2. Two-photon Direct Frequency Comb SpectroscopySpectroscopy
2.4. Two-Photon Frequency Comb Spectroscopy
To derive the steady state solutions for the frequency comb two-photon spectroscopy, we consider two equal counter propagating fields, which have the same carrier fre-quency ω0 and wave number k0 and are polarized along the z-axis.
E(t, z) = 1 2
E~01E(t+z
c)e−i(ω0t+k0z)+E~02E(t+z
c)e−i(ω0t−k0z)
+ c.c. , (2.30) with the normalized time-dependent field envelope E(t±zc) and electric field vectors E~0i. For a Gaussian pulse train we obtain:
E(t± z
c) = 1
√τ
4
s2 π
∞
X
m=−∞
e−(t±z/c−nTrep)2/τ2 (2.31) HereT = 2π/ωr is the repetition period. The field 1/e–pulse duration τ is related to the FWHM intensity pulse duration over τ1/2 = q2 ln(2)τ. We omitted for clarity other possible important characteristics of the pulse such as chirp and the transverse spatial dependence, which easily can be included into a numerical simulation. Here,
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2.4. Two-Photon Frequency Comb Spectroscopy
we are interested only in the general features of the solution. The Fourier transform of a Gaussian pulse envelope is given by the following expression.
E˜(mωr) =
√τ ωr
√4
2π
∞
X
m=−∞
e−(mωrτ)2/4−imωrz/cδ(mωr) (2.32) Thus the pulse train can be written as a sum of comb modes.
E(t, z) =E01√ τ ωr 2√4
2π
∞
X
m1=−∞
e−(m1ωrτ)2/4−i(ω0+m1ωr)t−ik0z−im1ωrz/c +E02
√τ ωr
2√4 2π
∞
X
m2=−∞
e−(m2ωrτ)2/4−i(ω0+m2ωr)t+ik0z−im2ωrz/c+ c.c.
(2.33)
Each pair of modes, which satisfies the resonance conditionωeg = 2ω0+ (m1+m2)ωr, contributes to the excitation rate. Due to the periodicity of the frequency comb, the transition will repeat itself for any multipleµωr of the repetition rate, however with reduced amplitude.
A(ω)∝e−((m−µ)ωrτ)2/2e−((m+µ)ωrτ)2/2 =e−(µωrτ)2 (2.34) Figure 2.5 shows the principle of the two-photon frequency comb spectroscopy. The spectrum of a frequency comb is depicted in the lower plot, with individual mode pairs contributing to the resonance transitions at ωeg+µωr.
The two-photon time-independent matrix element βeg can be easily extended for several different frequencies. For instance, for a frequency comb,βge calculates with the following expression.
βeg = e2 2hc0
τ ωr
√4π
∞
XZ
m=−∞
XZ
n
he|z|ni hn|z|ei
ωng−ω0 −ωrme−(mωrτ)2/2 (2.35) Here the sum runs over all mode number m. For a narrow frequency comb with spectral width (ω0 +mωr ≈ ω0) much smaller than the carrier frequency ω0, all terms in the sum are approximately equal. For a large number of modes, the sum can be approximated by an integral, such that eq. 2.35 reduces to the simple single frequency expression in eq. 2.15 for βeg. However care must be taken, if one of the comb modes is close to resonance to any of the intermediate states|ni. In this case βeg will be different and those terms have to be calculated separately. As explained above, this is not the case for the 1S–3S spectroscopy with a picosecond comb. The Rabi frequency Ω∝E1(t, z)E2(t, z) depends on the position of the atom. To see this, let the atom’s position be atz, where zero is the center of the pulse collision volume (PCV), the region where counter propagating pulses overlap (cf. fig. 3.9). For the Rabi frequency, assuming Gaussian pulses, we obtain the following dependence.
Ω(z)∝e−(z
0−z)2/(cτ)2
e−(z
0+z)2/(cτ)2 = Ω(0)e−2z2/(cτ)2 (2.36)
2. Two-photon Direct Frequency Comb Spectroscopy
Figure 2.5.:The principle of two-photon frequency spectroscopy is shown. For clarity only one frequency comb spectrum is displayed. All mode pairs, which satisfy the resonance condition, contribute to the total excitation. Due to the periodicity of the frequency comb, transitions appear at multiples of the repetition rate with an amplitude proportional to the square of the comb intensity at this frequency. In the right upper corner, an energy diagram for two-photon frequency comb excitation is shown.
Figure 2.6.:The dependence of the Rabi frequency on the position within the pulse collision volume for two-photon frequency comb spectroscopy is shown. At a position z0 = z the pulses pass the the atom asymmetrically, leading to a position dependence of Ω(z) ∝ e−2z2/(cτ)2.
Where Ω(0) is the Rabi frequency for an atom at the center of the pulse collision volume. Figure 2.6 illustrates the position dependence of the Rabi frequency. The steady state solution can be calculated using the CW solution in eq. 2.25, the linearity of the optical Bloch equations 2.43 and the dependencies on the position of the atom z in eq. 2.36 and the comb envelope detuning eq. 2.34. The Doppler free count rate for two-photon frequency comb excitation is therefore given by the
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2.4. Two-Photon Frequency Comb Spectroscopy
following sum.
RDFF C = ωrτ
√4π
∞
X
−∞
e−(mωrτ)2/2ΓΩ2/4
(ωeg−2ω0−2mωrvz/c)2 + Γ2/4e−4z20/(cτ)2e−(µωrτ)2 (2.37) Each resonant pair of modes produces a Doppler shifted Lorentzian line (∆ω = 2nωrvz/c). However, for any pair of the modes m = n there is always an identical line with m = −n, such that the overall line is only Doppler broadened, but there is no net shift. The total broadening is proportional to the width of the frequency comb and is the frequency domain description of the well known Time-of-flight bro-adening. It amounts to ΓT OF = 8 ln(2)v/cτ. For a large number of modes the sum is well approximated by an integral over the mode numberm. This is a convolution between a Lorentzian and a Gaussian, which upon integration results in a Voigt-like line shape. However for cryogenic temperatures and thus slow atoms, we can ap-proximately include the broadening into the natural line width, assuming that the simple Lorentzian line shape is preserved.
RDFF C = ΓΩ2/4
(ωeg−2ω0)2+ (Γ + ΓT OF)2/4e−4z20/(cτ)2e−(µωrτ)2 (2.38) Thus, if the Time-of-flight broadening is small compared to the natural line width Γ and additionally the central mode is on resonance with the transition frequency (µ = 0) and the atom is at the center of the pulse collision volume (z = 0), then the frequency comb two-photon Doppler-free count rate is the same as for the CW spectroscopy with the same average power.
The second order AC Stark shift for two-photon frequency comb excitation is analogous to the CW case in eq. 2.18 given by the following expression.
∆EACφ (t) = e2
~ ωrτ
√4π
∞
X
m=−∞
X
n,±
hφ|zˆ|ni hn|zˆ|φi
ωng±(ω0+mωr)e−(mωrτ)2/2E02 (2.39) It is linear in intensity and if the spectral band width of the comb is small compared to the carrier frequency and no resonant intermediate transitions are present, the sum over the mode number m can be approximated by an integral and we obtain the same AC Stark shift (cf. eq. 2.20) as for the CW spectroscopy with the same average intensity. The fourth order AC Stark shift for the frequency comb case is discussed in section B and constitutes a negligibly small correction for the 1S-3S comb spectroscopy of this work.
The Doppler-broadened count rate for frequency comb spectroscopy is different than for CW spectroscopy in several aspects. First, the spectral width of a frequency comb is typically much larger than CW Doppler width (2 ps comb, ∆ω ≈ 2π × 100 GHz ωD ≈ 2π ×3GHz), such that Doppler broadened line is given by the width of the frequency comb rather than the velocity distribution, as it is the case for the CW spectroscopy. This is advantageous as the Doppler broadened count rate
2. Two-photon Direct Frequency Comb Spectroscopy
is even less frequency dependent, reducing a possible asymmetric shift. Note, that due to the two-photon nature of the transition, the line width is proportional to the square of the intensity and thus the width of the comb reduces by a factor of √
2, assuming a Gaussian comb. Second, each velocity class is excited by the frequency comb as efficient as with the CW laser, but due to its large spectral width, a comb can simultaneously talk to many velocity classes, thus increasing the Doppler-broadened count rate by approximately the number of modes within the spectral bandwidth 2ωD/ωr(roughly 40 for 1S–3S transition). Therefore, the contrast of the Doppler-free to Doppler-broadened count rates is approximately given by the following expression.
Ccomb = 4kvp Γ√
π ωr
2ωD = ωr Γqπln(2)
(2.40) As in the CW case this equation does not take into the account the polarization dependent excitation of other allowed fine- and hyperfine components, which can also be resonant. Figure 2.4 shows the total theoretical and experimental line shapes of the 1S-3S transition in hydrogen for both CW and frequency comb excitation, which includes the polarization as well as detector properties (T = 7 K, main detector cf.
section 3). Third, since the width of the Doppler broadened line is given by the spectral width of the comb, both DB lines from counter propagating beams will be almost centered at the transition frequency, which is a disadvantage as compared to the CW case, where the use of an atomic beam significantly reduces the Doppler background and thus improves the contrast. Finally, as explained above, in the case of frequency comb excitation, the Doppler-free fluorescence is emitted only within the tiny region, where the pulses overlap. The Doppler-broadened fluorescence is emitted across the entire atomic beam and thus can be collected outside of the pulse collision volume using an independent detector. It than can be used as an almost perfect normalization signal. This is explained in detail in section 4.2. This normalization signal is not present in the CW case, as the Doppler-free and Doppler-broadened fluorescence can not be separated. In the next section, the Monte Carlo simulation for the 1S–3S experiment is explained.
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