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Soviet Journal of Low Temperature Physics

Radiation interaction of a system of two Josephson junctions

Wolfram Krech, Hans‐Helmut Grosse

Published May 1, 1979• 1 citation
Publisher

Abstract

Starting from the fundamental equations describing the interaction of a system consisting of two Josephson junctions with a single cavity mode, the creation of the coherent state is discussed in the framework of the adiabatic approximation. The condition of synchronization is given. Essentially, it expresses the relation between, on the one hand, the coupling intensity of the Josephson radiator and the Maxwellian field and, on the other, the separation between applied Josephson frequencies and the resonator damping.

DOI
10.1063/10.0029849

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Starting from the fundamental equations describing the interaction of a system consisting of two Josephson junctions with a single cavity mode, the creation of the coherent state is discussed in the framework of the adiabatic approximation. The condition of synchronization is given. Essentially, it expresses the relation between, on the one hand, the coupling intensity of the Josephson radiator and the Maxwellian field and, on the other, the separation between applied Josephson frequencies and the resonator damping.
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We analyze the behavior of systems of two and three qubits made by Josephson junctions, treated in the two level approximation, driven by a radiation mode in a cavity. The regime we consider is a strong coupling one recently experimentally reached for a single junction. Rabi oscillations are obtained with the frequency proportional to integer order Bessel functions in the limit of a large photon number, similarly to the case of the single qubit. A selection rule is derived for the appearance of Rabi oscillations. A quantum amplifier built with a large number of Josephson junctions in a cavity in the strong coupling regime is also described.
We have studied quantum mechanically a system of several small identical Josephson junctions in a lossless single-mode cavity for different initial states, under conditions such that the system is at resonance. This system is analogous to a collection of identical atoms in a cavity, which is described under appropriate conditions by the Dicke model. We find that our system can be well approximated by a reduced Hamiltonian consisting of two levels per junction. The reduced Hamiltonian is similar to the Dicke Hamiltonian, but contains an additional term resembling a dipole-dipole interaction between the junctions. This extra term can be understood as a natural consequence of degenerate second-order (L\"owdin) perturbation theory. For typical, physically reasonable values of the junction-cavity coupling, we find that this perturbation treatment is an adequate way to include the junction energy levels beyond the lowest two. As in the Dicke model, we find that, when N junctions are present in the cavity, the junction-cavity interaction is enhanced by N,\sqrt{N},N​, with a corresponding decrease in the Rabi oscillation period. We find that this enhancement survives even if the junctions differ slightly from one another, as expected in a realistic system. Since coherence effects thus reduce the Rabi period, it may become smaller than the decoherence time due to dissipation, making these oscillations observable.
We carry out a quantum-mechanical analysis of a small Josephson junction coupled to a single-mode resonant cavity. We find that the eigenstates of the combined junction-cavity system are strongly entangled only when the gate voltage applied at one of the superconducting islands is tuned to certain special values. One such value corresponds to the resonant absorption of a single photon by Cooper pairs in the junction. Another special value corresponds to a two-photon absorption process. Near the single-photon resonant absorption, the system is accurately described by a simplified model in which only the lowest two levels of the Josephson junction are retained in the Hamiltonian matrix. We noticed that this approximation does not work very well as the number of photons in the resonator increases. Our system shows also the phenomenon of ``collapse and revival'' under suitable initial conditions, and our full numerical solution agrees with the two level approximation result.