 Solution

ID: past-exam-of-the-superconductivity-and-quantum-coherence-course-of-the-university-of-cambridge/2014/paper-1/1/b/solution

∣ α ⟩ = e^{- ∣ α ∣^{2} /2} \sum_{N = 0}^{\infty} \frac{α ^{N}}{N !} ∣ N ⟩, a ∣ α ⟩ = α ∣ α ⟩ .

It has

⟨ N ⟩ = Var N = ∣ α ∣^{2}

, a selected complex amplitude, and Poisson-distributed particle number. A coherent-field condensate variational state extends this construction to a spatial field with

⟨ ψ (r)⟩ = Ψ (r)

. Evaluating a normal-ordered interacting Bose Hamiltonian in that state gives a classical-field energy functional; its variational equation is the Gross–Pitaevskii equation. This connects a macroscopic phase to interference, quantized circulation and neutral Josephson effects. A fixed-number condensate instead has vanishing one-point field expectation; off-diagonal long-range order identifies the condensate without requiring exact number fluctuations.

The BCS ground state has the pair-coherent form

∣ BCS ⟩ = \prod_{k} (u_{k} + v_{k} c_{k ↑}^{†} c_{- k ↓}^{†}) ∣0 ⟩ .

It is a product of Cooper-pair pseudospin coherent states: each paired momentum sector mixes its empty and doubly occupied configurations. The anomalous pair expectation is

⟨ c_{- k ↓} c_{k ↑} ⟩ = u_{k}^{*} v_{k}

, while an ordinary single-electron expectation vanishes by fermion parity. Pair operators are not unrestricted canonical boson operators, so this state must not be identified with a naive oscillator coherent state of individual electrons. Number projection gives a fixed-even-number description with pair correlations. Phase differences between condensates enter the Josephson effect; in a charged condensate that phase difference must be gauge invariant.

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