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from one laser beam an atom goes into an excited state (but only virtually since
there is strong frequency detuning) and then gets deexcited by a photon from
the other beam. By treating optical ﬁelds as classical, one can obtain eﬀective
Hamiltonian describing interaction of atoms with the laser ﬁelds
V0 † −iωt
(3.27)
(ρ e
+ ρ† q eiωt )
−
2q
Fermi’s golden then gives the rate with which excitations are created in the
system (this is linear response theory and applies only for exciting a relatively
small number of atoms)
Veﬀ = W = V02 S (q, ω ) (3.28) What is being measured in experiments is the number of atoms excited into a
state with ﬁnite momentum as a function of wavevector and frequency diﬀerences of the two laser beams (see ﬁgs 3.1 and 3.2).
To apply formulas (3.25), (3.27) to the BEC we write ρ† in equation (3.26)
q
using creation and annihilation operators of the Bogoliubov quasiparticles. The
1 /2
leading term in N0
1
ρ† = √ (b0 b† + b† b−q ) =
q
q
0
V N0
V 1/2
†
(uq − vq )(αq − α...
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 Fall '10
 EugeneDemler
 Physics, BoseEinstein condensation, Bogoliubov Theory, Bogoliubov quasiparticles

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