6. Heterodyning
6.4. Beat Node
6.4.1. Choice of Local Oscillator (LO) Frequency
The local oscillator in the heterodyne detection fulfills the main task of down-converting the probe beam into the radio frequency range, which can be recorded by standard electronic equipment. Furthermore, the detectable intensity of the down-mixed signal is proportional to the product of the probe and local oscillator amplitude, which hence permits to amplify the weak probe beam to a range so that it can be detected by a standard pin-photodiode28. In order
28 pin-diode: positive intrinsic negative diode
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0.01 0.1 1 10
spectral intensity (arb. units)
frequency (MHz) a)
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0.1
1 10 100 1000
b)
frequency - 27MHz (kHz)
Heterodyning 6.4 Beat Node
to accomplish these tasks, the choice of the best operating frequency is crucial and needs to obey certain boundary conditions. This will be eluded in this sub-section.
6.4.1.1. Scattered Light of LO
The heterodyne detection comprises a weak probe beam as well as a bright local oscillator.
While the probe beam is on the order of only a few pW, the intensity of the LO is significantly higher by a factor of about a billion. The optical setup which is required to overlap these two beams also causes a weak back-reflection of the local oscillator that impinges on the cavity.
This back-reflection cannot be attributed to a single component, but is rather due to very little reflections at each component in the optical path of the overlapped beams. Due to the grossly unbalanced intensity levels, part of the back-reflected local oscillator can enter the optical cavity and heat the atom. One way to quantify this effect is to store the atom inside the cavity and measure the probability of losing the atom by tuning the local oscillator to the frequency under study for a short time, which is on the order of a few hundred µs. Before and after each interval the local oscillator is tuned to its original frequency, which is detuned by
(
wLO-wc)
2p=27MHz to the probe beam. During this time the transmission is measured to check if the atom is well captured. This is determined by a suppression of the transmission probe beam, which is on resonance with the cavity, to 4% of its empty cavity value. This switching technique is exactly identical to the one which will be described in detail in Section 6.7 with a qualification factor of 4%. Scanning the frequency of the local oscillator during the probe intervals yields Figure 6.8.Here, a significant drop at a local oscillator frequency around the resonance frequency of the atom can be observed whereas for higher and also lower frequencies a steady value of slightly
-10 0 10 20 30 40 50 60 70
0 10 20 30 40 50 60 70
relative number of intervals with well-captured atoms (%)
(LO-c)/2 (MHz)
a/2
Figure 6.8: Heating of the atom by residual local oscillator power. The remaining number of atoms after intervals where the local oscillator is tuned to the value indicated on the x-axis is measured. The position of the atomic resonance is marked by the vertical dashed red line. The probe beam is set to be on resonance with the empty cavity, which is blue-detuned by 4 MHz to the atomic resonance. The error bars represent the 2 sigma confidence level.
6.4 Beat Node Heterodyning
below 50% is attained. The drop is explained by remnant light of the local oscillator impinging on the cavity and heating the atom. It is important to note that this light mainly enters the cavity by diffuse scattering on the coned facet of the mirrors. Coupling to the cavity mode would possess a strong dependency on any small misalignments of the local oscillator beam, which is not visible. The negative effects of back-reflection could be reduced by employing an optical faraday isolator at the detection port of the cavity. This, however, would be on the expense of probe beam intensity and would further require complex compensation of the generated magnetic field gradient. We decided to circumvent this effect by choosing local oscillator frequencies sufficiently far away from the atomic resonance so that heating effects are reduced to a minimum.
6.4.1.2. Frequency Response of Heterodyne Detection
Besides limitation in the choice of the heterodyne frequency imposed by the atomic resonance frequency, effects related to the heterodyne detection itself also need to be considered. As already mentioned in Section 6.3, noise in the LO beam in a frequency range of up to 15 MHz requires beat frequencies above this range. This way also higher LO powers can be used. For large frequency differences on the other hand, it needs to be ensured that bandwidth limitations of the digitizer do not minimize the frequency response. In order to measure the response curve, we scan the frequency difference between probe and LO, corresponding to the heterodyne frequency, while the probe is on resonance with the empty cavity. The detected probe heterodyne carrier is extracted and its magnitude is plotted as a function of the frequency difference as it is shown in Figure 6.9.
Figure 6.9: Frequency response of the heterodyne detection system. The heterodyne signal strength detected by the system is plotted as a function of its frequency. The Nyquist frequency (solid) of the digitizer as well as the -3dB bandwidth (dashed) of the low-pass filter in front of the digitizer are plotted by vertical red lines.
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0.01 0.1 1 10 100
heterodyne signal strength (arb. units)
heterodyne frequency fHd (MHz)
Nyquist frequency
3dB Bandwidth
Heterodyning 6.4 Beat Node
It can be seen that at 40 MHz the signal strength starts to drop. This is caused by the -3dB bandwidth filter at the analog input of the digitizer module as well as the bandwidth limitation of the digitizer. The latter one is given by the Nyquist frequency which corresponds to half the sampling frequency of the digitizer. From this we can deduct that the heterodyne frequency, i.e.
the frequency of the beat between probe beam and local oscillator, needs to stay below 40 MHz, which sets another limitation to the frequency choice.
6.4.1.3. Variable Beat‐Frequency
The various boundary conditions described above require using a variable heterodyne frequency which depends on the frequency of the probe beam. The best choice for the respective heterodyne frequency considering these effects is depicted in Figure 6.10. The AOM frequency of the probe beam (red) is indicated as a reference. The AOM frequency of the local oscillator (blue) is chosen such that it avoids the typical atomic resonance frequency as indicated by the horizontal, dashed line. The resulting absolute heterodyne frequency is always kept in the range between 18 MHz and 40 MHz where the heterodyne detection has a flat frequency response.
Only at probe-cavity detunings of
(
wp-wc)
2p<-30MHz this condition is weakened due to the missing tuning flexibility of the AOM which is centered at 150 MHz. The following scans unless otherwise noted are recorded with these frequency settings.Figure 6.10: Choice of the heterodyne frequency as a function of the probe beam frequency. The frequency of the local oscillator (blue) as well as the probe beam (red) is plotted as a function of the probe-cavity detuning. The resulting heterodyne frequency (green) is also plotted. The dashed horizontal line depicts the frequency where the local oscillator (and also the probe) is on resonance with the atom resonance. It is noteworthy to remember that the actual frequency difference is twice as high due to the double pass configuration of the AOM.
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6.4 Beat Node Heterodyning