= Solution
An <elementary excitation> is an approximately independent low-energy normal mode above a many-body equilibrium state. A <Bogoliubov transformation> mixes creation and annihilation operators to diagonalize a quadratic fluctuation Hamiltonian. For weakly interacting bosons,
$$
E_k=\sqrt{\varepsilon_k(\varepsilon_k+2gn_0)},\qquad \varepsilon_k=\frac{\hbar^2k^2}{2m},
$$
interpolating between sound and particle-like motion. Bosonic canonical commutators require $|u_k|^2-|v_k|^2=1$. In <BCS theory>, fermionic particle–hole mixtures instead have $|u_k|^2+|v_k|^2=1$ and excitation energies $E_k=\sqrt{\xi_k^2+|\Delta_k|^2}$. These fermionic quasiparticles must be distinguished from collective amplitude and phase oscillations of the pair field.
For the stated dimensionless first-order equation, a nonzero uniform static solution obeys $(|\psi_0|^2-1)\psi_0=0$. Thus \b[$\psi_0=e^{i\theta_0}$]. The zero field is also stationary, but is not the condensed minimum of the local potential $\beta(|\psi|^2-1)^2/2$.
Choose $\theta_0=0$. To linear order, $\psi=1+\rho/2+i\theta$, so $|\psi|^2-1=\rho$. Substitution and separation of real and imaginary parts give
$$
\frac i2\dot\rho-\dot\theta=-\frac12\rho''-i\theta''+\beta\rho,
\qquad
\boxed{\dot\rho=-2\theta'',\quad \dot\theta=\frac12\rho''-\beta\rho.}
$$
Differentiating the first relation and using the second yields
$$
\ddot\rho=-\rho''''+2\beta\rho''.
$$
The same elimination holds for $\theta$. A plane wave $e^{i(kx-\omega t)}$ therefore gives
$$
\boxed{\omega^2=k^4+2\beta k^2,\qquad \omega=|k|\sqrt{k^2+2\beta}.}
$$
The density and phase components share this frequency because of <density-phase conjugacy>. They are coupled components of one positive-frequency <Bogoliubov mode>, not two independent excitations that happen to have equal energies. At $k=0$, a constant phase change is a zero mode; the nonzero-$k$ elimination should not be used to assign an independent oscillation to a uniform density change.
Replacing the first-order time derivative by $-\gamma\ddot\psi$ changes the number and structure of dynamical degrees of freedom. The linearized real and imaginary equations now decouple:
$$
-\frac\gamma2\ddot\rho=-\frac12\rho''+\beta\rho,\qquad
-\gamma\ddot\theta=-\theta''.
$$
The <amplitude and phase modes of a second-order complex field> consequently have
$$
\boxed{\omega_\rho^2=\frac{k^2+2\beta}{\gamma},\qquad
\omega_\theta^2=\frac{k^2}{\gamma}.}
$$
The amplitude mode has a nonzero gap $\sqrt{2\beta/\gamma}$; the phase mode is gapless with speed $1/\sqrt\gamma$. Frequencies are nonnegative, so the printed expression $k/\sqrt\gamma$ is understood for $k\ge0$ or replaced by $|k|/\sqrt\gamma$.
In the complex plane, an amplitude oscillation moves radially relative to the chosen equilibrium $\psi=1$, and a phase oscillation moves tangentially, or along a short unit-circle arc beyond linear order. A first-order coupled mode combines them in quadrature. For a fixed spatial factor and $k\ne0$, one possible parametrization is
$$
\rho=R\cos\omega t,\qquad
\theta=-\frac{\omega R}{2k^2}\sin\omega t,
$$
so $\delta\psi=\rho/2+i\theta$ traces a small ellipse about $1$. Choose $R$ small enough that both components remain within the linear approximation.
\Image[/past-exam-of-the-superconductivity-and-quantum-coherence-course-of-the-university-of-cambridge/2016/paper-1-amplitude-phase-modes.png]
{title=Coupled density–phase motion versus separate amplitude and phase branches}
{description=Original linearized models with $\beta=\gamma=1$. The first-order trajectory is shown at a fixed position for $k=1$; the second-order trajectories are separate possible modes, not simultaneous motions of one excitation.}
A weak neutral Bose condensate realizes the first-order structure and its sound-like <Bogoliubov spectrum>. A particle–hole-symmetric effective complex field can realize the second-order structure, with a gapless <Goldstone mode> and a distinct <amplitude mode>. A neutral BCS superfluid likewise has collective phase and amplitude responses, although the amplitude response near pair-breaking energy need not be a sharp undamped particle; its fermionic quasiparticles are separate excitations.
In a charged bulk superconductor, long-range Coulomb interactions and electromagnetic dynamics remove the neutral sound-like phase branch from the low-frequency spectrum. The <Anderson-Higgs mechanism in a superconductor> produces a longitudinal plasma mode, schematically $\omega^2\simeq\omega_p^2+c_s^2k^2$ in an appropriate long-wavelength model. The amplitude scale is typically associated with $2\Delta/\hbar$ in a simple BCS regime, with damping and mixing dependent on material and symmetry. One should not simply retain the neutral Goldstone dispersion after charging the field, nor identify the static Ginzburg-Landau functional alone with a complete time-dependent microscopic theory.
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