7.4 Ramsey fringes 133as expected from the Fourier transform relationship of the frequency and
time domains.
We shall now consider what happens when an atom interacts with two
separate pulses of radiation, from timet=0toτpand again fromt=T
toT+τp. Integration of eqn 7.10 with the initial conditionc 2 =0at
t=0yields
c 2 (t)=Ω
2
∗{ 1 −exp[i(ω 0 −ω)τp]
ω 0 −ω+ exp[i(ω 0 −ω)T]1 −exp[i(ω 0 −ω)τp]
ω 0 −ω}
.
(7.51)
This is the amplitude excited to the upper level after both pulses (t>
T+τp). The first term in this expression is the amplitude arising from
the first pulse and it equals the part of eqn 7.14 that remains after
making the rotating-wave approximation.^21 Within this approximation,^21 Neglecting terms withω 0 +ωin the
interaction with the second pulse produces a similar term multiplied by denominator.
a phase factor of exp[i(ω 0 −ω)T]. Either of the pulses acting alone would
affect the system in the same way, i.e. the same excitation probability
|c 2 |^2 as in eqn 7.15. When there are two pulses the amplitudes in the
excited state interfere giving
|c 2 |^2 =∣
∣
∣
∣Ω
sin{(ω 0 −ω)τp/ 2 }
(ω 0 −ω)∣
∣
∣
∣
2
×|1 + exp[i(ω 0 −ω)T]|^2=
∣
∣
∣
∣
Ωτp
2∣
∣
∣
∣
2 [sin (δτ
p/2)
δτp/ 2] 2
cos^2(
δT
2)
, (7.52)
whereδ=ω−ω 0 is the frequency detuning. The double-pulse sequence
produces a signal of the form shown in Fig. 7.3. These are calledRamsey
fringesafter Norman Ramsey and they have a very close similarity to the
interference fringes seen in a Young’s double-slit experiment in optics—
Fraunhofer diffraction of light with wavevectorkfrom two slits of width
aand separationdleads to an intensity distribution as a function of
angleθgiven by^2222 See Section 11.1 and Brooker (2003).
I=I 0 cos^2(
1
2
kdsinθ)
sinc^2(
1
2
kasinθ)
. (7.53)
The overall envelope proportional to sinc^2 comes from single-slit diffrac-
tion. The cos^2 function determines the width of the central peak in both
eqns 7.53 and 7.52.^2323 In both quantum mechanics and op-
tics, the amplitudes of waves inter-
fere constructively, or destructively, de-
pending on their relative phase. Also,
the calculation of Fraunhofer diffrac-
tion as a Fourier transform of ampli-
tude in the plane of the object closely
parallels the Fourier transform relation-
ship between pulses in the time domain
and the frequency response of the sys-
tem.
For the atom excited by two pulses of radiation the excitation drops
from the maximum value atω=ω 0 to zero whenδT/2=π/2(orto
half the maximum atπ/4); so the central peak has a width (FWHM) of
∆ω=π/T,orequivalently
∆f=1
2 T
. (7.54)
This shows that Ramsey fringes from two interactions separated by time
T have half the width of the signal from a single long interaction of