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Journal of Modern Optics

Superconducting Josephson junction of finite capacitance in a lossless cavity: A plausible two-level system with a giant coupling constant

Y.B. Xie

Published Oct 1, 1998• 1 citation
Publisher

Abstract

We shall present our theoretical studies of a Josephson junction of small size in a lossless cavity. When the size of two superconductors are at order of micrometres and the Josephson coupling constant is adjusted to the correct magnitude, the Josephson junction is found to be reduced to a two-level system owing to the finite-size effect and the quenching of the low-lying single-electronic excitations by the superconducting gap. When such a two-level system is placed in a high-quality microwave cavity, the coupling constant with a single photon g may be extremely large, for instance of the order of 108 s−1. On the other hand, the damping rate of the cavity field is much less than g. Therefore, this two-level system may be used to study the quantum nature of photons and to control the dynamics of quantum fields in the framework of the Jayne-Cummings model much more easily.

DOI
10.1080/09500349808231740

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We shall present our theoretical studies of a Josephson junction of small size in a lossless cavity. When the size of two superconductors are at order of micrometres and the Josephson coupling constant is adjusted to the correct magnitude, the Josephson junction is found to be reduced to a two-level system owing to the finite-size effect and the quenching of the low-lying single-electronic excitations by the superconducting gap. When such a two-level system is placed in a high-quality microwave cavity, the coupling constant with a single photon g may be extremely large, for instance of the order of 108 s−1. On the other hand, the damping rate of the cavity field is much less than g. Therefore, this two-level system may be used to study the quantum nature of photons and to control the dynamics of quantum fields in the framework of the Jayne-Cummings model much more easily.
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 analyze the magnetic dipole coupling of an ensemble of spins to a superconducting microwave stripline structure, incorporating a Josephson junction based transmon qubit. We show that this system is described by an embedded Jaynes-Cummings model: in the strong coupling regime, collective spin-wave excitations of the ensemble of spins pick up the nonlinearity of the cavity mode, such that the two lowest eigenstates of the coupled spin wave-microwave cavity-Josephson junction system define a hybrid two-level system. The proposal described here enables new avenues for nonlinear optics using optical photons coupled to spin ensembles via Raman transitions. The possibility of strong coupling cavity QED with magnetic dipole transitions also opens up the possibility of extending quantum information processing protocols to spins in silicon or graphene, without the need for single-spin confinement.
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.
A single photon interacting with a single atom is the most fundamental form of light interacting with matter and has been extensively studied in the field of Cavity Quantum Electrodynamics (cavity QED). Here, a non-linearity like an atom is coupled to a single mode of the electromagnetic field in a cavity. Another field which explores the quantum mechanical nature of photons is Circuit Quantum Electrodynamics (cQED) where photons are the quantized excitations of a superconducting microwave resonator and non-linearity is introduced by the Josephson junction. Like this, setups analog to that of in cavity QED can be copied to cQED, with a number of differences. For example, the photons propagating in a transmission line are more confined and the circuits are made with conventional lithography techniques, allowing for more freedom in engineering the system parameters. In the first part of the thesis, we build a numerical model in order to examine the feasibility of quenching the ground state of a coplanar waveguide (CPW) interrupted by a tunable coupling element. Next, by means of experiments and simulations we considered the feasibility of observing experimentally a synchronization effect in a driven CPW with its central conductor interrupted by equally spaced capacitively shunted Josephson junctions (Josephson crystal) based on a recent proposal. Finally, we made a first step in understanding the synchronization from a classical perspective by modelling a Josephson crystal of two junctions as two degenerate non-linearly coupled Duffng oscillators. Concerning the quenching experiment, we found that the plasma frequency must be tuned faster than 1/f with f the resonance frequency of the CPW, which is in the sub-nanosecond regime and therefore unfeasible with current state the art electronics. We also found that, in contrast to what was claimed in the proposal, the synchronization effect cannot be observed for the parameters common in cQED. One way would be to push the limits of the capacitances to several picofarads. Finally, we found that the non-linear coupling causes the two degenerate non-linearly coupled Duffng oscillator to synchronize, which is a first step in understanding the proposed synchronization effect in a fully classical way.
We report a theoretical study of the macroscopic quantum dynamics in spatially extended Josephson systems. We focus on a Josephson tunnel junction of finite length placed in an externally applied magnetic field. In such a system, electromagnetic waves in the junction are excited in the form of cavity modes manifested by Fiske resonances, which are easily observed experimentally. We show that in the quantum regime, various characteristics of the junction such as its critical current Ic{I}_{c}Ic​, width of the critical current distribution $\ensuremath{\sigma}$, escape rate $\ensuremath{\Gamma}$ from the superconducting state to a resistive one, and the time-dependent probability P(t)P(t)P(t) of the escape are influenced by the number of photons excited in the junction cavity. Therefore, these characteristics can be used as a tool to measure the quantum states of photons in the junction, e.g., quantum fluctuations, coherent and squeezed states, entangled Fock states, etc.
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.