The position-selective manipulation of the trapped atoms requires precise knowledge about the respective atomic resonance frequencies. In preparation for all single atom addressing experiments presented in this chapter we therefore perform a calibration measurement which determines the resonance frequencies of the trapped atoms as a function of their position along the trap axis.
2.2.1 Magnetic field
Figure 2.1 shows an image of a string of five single atoms trapped in different potential wells of our standing wave dipole trap. It is the same image already depicted in Figure 1.11.
In order to spectroscopically resolve these atoms we apply an inhomogeneous magnetic field which introduces a position dependent |0i ↔ |1i transition frequency. For this purpose we use the MOT coils which produce a quadrupole magnetic fieldBquad(r) whose magnitude increases linearly from the center. In order to avoid state mixing between the atomic Zeeman levels, we additionally apply a homogeneous guiding field B0 along the trap axis, see Section 1.5.1. The total magnetic field therefore has the form
B(r) =
Bx By Bz
=B0+Bquad(r) =
B0
0 0
+B0·
x y
−2z
, (2.1)
where B0 denotes the magnitude of the gradient field. Since the dipole trap is aligned onto the MOT which runs close to the zero point of the quadrupole field (x=y=z= 0),
2.2 Calibration of the position dependent atomic resonance 49 the dipole trap is on the axis (y = 0, z = 0) so that the transition frequency between
|0i ≡ |F = 4, mF =−4i and |1i ≡ |F = 3, mF=−3i of the trapped atoms increases linearly with x:
ω0(x) = ∆HFS+δ0+ω0x . (2.2) Here, the frequency shift due to the homogeneous magnetic guiding field is labelled δ0, and the magnitude of the position dependent frequency shift is denoted by ω0 = (3g3 −4g4)µBB0/~. A guiding field of |B0| = 4 G causes a frequency shift of δ0/2π =
−9.8 MHz, while the gradient field of typically B0 −1.5 mG/µm results in a shift of ω0/2π = 3.7 kHz/µm.
2.2.2 Experimental sequence
For a precise calibration ofω0(x) we use trapped atoms as a probe. Initially, we determine δ0 by recording a spectrum in the homogeneous magnetic field B0, see Section 1.6.3.
Then, we measure ω0 in the gradient field by loading a large atom cloud into the dipole trap, initializing it in state |0i, and applying a microwave π pulse with a fixed frequency ωMW. After state-selective detection the atoms only remain trapped at the position that was resonant with the microwave pulse. By repetition of this measurement at different microwave frequencies we calibrate the position as a function of the microwave frequency, x(ωMW), from which we can inferω0 andδ0.
In the first step, we load a large number of about 50 atoms into the dipole trap. After the transfer from the MOT, their spatial distribution only extends over the MOT size of roughly 10 µm diameter. We broaden their distribution within the dipole trap to 2σtof = 60µm by switching off one of the two trap laser beams for 1 ms, see Section 1.3.4.
Under continuous illumination by the optical molasses we then image the trapped atoms with an integration time of 500 ms, see Figure 2.2 (a). Due to the large number of atoms, the average separation between individual atoms is too small to be resolved. In order to cool the trapped atoms as long as possible, the molasses lasers are switched off only after the CCD chip has been read out which takes 400 ms. Now we switch on the magnetic field B(r). To create the gradient field we run a current of 1.3 A through the MOT coils which corresponds to roughly one tenth of the current during MOT operation. After initializing the atoms in state |0i by optical pumping, we apply a microwave π pulse with a pulse duration of tpulse= 200µs at the frequency ∆HFS+δ0. The state-selective push-out laser then removes all atoms from the trap that have remained in |0i. A subsequently taken image with an exposure time of 500 ms, see Figure 2.2 (b), indeed reveals the presence of atoms in an area of 3µm diamater. Its center corresponds to the positionx= 0, according to Equation (2.2).
For a better signal-to-noise ratio we repeat this sequence 20 times and add up all initial and all final images. To analyze the resulting two images we follow the procedure introduced in Section 1.3.4. After suitably clipping the images to minimize the background noise, we bin the pixels of the pictures along the vertical z direction to obtain histograms for the intensity distribution along the trap axis x. The resulting histogram of the initial image shows a homogeneous distribution over the entire trapping region (Figure 2.2 (a)). In
Figure 2.2: Image analysis for frequency calibration. a) The picture shows 20 added images (exposure time: 0.5 s) of an atom cloud in the dipole trap of 50 atoms each. b) This picture shows 20 added images after application of a π pulse and state-selective atom removal from the trap. Only the atoms resonant with the microwave pulse are left. The histograms in (a) and (b) show the respective binned intensity distributions. c) Division of the second histogram (b) by the initial distribution (a) yields the normalized intensity distribution. The line is a Gaussian fit according to Equation (2.3).
2.2 Calibration of the position dependent atomic resonance 51 order to normalize the atom distribution after state-selective detection (Figure 2.2 (b)) onto the initial distribution of atoms, we divide the histogram of the final image by the one of the initial image.
The resulting intensity distribution (Figure 2.2 (c)), corresponds to a convolution of a Fourier-limitedπ pulse spectrum (Figure 1.21) with the intensity distribution of a single atom image (Figure 1.10 (b)). We determine its center xc by fitting the same Gaussian function as in Equation (1.15):
I(x) =B+Aexp
−(x−xc)2 2σx2
(2.3) The resulting fit parameters are listed in Table 2.1.
position xc 50.5± 0.1 pixel width σx 1.3± 0.1 pixel offset B 7.9± 0.4 % amplitude A 24± 1 %
Table 2.1: Fit results for the normalized intensity distribution of Figure 2.2 (c), accord-ing to Equation (2.3).
Theπ pulse time and the corresponding microwave power of the Fourier-limited pulse were chosen such that the width of the intensity distribution σx is not significantly broader than the width of a single atom image in the dipole trap σx(DT) = 1.24±0.16 pixel.
The 1/√
e−width of the respective pulse spectrum in frequency space is 1.9 kHz and corresponds towspec= 0.5 pixel using the calibration result below. The expected width of the intensity distribution σx =q
σx(DT)2+w2spec = 1.3±0.2 pixel agrees well with the measured width of σx= 1.3±0.1 pixel.
Due to the fact that the spectral pulse width is smaller than the width of a single atom image, the amplitude of the normalized intensity distribution is small. The resonance frequency of atoms trapped at a site which is, e. g., two pixels away from xc, is detuned from the microwave frequency far enough so that no spin flip occurs. However, their fluorescence still contributes to the histogram counts of Figure 2.2 (a) at the position xc. This additional background of the reference image significantly decreases the amplitude of the normalized intensity distribution.
2.2.3 Result
To measure the position as a function of the atomic resonance frequency we repeat the en-tire procedure described above for different microwave frequencies. The result is plotted in Figure 2.3 and reveals a purely linear dependence, as expected. From the slope of the linear fit we infer the position-dependent frequency shift to beω0/2π= 3.480±0.007 kHz/pixel = 3.71±0.03 kHz/µm. The magnetic field gradient is thereforeB0 =−1.51±0.01 mG/µm
Figure 2.3: Calibration of the atomic resonance frequencies. Each data point shows the center position of a fitted intensity distribution as in Figure 2.2 (c). The error bars are too small to be displayed here. From the slope of the linear fit we infer ω0/2π = 3.71±0.03 kHz/µm.
which is compatible with the expected gradient of−1.4±0.1 mG/µm, calculated from the applied current of 1.3±0.1 A [54].