5. Control of Atomic Motion
5.3. Measuring Atomic Motion
5.3.1. Correlation Measurement
The position of the atom determines its overlap with the probe-field, which in turn leads to a change in the effective coupling and hence alters the normal-mode spectrum. Since the frequency of the probe beam is fixed, variations in the normal-mode spectrum also cause a change in the transmitted probe beam.
The intracavity intensity of the probe-field (similar to Eqn. (5.1)) is given by:
( )
( )( )
We introduce z0 as the axial displacement between the antinode of the probe- and trap-field.
The effective coupling geff is proportional to the intensity of the probe-field at the position of the atom. Its behavior in axial and radial direction is plotted as dark blue line in Figure 5.3. For simplicity, z0=0. We assume a probe-cavity detuning of
(
wp-wc)
2p=0MHz and an effective atom-cavity detuning of(
wa-wc)
2p= -4.6MHz. The effective detuning includes a position dependent ac-Stark shift as it is caused by the trap depth of kB⋅850µK . In general, this configuration leads to a suppression of transmission, when the atom is well coupled, while the transmission through the cavity increases as the coupling of the atom decreases. Using Eqn.(2.33), the exact relative transmission of the probe beam compared to the empty-cavity case can be computed and is shown in the same graph as red line. Typical length scales, i.e. the waist
5.3 Measuring Atomic Motion Control of Atomic Motion
0 and /8 are indicated by vertical dashed lines. The position r=0 ,z=0 corresponds to the center of the antinode in the middle of the cavity.
Figure 5.3: Effective coupling geff (blue) and relative transmission of the probe beam (red) as a function of the radial (r) and axial (z) distance from an antinode of the intracavity probe field. Typical length scales, as the waist 0 and the wavelength , are marked by vertical dashed lines in the respective plots. The breakdown of the harmonic approximation is marked by vertical dotted lines.
Information about the dynamics of the atomic motion can be derived from a periodic pattern of the transmitted probe beam. One way of detecting this pattern consists in measuring correlation functions. Here, the second order correlation function g(2)() from data recorded by the SPCM is employed. The second order correlation function is a measure of how many photons arrive in pairs with a detection time difference of t. As it can be seen from Figure 5.3, the probability to detect a photon passing through the cavity becomes larger as the atom has a larger excursion from the center of the antinode, due to the decreasing coupling and thus increasing transmission.
An atom which is oscillating in the trap produces a transmission signal with peaks at its turning points. This subsequently leads to a modulation of the photon-stream at twice the atom’s oscillation frequency and hence, a high probability to detect both photons at the turning point or the second after half an oscillation period of the atom at the other turning point. The correlations have been measured for atoms stored in the cavity. The feedback, which will be described in Section 5.5, was switched off and the probe-laser was on resonance with the cavity as described before.
0 5 10 15 20 25
0.00 0.25 0.50 0.75 1.00
0 30 60 90 120 150 180 0 4 8 12 16
relative transmission
r (µm)
geff/
transmission
0
effective coupling g eff/ (MHz)
z (nm)
Control of Atomic Motion 5.3 Measuring Atomic Motion
Figure 5.4: Second order correlation function recorded for different probe powers. The graphs for different probe powers are stacked by a vertical offset of 0.25. The correlation time is plotted on the lower x-axis. The frequency corresponding to an oscillation with a period as indicated by the time on the lower axis, is given on the upper axis. This directly reflects the modulation frequency of the transmitted probe beam at the respective peak.
The correlation measurements obtained for different probe-powers are depicted in Figure 5.4.
A clear peak around 150 µs is visible, corresponding to a modulation between 6 and 7 kHz. The modulation is strongest for high intracavity photon numbers and becomes hardly visible if the number is reduced to as low as 0.01 intracavity photons on the empty cavity resonance. This dependency shows that high probe-power causes momentum diffusion which leads to an increase of the oscillation amplitude. As expected, the position of the peak, which is mainly determined by the depth of the dipole trap, remains almost unchanged. Typical probe-powers corresponding to 0.1 intracavity photons for an empty cavity (yellow) are employed during an experimental scan. In this regime two oscillation bumps are observed. The number of bumps portends to the coherence of the atomic oscillation. A low value thus indicates a low quality-factor of this oscillation. The detected modulation corresponds to twice the radial oscillation frequency of the atom, which hence amounts to frad =3.4 kHz. This value is clearly below the theoretically expected value of 4.8 kHz for the harmonic oscillation frequency computed in Section 5.2. A discussion of the discrepancy is given in Section 5.4.
Correlation measurements can also be used to detect the fast, axial component of the atomic oscillation. The two orders of magnitude smaller oscillation period in combination with their small modulation of the probe beam make them more difficult to observe. To counteract this,
0 100 200 300 400 500 600
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
g2 ()
(µs)
Intra-cavity photon number 0.190 0.038 0.114 0.019 0.095 0.010
10 8 7 6 5 4 3 2
1/ (kHz)
5.3 Measuring Atomic Motion Control of Atomic Motion
the duty-cycle of our system, i.e. the time an atom is captured in the cavity vs. the time it takes to load the atom, is increased by performing feedback cooling in the radial-direction. This so-called bang-bang feedback will be described in detail in Section 5.5.2 and shall here only be seen as a tool to increase the storage time. By doing so, the average storage time is increased by almost a factor of 50. The recorded correlation is plotted in Figure 5.5.
Figure 5.5: Second order correlation function recorded for the transmitted probe beam. As before, the correlation time is plotted on the lower x-axis with the corresponding frequency upper axis. Multiple peaks are visible with the first one being slightly below 500 kHz.
In this measurement the probe power was set to 0.11 intracavity photons on the empty cavity resonance. The trap power was set to 950 nW and only lowered to 400 nW for very short time intervals, if required by the outcome of the feedback. A clear modulation at 450 kHz is visible.
This corresponds to an axial atomic oscillation of faxial=225 kHz. This value is more than a factor of two away from the expected value of 520 kHz. The radial feedback causes a slight decrease in the average trap depth, which only lowers the expected value slightly. A different method to measure the oscillation, this time without radial feedback, is presented in the next section. However, the discrepancy between the detected and expected frequency has already been observed in previous works [77] and will be further elucidated in Section 5.4.