Experiments to Control Atom Number and Phase-Space Density in ...

(φl) effects into one factor, Tl = exp(−αl−iφl), where l = 0,±1 for the three frequency
components, the electric field of the probe beam after the spectroscopy cell is given by
EFM = E0
T0 e
2
iωt δ
+T1
2 ei(ω+ωm)t δ
−T1
2 ei(ω−ωm)t
+c.c.. (2.44)
The photodiode will thus record an intensity of the form
I ∝ e −2α0
e−iφ0 iωt δ
e +
To first order the signal is thus given by
2 e−i(α0−α1) −iφ1 i(ω+ωm)t δ
e e −
2 e−i(α0−α−1) 2
−iφ−1 i(ω+ωm)t
e e
. (2.45)
I ∝ e −2α0 (1+(α−1 −α1)δ cos(ωmt)+(φ1 −2φ0 +φ−1)δ sin(ωmt)). (2.46)
This form clearly shows two contributions to the overall signal: an in-phase signal pro-
portional to the difference in absorption of the low and the high-frequency sidebands,
and an out-of-phase contribution proportional to the phase differences. Mixing this sig-
nal in a lock-in amplifier with the driving frequency allows for generating an error-signal
showing either the absorption or the dispersion signal, where the phase has to be chosen
such that mixing between these two extreme cases does not occur.
The lineshape of an atomic transition is typically described by a Lorentzian.
Equation 2.46 suggests that the error signal consists of two symmetric lines centered
around ω ± ωm for the in-phase signal and of a superposition of the three dispersive
signals at ω−ωm, ω, and ω+ωm. The lineshape of an atomic transition, the associated
dispersion signal, the in-phase, and out-of-phase error signals are shown in figure 2.23.
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