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arXiv:2108.02549 · Physical Review Applied

Ultrastrong Capacitive Coupling of Flux Qubits

María Hita-Pérez, Gabriel Jaumà, M. Pino, Juan José García-Ripoll

Published Jan 24, 2022• 8 citations
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Abstract

A flux qubit can interact strongly when it is capacitively coupled to other circuit elements. This interaction can be separated into two parts, one acting on the qubit subspaces and one in which excited states mediate the interaction. The first term dominates the interaction between the flux qubit and an $LC$ resonator, leading to ultrastrong couplings of the form ${\ensuremath{\sigma}}^{y}(a+{a}^{\ifmmode\dagger\else\textdagger\fi{}})$, which complement the inductive ${\ensuremath{\sigma}}^{x}i({a}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}a)$ coupling. However, when coupling two flux qubits capacitively, all terms need to be taken into account, leading to complex nonstoquastic ultrastrong interaction of types ${\ensuremath{\sigma}}_{1}^{y}{\ensuremath{\sigma}}_{2}^{y}$, ${\ensuremath{\sigma}}_{1}^{z}{\ensuremath{\sigma}}_{2}^{z}$, and ${\ensuremath{\sigma}}_{1}^{x}{\ensuremath{\sigma}}_{2}^{x}$. Our theory explains all these interactions, describing them in terms of general circuit properties---coupling capacitances, qubit gaps, inductive, Josephson and capacitive energies---that apply to a wide variety of circuits and flux qubit designs.

DOI
10.1103/physrevapplied.17.014028

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A flux qubit can interact strongly when it is capacitively coupled to other circuit elements. This interaction can be separated into two parts, one acting on the qubit subspaces and one in which excited states mediate the interaction. The first term dominates the interaction between the flux qubit and an LCLCLC resonator, leading to ultrastrong couplings of the form \ensuremathσy(a+a\ifmmode†\else\textdagger\fi){\ensuremath{\sigma}}^{y}(a+{a}^{\ifmmode\dagger\else\textdagger\fi{}})\ensuremathσy(a+a\ifmmode†\else\textdagger\fi), which complement the inductive \ensuremathσxi(a\ifmmode†\else\textdagger\fi\ensuremath−a){\ensuremath{\sigma}}^{x}i({a}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}a)\ensuremathσxi(a\ifmmode†\else\textdagger\fi\ensuremath−a) coupling. However, when coupling two flux qubits capacitively, all terms need to be taken into account, leading to complex nonstoquastic ultrastrong interaction of types \ensuremathσ1y\ensuremathσ2y{\ensuremath{\sigma}}_{1}^{y}{\ensuremath{\sigma}}_{2}^{y}\ensuremathσ1y​\ensuremathσ2y​, \ensuremathσ1z\ensuremathσ2z{\ensuremath{\sigma}}_{1}^{z}{\ensuremath{\sigma}}_{2}^{z}\ensuremathσ1z​\ensuremathσ2z​, and \ensuremathσ1x\ensuremathσ2x{\ensuremath{\sigma}}_{1}^{x}{\ensuremath{\sigma}}_{2}^{x}\ensuremathσ1x​\ensuremathσ2x​. Our theory explains all these interactions, describing them in terms of general circuit properties---coupling capacitances, qubit gaps, inductive, Josephson and capacitive energies---that apply to a wide variety of circuits and flux qubit designs.
We analyze the coupling of two flux qubits with a general many-body projector into the low-energy subspace. Specifically, we extract the effective Hamiltonians that controls the dynamics of two qubits when they are coupled via a capacitor and/or via a Josephson junction. While the capacitor induces a static charge coupling tunable by design, the Josephson junction produces a magnetic-like interaction easily tunable by replacing the junction with a superconducting quantum interference device. Those two elements allow to engineer qubits Hamiltonians with XX, YY, and ZZ interactions, including ultrastrongly coupled ones. We present an exhaustive numerical study for two three-Josephson junctions flux qubit that can be directly used in experimental work. The method developed here, namely, the numerical tool to extract qubit effective Hamiltonians at strong coupling, can be applied to replicate our analysis for general systems of many qubits and any type of coupling.
Cavity quantum electrodynamics (QED) investigates the interaction between resonator-confined radiation and natural atoms. Similar phenomena can be explored using superconducting artificial atoms coupled to microwave resonators. Here, we examine incoherent and coherent spectra in a circuit QED system consisting of a flux qubit coupled to an LC resonator, considering different coupling configurations to the output transmission line. Although such systems can be effectively described by the quantum Rabi model, we demonstrate that the choice of the coupling between the resonator and the input-output channel can significantly affect the emission properties, particularly in the ultrastrong coupling regime. Specifically, we show that when the resonator is capacitively coupled with the input-output transmission line, the spectral features at zero flux offset can be mapped onto those of the cavity QED model. However, this correspondence breaks down for the mutual inductive coupling case. Extending our analysis to the parity-symmetry broken regime of the flux qubit, we further highlight how capacitive and inductive coupling mechanisms lead to distinct spectral features, emphasizing the necessity of a careful input-output treatment for an accurate description of the emission processes.
Demonstration of a $n\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}q\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c$ Hamiltonian---one for which there exists no local basis in which all off-diagonal elements are nonpositive---is an important step toward the development of quantum annealers that are more computationally powerful, or even universal. (This condition is tied to the ``sign problem'' in quantum Monte Carlo techniques.) This work presents the implementation of such a Hamiltonian by coupling two flux qubits both inductively and capacitively. The signature of nonstoquastic behavior is observed through destructive interference in quantum coherent oscillations. This result would seem to bear strong implications for scalable quantum computing.
We theoretically study a circuit QED architecture based on a superconducting flux qubit directly coupled to the center conductor of a coplanar waveguide transmission-line resonator. As already shown experimentally [A. A. Abdumalikov, Jr. et al., Phys. Rev. B 78, 180502(R) (2008)], the strong coupling regime of cavity QED can readily be achieved by optimizing the local inductance of the resonator in the vicinity of the qubit. In addition to yielding stronger coupling with respect to other proposals for flux qubit based circuit QED, this approach leads to a qubit-resonator coupling strength ggg which does not scale as the area of the qubit but is proportional to the total inductance shared between the resonator and the qubit. Strong coupling can thus be attained while still minimizing sensitivity to flux noise. Finally, we show that by taking advantage of the large kinetic inductance of a Josephson junction in the center conductor of the resonator can lead to coupling energies of several tens of percent of the resonator frequency, reaching the ultrastrong coupling regime of cavity QED where the rotating-wave approximation breaks down. This should allow an on-chip implementation of the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ Jahn-Teller model.
Conquer with skill, not with numbers. A. V. Suvorov, from his military maxims, ca. 1795, translated from Russian by the author Physical implementations of qubit coupling Coupling by linear passive elements. Capacitive coupling To have good basic elements is not enough – it is necessary to be able to connect them in a controllable way, without losing quantum coherence. Any simple effective coupling Hamiltonian (like in Eqs. (3.81, 3.82)) must be somehow implemented “in metal”. Here superconducting circuits provide a wide variety of coupling schemes to choose from (see, e.g., Wendin and Shumeiko, 2005). We will begin with the simplest case, when the interaction between the qubits is realized using linear elements (conventional capacitances and inductances), the coupling circuit stays in its ground state and adiabatically follows the evolution of the qubits (Averin and Bruder, 2003) – that is, it remains “passive”. For this to happen, the excitation energy of the coupler, ħω res , must be much higher than the interlevel spacing in the qubits (where ω res is the resonance frequency of the coupler). In other words, the evolution of the coupler is much faster than that of the qubits, and the coupler can indeed adjust to changes in the state of the latter. In the case of a purely capacitive or purely inductive coupling this condition is automatically satisfied, as then ω res → ∞.
We derive the Hamiltonian of a superconducting circuit that comprises a single-Josephson-junction flux qubit inductively coupled to an LC oscillator, and we compare the derived circuit Hamiltonian with the quantum Rabi Hamiltonian, which describes a two-level system coupled to a harmonic oscillator. We show that there is a simple, intuitive correspondence between the circuit Hamiltonian and the quantum Rabi Hamiltonian. While there is an overall shift of the entire spectrum, the energy level structure of the circuit Hamiltonian up to the seventh excited states can still be fitted well by the quantum Rabi Hamiltonian even in the case where the coupling strength is larger than the frequencies of the qubit and the oscillator, i.e., when the qubit-oscillator circuit is in the deep-strong-coupling regime. We also show that although the circuit Hamiltonian can be transformed via a unitary transformation to a Hamiltonian containing a capacitive coupling term, the resulting circuit Hamiltonian cannot be approximated by the variant of the quantum Rabi Hamiltonian that is obtained using an analogous procedure for mapping the circuit variables onto Pauli and harmonic oscillator operators, even for relatively weak coupling. This difference between the flux and charge gauges follows from the properties of the qubit Hamiltonian eigenstates.