  When trapped atoms are illuminated by weak lasers, off-resonant transitions
cause shifts in the frequencies of the vibrational-sideband resonances. These
frequency shifts may be understood in terms of Stark-shifts of the individual
levels or, as proposed here, as a vibrational Bloch-Siegert shift, an effect
closely related to the usual (radio-frequency or optical) Bloch-Siegert shift
and associated with rapidly oscillating terms when the Rotating Wave
Approximation is not made. Explicit analytic expressions are derived and
compared to numerical results, and the similarities and differences between the
usual and the vibrational Bloch-Siegert shifts are also spelled out.
