Past exam of the superconductivity and Quantum Coherence course of the University of Cambridge / 2012 / Paper 1 / 1

A bosonic coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$ and has expansion $e^{-|\alpha {|}^2/2}{\sum }_n{\alpha }^n|n⟩/\sqrt{n!}$. Its occupation is Poisson distributed, with mean and variance $|\alpha {|}^2$. Its nonzero field amplitude makes phase coherence natural, although physically observable phase comparisons require a reference.

For paired fermions, the corresponding Cooper-pair pseudospin coherent state isIt coherently superposes empty and occupied pair states and has an anomalous average $⟨c_{-\mathbf{k}\downarrow }{c}_{\mathbf{k}\uparrow }⟩\ne 0$. It is not an ordinary complex-eigenvalue coherent state of a single fermion annihilation operator. The broken-symmetry state has indefinite particle number; number projection can restore a fixed-number description without discarding the underlying pairing correlations.

Hartree-Fock-Bogoliubov theory approximates a two-body interaction using both the normal density ${\rho }_{ij}=⟨c_j^{†}{c}_i⟩$ and anomalous pairing tensor ${\kappa }_{ij}=⟨c_j{c}_i⟩$. Direct and exchange contractions determine a normal one-particle field, while anomalous contractions determine the pair potential. The resulting quadratic Hamiltonian is diagonalized by a fermionic Bogoliubov transformation. In a homogeneous singlet case its Nambu block isThe fields are determined self-consistently, rather than selected independently of the quasiparticle state. For example,together with the particle-number equation. The pairing field extends ordinary Hartree-Fock theory to states with pair correlations. This mean-field treatment omits fluctuations beyond its self-consistent quadratic description.

For bosons, write the field as a coherent condensate plus fluctuations, $\hat{\psi }=\varphi +\delta \hat{\psi }$. The same normal/anomalous strategy retains $\tilde{n}=⟨\delta {\hat{\psi }}^{†}\delta \hat{\psi }⟩$ and $\tilde{m}=⟨\delta \hat{\psi }\delta \hat{\psi }⟩$, with a bosonic, commutation-preserving Bogoliubov transformation. In the leading weakly depleted contact-gas limit, the condensate density $n_0$ and effective coupling $g>0$ give $E_k=\sqrt{{ϵ}_k({ϵ}_k+2g{n}_0)}$, with ${ϵ}_k={\hslash }^2{k}^2/(2m)$. This Bogoliubov theory connects low-momentum sound to high-momentum particle excitations. More elaborate self-consistent truncations must respect the symmetry constraints needed for a gapless condensate mode.

The electron-phonon interaction can produce a retarded attraction. A carrier distorts the ionic lattice; a second carrier can lower its energy in that distortion. Integrating out a phonon gives a frequency-dependent interaction with an attractive low-frequency region. A schematic Matsubara-frequency contribution is $V_{\mathrm{ph}}(\mathbf{q},i{\omega }_n)\propto -|g_{\mathbf{q}}{|}^2{\omega }_{\mathbf{q}}/({\omega }_n^2+{\omega }_{\mathbf{q}}^2)$. It must be combined with the screened Coulomb repulsion, not treated as a statement that the bare electron-electron force is attractive.

Retardation separates the electronic and phonon energy scales and reduces the repulsive contribution in the pairing channel to a Coulomb pseudopotential. In the simplest weak-coupling approximation, a positive net dimensionless pairing strength givesThe formula assumes a phonon cutoff and simplified coupling; strong-coupling calculations need the spectral interaction rather than this one-number replacement. The phonon frequency's dependence on ionic mass motivates an isotope effect, whose actual exponent can be modified by other physics.

An interaction need not be attractive in every momentum channel to support pairing. Spin fluctuations or other electronic correlations can favor a sign-changing gap: in the gap equation, repulsive scattering between two regions can contribute constructively when their gap signs differ. Such mechanisms are candidates for unconventional superconductivity. Observing a d-wave gap identifies symmetry information but does not, by itself, establish a unique microscopic pairing mechanism in a cuprate.

Write the condensate order parameter as $\Psi =|\Psi |e^{i\theta }$. Away from vortex cores, single-valuedness around a closed loop gives $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$. For neutral condensed objects of mass $M$, ${\mathbf{v}}_s=\hslash \nabla \theta /M$, hence the quantized circulationHere $M$ is the atomic mass for a simple bosonic atomic condensate, or the pair mass for a paired-fermion condensate. A nonzero winding can surround a vortex core where the order-parameter amplitude falls to zero. The result concerns collective condensate phase winding, not separate orbit quantization of every constituent particle.

For charge $q$ per condensed object, the gauge-invariant relation is $M{\mathbf{v}}_s=\hslash \nabla \theta -q\mathbf{A}$. Integrating givesThis is fluxoid and flux quantization in a superconductor. With Cooper-pair charge magnitude $2e$, the flux quantum's magnitude isThe orientation fixes its sign. A loop taken deep in a thick superconductor has negligible superflow and therefore encloses an integer flux quantum. In a thin loop or one with appreciable screening current, it is the fluxoid, including the circulation term, that is quantized; the magnetic flux alone need not be exactly an integer multiple.

A Josephson junction weakly couples two superconducting condensates. Its coupling energy is $-E_J\cos \phi$, with a gauge-invariant phase difference across the junction. Choose current and voltage orientations consistently so thatThe DC Josephson effect permits a stationary zero-voltage supercurrent up to $I_c$. A constant nonzero voltage produces the AC Josephson effect, with frequency $f=2e|V|/h$. Resistive and capacitive currents are additional circuit contributions, not part of the ideal sinusoidal supercurrent alone.

In a symmetric DC SQUID with negligible loop self-inductance, the two junction phases differ by $2\pi \Phi /{\Phi }_0$. Adding their currents and maximizing over the common phase givesFlux changes therefore modulate the critical current or a suitably biased output voltage. This converts phase interference into sensitive flux measurement. Finite loop inductance, unequal junctions, noise and readout conditions alter the simple modulation; the loop flux then includes screening currents. A SQUID is useful in magnetometry, but achieving sensitivity requires both a suitable circuit and controlled environmental noise.