Circuit QED with fluxonium qubits: Theory of the dispersive regime

Abstract

In circuit QED, protocols for quantum gates and readout of superconducting qubits often rely on the dispersive regime, reached when the qubit-photon detuning ∆ is large compared to the mutual coupling strength. For qubits including the Cooper-pair box and transmon, selection rules dramatically restrict the contributions to dispersive level shifts χ. By contrast, in the absence of selection rules many virtual transitions contribute to χ and can produce sizable dispersive shifts even at large detuning. We present theory for a generic multi-level qudit capacitively coupled to one or multiple harmonic modes, and give general expressions for the effective Hamiltonian in second and fourth order perturbation theory. Applying our results to the fluxonium system, we show that the absence of strong selection rules explains the surprisingly large dispersive shifts observed in experiments and leads to the prediction of a two-photon vacuum Rabi splitting. Quantitative predictions from our theory are in good agreement with experimental data over a wide range of magnetic flux and reveal that fourth-order resonances are important for the phase modulation observed in fluxonium spectroscopy.

Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics

“…We consider a Hamiltonian in which each uncoupled harmonic mode of the resonator, with resonance frequency ω m and annihilation operatorâ m , is coupled to the transition between the bare atomic states |i , |j with energiesh i ,h j through a coupling strengthhg m,i,j [24,25]. We derive such a Hamiltonian by constructing a lumped element equivalent circuit, or Foster decomposition, of the transmission line resonator as represented in Fig.…”

Circuit QED with qutrits: Coupling three or more atoms via virtual-photon exchange

“…(4) is also obtained by using RWA, the Hamiltonian does not conserve the total excitation number N T = a † a+ N q=1 (|e q e|+ 2|i q i|) [37]. Therefore, the physical process governed by this Hamiltonian can violate the conservation law of the excitation number.…”

“…(7) with respect to the renormalization form of the bare Hamiltonian. Following the recent work of Guanyu Zhu et al [37], the renormalization form of the bare Hamiltonian H 0 in the second-order perturbation theory can be given as…”

“…Therefore, the eigenstates of the Hamiltonian in Eq. (4) (with N = 3) can be well approximated by the bare states [29,37], which are the eigenstates of the bare Hamiltonian H 0 . As shown in Fig.…”

“…The occurance of this process indicates a resonant interaction among multiple atoms via the exchange of virtual photons. The physics behind this virtual photon mediated interaction is that the auxiliary third level of the cyclic qutrits allows a large number of virtual transitions which do not conserve the excitation number contribute to the effective coupling [37]. Furthermore, we have done numerical simulation with experimentally feasible parameters in circuit QED systems.…”