5. Control of Atomic Motion
5.4. Anharmonic Oscillator
frequency corresponds to twice the atom’s oscillation frequency, we can extract the atom’s harmonic oscillation frequency from the sharp onset of the parametric driving on the high-frequency side, as it is shown in Figure 5.9.
Figure 5.9: Onset of parametric heating as function of the trap depth. The experimental data points are marked by black dots.
A red line shows a fitted square-root curve. The theoretically expected behavior is plotted as dashed green line.
The obtained data points follow nicely the expected square-root behavior, cf. Eqn. (5.5). The obtained fit is indicated by the red line. In contrast to the previous measurements, the theoretically calculated values, indicated by the dashed green line, agree now well with the measured data points. The small remaining discrepancy can be explained due to small deviations associated in gauging the trap depth. This gauging was performed by a heterodyne measurement as it will be described later in Section 7.6. The parametric heating technique hence is a useful and simple method to gain information about the harmonic oscillation frequency of an atom. On the other hand, providing that this frequency is known, it can also serve for gauging the trap depth. In the next section an explanation will be developed showing where the discrepancy between the parametric heating method and methods relying on a frequency analysis of the transmitted photon stream stems from.
5.4. Anharmonic Oscillator
The break-down of the harmonic approximation for increasing atomic excursions is the key driver for deviations of the observed atomic oscillation frequency, obtained by analyzing the transmitted photon stream, from its expected value. These values on the other hand are successfully confirmed by a parametric heating measurement. The harmonic approximation was already described in Section 5.2. As it can be seen in Figure 5.2, the actual trapping potential yields substantially shallower trapping potentials than the harmonic approximation starting from atomic excursion beyond 7 µm for the radial and 70 nm for the axial direction,
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700 experimental data square root fit theory
f axial (MHz)
trap depth Û/kB (µK)
Control of Atomic Motion 5.4 Anharmonic Oscillator
respectively. These excursion levels are reached once the atom possesses a kinetic energy around 300 µK, assuming typical depth of the dipole trap potential of 850 µK. When this level is reached, the deviations from the harmonic approximation cause an actual oscillation frequency which is below the corresponding harmonic value. This anharmonic oscillation frequency is computed for different maximum excursion levels of the atom with a numerical simulation. The result is plotted as a function of the axial and radial excursion separately and is depicted in Figure 5.10. The harmonic boundary values frad,hm and faxial,hm are indicated by the solid red lines.
Figure 5.10: Deviation of the oscillation frequency from its harmonic value, once the harmonic approximation breaks down.
The behavior of the oscillation frequency is shown for excursions of the atom in the radial direction r a) and in the axial direction z b). Each left vertical axis shows the frequency in kHz while the right axis features its relative value compared to the respective harmonic boundary value. The excursion is given once in absolute values for our cavity parameters (lower axis) and once with respect to the respective typical length scales (upper axis), as they are also indicated by the dashed, vertical, green lines. A decrease of the oscillation frequency down to 0 Hz is visible in both cases.
As expected, a significant drop in the actual oscillation frequency is visible as the excursion of the atom increases. This occurs when the atom is heated up and hence increases its oscillation amplitude. When interpreting the measurements of the oscillation of the atom by means of a spectral analysis of the transmitted probe power (cf. 5.3.1,5.3.2), it is important to examine the modulation strength of the probe beam corresponding to different excursion amplitude of the atom. In order to determine the modulation strength, one has to consider the probe-light transmitted through the cavity. This light depends on the position-dependent coupling constant geff(r,z) as given by Eqn. (5.6) and shown in Figure 5.3. The motion of the atom hence leads to
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5.4 Anharmonic Oscillator Control of Atomic Motion
a modulation in the transmission. The strength of this modulation is given by the difference of the minimal and maximal transmission per oscillation period. Since the oscillation amplitude corresponds to its maximum excursion depicted in Figure 5.10, one can assign a relative modulation strength of the transmitted probe beam for every anharmonic oscillation frequency.
These values are depicted in Figure 5.11 and indicate how well each frequency component will be visible in the transmitted probe beam spectrum. As visible from Eqn. (5.6), a mismatch of the center of the trapping and probing mode will result in a different effective coupling strength.
This is why the behavior is evaluated for two different scenarios. In the first both modes overlap perfectly, causing z0=0nm and in the second both modes are offset by z0 =76nm. This has been proven to be a typical average value for our system, cf. [81].
Figure 5.11: Visibility of the modulation of the transmitted probe signal for different oscillation frequencies of the atom in the cavity mode. The anharmonicity of the trapping potential causes the oscillation frequencies of the atom to depend on its oscillation amplitude. The relationship is plotted separately for the radial oscillations a) and for the axial oscillations b). Two scenarios are depicted, once for the case that the dipole trap and probing mode exactly overlap z0=0 nm (brown curve) and once for the case that both modes are offset by a typical value of z0=76 nm (green curve). The dashed vertical line indicates the measured atomic oscillation frequencies from the correlation measurements.
The plots show that hardly any information about an atom oscillating at its harmonic oscillation frequency is encoded in the transmitted probe light. Only when the atom has acquired a certain kinetic energy, which results in a larger oscillation amplitude and reduced frequency, its motion becomes “visible”. This is also the reason why any measurement of the transmitted probe beam will output drastically smaller oscillation frequencies. The frequencies measured via the second order correlation function are also indicated by the vertical dashed lines in Figure 5.11. These frequencies exactly mark the border of the region above which the visibility of the frequency components is significantly reduced. Thus, this confirms our initial hypothesis, that when measuring the spectral properties of the transmitted probe beam we will mainly detect the