Dispersive Multiphoton Qubit-Oscillator Interactions for Qubit Readout

Multiphoton qubit-oscillator interactions have recently become a driving force in improving quantum gate operations and readout for quantum information implementations. For the goal of improving qubit readout by exploiting nonperturbative nonlinear qubit-oscillator interactions, the dispersive regime is employed. In this work, the dispersive regime of multiphoton ($n$-photon) qubit-oscillator interactions is analyzed using first-order perturbation theory. An effective Hamiltonian is derived where higher-order qubit-oscillator cross-Kerr terms and oscillator self-Kerr terms are obtained. The cross-Kerr terms are expressed as a product of a qubit Pauli operator and a polynomial in the oscillator number operator of degree $n$. Meanwhile, the self-Kerr terms are expressed as a polynomial in the oscillator photon number operator of degree $n-1$. In addition to the higher-order Kerr terms, for stronger qubit-oscillator coupling beyond the typical rotating-wave approximation, a qubit-conditional $2n$-photon squeezing term is obtained in the effective Hamiltonian. This theory is then used to propose a qubit readout scheme based on a dispersive two-photon interaction. The efficacy of the readout is evaluated for different ratios of the two-photon dispersive shift to the oscillator's single-photon loss rate, using signal-to-noise ratio and assignment error probability as performance metrics. The readout is compared with commonly used schemes, such as the one-photon dispersive and longitudinal readout schemes. The proposed readout is shown to be more advantageous in parameter regimes where these schemes typically excel, with an order-of-magnitude improvement in SNR observed for the same measurement time and drive strengths.