# Full simulation of the cavity mode using reconstructed mirror profiles

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## Trapped atom clock on a chip (TACC)

The TACC experiment is presented in great detail in [89, 91, 92] but as it is the foundation of TACC2, a brief overview is provided in the following. The characteristic feature of TACC is that the atoms are trapped during the Ramsey time. This allows to increase the interrogation time without increasing the system size, as it is the case for fountain clocks. Since it is the main diﬀerence compared to conventional atom clocks, we begin this section with a description of the eﬀects that arise from the trapping.

Magic traps

Using trapped atoms as a frequency reference might appear contradictory at first glance. Trapping, independent of the force or the object, requires a position dependent potential with a local minimum. Ideally, a frequency reference should not couple to external fields and if it does, the shift of the frequency should not be position or time dependent. Since in our case the trapping is realized by a magnetic field, which shifts the atomic energy levels via the Zeeman-eﬀect, we violate both points. Fortunately, the level structure of 87Rb holds a solution to this contradiction. Here, the energy shift of the two clock states is the same to first order for a specific magnetic field strength, called “magic field” Bm. A similar approach is taken for dipole traps for optical clocks where specific wavelengths (magic wavelengths) are used for trapping [93].

Density shifts in trapped atom clouds

A second position dependent frequency shift is experienced by the atoms due to atom-atom interactions. The interactions are strongly enhanced by the high densities of trapped clouds compared to the typical densities in fountain clocks. By using a mean field approach, the frequency shift due to s-wave collisions can be written as [95] with f = (n1 − n2) /n and n = n1 + n2 being the atom number density, m the atom mass, and a11 = 100.44 a0, a22 = 95.47 a0 and a12 = 98.09 a0 the scattering lengths of the respective states with a0 the Bohr radius. As the density profile of the trapped cloud has Gaussian shape, the collisional shift is position dependent, as depicted in
6 Chapter 1.
Figure 1.4: Mean field and second order Zeeman shift. a) shows the near cancellation of the total shift (solid black line) by the addition of the negative mean field shift (dotted red line) and the positive second order Zeeman shift (blue dashed line) in zero gravity. b) shows the same situation including the gravitational sag (cf. Sec. 1.4.4). Adapted from [97].
Fig. 1.4. The spatial variation of the clock frequency can be canceled to first order by detuning the magnetic field minimum slightly from Bmin = Bm, as sketched in Fig. 1.4 and shown in [98]. This reduced spatial inhomogeneity is of great assistance for atomic clocks with magnetically trapped atoms [97].

### 87Rb for TACC

Due to several reasons 87Rb is an attractive choice for TACC:
• 87Rb is the work horse of the cold atom community and thus a lot of methods and experience are available to cool, trap and manipulate the atoms.
• As shown in Sec. 1.2.1, it is possible to realize a magic trap for 87Rb, an indis-pensable tool for the operation of a magnetically trapped frequency reference. Furthermore, it allows the first order cancellation of line broadening due to den-sity shifts, as discussed in Sec. 1.2.2.
• The hyperfine splitting is relatively large, yielding with Eq. 1.1.5 a low relative frequency error for a microwave frequency standard.
• The clock states we use have similar scattering lengths [95] which leads to spin-self rephasing, increasing the coherence time, as discussed in [65].
• The collisional shift for 87Rb atoms is at least 50 times lower than for 133Cs [99].
The listed favorable properties of 87Rb lead to the endorsement of the unperturbed ground-state hyperfine transition of 87Rb by the Comité international des poids et mesures (CIPM) as a secondary representation of the second [100].

TACC clock transition

Only particular Zeemann sub-levels can be trapped magnetically, as will be seen in the next section  clocks, where |F = 2, mF = 0i , |F = 1, mF = 0i are used.
To be able to trap the atoms and to benefit from the magic field (cf. Sec. 1.2.1), we use as clock states |1i = |F = 1, mF = −1i and |2i = |F = 2, mF = 1i. These states cannot be coupled with a single photon transition since ΔmF = 2. Therefore we use a two-photon transition consisting of a radio-frequency (RF) photon and a micro-wave (MW) photon, leading to an eﬀective Rabi frequency [91]
Ω = ΩRF ΩMW , (1.2.3) where Δ ≈ 500 kHz is the common detuning of both frequencies to the intermediate state |F = 2, mF = 0i. The two-photon transition and the clock states are presented in the level diagram in Fig. 1.5. Further information on the 87Rb D-line can be found in [101].
Figure 1.5: Zeeman sub-level diagram of the 52S1/2 manifold of 87Rb. The degeneration of the hyperfine levels is canceled by an external quantization magnetic field and the Zeeman sub-levels are visible (grey lines). The clock-states used in TACC are marked green and the RF (MW) transition is indicated by a blue (red) arrow.

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Magnetic trapping and manipulation of neutral atoms

The unique properties of 87Rb allow clock operation in a magnetic trap, as we have seen in the previous section. In this section we provide the general principles of magnetic trapping of neutral atoms, to prepare for the discussion of the atom chip technology.

Basic principle

if U is smaller than the hypferfine splitting of the atomic structure. In our experiment we will work with 87Rb, where this approximation holds for |B| < 200 Gauss [102], met by our chip traps. Here, F is the total angular momentum of the atom, µB the Bohr magneton and gF the Landé factor. Since the Maxwell equations forbid the existence of local maxima of the magnetic field [103], high-field-seekers with a magnetic moment parallel to the magnetic field cannot be trapped. The two clock states used in the experiment, |1i = |F = 1, mF = −1i and |2i = |F = 2, mF = 1i, are thus low-field-seekers with their magnetic moment antiparallel to the magnetic field and can therefore be trapped in magnetic minima. The creation of magnetic minima will be discussed in Sec. 1.4.1.

#### Majorana losses

Spin-flips, transforming low-field-seekers into untrapped atoms (high-field-seekers or to states with mF = 0), have to be avoided, since they lead to atom loss. To ensure this, the magnetic field orientation has to change much slower than the precession of the magnetic moment around the magnetic field axis [104]:
where ωL is defined as the Larmor frequency.
The Majorana losses can become the main loss channel particularly for magnetic traps with a vanishing magnetic field in the center, like the quadrupol trap (see Sec. 1.4.1). The loss rate for 87Rb atoms with a magnetic moment µ = gF mF µB scales with [105] where B0 is the gradient close to the trap center, T the temperature of the atom cloud and m the mass of the atom. Bringing the atoms closer to the trap center by steepening the gradient or lowering the temperature increases the loss rate. In [105] the atoms are prepared in the same state as in our experiment (|F = 1, mF = −1i) and they find experimentally the lifetime as an empirical formula. Note that in [106] the proportionality factor is smaller by a factor of 2, showing that theses formulas can only be used as a first estimation, but they will supply an orientation for choosing the trap parameters in the following chapters. For Ioﬀe-Pritchard traps (see Sec. 1.4.1) with non-vanishing magnetic field B0 in the center, condition 1.3.2 restricts the choice of the transversal trapping frequency ω⊥ [102]. This can be expressed with the adiabaticity coeﬃcient.

1 Trapped Atom Clock on a Chip
1.1 Atomic clocks
1.1.1 Rabi and Ramsey spectroscopy
1.1.2 Clock stability
1.2 Trapped atom clock on a chip (TACC)
1.2.1 Magic traps
1.2.2 Density shifts in trapped atom clouds
1.2.3 87Rb for TACC
1.2.4 TACC clock transition
1.3 Magnetic trapping and manipulation of neutral atoms
1.3.1 Basic principle
1.3.2 Majorana losses
1.4 Atom chip technology
1.4.1 Basic principle
1.4.2 Trap types
1.4.3 Trap depth
1.4.4 Gravitational sag and rotation
2 Spin-Squeezing for Metrology
2.1 Spin-squeezing
2.2 Atom-light interaction
2.3 Spin-squeezing for TACC 2
2.3.1 Cavity feedback squeezing
2.3.2 Squeezing by QND measurement
2.4 Requirements for the new experiment
2.5 Conclusion
3 CO2 Machining with Multiple Pulses
3.1 CO2 machining of fused silica
3.2 Limiting factors in single-pulse machining
3.3 Dot milling setup
3.3.1 In situ profilometry
3.3.2 Multi-fiber holder for mass production
3.4 Dot milling with CO2 laser pulses on fiber end facets
3.4.1 Fiber preparation
3.4.2 Multiple shots
3.4.3 Machining large spherical structures by CO2 dot milling
3.5 Conclusion
4 Long Fiber Fabry-Pérot Resonators
4.1 Optical Resonators
4.1.1 Resonance, transmission and finesse
4.1.2 Mode geometry
4.1.3 Fiber Fabry-Pérot resonators
4.1.4 Resonator losses
4.1.5 Coupling to the resonator
4.1.6 Photonic-crystal fibers for FFP cavities
4.1.7 Conclusion
4.2 Experimental realization and results
4.2.1 Finesse and transmission measurements
4.2.2 Analytical model: clipping loss
4.2.3 Full simulation of the cavity mode using reconstructed mirror profiles
4.2.4 Cavity transmission
4.2.5 Higher order modes
4.2.6 Double -Resonators
4.2.7 A compact FFP resonator mount
4.2.8 Resonators for TACC2
4.3 Conclusion
5 Atom Chip for TACC 2
5.1 Requirements catalogue
5.2 Layout of the atom chip
5.2.1 Atom transport
5.2.2 Three-wire trap
5.2.3 Effect of the bias field inhomogeneity on the trap position
5.2.4 Stand-alone chip trap
5.2.5 MW coplanar wave guide
5.3 Fabrication
5.3.1 Base chip
5.3.2 Science chip
5.3.3 “Marriage” of chip and resonator
Conclusion and Outlook

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