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arXiv:cond-mat/0509570 · Physical Review Letters

Theory of Microwave Parametric Down-Conversion and Squeezing Using Circuit QED

Kyungsun Moon, S. M. Girvin

Published Sep 30, 2005• 97 citations
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Abstract

We study theoretically the parametric down-conversion and squeezing of microwaves using cavity quantum electrodynamics of a superconducting Cooper-pair box (CPB) qubit located inside a transmission line resonator. The nonlinear susceptibility chi2 describing three-wave mixing can be tuned by dc gate voltage applied to the CPB and vanishes by symmetry at the charge degeneracy point. We show that the coherent coupling of different cavity modes through the qubit can generate a squeezed state. Based on parameters realized in recent successful circuit QED experiments, squeezing of 95% approximately 13 dB below the vacuum noise level should be readily achievable.

DOI
10.1103/physrevlett.95.140504

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We study theoretically the parametric down-conversion and squeezing of microwaves using cavity quantum electrodynamics of a superconducting Cooper-pair box (CPB) qubit located inside a transmission line resonator. The nonlinear susceptibility chi2 describing three-wave mixing can be tuned by dc gate voltage applied to the CPB and vanishes by symmetry at the charge degeneracy point. We show that the coherent coupling of different cavity modes through the qubit can generate a squeezed state. Based on parameters realized in recent successful circuit QED experiments, squeezing of 95% approximately 13 dB below the vacuum noise level should be readily achievable.
We study the theory of degenerate three-wave mixing and the generation of squeezed microwaves using circuit quantum electrodynamics in solid state circuits. The Hamiltonian for degenerate three-wave mixing, which seemed to be given phenomenologically in quantum optics, is derived by quantum mechanical calculations. The nonlinear medium needed in three-wave mixing is composed of a series of superconducting charge qubits which are located inside two superconducting transmission-line resonators. Here, the multiqubit ensemble is present to enhance the effective coupling constant between the two modes in the transmission-line resonators. In the squeezing process, the qubits are kept in their ground states so that their decoherence does not corrupt the squeezing. The main obstacle preventing a large squeezing efficiency is the decay rate of the transmission-line resonator.
We describe a scheme for the efficient generation of microwave photon pairs by parametric down-conversion in a superconducting transmission line resonator coupled to a Cooper-pair box serving as an artificial atom. By properly tuning the first three levels with respect to the cavity modes, the down-conversion probability may reach the percentage level at good fidelity. We show this by numerically simulating the dissipative quantum dynamics of the coupled cavity-box system and discussing the effects of dephasing and relaxation in the solid state environment. The setup analyzed here might form the basis for a future on-chip source of entangled microwave photons, e.g., using Franson's idea of energy-time entanglement.
We propose a scheme for generating squeezed states in solid state circuits which consist a superconducting transmission line resonator (STLR), a superconducting Cooper-pair box (CPB) and a nanoelectromechanical resonator (NMR). The nonlinear interaction between the STLR and the CPB can be implemented by setting the external biased flux of the CPB at some certain points. The interaction Hamiltonian between the STLR and the NMR is derived by performing Frohlich transformation on the total Hamiltonian of the combined system. Just by adiabatically keeping the CPB at the ground state, we get the standard parametric down-conversion Hamiltonian, and the squeezed states of the STLR can be easily generated, which is similar to the three-wave mixing in quantum optics.
We propose a scheme for generating squeezed states in solid state circuits consisting of a nanomechanical resonator (NMR), a superconducting Cooper-pair box (CPB) and a superconducting transmission line resonator (STLR). The nonlinear interaction between the NMR and the STLR can be implemented by setting the external biased flux of the CPB at certain values. The interaction Hamiltonian between the NMR and the STLR is derived by performing Fro¨\rm\ddot oo¨hlich transformation on the total Hamiltonian of the combined system. Just by adiabatically keeping the CPB at the ground state, we get the standard parametric down-conversion Hamiltonian. The CPB plays the role of ``nonlinear media", and the squeezed states of the NMR can be easily generated in a manner similar to the three-wave mixing in quantum optics. This is the three-wave mixing in a solid-state circuit.
With the assistance of a cyclic three-level artificial atom of a superconducting flux quantum circuit interacting with a two-mode superconducting transmission-line resonator, we study theoretically the degenerate microwave parametric down-conversion (PDC) in a circuit-QED system. By adiabatically eliminating the excited states of the three-level artificial atom, we obtain an effective microwave PDC Hamiltonian for the two resonator modes (i.e., the fundamental and second-harmonic modes). The corresponding PDC efficiency in our model can be much larger than that in the similar circuit-QED system based on a single two-level superconducting qubit [K. Moon and S. M. Girvin, Phys. Rev. Lett. 95, 140504 (2005)]. Furthermore, we investigate the squeezing and bunching behavior of the fundamental resonator mode resulting from the coherent driving to the second-harmonic one.