Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
and η are self-adjoint (Hermitian), their product is not, (ηQ) † = Qη �= ηQ.
But ηQ is intimately connected with its adjoint: (ηQ) † = Qη = η(ηQ)η =
η(ηQ)η −1 . This property is called pseudo-Hermiticity with respect to η
[108].
In presence of an arbitrary potential with a corresponding ground-state
density, the determination of the eigenstates is possible only numerically.
However, some analytical statements on their properties can be made.
2.5.3. Zero-frequency mode
The relation of the excitations to the ground state Φ(r) is of particular
interest. A special solution of (2.68) is given by an excitation, where both
u(r) and v(r) are proportional to the ground state, u0(r) = v0(r) = αΦ(r),
α ∈ C. This solution has zero frequency, as can be seen from (L − G)Φ = 0,
because Φ is a stationary solution of the Gross-Pitaevskii equation (2.16).
The contribution of this “excitation” to the field operator (2.67) is given
as 2iIm(α)Φ(r), i.e. it consists only in a global phase shift of the wave
function. Due to the gauge symmetry, there is no restoring force, so it is
not an excitation in the strict sense. This zero-mode is the Goldstone mode
related to the U(1) symmetry breaking of the order parameter [72, 89].
The fact that the zero-mode takes place only in the phase of the condensate
and not in the density is perfectly compatible with the observation
made in subsection 2.3.1 that the low-energy Bogoliubov-excitations oscillate
mainly in the phase and hardly in the density (figure 2.3).
2.5.4. Eigenstates of non self-adjoint operators
Before considering the proper excitations with finite frequencies, it is useful
to discuss some general properties of the eigenstates of non-Hermitian
operators [109]. Let us assume the eigenstates of a non-Hermitian operator
U �= U † to be known and investigate their orthogonality relations. In the
following, the orthonormal basis ek, which will in general not be the eigenbasis
of the operator U, will be employed. The standard scalar product
(a, b) = a † b is defined by the conventional matrix product of the adjoint
vector a † with b. The orthonormality then reads e †
jek = δj,k. Now we
consider the effect of U on the basis vectors:
50
U ek = �
j
Ujkej
U = �
j,k
ejUjke † k
Ujk = e †
jU ek (2.69)