Using an asymptotic phase representation of the particle density operator ρ̂(z) in the one-dimensional harmonic trap, the part δ ρ̂F(z) which describes the Friedel oscillations is extracted. The expectation value 〈δ ρ̂F(z)〉 with respect to the interacting ground state requires the calculation of the mean square average of a properly defined phase operator. This calculation is performed analytically for the Tomonaga-Luttinger model with harmonic confinement. It is found that the envelope of the Friedel oscillations at zero temperature decays with the boundary exponent ν = (K + 1)/2 away from the classical boundaries. This value differs from that known for open boundary conditions or strong pinning impurities. The soft boundary in the present case thus modifies the decay of Friedel oscillations. The case of two components is also discussed.
|Journal||Journal of Physics B: Atomic, Molecular and Optical Physics|
|Publication status||Published - 14 Apr 2004|
ASJC Scopus subject areas
- Atomic and Molecular Physics, and Optics
- Condensed Matter Physics