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arXiv:cond-mat/9909380 · Physical review. B, Condensed matter

Model for a Josephson junction array coupled to a resonant cavity

J. Kent Harbaugh, D. Stroud

Published Jun 1, 2000• 15 citations
arXivHTMLPDF

Abstract

We describe a simple Hamiltonian for an underdamped Josephson array coupled to a single photon mode in a resonant cavity. Using a Hartree-like mean-field theory, we show that, for any given strength of coupling between the photon field and the Josephson junctions, there is a transition from incoherence to coherence as a function of N, the number of Josephson junctions in the array. Above that value of N, the energy in the photon field is proportional to ${N}^{2},$ suggestive of coherent emission. These features remain even when the junction parameters have some random variation from junction to junction, as expected in a real array. Both of these features agree with recent experiments by Barbara and co-workers.

DOI
10.1103/physrevb.61.14765

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Topics in the physics of underdamped Josephson systems

I. Tornes

We describe a variety of models to characterize the behavior of several types of underdamped Josephson systems.We also present numerical results based on these models.We first discuss a stack of small junctions in a single mode resonant cavity.Such a stack is believed to be appropriate to model some high temperature cuprate superconductors such as Bi 2 Sr 2 CaCu 2 O 8+x .Our results show self-induced-resonant-steps in the current voltage (IV) curve.These steps represent a coherent phase locking between the junctions in the stack and the resonant cavity.Next we generalize these results to treat a single long junction coupled to a single mode resonant cavity.We find a number of features not seen in stacks of small junctions.Our numerical results show that if the electromagnetic fields of the cavity mode are spatially periodic, we again see steps in the IV curve and furthermore, for sufficiently large coupling, the cavity frequency is shifted by coupling to the long junction.We also discuss how a stack of long junctions could be used to model a layered high temperature superconductor.We then turn to some properties of single junctions in the "quantum" regime where the phase/number non-commutation is important.This quantum regime is crucial to all proposals to use such junctions as solid state quantum bits (qubits).We show that such qubits, when subjected to external a.c.perturbations, exhibit not only . . . . . . . . . .

•Jan 1, 2006•0 citations
We describe a simple Hamiltonian for an underdamped Josephson array coupled to a single photon mode in a resonant cavity. Using a Hartree-like mean-field theory, we show that, for any given strength of coupling between the photon field and the Josephson junctions, there is a transition from incoherence to coherence as a function of N, the number of Josephson junctions in the array. Above that value of N, the energy in the photon field is proportional to N2,{N}^{2},N2, suggestive of coherent emission. These features remain even when the junction parameters have some random variation from junction to junction, as expected in a real array. Both of these features agree with recent experiments by Barbara and co-workers.
We derive dynamical equations for a Josephson array coupled to a resonant cavity by applying the Heisenberg equations of motion to a model Hamiltonian described by us earlier [Phys. Rev. B 63, 144522 (2001); 64, 179902(E) (2001)]. By means of a canonical transformation, we also show that, in the absence of an applied current and dissipation, our model reduces to one described by Shnirman et al [Phys. Rev. Lett. 79, 2371 (1997)] for coupled qubits, and that it corresponds to a capacitive coupling between the array and the cavity mode. From extensive numerical solutions of the model in one dimension, we find that the array locks into a coherent, periodic state above a critical number of active junctions; that the current-voltage characteristics of the array have self-induced resonant steps (SIRS's); that when Na{N}_{a}Na​ active junctions are synchronized on a SIRS, the energy emitted into the resonant cavity is quadratic in Na;{N}_{a};Na​; and that when a fixed number of junctions is biased on a SIRS, the energy is linear in the input power. All these results are in agreement with recent experiments. By choosing the initial conditions carefully, we can drive the array into any of a variety of different integer SIRS's. We tentatively identify terms in the equations of motion which give rise to both the SIRS's and the coherence threshold. We also find higher-order integer SIRS's and fractional SIRS's in some simulations. We conclude that a resonant cavity can produce threshold behavior and SIRS's even in a one-dimensional array with appropriate experimental parameters, and that the experimental data, including the coherent emission, can be understood from classical equations of motion.
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.
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 study the radiative properties---the Lamb shift, Purcell decay rate, and spontaneous emission dynamics---of an artificial atom coupled to a long, multimode cavity formed by an array of Josephson junctions. Introducing a tunable coupling element between the atom and the array, we demonstrate that such a system can exhibit a crossover from a perturbative to a nonperturbative regime of light-matter interaction as one strengthens the coupling between the atom and the Josephson-junction array. As a consequence, the concept of spontaneous emission as the occupation of the local atomic site being governed by a single complex-valued exponent breaks down. This breakdown, we show, can be interpreted in terms of formation of hybrid atom-resonator modes with radiative losses that are nontrivially related to the effective coupling between individual modes. We develop a singular function expansion approach for the description of the open quantum system dynamics in such a multimode nonperturbative regime. This modal framework generalizes the normal-mode description of quantum fields in a finite volume, incorporating exact radiative losses and incident quantum noise at the delimiting surface. Our results are pertinent to recent experiments with Josephson atoms coupled to high-impedance Josephson-junction arrays.
We report measurements of mm-wavelength radiation from two-dimensional arrays of underdamped Josephson junctions. All of our samples emit coherently in a novel synchronized state, which is triggered by a resonance in the array structure. Measurements of the detected power as a function of the number NNN of active junctions show a threshold, suggesting population inversion. Above threshold, the power scales with N2{N}^{2}N2 up to an array size bigger than the free-space radiation wavelength. The highest measured conversion efficiency from dc to ac power is about 17%. Our data are consistent with stimulated emission causing coherence and cannot be explained by existing classical coupling mechanisms.
We describe a variety of models to characterize the behavior of several types of underdamped Josephson systems.We also present numerical results based on these models.We first discuss a stack of small junctions in a single mode resonant cavity.Such a stack is believed to be appropriate to model some high temperature cuprate superconductors such as Bi 2 Sr 2 CaCu 2 O 8+x .Our results show self-induced-resonant-steps in the current voltage (IV) curve.These steps represent a coherent phase locking between the junctions in the stack and the resonant cavity.Next we generalize these results to treat a single long junction coupled to a single mode resonant cavity.We find a number of features not seen in stacks of small junctions.Our numerical results show that if the electromagnetic fields of the cavity mode are spatially periodic, we again see steps in the IV curve and furthermore, for sufficiently large coupling, the cavity frequency is shifted by coupling to the long junction.We also discuss how a stack of long junctions could be used to model a layered high temperature superconductor.We then turn to some properties of single junctions in the "quantum" regime where the phase/number non-commutation is important.This quantum regime is crucial to all proposals to use such junctions as solid state quantum bits (qubits).We show that such qubits, when subjected to external a.c.perturbations, exhibit not only . . . . . . . . . .