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arXiv:1012.5404 · Physical Review A

Cross-Kerr-effect induced by coupled Josephson qubits in circuit quantum electrodynamics

Yong Hu, Guo-Qin Ge, Shi Chen, Xiaofei Yang, You-Ling Chen

Published Jul 25, 2011• 49 citations
arXivHTMLPDF

Abstract

We propose a scheme for implementing cross-Kerr nonlinearity between two superconducting transmission line resonators (TLRs) via their interactions with a coupler constructed by two superconducting transmon qubits connected to each other through a superconducting quantum interference device. When suitably driven, the coupler can induce very strong cross phase modulation (XPM) between the two TLRs due to its $N$-type level structure and the consequent electromagnetically induced transparency in its lowest states. The flexibility of our design can lead to various inter-TLR coupling configurations. The obtained cross-Kerr coefficient is large enough to allow many important quantum operations in which only few photons are involved. We further show that this scheme is very robust against fluctuations in solid-state quantum circuits. Our numerical calculations imply that the absorption and the dispersion of the TLRs resulting from the decoherence of the coupler are very small compared with the proposed XPM strength.

DOI
10.1103/physreva.84.012329

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We propose a scheme for implementing cross-Kerr nonlinearity between two superconducting transmission line resonators (TLRs) via their interactions with a coupler constructed by two superconducting transmon qubits connected to each other through a superconducting quantum interference device. When suitably driven, the coupler can induce very strong cross phase modulation (XPM) between the two TLRs due to its NNN-type level structure and the consequent electromagnetically induced transparency in its lowest states. The flexibility of our design can lead to various inter-TLR coupling configurations. The obtained cross-Kerr coefficient is large enough to allow many important quantum operations in which only few photons are involved. We further show that this scheme is very robust against fluctuations in solid-state quantum circuits. Our numerical calculations imply that the absorption and the dispersion of the TLRs resulting from the decoherence of the coupler are very small compared with the proposed XPM strength.
We propose a scheme to enable a controllable cross-Kerr interaction between microwave photons in a circuit quantum electrodynamics (QED) system. In this scheme we use two transmission-line resonators (TLRs) and one superconducting quantum interference device (SQUID) type charge qubit, which acts as an artificial atom. It is shown that in the dispersive regime of the circuit-QED system, a controllable cross-Kerr interaction can be obtained by properly preparing the initial state of the qubit, and a large cross-phase shift between two microwave fields in the two TLRs can then be reached. Based on this cross-Kerr interaction, we show how to create a macroscopic entangled state between the two TLRs.
Abstract We theoretically investigate the photon statistical properties of a weakly-driven circuit quantum electrodynamic system with a cross-Kerr nonlinearity, which is built from two coupled transmon qubits through a superconducting quantum interference device (SQUID). We show that when two resonators are resonant with each other, photon blockade and sub-Poissonian photon statistics in the first superconducting transmission line resonator (TLR), TLR1, can be achieved by modulating the resonator-driving detunings and the cross-Kerr nonlinearity. Moreover, the optimal conditions for obtaining sub-Poissonian statistics are put forward through analytical derivations, which are in good agreement with those obtained by numerical simulations. We also examine the role of the resonator–resonator hopping interaction for photon statistics in TLR1 by numerically analyzing the case where the frequencies of two resonators are far detuned from each other. Due to the cutoff of the indirect transition paths in this case, the output field always acquires super-Poissonian statistics. In addition, a physical model with a SQUID embedded in TLR1 is proposed to tune the detuning between two TLRs, making it possible to demonstrate the importance of the resonator–resonator hopping interaction in obtaining a single-photon blockade. Thereby, our model of two coupled resonators with cross-Kerr nonlinearity has the potential to engineer the photon statistics through operating on the system parameters.
In recent years, superconducting qubits have shown great potential in quantum computation. Hence, microwave photons become very interesting qubits for quantum information processing assisted by superconducting quantum computation. Here, we present a protocol for the entanglement concentration on microwave photons, resorting to the cross-Kerr effect in circuit quantum electrodynamics (QED). Two superconducting transmission line resonators (TLRs) coupled to superconducting molecule with the NNN-type level structure induce the effective cross-Kerr effect for realizing the quantum nondemolition (QND) measurement on microwave photons. With this device, we present a two-qubit polarization parity QND detector on the photon states of the superconducting TLRs, which can be used to concentrate the nonlocal nonmaximally entangled states of microwave photons assisted by several linear microwave elements efficiently. This protocol has a high efficiency and it may be useful for solid-state quantum information processing assisted by microwave photons.
Superconducting quantum circuits typically use capacitive charge-based linear coupling schemes to control interactions between elements such as qubits. While simple and effective, this coupling scheme makes it difficult to satisfy competing circuit design requirements such as maintaining large qubit anharmonicity and coherence along with a high degree of qubit connectivity and packing density. Moreover, tunable interactions using linear coupling elements produce dynamical variations in mode hybridization, which can induce non-adiabatic transitions, resulting in leakage errors and limiting gate speeds. In this work we attempt to address these challenges by proposing a junction-based coupling architecture based on SQUID (superconducting quantum interference device) couplers with relatively small Josephson energies. SQUID couplers provide intrinsic cross-Kerr interactions that can be controlled by external fluxes and that do not rely on mode hybridization. The small Josephson energies of the coupler maintain the interaction at a perturbative scale, which limits undesired higher-order mixing between coupled elements while achieving a sufficiently strong cross-Kerr interaction originating from diagonal coupling elements. Based on these properties, we show that a SQUID coupler can be used to implement a fast, adiabatic, and high-fidelity controlled-Z gate without introducing extra modes, and the operation is robust against junction asymmetry for high-frequency qubits. Although unconventional crosstalk may arise due to junction asymmetries and parasitic hybridization with spectator qubits, we show that these effects are sufficiently small for realistic circuit parameters. As an example of the utility of such junction-based coupling schemes, we present a scalable tiling strategy for a miniaturized superconducting quantum processor based on merged-element transmon qubits.
We provide a general method to find the Hamiltonian of a linear circuit in the presence of a nonlinearity. Focusing on the case of a Josephson junction embedded in a transmission-line resonator, we solve for the normal modes of the system by taking into account exactly the effect of the quadratic (i.e., inductive) part of the Josephson potential. The nonlinearity is then found to lead to self and cross-Kerr effects, as well as beam-splitter-type interactions between modes. By adjusting the parameters of the circuit, the Kerr coefficient KKK can be made to reach values that are weak ($K<\ensuremath{\kappa}$), strong ($K>\ensuremath{\kappa}$), or even very strong ($K\ensuremath{\gg}\ensuremath{\kappa}$) with respect to the photon-loss rate $\ensuremath{\kappa}$. In the latter case, the resonator+junction\mathrm{resonator}+\mathrm{junction}resonator+junction circuit corresponds to an in-line version of the transmon. By replacing the single junction by a SQUID, the Kerr coefficient can be tuned in situ, allowing, for example, the fast generation of Schr\"odinger cat states of microwave light. Finally, we explore the maximal strength of qubit-resonator coupling that can be reached in this setting.
Although the multi-level structure of superconducting qubits may result in calculation errors, it can be rationally used to effectively improve the speed of gate operations. Utilizing a current-biased Josephson junction (λ-type rf-SQUID) as a tunable coupler for superconducting transmission line resonators (TLRs), under the large detuning condition, we demonstrate the controllable generation of entangled coherent states in circuit quantum electrodynamics (circuit QED). The coupling between the TLRs and the qubit can be effectively regulated by an external bias current or coupling capacitor. Further investigations indicate that the maximum entangled state can be obtained through measuring the excited state of the superconducting qubits. Then, the influence of the TLR decay on the prepared entangled states is analyzed.