Past exam of the superconductivity and Quantum Coherence course of the University of Cambridge / 2016 / Paper 1 / 1

An Abrikosov vortex combines a core of suppressed superconducting order parameter, circulating electric current and a $2\pi$ winding of the quantum phase. In the bulk limit a singly wound vortex carries the superconducting flux quantum ${\Phi }_0=h/(2e)$. More precisely it is the fluxoid that is quantized; appreciable current on the integration contour prevents identifying enclosed flux alone with the integer.

For a type-II superconductor, the Ginzburg-Landau parameter is $\kappa =\lambda /\xi >1/\sqrt{2}$. The superconducting mixed state allows vortices between the lower critical magnetic field and upper critical magnetic field. A short superconducting coherence length permits a high orbital upper field,Thus surviving a large applied field generally requires accommodating many vortices, not excluding flux everywhere. The Abrikosov vortex lattice provides their equilibrium arrangement in the simplest isotropic model. Material anisotropy, disorder and thermal fluctuations can change that arrangement and its stability. Spin-polarization limits can also constrain an actual upper field, beyond this orbital estimate.

Carrying a large transport current density is a separate problem. For a straight vortex with tangent $\hat{\mathbf{t}}$, the driving Lorentz force per unit length is ${\mathbf{f}}_L=\mathbf{J}\times {\Phi }_0\hat{\mathbf{t}}$. Vortex motion produces a mean electric field $\mathbf{E}=\mathbf{B}\times {\mathbf{v}}_L$ and hence dissipative power density $\mathbf{J}\cdot \mathbf{E}$. A large $B_{c2}$ alone therefore does not imply a large usable superconducting critical current.

Flux pinning holds vortices at favorable defects: suppressing the order parameter costs less where superconductivity is already weakened. Precipitates, dislocations, grain boundaries and deliberately introduced damage can provide pinning, subject to material-specific trade-offs. A defect scale comparable with a core size can be effective; a defect that destroys connected superconducting paths is harmful. In a coarse force balance, a maximum pinning force density $F_p$ gives $J_{c}B\simeq F_p$ for transverse transport. Vortex depinning and thermally activated flux creep can produce a voltage below an ideal static threshold.

High-field magnet conductors consequently need both a large upper field and strong, persistent pinning while retaining mechanical integrity and electrical connectivity. Niobium-titanium and niobium-tin conductors illustrate why superconducting composition and microstructural processing are developed together. A critical-current measurement should specify field, temperature and voltage criterion; it is not simply a measurement of the temperature at which resistance first vanishes.

The superconducting gap function is a momentum-dependent pairing amplitude $\Delta (\mathbf{k})$, including a phase as well as a magnitude. A fully gapped isotropic state and a state with symmetry-required nodes can have different low-temperature responses. Heat capacity, thermal transport, London penetration depth and tunnelling spectra probe the low-energy quasiparticle spectrum. Activated behavior suggests a nonzero minimum gap; power laws can indicate nodes. For example, clean line nodes give a penetration-depth correction linear in temperature, whereas disorder can change the apparent power.

Such magnitude-sensitive probes require care: impurities, several bands, anisotropic full gaps and surface states can imitate or modify simple expectations. A node does not by itself establish a particular sign pattern, and the absence of a coherence peak is not a complete symmetry determination. The strongest identification combines momentum-sensitive spectra with phase-sensitive superconducting gap experiments.

For an illustrative $d_{x^2-y^2}$ state, $\Delta (\phi )\propto \cos 2\phi$ changes sign under a $90^{\circ }$ rotation. A Josephson junction to a conventional s-wave reference can turn that sign into an intrinsic phase shift. In a corner DC SQUID with junctions on orthogonal crystal faces, one branch can acquire an extra $\pi$ relative to the other. For equal junctions and negligible loop inductance,Changing ${\delta }_{\mathrm{int}}$ from zero to $\pi$ shifts the interference pattern by half a flux period. The 1993 YBCO–Pb experiment compared phase-coherent junction geometries and found evidence for that sign reversal, while explicitly discussing SQUID asymmetry and trapped vortices. Those controls matter: an arbitrary shifted trace is not by itself a proof of pairing symmetry.^[ref]

A tricrystal half-flux experiment arranges grain-boundary junctions so that a sign-changing order parameter produces an odd total $\pi$ shift around a ring. Fluxoid consistency then favors a half-integer flux state, with the detailed spontaneous flux depending on loop inductance and junction strengths. Choosing comparison geometries without that frustration and controlling trapped external flux separates the pairing-phase effect from ordinary flux trapping.^[ref]

The conclusion is strongest when the same candidate symmetry explains nodes, crystal-axis dependence and these phase-sensitive interference results. The phase-sensitive observation distinguishes a sign-changing gap from a merely anisotropic positive gap; no single temperature power law supplies that distinction.

The two-fluid model of superfluidity separates mass current into a superfluid fraction and a normal component of a superfluid:The normal component carries entropy and responds to shear viscosity; an ideal superfluid flow is entropy-free. These are interpenetrating hydrodynamic responses, not two permanently distinguishable sets of atoms. Thermal elementary excitations carry momentum relative to the condensate and generate the normal response.

The Andronikashvili experiment gives an operational picture. A closely spaced stack of disks oscillates torsionally in He-II. Viscous coupling entrains the normal response between disks, changing the oscillator's inertia, while vortex-free superfluid need not follow the rotation. The temperature-dependent inertia measures the normal component under the apparatus's coupling conditions. Counterflow also permits second sound, primarily an entropy/temperature wave rather than the ordinary density wave of first sound.

Condensate depletion is a different quantity. The condensate fraction counts occupation of a single-particle mode, while ${\rho }_s$ measures the response to phase gradients or motion. In a weak three-dimensional homogeneous Bose gas at zero temperature,so interactions produce quantum depletion even without thermal excitations. In a translation-invariant Galilean system at zero temperature, the superfluid mass density can nevertheless equal the total density. He-II has strong interaction-induced depletion but a nearly complete zero-temperature superfluid response. Lattices, disorder and finite temperature require their own response calculation; condensate and superfluid fractions should not be equated in general.

The Landau criterion follows by considering an excitation of momentum $\mathbf{p}$ in a moving fluid. Its laboratory-frame creation cost can be $ϵ(\mathbf{p})-\mathbf{v}\cdot \mathbf{p}$. A necessary energetic condition against this decay channel isFor the low-momentum Bogoliubov spectrum of a weakly interacting Bose gas, $ϵ\simeq c_{s}p$, giving the sound-speed threshold for that channel. The phonon–roton spectrum of He-II can instead have its minimum ratio near the roton region. Vortex nucleation, boundaries and a moving obstacle can lower a measured critical velocity; the ideal excitation criterion is not a guarantee of dissipation-free flow at every smaller speed in every apparatus. An ideal noninteracting Bose gas has a condensate but quadratic excitations with $v_L=0$, another distinction between condensation and robust superfluidity.

A cold-gas test moved a blue-detuned laser beam through a trapped condensate and measured heating as its speed changed. The repulsive optical obstacle supplies a controlled perturbation; strong heating appeared above a threshold. Its interpretation must account for the inhomogeneous cloud and excitation/nucleation mechanism, rather than identify every observed threshold with the central sound speed.^[ref]

A Josephson junction weakly couples two superconducting condensates. In a consistent orientation convention, its Gauge-invariant Josephson phase $\delta$ obeysThe DC Josephson effect permits a current at zero voltage; the AC Josephson effect gives oscillations at frequency $2eV/h$. The junction's energy is $-E_J\cos \delta$. Adding a capacitance gives a charging scale $E_C=e^2/(2C)$ and, in the simplest isolated-island model, Hamiltonian $4E_C(n-n_g{)}^2-E_J\cos \delta$. Competition between phase locking and charge fluctuations permits quantized circuit dynamics, rather than an ordinary linear resistor.

Two junctions in a loop form a DC SQUID. When junctions are equal and loop self-inductance is negligible, their phase difference is fixed by external flux and their currents interfere:This supplies sensitive magnetometry and flux-tunable circuit elements. Unequal junctions leave nonzero minima; finite inductance and screening change the simple pattern. The phase-sensitive gap test in the preceding alternative adds an internal pairing-phase offset to this same interference principle.

Matter-wave interference also compares coherent complex amplitudes. For overlapping condensates with local densities $n_1,n_2$,An atomic interferometer can split a trapped condensate, evolve its two components, release them and image the overlapping density by absorption of a probe beam. For two initially separated sources a distance $d$ apart, free expansion for time $t$ gives an approximate far-field fringe spacing $ht/(md)$. The measured fringe phase encodes relative phase and accumulated dynamical phase; population imbalance lowers visibility. Interference between expanding condensates was an important demonstration of atomic coherence.^[ref]

Josephson oscillation requires an ongoing weak link that exchanges particles; interference after release need not have such a link. Independent condensates can show fringes with a different relative phase in each realization, so a single image is not evidence of a reproducible phase reference over many runs. In both settings the useful object is relative phase, but superconducting circuits also couple electrically to magnetic flux and charge, while neutral-atom devices accumulate matter-wave phase from motion and external potentials.