where ${g}_{\ensuremath{\parallel}}$ is generated by the static spectrum curvature (or quantum capacitance)---with the diabatic limit, where ${\ensuremath{\omega}}_{\mathrm{r}}$ is large and the static spectrum plays no role. We derive analytical results supported by exact numerical simulations. We apply them to superconducting and spin-hybrid cQED systems, showcasing the flexibility of faster-than-dispersive longitudinal readout.
We devise a Floquet theory of longitudinal and dispersive readout in circuit quantum electrodynamics (cQED). By studying qubits coupled to cavity photons and driven at the resonance frequency of the cavity \ensuremathωr, we establish a universal connection between the qubit ac Stark shift and the longitudinal and dispersive coupling to photons. We find that the longitudinal coupling g\ensuremath∥ is controlled by the slope of the ac Stark shift as function of the driving strength Aq, while the dispersive shift \ensuremathχ depends on its curvature. The two quantities become proportional to each other in the weak drive limit (Aq\ensuremath→0). Our approach unifies the adiabatic limit (\ensuremathωr\ensuremath→0)---where g\ensuremath∥ is generated by the static spectrum curvature (or quantum capacitance)---with the diabatic limit, where \ensuremathωr is large and the static spectrum plays no role. We derive analytical results supported by exact numerical simulations. We apply them to superconducting and spin-hybrid cQED systems, showcasing the flexibility of faster-than-dispersive longitudinal readout.
Robust manipulation and control of quantum systems require a fast and efficient way to read out their state. After cavity quantum electrodynamics (cavity QED) pioneered the manipulation of neutral atoms and ions with light, circuit quantum electrodynamics (circuit QED) became the dominant readout paradigm for artificial atoms such as superconducting circuits, with dispersive readout widely adopted for its stability and efficiency.Recently, longitudinal readout has emerged as a potentially faster readout protocol, but relies on sophisticated parametric modulation, whose underlying mechanisms have remained unclear so far.Longitudinal interactions have been demonstrated across platforms ranging from superconducting qubits to quantum dots, and rely on driving the qubit at the resonator frequency. In different regimes, the coupling emerges parametrically in the drive strength, growing linearly as the drive increases.In this manuscript, we experimentally demonstrate the breakdown of this parametric regime in a two-level quantum dot charge qubit and devise a unifying Floquet theory to establish a universal connection between AC Stark shift, longitudinal coupling and dispersive readout in cQED.We derive analytical results supported by exact numerical simulations that we apply to superconducting and spin-hybrid cQED systems, showcasing the flexibility of faster-than-dispersive longitudinal readout.Inspired by cQED’s readout capabilities in single qubit measurements, we address many-body state readout in quantum simulators.Since the dynamical structure factor (DSF) encodes key information about density excitations in quantum systems, we propose Many-Body Admittance Spectroscopy (MADS) as a weakly invasive method to probe the DSF by analyzing the transmission coefficients of microwave signals coupled to the quantum simulator.We demonstrate the approach on a one-dimensional spinless fermionic chain, representative of quantum dot implementations of the Heisenberg model, and we discuss, supported by numerical simulations obtained using exact diagonalization and density matrix renormalization group (DMRG), the possibility to measure the qualitative modifications of the DSF caused by interactions. Our results highlight the potential of MADS for characterizing the dynamical properties in highly tunable solid-state quantum simulators, extending the toolbox for the readout of quantum simulators.
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
We propose a protocol for the measurement of adiabatic phases of periodically driven quantum systems coupled to an open cavity that enables dispersive readout. It turns out that the cavity transmission exhibits peaks at frequencies determined by a resonance condition that involves the dynamical and the geometric phase. Since these phases scale differently with the driving frequency, one can determine them by fitting the peak positions to the theoretically expected behavior. For the derivation of the resonance condition and for a numerical study, we develop a Floquet theory for the dispersive readout of ac driven quantum systems. The feasibility is demonstrated for two test cases that generalize Landau-Zener-Stückelberg-Majorana interference to two-parameter driving.
Longitudinal (curvature) couplings of an N-level qudit to a superconducting resonator at the adiabatic limit and beyond
Understanding how and to what magnitude solid-state qubits couple to metallic wires is crucial to the design of quantum systems such as quantum computers. Here, we investigate the coupling between a multi-level system, or qudit, and a superconducting (SC) resonator's electromagnetic field, focusing on the interaction involving both the transition and diagonal dipole moments of the qudit. Specifically, we explore the effective dynamical (time-dependent) longitudinal coupling that arises when a solid-state qudit is adiabatically modulated at small gate frequencies and amplitudes, in addition to a static dispersive interaction with the SC resonator. For the first time, we derive Hamiltonians describing the longitudinal multi-level interactions in a general dispersive regime, encompassing both dynamical longitudinal and dispersive interactions. These Hamiltonians smoothly transition between their adiabatic values, where the couplings of the n-th level are proportional to the level's energy curvature concerning a qudit gate voltage, and the substantially larger dispersive values, which occur due to a resonant form factor. We provide several examples illustrating the transition from adiabatic to dispersive coupling in different qubit systems, including the charge (1e DQD) qubit, the transmon, the double quantum dot singlet-triplet qubit, and the triple quantum dot exchange-only qubit. In some of these qubits, higher energy levels play a critical role, particularly when their qubit's dipole moment is minimal or zero. For an experimentally relevant scenario involving a spin-charge qubit with magnetic field gradient coupled capacitively to a SC resonator, we showcase the potential of these interactions. They enable close-to-quantum-limited quantum non-demolition (QND) measurements and remote geometric phase gates, demonstrating their practical utility in quantum information processing.
We present a unified picture of dispersive readout of quantum systems in and out of equilibrium. A cornerstone of the approach is the backaction of the measured system to the cavity obtained with nonequilibrium linear-response theory. It provides the dispersive shift of the cavity frequency in terms of a system susceptibility. It turns out that already effortless computations of the susceptibility allow one to generalize former results beyond a rotating-wave approximation. Examples are the readout of detuned qubits and thermally excited multilevel systems. For ac-driven quantum systems, we identify the relevant Fourier component of the susceptibility and introduce a computational scheme based on Floquet theory. The usefulness is demonstrated for two-tone spectroscopy and interference effects in driven two-level systems. This also reveals that dispersive readout does not necessarily measure excitation probabilities.
Circuit quantum electrodynamics (cQED), exemplified by a coupled system of a superconducting qubit and a microwave cavity, is a powerful framework for the quantum control and measurement of macroscopic quantum states in solid-state circuits. Quantum circuits based on superconducting qubits have emerged as one of the leading platforms in the pursuit of fault-tolerant quantum computing. Despite significant progress in extending qubit lifetimes, refining experimental protocols, and scaling up quantum processors, errors arising from the control and measurement of superconducting qubits continue to hinder the realization of a universal quantum computer. To investigate the error mechanisms for state-of-the-art qubits, we realize long-lived superconducting transmon qubits that exhibit fluctuating lifetimes, with an average of 0.2 ms and a maximum exceeding 0.4 ms. We then investigate their dominant error mechanism. By introducing novel time-resolved error measurements that are synchronized with the operation of the pulse tube cooler in a dilution refrigerator, we find that mechanical vibrations from the pulse tube induce nonequilibrium dynamics in highly coherent qubits, leading to their correlated bit-flip errors. Our findings not only deepen our understanding of qubit error mechanisms but also provide valuable insights into potential error-mitigation strategies for achieving fault tolerance by decoupling superconducting qubits from their mechanical environments. We also present a novel scheme for operating a cQED system in a regime where the cavity is largely displaced. We find that a qubit drive at the bare cavity frequency can effectively displace the cavity field while minimizing the cavity-drive-induced dephasing and the ac Stark shift. Furthermore, we demonstrate that standard cQED protocols, such as qubit control, readout, and cavity single-photon generation, remain compatible with largely displaced cavity fields, even beyond the critical photon number. These findings open new pathways for exploring cQED dynamics in large-photon-number regimes, including the study of nonlinear phenomena in microwave cavities, the saturation of two-level system baths for extending the coherence of microwave quantum states, and the acceleration of the dispersive readout. Finally, mechanical resonators have emerged as integral components in quantum computing for storing and transducing quantum information. Low-frequency mechanical resonators, with extremely low energy dissipation rates, are particularly interesting for extending the lifetimes of stored quantum information. However, integrating low-frequency mechanical resonators with qubits is challenging because their small energy gaps make coherent energy exchange and ground-state preparation difficult. In this thesis, we report the first steps towards the characterization of a hybrid quantum device, consisting of a Transmon qubit and a circuit optomechanical system, both coupled to a 3D microwave cavity.