Internal Josephson Effect Research Articles

Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects,πoscillations, and macroscopic quantum self-trapping

We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The boson Josephson junction (BJJ) dynamics is described by the two-mode nonlinear Gross-Pitaevskii equation that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of dynamical regimes for the phase difference across the junction and for the population imbalance that are not accessible with superconductor Josephson junctions (SJJ's). These include oscillations with either or both of the following properties: (i) the time-averaged value of the phase is equal to $\ensuremath{\pi} (\ensuremath{\pi}$-phase oscillations); (ii) the average population imbalance is nonzero, in states with macroscopic quantum self-trapping. The (nonsinusoidal) generalization of the SJJ ac and plasma oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and the total number of condensate atoms) onto a single universal curve for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of ${}^{3}\mathrm{He}\ensuremath{-}B$ and the internal Josephson effect in ${}^{3}\mathrm{He}\ensuremath{-}A$ are also discussed.

Coupled s-wave and d-wave states in the heavy-fermion superconductor U1-xThxBe13.

In the heavy-fermion superconductor ${\mathrm{U}}_{1\ensuremath{-}x}{\mathrm{Th}}_{x}{\mathrm{Be}}_{13}$, superconducting states coexist for thorium concentrations $0\ensuremath{\le}x\ensuremath{\le}0.06$. Assuming $s$-wave and $d$-wave symmetries for these states, we derive a Ginzburg-Landau free-energy expression which couples $s$- and $d$-wave states and is rotationally invariant, in contrast to the free-energy expression proposed by P. Kumar and P. W\"olfle Phys. Rev. Lett. 59 1954 1987. We discuss in detail the consequences that follow from our free-energy relation. In particular, we predict that in the above system there are two eigenfrequencies associated with the dynamics of phase oscillations (internal Josephson effect) which are characteristic of the $s$-wave and $d$-wave states.

Hydrodynamic description of collective modes in heavy-fermion superconductors.

We present a hydrodynamical theory for the collective modes of a p-wave superconductor that is described by an order parameter d which transforms as a vector under rotations generated by the spin angular-momentum operator J=L+S. Such superconducting order is possible in metals with cubic, hexagonal, or tetragonal crystal symmetry whose pairing electrons experience strong spin-orbit coupling. This is true of the known heavy-fermion superconductors. The variables J and d follow a Hamiltonian dynamics that results in an internal Josephson effect between the populations of spin-up and spin-down Cooper pairs. The effect is similar to the longitudinal spin-resonance behavior observed in the superfluid phases of $^{3}\mathrm{He}$. Viscous corrections to the spin angular-momentum conservation of the pairs and to the momentum conservation of the bulk are introduced, serving to couple this Josephson mode to sound propagation. The coupling, which exists for both longitudinal and transverse sound, results in a resonant absorption contribution to the sound attenuation constant with a Lorentzian shape centered at the longitudinal spin-orbit resonance frequency. The height of this Lorentzian varies as ${\ensuremath{\omega}}^{2}$ with frequency, characteristic of hydrodynamics, and its width depends upon frequency and the amount of quasiparticle scattering. It is suggested that this mechanism accounts for the sound absorption peak recently observed in the heavy-fermion superconductor ${\mathrm{UBe}}_{13}$.

Dynamic magnetism of an itinerant-electron antiferromagnet as an internal Josephson effect

The itinerant-electron antiferromagnet is a condensate of electron-hole pairs which have pair spin S=1. In chromium, the order parameter is a sinusoidal wave with spin polarisation in a single direction. This means that two equal-spin-pairing states occur with Sz=+or-1, and the Sz=0 state is missing. This corresponds to the Anderson-Brinkman-Morel state believed to describe the A phase condensate of helium three. It is shown that the dynamic magnetism of an itinerant-electron antiferromagnet such as chromium can be expressed in a formalism identical to the one used by Leggett for the condensed phases of helium three. Antiferromagnetic resonances of an itinerant-electron antiferromagnet are the same as the 'internal Josephson effect' in helium three discussed by Leggett, Maki and Tsuneto (1974), and others.

Orientational dynamics of superfluid 3He: A “two-fluid” model. I. Spin dynamics with relaxation

This is the first of two papers devoted to the dynamics of the orientation of the Cooperpair wavefunction in superfluid 3He. The present paper deals with the dynamics of the spin orientation, including relaxation; a second paper will discuss the dynamics of the l -vector (orbital dynamics) in the A or A 1 phase. The theory developed here is claimed to be approximately valid in both the A and B phases for spatially homogeneous situations at all frequencies much less than the “gap frequency” Δ h ̷ ; it embraces nonlinear as well as linear phenomena. It is believed (but not proved) that it is also valid for the A 1 phase provided that the deviation from equilibrium is not too large. The output of the theory is three simple equations of spin dynamics (Eqs. (4.20)–(4.22)) which generalize the earlier “hydrodynamic” results of one of us and may be solved for a great many cases of practical interest. The theory used is a “two-fluid” model, in which the changes in macroscopic physical quantities arising from variation of the Cooper-pair wavefunction are associated with the “superfluid” component, while those arising from changes in the statistical occupation factors are attributed to the “normal” components. We calculate the “susceptibilities” associated with the two components separately, and show that the superfluid susceptibility is proportional to (T c − T) 1 2 in the limit T → T c . The physics of the model is illustrated by considering the longitudinal dipole resonance (internal Josephson effect): We observe that in the “tunneling” between the up- and down-spin bands it is only the superfluid component which is involved, so that the result of the tunneling is to produce an imbalance between the superfluid and normal components in the two bands separately; this imbalance simultaneously relaxes by spin-conserving collisions, with some relaxation time τ( T) which is of the order of the quasi-particle lifetime. For ωτ ⪡ 1 the theory reduces to the hydrodynamic theory, but for ωτ ⪢ 1 the hydrodynamic susceptibility χ is replaced by the superfluid susceptibility χ p ; for ωτ ∼ 1 the relaxation between the superfluid and normal components gives rise to damping of the NMR. After explaining the physical picture we derive general equations of motion with the help of operator definitions of the superfluid spin, etc. A comparison of our theory with the results of microscopic calculations shows that exact agreement is obtained for longitudinal motion, linear or nonlinear, at arbitrary frequencies. For transverse motion our theory is valid to first order in ωτ; it is not exact for ωτ ≳ 1 but gives surprisingly good agreement with the microscopic results when explicit comparison is possible. It is pointed out that a simple single-relaxation-time approximation to the solution of the kinetic equation may ignore important effects, and in particular in the region of moderate magnetic fields near T c probably actually gives worse results than our phenomenological theory. Applications considered include cw resonance linewidths, recovery of the magnetization following a field step, recovery after a large-angle pulse, the ringing-down of the B-phase “wall-pinned” mode, and the general theory of nonlinear longitudinal resonance. A useful general equation for energy dissipation in the limit ωτ ⪡ 1 is also given. An Appendix discusses the “anomalous” case where susceptibility anisotropy is important.

Textures and spin currents in the B phase of3He

The effects of boundaries and magnetic fields on the textures of the B phase of liquid3He are considered. It is found that the liquid undergoes a change from an unbended into a bended state analogously to a second order phase transition. A measurement of the phase boundary would give the coefficients of the relevant free energies. Spin currents in bended states are given. It is shown, using minimization of the free energy, that spin conservation in the presence of supercurrents is satisfied by the internal Josephson effect.

Magnetic resonance and spin waves in theAphase of superfluidHe3

Extending an idea due to Leggett, that the longitudinal resonance in the $A$ phase of superfluid $^{3}\mathrm{He}$ is considered as an internal Josephson effect, we discuss the nuclear magnetic resonance as well as spin waves in the $A$ phase of superfluid $^{3}\mathrm{He}$. The spin-wave dispersion in the hydrodynamic regime is determined explicitly.

Longitudinal Nuclear Magnetic Resonance in Superfluid 3He as Internal Josephson Effect

Longitudinal Nuclear Magnetic Resonance in Superfluid 3He as Internal Josephson Effect

Nonlinear phenomena in the dynamic parallel magnetization of superfluid 3He-A

Nonlinearity in parallel ringing of magnetization in superfluid 3He-A has been observed, and is interpreted via the Maki-Tsuneto theory of an internal Josephson effect in a two-component superfluid.