 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,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 ${\psi }_0=e^{i{\theta }_0}$. 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 giveDifferentiating the first relation and using the second yieldsThe same elimination holds for $\theta$. A plane wave $e^{i(kx-\omega t)}$ therefore givesThe 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:The amplitude and phase modes of a second-order complex field consequently haveThe 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 $k\ge 0$ 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 $k\ne 0$, one possible parametrization isso $\delta \psi =\rho /2+i\theta$ traces a small ellipse about $1$. Choose $R$ small enough that both components remain within the linear approximation.

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 ${\omega }^2\simeq {\omega }_p^2+c_s^2{k}^2$ in an appropriate long-wavelength model. The amplitude scale is typically associated with $2\Delta /\hslash$ 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.