Josephson Plasma Research Articles

Hysteretic jumps in the response of layered superconductors to electromagnetic fields

We consider here a layered superconductor subject to an externally applied moderately strong electromagnetic field. We predict hysteretic jumps in the dependence of the surface reactance of the superconductor on the amplitude ${H}_{0}$ of the incident electromagnetic wave. This very unusual nonlinear phenomenon can be observed in thin superconducting slabs at not very strong ac amplitudes if the frequency of the irradiating field is close to the Josephson plasma frequency. Using the set of coupled sine-Gordon equations, we derive the expression for the phase shift $\ensuremath{\chi}$ of the reflected wave and obtain the conditions for the appearance of hysteresis in the $\ensuremath{\chi}({H}_{0})$ dependence.

Quantum electrodynamics and photon-assisted tunneling in long Josephson junctions

We describe the interaction between an electromagnetic field and a long Josephson junction driven by a dc current. We calculate the amplitudes of emission and absorption of light via the creation and annihilation of quantized Josephson plasma waves (JPWs). Both the energies of JPW quanta and the amplitudes of light absorption and emission strongly depend on the junction's length and can be tuned by an applied dc current. Moreover, photon-assisted macroscopic quantum tunneling in long Josephson junctions shows resonances when the frequency of the outside radiation coincides with the current-driven eigenfrequencies of the quantized JPWs.

Nonlinear electrodynamics in layered superconductors

We analyze theoretically the effect of a weak nonlinearity on the propagation of Josephson plasma waves in layered superconductors. The nonlinearity originates from the Josephson relation between the current density across superconducting layers and gauge-invariant phase difference of the order parameter. We show that strong nonlinear effects can be observed for electromagnetic waves with frequency slightly above or slightly below the plasma frequency. We study the nonlinear plasma resonance accompanied by the hysteretic dependence of the wave amplitude on the frequency. This hysteresis transforms the continuous terahertz radiation into a series of short electromagnetic high-amplitude pulses. We also consider the propagation of a nonlinear terahertz beam localized in the direction across the superconducting layers. This phenomenon is an analog of the self-focusing effect in nonlinear optics. The nonlinear phenomena in layered superconductors considered here can be potentially useful for the design of a new generation of terahertz devices.

Entanglement of Solid Vortex Matter: A Boomerang-Shaped Reduction Forced by Disorder in Interlayer Phase Coherence inBi2Sr2CaCu2O8+y

We present evidence for entangled solid vortex matter in a glassy state in a layered superconductor Bi2Sr2CaCu2O8+y containing randomly splayed linear defects. The interlayer phase coherence--probed by the Josephson plasma resonance--is enhanced at high temperatures, reflecting the recoupling of vortex liquid by the defects. At low temperatures in the vortex solid state, the interlayer coherence follows a boomerang-shaped reentrant temperature path with an unusual low-field decrease in coherence, indicative of meandering vortices. We uncover a distinct temperature scaling between in-plane and out-of-plane critical currents with opposing dependencies on field and time, consistent with the theoretically proposed "splayed-glass" state.

Quantum metamaterials: Electromagnetic waves in a Josephson qubit line

We consider the propagation of a classical electromagnetic wave through a transmission line, formed by identical superconducting charge qubits inside a superconducting resonator. Since the qubits can be in a coherent superposition of quantum states, we show that such a system demonstrates interesting effects, such as a ``breathing'' photonic crystal with an oscillating band gap and a ``quantum Archimedean screw'' that transports, at an arbitrary controlled velocity, Josephson plasma waves through a transmission line. The key ingredient of these effects is that the optical properties of the Josephson transmission line are controlled by the quantum coherent state of the qubits.

Subterahertz Josephson plasma emission in layered high-TC superconducting tunnel junctions

We investigated the emission of subterahertz frequency electromagnetic radiation from high-TC superconducting c-axis NdBa2Cu3O7−δ∕PrBa2Cu3O7−δ∕NdBa2Cu3O7−δ trilayer thin film tunneling junctions when external electric and magnetic fields are applied. The current-voltage characteristics under applied ab-plane magnetic fields H (up 8T) exhibit well defined steps, Vn, such that eVn≈hωp∕(2nπ), where the plasma frequency ωp≈0.4 THz and n=1,2,3,….. These steps may be interpreted using Josephson plasma dynamics. The applied voltage creates oscillating currents via the Josephson coupling energy EJ (EJ=hIJ∕4πe, where IJ is the Josephson current and h is Planck’s constant) and the charge energy EC (EC=e2∕2C, where C is the junction capacitance). Thus the Josephson plasma becomes excited by the tunneling current, with some of the energy being emitted as subterahertz frequency radiation. Our results provide a new insight into a solid-state quantum system with considerable potential for new solid-state terahertz emission sources.

Emission of terahertz electromagnetic waves driven by an external current in intrinsic Josephson junctions

We propose a mechanism for the strong emission of terahertz wave recently observed in current-biased mesa-shaped Bi2Sr2CaCu2O8 single crystals by Ozyuzer et al. When the mesa length is approximately equal to the c-axis penetration depth and an external current is applied along the c-axis of the mesa-crystal, the Josephson oscillation can cause the resonance excitation of cavity-mode of the transverse Josephson plasma in the voltage state. When the oscillating electric and magnetic fields excited inside the mesa have antinodes at the surfaces, the electromagnetic wave with Josephson frequency is strongly emitted into the vacuum. We evaluate the frequency and power of emitted electromagnetic wave to be respectively of the orders of THz and 103W/cm3 near the retrapping region.

C-axis coupling in underdopedBi2Sr2CaCu2O8+δwith varying degrees of disorder

The dependence of the Josephson plasma resonance (JPR) frequency in heavily underdoped ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ on temperature and controlled pointlike disorder, introduced by high-energy electron irradiation, is cross-correlated and compared to the behavior of the $ab$-plane penetration depth. It is found that the square of the zero-temperature plasma frequency, representative of the superfluid component of the $c$-axis spectral weight, decreases proportionally with ${T}_{c}$ when the disorder is increased. The temperature dependence of the JPR frequency is the same for all disorder levels, including pristine crystals. The reduction of the $c$-axis superfluid density as function of disorder is accounted for by pair breaking induced by impurity scattering in the $\mathrm{Cu}{\mathrm{O}}_{2}$ planes, rather than by quantum fluctuations of the superconducting phase. The reduction of the $c$-axis superfluid density as function of temperature follows a ${T}^{2}$ law and is accounted for by quasiparticle hopping through impurity-induced interlayer states.

Nonlinear Josephson plasma waves in slabs of layered superconductors

For layered superconductors, we study the specific nonlinear Josephson plasma waves (NJPWs) propagating along thin superconducting slabs and damping away from them. Two cases are considered, when the superconductor is surrounded by either vacuum or metals. The magnetic field of the NJPW is distributed symmetrically with respect to the middle of the sample and can change its sign inside the slab. The impedance ratio of the tangential electric and magnetic field amplitudes for NJPWs can be of the order of unity. For the case of a superconductor surrounded by the vacuum, this results in a non-monotonic dispersion relation, ω(k), strongly sensitive to the NJPW amplitudes. Therefore, the “stopping light” phenomenon can be observed at frequencies where dω(k)/dk=0. Resonance excitations of the NJPWs should produce anomalies in the amplitude dependence of the reflectivity and transmissivity of the incident THz waves, which could be useful for THz devices. Animations illustrating the results presented here, are available online at dml.riken.go.jp/nonlinear.

Analytical study of dissipative solitary waves

In this paper, the analytical solution to a new class of nonlinear solitons is presented with cubic nonlinearity, subject to a dissipation term arising as a result of a first-order derivative with respect to time, in the weakly nonlinear regime. Exact solutions are found using the combination of the perturbation and Green's function methods up to the third order. We present an example and discuss the asymptotic behavior of the Green's function. The dissipative solitary equation is also studied in the phase space in the non-dissipative and dissipative forms. Bounded and unbounded solutions of this equation are characterized, yielding an energy conversation law for non-dissipative waves. Applications of the model include weakly nonlinear solutions of terahertz Josephson plasma waves in layered superconductors and ablative Rayleigh–Taylor instability.

Nonlocal macroscopic quantum tunneling and quantum terahertz electrodynamics in layered superconductors: Theory and simulations

We derive a quantum field theory of Josephson plasma waves (JPWs) in layered superconductors, which describes two types of interacting JPW bosonic quanta (one heavy and one lighter). We propose a mechanism of enhancement of macroscopic quantum tunneling (MQT) in stacks of intrinsic Josephson junctions. Due to the long-range interaction between junctions in layered superconductors, the calculated MQT escape rate $\ensuremath{\Gamma}$ has a nonlinear dependence on the number of junctions in the stack. We develop a numerical procedure, based on quantum field tunneling theory, to calculate $\ensuremath{\Gamma}$ for the stack of Josephson junctions. We also propose a simple analytical formula to estimate the MQT escape rate. Moreover, we demonstrate that the direct analogy between fluxon tunneling and tunneling of a quantum particle fails even for very thin junction stacks (about $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$ for ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$) and a field-theoretical approach is necessary. The theory developed here allows to quantitatively describe striking recent experiments in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ stacks.

Excitation of surface Josephson plasma waves in layered superconductors

This is a theoretical study of the resonant suppression of the specular reflection of terahertz waves in layered superconductors due to the excitation of surface Josephson plasma waves (SJPWs). Here, we consider in detail the specific case of SJPW excitations by evanescent electromagnetic waves via the attenuated total reflection of incident waves in a dielectric prism. We also derive the dispersion relation for surface waves propagating along the vacuum-superconductor interface parallel to the $\mathbf{a}\mathbf{b}$ plane. We show that, due to the SJPW excitation, the reflectivity of the incident wave depends resonantly on both its frequency and incident angle. We find the optimal conditions for the best matching of the incident wave and SJPWs, as well as for the total suppression of the specular reflection.

In-plane field-induced vortex liquid correlations in underdopedBi2Sr2CaCu2O8+δ

The effect of a magnetic field component parallel to the superconducting layers on longitudinal Josephson plasma oscillations in the layered high temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ is shown to depend on the thermodynamic state of the underlying vortex lattice. Whereas the parallel magnetic field component depresses the Josephson Plasma Resonance (JPR) frequency in the vortex solid phase, it may enhance it in the vortex liquid. There is a close correlation between the behavior of microwave absorption near the JPR frequency and the effectiveness of pancake vortex pinning, with the enhancement of the plasma resonance frequency occurring in the absence of pinning, at high temperature close to the vortex melting line. An interpretation is proposed in terms of the attraction between pancake vortices and Josephson vortices, apparently also present in the vortex liquid state.

Interlayer electrodynamics and unconventional vortex state inYBa2Cu3Oy

We report on the $c$-axis magneto-optical response of $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$ ($y=6.65$ and 6.75) single crystals, with magnetic fields oriented both parallel and perpendicular to the $\mathrm{Cu}{\mathrm{O}}_{2}$ planes. The dominant characteristic of the $c$-axis electrodynamics in the superconducting state, the Josephson plasma resonance (JPR), is remarkably sensitive to fairly modest magnetic fields below $8\phantom{\rule{0.3em}{0ex}}\mathrm{T}$. Fields oriented perpendicular to the $\mathrm{Cu}{\mathrm{O}}_{2}$ planes are shown to shift the edge of the JPR and also reduce the weight of the so-called ``$400\text{\ensuremath{-}}{\mathrm{cm}}^{\ensuremath{-}1}$ mode,'' shedding light on this enigmatic feature. In the $H\ensuremath{\Vert}\mathrm{Cu}{\mathrm{O}}_{2}$ geometry, where the magnetic field initiates Josephson vortices, we observed a strong mode in the far-infrared which hardens with increasing field. The field dependence of the low-frequency resonance behavior is contrasted to that of two other cuprate materials: ${\mathrm{La}}_{2\ensuremath{-}x}\mathrm{Sr}\mathrm{Cu}{\mathrm{O}}_{4}$ compounds that we have investigated earlier, and ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8\ensuremath{-}\ensuremath{\delta}}$. Specifically, there exist disparities in the number and field dependence of longitudinal modes measured for each system. Many of these differences can be explained through a new numerical solution of the interlayer phase equations which includes effects of both in-plane and $c$-axis dissipation parameters. Support for this approach is given by calculations of the Josephson vortex lattice ground state configuration, and further insight is gained through the phenomenological framework of the transverse JPR model, as well as a classical model of vortex dynamics.

Fractional Josephson vortices: oscillating macroscopic spins

Fractional Josephson vortices carry a magnetic flux Φ, which is a fraction of the magnetic flux quantum Φ0≈2.07×10-15 Wb. We consider a fractional vortex which spontaneously appears at a phase discontinuity. Its properties are very different from the properties of the usual integer fluxon. In particular, a fractional vortex is pinned and may have one of two possible polarities – just like a usual spin 1/2 particle. The fractional vortex may also oscillate around its equilibrium position with an eigenfrequency which is expected to be within the Josephson plasma gap. Using microwave spectroscopy, we investigate the dependence of the eigenfrequency of a fractional Josephson vortex on its magnetic flux Φ and on the bias current. The experimental results are in good agreement with theoretical predictions. Positive result of this experiment is a cornerstone for further investigation of more complex fractional vortex systems such as fractional vortex molecules and tunable bandgap materials.

Nonlinear dynamics of intrinsic Josephson junctions under an applied current

Nonlinear dynamics of Josephson plasma oscillations in intrinsic Josephson junctions formed in high Tc superconductors is studied based on numerical simulations. The dynamics is described by a differential equation similar to sine-Gordon equation. We discuss within classical dynamics that the eigenmodes, well defined in linearized equation, are modified by the nonlinear effects. Especially we concentrate on the formation of the so-called “breather modes”, which gives rise to the mixing of eigenmodes.

Interlayer coherence in Bi2Sr2CaCu2O8+y with splayed columnar defects

By using the Josephson plasma resonance, we investigate the interlayer phase coherence (IPC) in the vortex states of GeV-proton-irradiated Bi2Sr2CaCu2O8+y crystals containing splayed columnar defects, which are introduced by Bi fission. At high temperatures in the vortex liquid state, we observe an enhancement of IPC in a field range beginning well below the matching field BΦ, in close correspondence with the recoupling crossover observed near BΦ/3 in the heavy-ion irradiated Bi2Sr2CaCu2O8+y. At low temperatures in the vortex solid state, a reentrant behavior of the resonance field is found in the zero-field-cooling condition, which has not been observed in pristine samples. This behavior suggests strongly suppressed IPC in this system, which may be explained by the vortex entanglement induced by the columnar defects in the splayed glass state.

Layered superconductors as nonlinear waveguides for terahertz waves

We show that unusual nonlinear self-sustained waves, called nonlinear waveguide modes NWGMs, can propagate along thin slabs of layered superconductors. We show that these waves are stable in the main approximation for extremely anisotropic superconductors if the nonlinear wave amplitude is smaller than a critical value. These modes have no analogs among linear Josephson plasma waves and do not exist in thick samples. The magnetic field of the NWGM is distributed symmetrically with respect to the middle of the slab, decays far from the sample, and can change its sign inside. The impedance ratio of the tangential electric- and magnetic-field amplitudes for NWGMs can be of the order of unity, resulting in a nonmonotonic dispersion relation, k, strongly sensitive to the NWGM amplitudes. Thus, the “stopping light” phenomenon, now controlled by the magnetic-field amplitude, can be observed. Resonance excitations of the NWGMs should produce anomalies in the amplitude dependence of the reflectivity and transmissivity of the incident terahertz waves, which could be useful for terahertz devices.

Vortex liquid correlations induced by in-plane field in underdoped Bi 2Sr 2CaCu 2O 8+ δ

By measuring the Josephson plasma resonance, we have probed the influence of an in-plane magnetic field on the pancake vortex correlations along the c-axis in heavily underdoped Bi 2Sr 2CaCu 2O 8+ δ ( T c = 72.4 ± 0.6 K) single crystals both in the vortex liquid and in the vortex solid phase. Whereas the in-plane field enhances the interlayer phase coherence in the liquid state close to the melting line, it slightly depresses it in the solid state. This is interpreted as the result of an attractive force between pancake vortices and Josephson vortices, apparently also present in the vortex liquid state. The results unveil a boundary between a correlated vortex liquid in which pancakes adapt to Josephson vortices, and the usual homogeneous liquid.

Role of pair-breaking and phase fluctuations in c-axis tunneling in underdoped Bi 2Sr 2CaCu 2O 8+ δ

The Josephson plasma resonance is used to study the c-axis supercurrent in the superconducting state of underdoped Bi 2Sr 2CaCu 2O 8+ δ with varying degrees of controlled point-like disorder, introduced by high-energy electron irradiation. As disorder is increased, the Josephson Plasma frequency decreases proportionally to the critical temperature. The temperature dependence of the plasma frequency does not depend on the irradiation dose, and is in quantitative agreement with a model for quantum fluctuations of the superconducting phase in the CuO 2 layers.